From 8f027176d576ab23dee22031626032f8e361d2ff Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Zden=C4=9Bk=20Hur=C3=A1k?= Date: Sat, 14 Dec 2024 22:13:07 +0100 Subject: [PATCH] Built site for gh-pages --- .nojekyll | 2 +- classes_reset.html | 136 +- classes_switched.html | 58 +- complementarity_constraints.html | 4 +- complementarity_figures/coulomb_friction.png | Bin 0 -> 4069 bytes complementarity_references 24.html | 1105 ++ complementarity_simulations.html | 1506 +- .../fig-state-portrait-output-1.svg | 798 + des_automata 23.html | 3336 ++++ des_automata.html | 14 +- hybrid_automata_software 23.html | 1087 ++ hybrid_equations.html | 154 +- hybrid_equations_references 24.html | 1093 ++ intro_outline 25.html | 1097 ++ max_plus_algebra 13.html | 8168 ++++++++++ max_plus_algebra.html | 13008 ++++++++-------- max_plus_systems 19.html | 1397 ++ mld_intro 24.html | 1444 ++ mpc_mld_software 24.html | 1059 ++ petri_nets 11.html | 1679 ++ petri_nets_timed 24.html | 1316 ++ search.json | 6 +- ...p-2be10d9e998f81ff6e49e26833438aa5.min.css | 12 + ...p-8c89492777ddc1c6ab649c9c2f2ee7ae.min.css | 12 + site_libs 27/bootstrap/bootstrap-icons.css | 2078 +++ site_libs 27/bootstrap/bootstrap-icons.woff | Bin 0 -> 176200 bytes site_libs 27/bootstrap/bootstrap.min.js | 7 + site_libs 27/clipboard/clipboard.min.js | 7 + site_libs 27/quarto-html/anchor.min.js | 9 + site_libs 27/quarto-html/popper.min.js | 6 + ...hting-e26003cea8cd680ca0c55a263523d882.css | 205 + site_libs 27/quarto-html/quarto.js | 911 ++ site_libs 27/quarto-html/tippy.css | 1 + site_libs 27/quarto-html/tippy.umd.min.js | 2 + site_libs 27/quarto-nav/headroom.min.js | 7 + site_libs 27/quarto-nav/quarto-nav.js | 325 + .../quarto-search/autocomplete.umd.js | 3 + site_libs 27/quarto-search/fuse.min.js | 9 + site_libs 27/quarto-search/quarto-search.js | 1290 ++ sitemap.xml | 6 +- ...ity_via_multiple_lyapunov_function 25.html | 1550 ++ stability_via_multiple_lyapunov_function.html | 2 +- verification_barrier.html | 2774 ++-- verification_references 24.html | 1126 ++ 44 files changed, 40178 insertions(+), 8631 deletions(-) create mode 100644 complementarity_figures/coulomb_friction.png create mode 100644 complementarity_references 24.html create mode 100644 complementarity_simulations_files 12/figure-html/fig-state-portrait-output-1.svg create mode 100644 des_automata 23.html create mode 100644 hybrid_automata_software 23.html create mode 100644 hybrid_equations_references 24.html create mode 100644 intro_outline 25.html create mode 100644 max_plus_algebra 13.html create mode 100644 max_plus_systems 19.html create mode 100644 mld_intro 24.html create mode 100644 mpc_mld_software 24.html create mode 100644 petri_nets 11.html create mode 100644 petri_nets_timed 24.html create mode 100644 site_libs 27/bootstrap/bootstrap-2be10d9e998f81ff6e49e26833438aa5.min.css create mode 100644 site_libs 27/bootstrap/bootstrap-8c89492777ddc1c6ab649c9c2f2ee7ae.min.css create mode 100644 site_libs 27/bootstrap/bootstrap-icons.css create mode 100644 site_libs 27/bootstrap/bootstrap-icons.woff create mode 100644 site_libs 27/bootstrap/bootstrap.min.js create mode 100644 site_libs 27/clipboard/clipboard.min.js create mode 100644 site_libs 27/quarto-html/anchor.min.js create mode 100644 site_libs 27/quarto-html/popper.min.js create mode 100644 site_libs 27/quarto-html/quarto-syntax-highlighting-e26003cea8cd680ca0c55a263523d882.css create mode 100644 site_libs 27/quarto-html/quarto.js create mode 100644 site_libs 27/quarto-html/tippy.css create mode 100644 site_libs 27/quarto-html/tippy.umd.min.js create mode 100644 site_libs 27/quarto-nav/headroom.min.js create mode 100644 site_libs 27/quarto-nav/quarto-nav.js create mode 100644 site_libs 27/quarto-search/autocomplete.umd.js create mode 100644 site_libs 27/quarto-search/fuse.min.js create mode 100644 site_libs 27/quarto-search/quarto-search.js create mode 100644 stability_via_multiple_lyapunov_function 25.html create mode 100644 verification_references 24.html diff --git a/.nojekyll b/.nojekyll index 5b4a740..a22e8f1 100644 --- a/.nojekyll +++ b/.nojekyll @@ -1 +1 @@ -8e6a24a6 \ No newline at end of file +ab5b338b \ No newline at end of file diff --git a/classes_reset.html b/classes_reset.html index 3190787..6bf3d64 100644 --- a/classes_reset.html +++ b/classes_reset.html @@ -770,49 +770,49 @@

Reset systems

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Clegg’s integrator - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/classes_switched.html b/classes_switched.html index 3cfdfa1..0ef4141 100644 --- a/classes_switched.html +++ b/classes_switched.html @@ -819,46 +819,46 @@

State-dependent - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/complementarity_constraints.html b/complementarity_constraints.html index de943a1..ec344ad 100644 --- a/complementarity_constraints.html +++ b/complementarity_constraints.html @@ -705,8 +705,8 @@

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Linear complementarity problem (LCP)

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Linear complementarity problem (LCP)

For a given square matrix \mathbf M and a vector \mathbf q , the linear complementarity problem (LCP) asks for finding two vectors \bm w and \bm z satisfying \begin{aligned} \bm w-\mathbf M\bm z &= \mathbf q \\ diff --git a/complementarity_figures/coulomb_friction.png b/complementarity_figures/coulomb_friction.png new file mode 100644 index 0000000000000000000000000000000000000000..fde23d2246c7c27a19d5d1a92a499125736e9e06 GIT binary patch literal 4069 zcmbVPc{o(>+dnhI*akynlr^KpHj^dF8sp2zmTZ+ZiL&p?U`DpkHc=8%Bzr?f_D;4e z-zbcoL{YK~+0x*5{NC%m-hbZZ_j}Jd*K?h7uIIj=``qXAdG61B%hKG4m-_%W006v1 zV}ca`z!A`WIVTbt&;Qj3jR5k}86yI)yLaX^ynFxv{24^TY3uOJh4ILHHfxdZmp6v% zISi5WNy74RHN=9a8ELLkru?eWlTX1(jMJA6-44E# za%=gSnog^cloTf>ra;fznEeb)!x@WrT zQS(w%P{!2MfOr^@s@#p2B3iYHBC*6yjb<{IDC&W^hdnm^pOX_9gM+WVhm|ujGWec) zUAuOvIiA;hw5pV`;XPiP#(VGtTK}l^V`^UL6-;||$j^b}9^GQsyu1p_%C0~mq^6~P z8X9UJs|gZSzRWq@cH3=}`9*7&^(*1?8_!RN+U|&VB?`!3)P2S1bh=R-C(-H$9(8$a z*yHJk6j7C-nxG>b{1TUY^N91quM@g(Z!;ydZ_xa|=tC=TNi$cva0vg6@vvX*GIyCE zf}hE3=10(=U{Cl>WX?z$QdhZt_cx^!my~>+f8BFsV{sgSyvF}5tGjd2w}5{WehC!fQeVDV{l-KD3WFzmWM7Ye ztl}ZT2M6SBOMsk`k`yQl{yo|oV!jaz^g#fl;V<+K)TlTz;|#2@t4 zDDwHe@@;h> zr>Yvk-!1b^9|u~G_P#h%SX`Xz{5A@8@I(oo{S0*^%T;Q6pyYfrH2EkF=h2oTsyN~t zxn=_>)y-Gz_xUXyb8FubLBq2N1ILJS8u65rE`EU&am}h$Hc;_m6)kG}hMb&Sx=!TQ zSs&@x=AV3~*;1_1D?euW#zHM*Pc(UWaNZ-79$ zha3kbBQrDJpxJJB=D}%upi&$##sp@&G^72DfXQ#U%aTN zTT-nB;iLl^3$Y>+AF9sRTMOo1hZt?rVc5~gaH0S* zh*dck?W`Dx>((YK+6cuaJ+cyLL1~Mz?dQ@kl_JV8e zUC(rody!gyjbNXn1au&m#gVhJqZ{W6iofn{5HAi%B5euyhU*4o&P_Gug?GhsY?&&M zH{suS+Uig|C!B^E+`mbf7rloSZb0L3tRG+9lkouh-Di=q=iRN@Feq?L>;2CqcU6VZ zlYR(qZhs?k4Qw5`8TdZUzIM+}TvkubUxyhi@Zs;CS2*Td0h0goWJC(Gk>l?am#9GB zqfDPE4osnF1QgHvEoCO2rULfC8u#ZIb*kMqU+>bHT6+uIfrS?g*OZd#jifx{p(F&C zaC#n!GG)$EqL1()aqQJ{dOiBWf;n0(cd?BcCm9R5q-eGXnQHMwIO%vx10wfS#emfe zp3)ybHc+~{-TSCl$pG94ou53BqsuDLoIg{SjnYWVjh3vH{geEl^cXmFG`r!mtG<>i zQ{v9p3C@oPH+z6lnF8(Z2zg30;xG^f{|sN`XbZjk;YBC3TR$tbkPID*jvjI6Oouic z3+*+bM_|~Jx7*>G5r*j$gO#!Sml)RRho3AHL#xkCPMms=b|y9PT-;Cx6HPm8=0<6l|7y$g$u`>Y3z{gBU352)6w7%9@ zPAnIZ8l1{0ab&{8mQDA}3@-^*owRv1{sCpUCG0Rul9{k7r=SE?6RVd(tp*2q?wlN- z$!k^!O8@st&A(cg`hUcEN!9a8tB&f#(vAX4z994Ax0w4r!MWi(Z{f@nmLK#P=6yY{7|#>xY9$mf1Mvwc+SG3zhFn~RDZ?x zJC3@b<<{-$<+h19oVis5M9x2ChT;7c~s#WFaxT{=NOvG za3~@K#F53(KECVOx6l$XOl0gMsa5&#VL40$Qh71#ZCAO^MbWct%?WQ<<<2$AeB2@q zmKHU;bmJFuJ(Qy-bwr-?-&Kv10HXo358YXOEW4FPAFvNR-%)-_diO?8HR%ExBWeLCm2INR@w=(GfkOjjx106c znhc^ z@_=Kh+D*hl>ud*vD-!dxsEWtpx4Ghtir}jv*-^DrpV2_mr;SfvMTIzb3X2SMXSt7E z_voqYLNm91+U?VIn++Olp2X!l_5`eQ$Wf;|FFW(HRG5yOA)rE{#KMrWK4195aDiv9 zsSj>SdjnN~okT-ExuP4YqQCKY*Ctrw&RT=wi0+@^naxu6?fIoV$3y^wV19jIQ4zW@Qt5R+sjbMS zcqJj4xUzdIuX#W4(|m5Sp}EcNEntZEbDi(y_#=B8P}noyLla3iaNy(^M?+-@Jq$i+xla zR?l}x#u80V^QCj3ko#Z$$HVy%c4KN@@PBnMt;MKKU{lzi{p0K0MgkQp{8H>Ae>%WI z5AoF-i=l4N2Op&#aaI*fXE;g95ve1FWy1mP6_Wfc!`>=bHIKc&va=N*0lW^%9{B}@ zS~yN;+;rw+o3x+=WM0Mq;Xw7;WnIWGy^ecIavM;EMNqauH&BFJwy-SwZDCm&vN8C2 z6Ffcn2s8kHfs}QY;9h6dS}cI!E6L45=mqU{UP+^KcXr@@im6by6$t(T4@@?vdfopY zG9%e_3?#TH(YQ2To*zX|+PHA|+wA1YqfMy&*H<~B^jTUrlgW!9X3En4FcWT><~^Hd z^$bgtOCL=%cz07z#O3(%^qnBkMatxrDv@fq35py{OHJPNYv6J;xbO&}USeB$B$Mt%y@ycL1k5~@G-Po!A4>_}#;J4P9E7nkid O?nFa#LYclx%s&A7|6u0; literal 0 HcmV?d00001 diff --git a/complementarity_references 24.html b/complementarity_references 24.html new file mode 100644 index 0000000..9251a29 --- /dev/null +++ b/complementarity_references 24.html @@ -0,0 +1,1105 @@ + + + + + + + + + +Literature – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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Literature

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Concise (short and yet sufficient for our purposes) introduction to (linear) complementarity problems and systems is in [1]. Besides describing the idea of complementarity in dynamical systems, it also shows how it is related to other modeling frameworks for hybrid dynamical systems. More detailed and yet very accessible introduction is in the thesis [2]. Condensed treatment is in the papers [3] and [4].

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A readable introduction to the Extended Linear Complementarity Problem is in [5] (it is also freely available as a technical report).

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The topics of complementarity constraints in dynamical systems and optimization is still being actively researched. A recent publication on QP optimization with complementarity constraints (LCQP) is [6].

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Numerical methods for nonsmooth dynamical systems that are based on complementary constraints (and implemented in SICONOS software) are comprehensively presented in [7].

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References

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[1]
W. P. M. H. Heemels, B. De Schutter, and A. Bemporad, “Equivalence of hybrid dynamical models,” Automatica, vol. 37, no. 7, pp. 1085–1091, Jul. 2001, doi: 10.1016/S0005-1098(01)00059-0.
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[2]
M. Heemels, “Linear complementarity systems: a study in hybrid dynamics,” PhD thesis, Technische Universiteit Eindhoven, Eindhoven, NL, 1999. Available: https://heemels.tue.nl/content/papers/Hee_TUE99a.pdf
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[3]
W. P. M. H. Heemels, J. M. Schumacher, and S. Weiland, “Linear Complementarity Systems,” SIAM Journal on Applied Mathematics, vol. 60, no. 4, pp. 1234–1269, Jan. 2000, doi: 10.1137/S0036139997325199.
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[4]
A. J. van der Schaft and J. M. Schumacher, “Complementarity modeling of hybrid systems,” IEEE Transactions on Automatic Control, vol. 43, no. 4, pp. 483–490, Apr. 1998, doi: 10.1109/9.664151.
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[5]
B. De Schutter and B. De Moor, “The Extended Linear Complementarity Problem and the Modeling and Analysis of Hybrid Systems,” in Hybrid Systems V, P. Antsaklis, M. Lemmon, W. Kohn, A. Nerode, and S. Sastry, Eds., in Lecture Notes in Computer Science. Berlin, Heidelberg: Springer, 1999, pp. 70–85. doi: 10.1007/3-540-49163-5_4.
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[6]
J. Hall, A. Nurkanovic, F. Messerer, and M. Diehl, “LCQPow – A Solver for Linear Complementarity Quadratic Programs.” arXiv, Nov. 2022. Accessed: Dec. 03, 2022. [Online]. Available: http://arxiv.org/abs/2211.16341
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[7]
V. Acary and B. Brogliato, Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. in Lecture Notes in Applied and Computational Mechanics, no. 35. Berlin Heidelberg: Springer, 2008. Available: https://doi.org/10.1007/978-3-540-75392-6
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Copyright 2024, Zdeněk Hurák

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Simulations of complementarity systems using time-stepping

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Simulations of complementarity systems using time-stepping

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All nine regions

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Solutions usi - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Figure 8: Solution trajectory for a discontinuous system obtained by the MCP solver @@ -1610,255 +1610,255 @@

Solutions usi - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Figure 9: Solution trajectory for a discontinuous system obtained by the MCP solver with a smaller step size @@ -1945,56 +1945,260 @@

Solutions usi t = range(0.0, step=h, stop=tfinal) X = [getindex.(x, i) for i in 1:length(x[1])] -Plots.plot(t,X[1],label="x₁",lw=3,xlabel="t",ylabel="Positions") -Plots.plot!(t,X[2],label="x₂",lw=3) +Plots.plot(t,X[1],label="x₁",lw=3,xlabel="t",ylabel="Positions",markershape=:circle) +Plots.plot!(t,X[2],label="x₂",lw=3,markershape=:circle)
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Note, once again, the amazingly small number of steps

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Solutions usi

while the shape of the trajectory is already quite accurate (you can try reducing the step lenght by yourself)and whatever chattering is absent in the solution.

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Example 3 (Time stepping for a mechanical system with Coulomb friction modelled as complementarity constraints) We consider an object of mass m moving on a horizontal surface. The object is subject to an applied force F_\mathrm{a} and a friction force F_\mathrm{f}, both oriented in the positive direction of position and velocity, as in Fig. 10.

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+Figure 10: Coulomb friction to be modelled using complementarity constraints +
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The friction force is modelled as a Coulomb friction model +F_\mathrm{f}(t) = -m g \mu\, \mathrm{sgn}(v(t)), + where \mu is a coefficient that relates the frictional force to the normal force mg.

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The motion equations are +\begin{aligned} +\dot x(t) &= v(t)\\ +\dot v(t) &= \frac{1}{m}(F_\mathrm{a}(t) + F_\mathrm{f}(t)). +\end{aligned} +

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We now express the friction force as a difference of two nonnegative variables +F_\mathrm{f}(t) = F^+_\mathrm{f}(t) - F^-_\mathrm{f}(t),\quad F^+_\mathrm{f}(t), \,F^-_\mathrm{f}(t) \geq 0. +

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And we also introduce another auxiliary variable \nu(t) that is just the absolute value of the velocity +\nu(t) = |v(t)|. +

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With the three new variables we formulate the Coulomb friction model as +\begin{aligned} +0 \leq v + \nu &\perp F^+_\mathrm{f} \geq 0,\\ +0 \leq -v + \nu &\perp F^-_\mathrm{f} \geq 0, \\ +0 \leq \mu m g - F^+_\mathrm{f} - F^-_\mathrm{f} &\perp \nu \geq 0. +\end{aligned} +

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If F^-_\mathrm{f} > 0, that is, if the friction force is negative, the second complementarity constraint implies that the velocity is equal to its absolute value, in other words, it is nonnegative, the object moves to the right (or stays still). If the velocity is positive, the first constraint implies that F^+_\mathrm{f} = 0, and the third constraint then implies that F^-_\mathrm{f} = mg\mu. Feel free to explore other cases.

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We now proceed to implement the time-stepping method for this system. Once again, we use the semi-explicit Euler scheme +\begin{aligned} +x_{k+1} &= x_k + h v_{k+1},\\ +v_{k+1} &= v_k + \frac{h}{m} (F^+_{\mathrm{f},k} - F^-_{\mathrm{f},k}), +\end{aligned} + where the velocity is subject to the complementarity constraints +\begin{aligned} +0 \leq v_{k+1} + \nu_{k+1} &\perp F^+_{\mathrm{f},k} \geq 0,\\ +0 \leq -v_{k+1} + \nu_{k+1} &\perp F^-_{\mathrm{f},k} \geq 0, \\ +0 \leq \mu m g - F^+_{\mathrm{f},k} - F^-_{\mathrm{f},k} &\perp \nu_{k+1} \geq 0, +\end{aligned} + into which we substitute for v_{k+1} \boxed{ +\begin{aligned} +0 \leq v_k + \frac{h}{m} ({\color{blue}F^+_{\mathrm{f},k}} - {\color{blue}F^-_{\mathrm{f},k}}) + {\color{blue}\nu_{k+1}}\, &\perp\, {\color{blue}F^+_{\mathrm{f},k}} \geq 0,\\ +0 \leq -\left(v_k + \frac{h}{m} ({\color{blue}F^+_{\mathrm{f},k}} - {\color{blue}F^-_{\mathrm{f},k}})\right) + {\color{blue}\nu_{k+1}}\, &\perp\, {\color{blue}F^-_{\mathrm{f},k}} \geq 0, \\ +0 \leq \mu m g - {\color{blue}F^+_{\mathrm{f},k}} - {\color{blue}F^-_{\mathrm{f},k}} \,&\perp\, {\color{blue}\nu_{k+1}} \geq 0, +\end{aligned} +} + in which we highlighted the unknowns by blue color for convenience.

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+Show the code +
using JuMP
+using PATHSolver
+using Plots
+
+m = 100.0
+g = 9.81
+μ = 10.0
+
+h = 2e-1
+
+x0 = 0.0
+v0 = 100.0
+
+x = [x0]
+v = [v0]
+
+tfinal = 5.0
+N = tfinal/h
+
+for i = 1:N
+    model = Model(PATHSolver.Optimizer)
+    set_optimizer_attribute(model, "output", "no")
+    set_silent(model)
+    @variable(model, vabs >= 0)
+    @variable(model, F⁺ >= 0)
+    @variable(model, F⁻ >= 0)
+    @constraint(model, (v[end] + h/m*(F⁺-F⁻) + vabs) ⟂ F⁺)
+    @constraint(model, (-(v[end] + h/m*(F⁺-F⁻)) + vabs) ⟂ F⁻)
+    @constraint(model, (μ*m*g - (F⁺+F⁻)) ⟂ vabs)
+    optimize!(model) 
+    push!(v, v[end] + h/m*(value(F⁺)-value(F⁻)))
+    push!(x, x[end] + h*v[end])         # Here v[end] on the left already equals v_k+1
+end
+
+t = range(0.0, step=h, stop=tfinal)
+Plots.plot(t,v,label="v",lw=3,markershape=:circle)
+Plots.plot!(t,x,label="x",lw=3,markershape=:circle,xlabel="t")
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+

Once again, we can see that the method correctly simulates the system coming to a complete stop due to friction.

+
@@ -2019,7 +2427,7 @@

Solutions usi

-

We have not certainly presented the most advanced and state-or-the-art methods. But the essence of using complementarity concepts in numerical simulation is a bit clearer now, hopefully.

+

While concluding this section about time-stepping methods, we have to emphasize that this was really just an introduction. But the essence of using complementarity concepts in numerical simulation is perhaps a bit clearer now.

diff --git a/complementarity_simulations_files 12/figure-html/fig-state-portrait-output-1.svg b/complementarity_simulations_files 12/figure-html/fig-state-portrait-output-1.svg new file mode 100644 index 0000000..174d8bc --- /dev/null +++ b/complementarity_simulations_files 12/figure-html/fig-state-portrait-output-1.svg @@ -0,0 +1,798 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/des_automata 23.html b/des_automata 23.html new file mode 100644 index 0000000..628bd4b --- /dev/null +++ b/des_automata 23.html @@ -0,0 +1,3336 @@ + + + + + + + + + +State automata – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + +
+ + + +
+ +
+
+

State automata

+
+ + + +
+ + + + +
+ + + +
+ + +

Having just discussed the concept of a discrete-event system, we now introduce the most popular modeling framework for such systems: a state automaton, or just an automaton (plural automata). It is also known as a state machine or a (discrete) transition system.

+
+

Definition 1 (Automaton) Automaton is a tuple \boxed{ +G = \{\mathcal X,\mathcal X_0,\mathcal E,\mathcal F\},} +
+where

+
    +
  • \mathcal X is the set of states (also called modes or locations).
    • +
  • \mathcal X_0 \subseteq \mathcal X is the set of initial states.
    • +
  • \mathcal E is the set of events (also actions, transition labels, symbols). It is also called alphabet.
    • +
  • \mathcal F\subseteq \mathcal X \times \mathcal E \times \mathcal X is the set of transitions. In the deterministic case it can also be narrowed down to a transition function f:\mathcal X \times \mathcal E \rightarrow \mathcal X. Note that f is then is a partial function, it is not necessarily defined for all combinations of states and events. Sometimes f is used even for multivalued functions: f:\mathcal X \times \mathcal E \rightarrow 2^\mathcal{X}, where 2^\mathcal{X} is a power set (a set of all subsets of X).
    • +
+
+
+
+
+ +
+
+Some comments on the notation +
+
+
+
    +
  • The set of states is often denoted by \mathcal Q to spare the letter \mathcal X for the continuous valued state space of hybrid systems.
    • +
  • The set of events is often denoted by \mathcal A to spare the letter \mathcal E for the set of transitions (edges in the corresponding graph), because F and f may also need to be spared for the continuous-valued transitions. But then the letter \mathcal A actually fits this purpose nicely because the event set is also called the alphabet.
    • +
+
+
+
+

Marked states

+

In some literature, the definition of the automaton also includes a set \mathcal X_\mathrm{m} \subseteq \mathcal X of marked or accepting states, in which case the definition of an automaton now includes three (sub)sets of states: \mathcal X, \mathcal X_0 and \mathcal X_\mathrm{m}. \boxed{ +G = \{\mathcal X,\mathcal X_0,\mathcal E,\mathcal F, \mathcal X_\mathrm{m}\}.} +

+

The marked states are just some states with special roles in the system. Namely, these are the states into which the system should be controlled. I do not particularly like this idea of mixing the model of the system with the requirements, but some part of the community likes it this way.

+
+
+

Automaton as a (di)graph (also a state transition diagram)

+

So far the definition of an automaton was not particularly visual. This can be changes by viewing the automaton as a directed graph (digraph) with. These are the basic rules

+
    +
  • State is represented as a node of the graph.
    • +
  • Transition from a given state to another state is represented as an edge connecting the two nodes.
    • +
  • Events (actions) are the labels attached to the edges. It is not necessary that each edge has its unique label.
    • +
+
+

Example 1 (Automaton as a digraph) Consider an automaton defined by these sets: \mathcal X = \{x_1,x_2,x_3\}, \mathcal X_0 = \{x_1\}, \mathcal E = \{e_1,e_2,e_3\}, \mathcal F = \{(x_1,e_1,x_2),(x_2,e_2,x_1),(x_1,e_3,x_3),(x_2,e_2,x_3)\}.

+

The corresponding digraph is in Fig 1.

+
+
+
+
+
+
+ + +G + + +init +init + + + +x₁ + +x₁ + + + +init->x₁ + + + + + +x₂ + +x₂ + + + +x₁->x₂ + + +e₁ + + + +x₃ + +x₃ + + + +x₁->x₃ + + +e₂ + + + +x₂->x₁ + + +e₂ + + + +x₂->x₃ + + +e₃ + + + +
+
+
+Figure 1: An example automaton as a digraph +
+
+
+
+
+
+

We may also encounter the following term.

+
+

Definition 2 (Active event function and set) Active event function (actually a multivalued function) \Gamma: \mathcal X \rightarrow 2^\mathcal{E} assigns to each state a set of active events. Active event set \Gamma(x) is the set of active events in a particular state x.

+
+
+
+

Finite state automaton (FSA)

+

This may be regarded as a rather superfluous definition – a finite state automaton (FSA) is a state automaton with a finite set \mathcal X of states. It is also known as a finite state machine (FSM).

+
+
+

Execution of an automaton

+
    +
  • x_1\xrightarrow{e_1} x_2\xrightarrow{e_2} x_1 \xrightarrow{e_1} x_2 \xrightarrow{e_4} x_3\ldots
    • +
  • Sometimes also written as x_1,e_1,x_2,e_2,\ldots
    • +
+
+
+
+
+ +
+
+Notational confusion +
+
+
+

Here x_k for some k is the name of a particular state. It is not the name of a (yet to be introduced) state variable; In fact, it can be viewed as its value (also valuation).

+
+
+
    +
  • Some authors strictly distinguish between the state variable and the state (variable valuation), +
      +
    • similarly as in probability theory random variable X vs its value x, as in F(x) = P(X\leq x);
      • +
    • +
  • some do not, but then it may lead to confusion;
    • +
  • yet some others avoid the problem by not introducing state variables and only working with enumerated states.
    • +
+
+
+
+
+ +
+
+Notational confusion 2 +
+
+
+

Even worse, it is also tempting to interpret the lower index k as (discrete) time, but nope, in the previous k is not the time index.

+

Again, some authors do not distinguish…

+
+
+
+
+

Path of an automaton

+

Corresponding to the execution

+

x_1\xrightarrow{e_1} x_2\xrightarrow{e_2} x_1 \xrightarrow{e_1} x_2 \xrightarrow{e_4} x_3\ldots

+

the path is just the sequence of visited states:

+

x_1,x_2,x_1,x_2,x_3,\ldots

+
+

In continuous-valued dynamical systems, we have a state trajectory, but then time stamps are attached to each visited state.

+
+
+

Example 2 (Beverage vending machine)  

+
+
+
+
+
+
+ + +G + + +init +init + + + +waiting + +waiting + + + +init->waiting + + + + + +swiped + +swiped + + + +waiting->swiped + + +swipe card + + + +swiped->waiting + + +reject payment + + + +paid + +paid + + + +swiped->paid + + +accept payment + + + +coke_dispensed + +coke_dispensed + + + +paid->coke_dispensed + + +choose coke + + + +fanta_dispensed + +fanta_dispensed + + + +paid->fanta_dispensed + + +choose fanta + + + +coke_dispensed->waiting + + +take coke + + + +fanta_dispensed->waiting + + +take fanta + + + +
+
+
+Figure 2: Example of a digraph representation of the automaton for a beverage vending machine +
+
+
+
+
+
    +

  • State sequence (path): waiting, swiped, paid, coke_dispensed, waiting

    • +

  • Events sequence: swipe card, accept payment, choose coke, take coke

    • +

  • Indeed, the two states coke_dispensed and fanta_dispensed can be merged into just beverage_dispensed.

    • +

  • How about other paths? Longer? Shorter?

    • +
+
+
+
+
+
+
+ + +G + + +init +init + + + +waiting + + +waiting + + + +init->waiting + + + + + +swiped + +swiped + + + +waiting->swiped + + +swipe card + + + +swiped->waiting + + +reject payment + + + +paid + +paid + + + +swiped->paid + + +accept payment + + + +coke_dispensed + +coke_dispensed + + + +paid->coke_dispensed + + +choose coke + + + +fanta_dispensed + +fanta_dispensed + + + +paid->fanta_dispensed + + +choose fanta + + + +coke_dispensed->waiting + + +take coke + + + +fanta_dispensed->waiting + + +take fanta + + + +
+
+
+Figure 3: Example of a digraph representation of the automaton for a beverage vending machine with a marked state +
+
+
+
+
+

The waiting state can be marked (is accepting).

+
+
+

Example 3 (Longitudinal control of a ground vehicle)  

+
+
+
+
+
+
+ + +G + + +init +init + + + +still + +still + + + +init->still + + + + + +accelerating + +accelerating + + + +still->accelerating + + +push acc + + + +cruising + +cruising + + + +accelerating->cruising + + +cruise ON + + + +coasting + +coasting + + + +accelerating->coasting + + +rel acc + + + +cruising->accelerating + + +push acc + + + +cruising->coasting + + +rel acc + + + +braking + +braking + + + +cruising->braking + + +push brake + + + +coasting->braking + + +push brake + + + +braking->still + + +zero vel + + + +braking->cruising + + +cruise ON + + + +braking->coasting + + +rel brake + + + +
+
+
+Figure 4: Example of a digraph representation of the automaton for a longitudinal control of a ground vehicle +
+
+
+
+
+
+
+
    +
  • By cruise on I mean switching on some kind of a cruise control system, which keeps the velocity constant.
    • +
  • It turns out the optimal control strategy for trains (under some circumstances).
    • +
  • Note that some of the events are indeed actions started by the driver, but some are just coming from the physics of the vehicle (transition from braking to zero velocity).
    • +
+
+
+

Example 4 (Corridor switch)  

+
+
+
+
+
+
+ + +G + + +init +init + + + +OFF + +OFF + + + +init->OFF + + + + + +ON + +ON + + + +OFF->ON + + +switch₁,switch₂ + + + +ON->OFF + + +switch₁,switch₂ + + + +
+
+
+Figure 5: Example of a digraph representation of the automaton for a corridor switch +
+
+
+
+
+

Two events associated with one transitions can be seen as two transitions, each with a single event, both sharing the starting and ending states.

+
+
+

Example 5 (JK flip-flop) We now consider the classical JK flip-flop logical circuit. It symbol is in Fig 7 and the truth table follows. Our goal is to represent its functionality using a state automaton.

+
+
+
+ +
+
+Figure 6: Symbol for a JK flip-flop logical circuit +
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
JKQ_kQ_{k+1}Description
0000No change
0011No change
0100Reset
0110Reset
1001Set
1011Set
1101Toggle
1110Toggle
+
+
+
+
+
+
+ + +G + + +init +init + + + +Low + +Low + + + +init->Low + + + + + +Low->Low + + +¬J ∧ ¬K ∧ clk + + + +High + +High + + + +Low->High + + +J ∧ clk + + + +High->Low + + +K ∧ clk + + + +High->High + + +¬J ∧ ¬K ∧ clk + + + +
+
+
+Figure 7: JK flip-flop as an automaton +
+
+
+
+
+
+
+

Example 6 (Double intensity switching)  

+
+
+
+
+
+
+ + +G + + +init +init + + + +OFF + +OFF + + + +init->OFF + + + + + +ON + +ON + + + +OFF->ON + + +push + + + +ON->OFF + + +push + + + +ON2 + +ON2 + + + +ON->ON2 + + +push + + + +ON2->OFF + + +push + + + +
+
+
+Figure 8: Example of a digraph representation of the automaton for double intensity switching +
+
+
+
+
+

Obviously we need to introduce time into the automaton…

+
+
+
+

State as the value of a state variable

+

Definition of the state space by enumeration (such as \mathcal X = \{0,1,2,3,4,5\}) doesn’t scale well. As an alternative, a state can be characterized by the value (sometimes also valuation) of a state variable. A state variable is then given by

+
    +
  • the name (for example, x),
    • +
  • the “type” (boolean, integer, vector, …).
    • +
+
+

Example 7 (Examples of state variables)  

+
    +
  • Corridor switch: x \in \{\mathrm{false},\mathrm{true}\} (possibly also \{0,1\}).
    • +
  • Double intensity switching: +
      +
    • x \in \{0,1,2\} \subset \mathbb Z,
      • +
    • or \bm x = \begin{bmatrix}x_1\\ x_2 \end{bmatrix}, where x_1,x_2 \in \{0,1\}.
      • +
    • +
+
+
+
+

State (transition) equation

+

Denoting a new state after a transition as x^+, the state equation reads \boxed{x^+ = f(x,e)}

+

Upon introduction of discrete-time (index) k, it can also be rewritten as x_{k+1} = f(x_k,e_k) or also x[k+1] = f(x[k],e[k]).

+
+
+
+ +
+
+Note +
+
+
+
    +
  • The function f can be defined by a computer code rather than a clean mathematical formula.
    • +
  • The discrete-time index of the event is sometimes considered shifted, that is x_{k+1} = f(x_k,e_{k+1}). You should be aware of this.
    • +
+
+
+
+
+

Extensions

+

The concept of an automaton can be extended in several ways. In particular, the following two extensions introduce the concept of an output to an automaton.

+
+

Moore machine

+

One extension of an automaton with outputs is Moore machine. The outputs assigned to the states by the output function y = g(x).

+

The output is produced (emitted) when the (new) state is entered.

+

Note, in particular, that the output does not depend on the input. This has a major advantage when a feedback loop is closed around this system, since no algebraic loop is created.

+

Graphically, we make a conventions that outputs are the labels of the states.

+
+

Example 8 (Moore machine) The following automaton has just three states, but just two outputs (FLOW and NO FLOW).

+
+
+
+
+
+
+ + +G + + +init +init + + + +closed + +NO FLOW +Valve +closed + + + +init->closed + + + + + +partial + +FLOW +Valve +partially +open + + + +closed->partial + + +open valve one turn + + + +partial->closed + + +close valve one turn + + + +full + +FLOW +Valve +fully open + + + +partial->full + + +open valve one turn + + + +full->closed + + +emergency shut off + + + +full->partial + + +close valve one turn + + + +
+
+
+Figure 9: Example of a digraph representation of the Moore machine for a valve control +
+
+
+
+
+
+
+
+

Mealy machine

+

Mealy machine is another extension of an automaton. Here the outputs are associated with the transitions rather than the states.

+

Since the events already associated with the states can be viewed as the inputs, we now have input/output transition labels. The transition label e_\mathrm{i}/e_\mathrm{o} on the transion from x_1 to x_2 reads as “the input event e_\mathrm{i} at state x_1 activates the transition to x_2, which outputs the event e_\mathrm{o}” and can be written as x_1\xrightarrow{e_\mathrm{i}/e_\mathrm{o}} x_2.

+

It can be viewed as if the output function also considers the input and not only the state y = e_\mathrm{o} = g(x,e_\mathrm{i}).

+

In contrast with the Moore machine, here the output is produced (emitted) during the transition (before the new state is entered).

+
+

Example 9 (Mealy machine) Coffee machine: coffee for 30 CZK, machine accepting 10 and 20 CZK coins, no change.

+
+
+
+
+
+
+ + +G + + +init +init + + + +0 + +No coin + + + +init->0 + + + + + +10 + +10 CZK + + + +0->10 + + +insert 10 CZK / no coffee + + + +20 + +20 CZK + + + +0->20 + + +insert 20 CZK / no coffee + + + +10->0 + + +insert 20 CZK / coffee + + + +10->20 + + +insert 10 CZK / no coffee + + + +20->0 + + +insert 10 CZK / coffee + + + +20->10 + + +insert 20 CZK / coffee + + + +
+
+
+Figure 10: Example of a digraph representation of the Mealy machine for a coffee machine +
+
+
+
+
+
+
+

Example 10 (Reformulate the previous example as a Moore machine) Two more states wrt Mealy

+
+
+
+
+
+
+ + +G + + +init +init + + + +0 + +NO COFFEE +No +coin + + + +init->0 + + + + + +10 + +NO COFFEE +10 +CZK + + + +0->10 + + +insert 10 CZK + + + +20 + +NO COFFEE +20 +CZK + + + +0->20 + + +insert 20 CZK + + + +10->20 + + +insert 10 CZK + + + +30 + +COFFEE +10+20 +CZK + + + +10->30 + + +insert 20 CZK + + + +20->30 + + +insert 10 CZK + + + +40 + +COFFEE +20+20 +CZK + + + +20->40 + + +insert 20 CZK + + + +30->0 + + + + + +30->10 + + +insert 10 CZK + + + +30->20 + + +insert 20 CZK + + + +40->10 + + + + + +40->20 + + +insert 10 CZK + + + +40->30 + + +insert 20 CZK + + + +
+
+
+Figure 11: Example of a digraph representation of the Moore machine for a coffee machine +
+
+
+
+
+
+
+
+
+ +
+
+Note +
+
+
+

There are transitions from 30 and 40 back to 0 that are not labelled by any event. This does not seem to follow the general rule that transitions are always triggered by events. Not what? It can be resolved upon introducing time as the timeout transitions.

+
+
+
+

Example 11 (Dijkstra’s token passing) The motivation for this example is to show that it is perhaps not always productive to insist on visual description of the automaton using a graph. The four components of our formal definition of an automaton are just enough, and they translate directly to a code.

+

The example comes from the field of distributed computing systems. It considers several computers that are connected in ring topology, and the communication is just one-directional as Fig 12 shows. The task is to use the communication to determine in – a distributed way – which of the computers carries a (single) token at a given time. And to realize passing of the token to a neighbour. We assume a synchronous case, in which all the computers are sending simultaneously, say, with some fixed sending period.

+
+
+
+
+
+
+ + +G + + +0 + +0 + + + +1 + +1 + + + +0->1 + + + + + +2 + +2 + + + +1->2 + + + + + +3 + +3 + + + +2->3 + + + + + +3->0 + + + + + +
+
+
+Figure 12: Example of a ring topology for Dijkstra’s token passing in a distributed system +
+
+
+
+
+

One popular method for this is called Dijkstra’s token passing. Each computer keeps a single integer value as its state variable. And it forwards this integer value to the neighbour (in the clockwise direction in our setting). Upon receiving the value from the other neighbour (in the counter-clockwise direction), it updates its own value according to the rule displayed in the code below. At every clock tick, the state vector (composed of the individual state variables) is updated according to the function update!() in the code. Based on the value of the state vector, an output is computed, which decodes the informovation about the location of the token from the state vector. Again, the details are in the output() function.

+
+
+Show the code +
struct DijkstraTokenRing
+    number_of_nodes::Int64
+    max_value_of_state_variable::Int64
+    state_vector::Vector{Int64}
+end
+
+function update!(dtr::DijkstraTokenRing)                        
+    n = dtr.number_of_nodes
+    k = dtr.max_value_of_state_variable
+    x = dtr.state_vector
+    xnext = copy(x)
+    for i in eachindex(x)   # Mind the +1 shift. x[2] corresponds to x₁ in the literature.
+        if i == 1                                              
+            xnext[i] = (x[i] == x[n]) ? mod(x[i] + 1,k) : x[i]  # Increment if the left neighbour is identical.
+        else                                                    
+            xnext[i] = (x[i] != x[i-1]) ? x[i-1] : x[i]         # Update by the differing left neighbour.
+        end
+    end
+    dtr.state_vector .= xnext                                              
+end
+
+function output(dtr::DijkstraTokenRing)     # Token = 1, no token = 0 at the given position. 
+    x = dtr.state_vector
+    y = similar(x)
+    y[1] = iszero(x[1]-x[end])
+    y[2:end] .= .!iszero.(diff(x))
+    return y
+end
+
+
+
output (generic function with 1 method)
+
+
+

We now rund the code for a given number of computers and some initial state vector that does not necessarily comply with the requirement that there is only one token in the ring.

+
+
+Show the code +
n = 4                           # Concrete number of nodes.
+k = n                           # Concrete max value of a state variable (>= n).
+@show x_initial = rand(0:k,n)   # Initial state vector, not necessarily acceptable (>1 token in the ring).
+dtr = DijkstraTokenRing(n,k,x_initial)
+@show output(dtr)               # Show where the token is (are).
+
+@show update!(dtr), output(dtr) # Perform the update, show the state vector and show where the token is.
+@show update!(dtr), output(dtr) # Repeat a few times to see the stabilization.    
+@show update!(dtr), output(dtr)
+@show update!(dtr), output(dtr)
+@show update!(dtr), output(dtr)
+
+
+
x_initial = rand(0:k, n) = [1, 0, 4, 3]
+output(dtr) = [0, 1, 1, 1]
+(update!(dtr), output(dtr)) = ([1, 1, 0, 4], [0, 0, 1, 1])
+(update!(dtr), output(dtr)) = ([1, 1, 1, 0], [0, 0, 0, 1])
+(update!(dtr), output(dtr)) = ([1, 1, 1, 1], [1, 0, 0, 0])
+(update!(dtr), output(dtr)) = ([2, 1, 1, 1], [0, 1, 0, 0])
+(update!(dtr), output(dtr)) = ([2, 2, 1, 1], [0, 0, 1, 0])
+
+
+
([2, 2, 1, 1], [0, 0, 1, 0])
+
+
+

We can see that although initially the there can be more tokens, after a few iterations the algorithm achieves the goal of having just one token in the ring.

+
+
+
+

Extended-state automaton

+

Yet another extension of an automaton is the extended-state automaton. And indeed, the hyphen is there on purpose as we extend the state space.

+

In particular, we augment the state variable(s) that define the states/modes/locations (the nodes in the graph) by additional (typed) state variables: Int, Enum, Bool, …

+

Transitions from one mode to another are then guarded by conditions on theses new extra state variables.

+

Besides being guarded by a guard condition, a given transition can also be labelled by a reset function that resets the extended-state variables.

+
+

Example 12 (Counting up to 10) In this example, there are two modes (on and off), which can be captured by a single binary state variable, say x. But then there is an additional integer variable k, and the two variables together characterize the extended state.

+
+
+
+
+
+
+ + +G + + +init +init + + + +OFF + +OFF + + + +init->OFF + + +int k=0 + + + +ON + +ON + + + +OFF->ON + + +press + + + +ON->OFF + + +(press ⋁ k ≥ 10); k=0 + + + +ON->ON + + +(press ∧ k < 10); k=k+1 + + + +
+
+
+Figure 13: Example of a digraph representation of the extended-state automaton for counting up to ten +
+
+
+
+
+
+
+
+
+

Composing automata

+

Any practically useful modelling framework should support decomposition of a large system into smaller subsystems. These should then be able to communicate/synchronize with each other. In automata such synchronization can be realized by sending (or generating) and receiving (or accepting) events. A common choice of symbols for the two is !,?, as illustrated in the following example. But these symbols are just one possible convention, and any other symbols can be used.

+
+

Example 13 (Composing automata)  

+
+
+
+
+
+
+ + +G + + +init +init + + + +1 + +1 + + + +init->1 + + + + + +2 + +2 + + + +1->2 + + +press? + + + +3 + +3 + + + +3->3 + + +press! + + + +
+
+
+Figure 14: Example illustrating how two automata can be synchronized by sending and receiving events +
+
+
+
+
+
+
+
+

Languages and automata

+

When studying automata, we often encounter the concept of a language. Indeed, the concept of an automaton is heavily used in the formal laguage theory. Although in our course we are not going to refer to these results, some resources we recommend for our courses do, and so it is useful to understand how automata and languages are related.

+

First, we extend the definition of a transition function in that it accepts the current state and not just a single event but a sequence of events, that is

+

+f: \mathcal X \times \mathcal E^\ast \rightarrow \mathcal X, + where \mathcal E^\ast stands for the set of all possible sequences of events.

+

Language generated by the automaton is +\mathcal L(\mathcal G) = \{s\in\mathcal E^\ast \mid f(x_0,s) \;\text{is defined}\} +

+

Language marked by the automaton (the automaton is accepting or recognizing that language) +\mathcal L_\mathrm{m}(\mathcal G) = \{s\in\mathcal L(\mathcal G) \mid f(x_0,s) \in \mathcal{X}_\mathrm{m}\} +

+
+

Example 14 (Language accepted by automaton) +\mathcal{E} = \{a,b\}, \mathcal{L} = \{a,aa,ba,aaa,aba,baa,bba,\ldots\} +

+
+
+
+
+
+
+ + +G + + +init +init + + + +0 + +0 + + + +init->0 + + + + + +1 + + +1 + + + +1->1 + + +a + + + +1->0 + + +b + + + +0->1 + + +a + + + +0->0 + + +b + + + +
+
+
+Figure 15: Example of an automaton generating the language \mathcal{L} = \{a,aa,ba,aaa,aba,baa,bba,\ldots\} +
+
+
+
+
+

What if we remove the self loop at state 0? The automaton then accepts languages starting with a and with b being the last event or immediately followed by a.

+
+
+

What is the language view of automata good for?

+
    +
  • Definitions, analysis, synthesis.
    • +
  • We then need language concepts such as +
      +
    • concatenation of strings: \quad c = ab
      • +
    • empty string \varepsilon: \quad\varepsilon a = a \varepsilon = a
      • +
    • prefix, suffix
      • +
    • prefix closure \bar{\mathcal{L}} (of the language \mathcal L)
      • +
    • …
      • +
    • +
+
+
+
+

Blocking

+

An important concept in automata is blocking. A state is blocking if there is no transition out of it. An example follows.

+
+

Example 15 (Blocking states) In the automaton in Fig 16, state 2 is blocking. It is a deadlock state. States 3 and 4 are livelock states.

+
+
+
+
+
+
+ + +G + + +init +init + + + +0 + +0 + + + +init->0 + + + + + +2 + + +2 + + + +2->0 + + +g + + + +1 + +1 + + + +0->1 + + +a + + + +1->2 + + +b + + + +5 + +5 + + + +1->5 + + +g + + + +3 + +3 + + + +1->3 + + +a + + + +4 + +4 + + + +3->4 + + +b + + + +4->3 + + +a + + + +4->4 + + +g + + + +
+
+
+Figure 16: Example of an automaton with blocking states +
+
+
+
+
+

Language characterization: \bar{\mathcal{L}}_\mathrm{m}(\mathcal G) \sub \mathcal L(\mathcal G).

+
+
+
+

Queueing systems

+

Queueing systems are a particular and very useful class of discrete-event systems. They consist of these three components:

+
    +
  • entities (also customers, jobs, tasks, requests, etc.)
    • +
  • resources (also servers, processors, etc.): customers are waiting for them
    • +
  • queues (also buffers): where waiting is done
    • +
+

A common graphical representation that contains all these three compoments is in Fig 17.

+
+
+
+ +
+
+Figure 17: Queueing system +
+
+
+
+

Examples of queueing systems

+
    +
  • entities: people waiting for service in a bank or at a bust stop
    • +
  • resources: people (again) in a bank at the counter
    • +
  • queues: bank lobbies, bus stops, warehouses, …
    • +
+
+
+
+ +
+
+Note +
+
+
+

What are other examples?

+
    +
  • entities: packets, …
    • +
  • resources: processor, computer periphery, router, …
    • +
  • queues: …
    • +
+
+
+
+
+

Why shall we study queueing systems?

+
    +
  • Resources are not unlimited
    • +
  • Tradeoff needed between customer satisfaction and fair resources allocation
    • +
+
+
+

Networks of queueing systems

+

Queueing systems can be interconnected into networks.

+
+
+
+ +
+
+Figure 18: Example of a network of queueing systems +
+
+
+
+
+

Queueing systems as automata

+

The reason why we mentioned queueing systems in this part of our course is that they can be modelled as automata. And we already know that in order to define and automaton, we must characterize the key components defining the automaton – three in this case:

+
    +

  • events: \mathcal E = \{\text{arrival},\text{departure}\};

    • +

  • states: number of customers in the queue +\mathcal X = \{0,1,2,3,\ldots\}, \quad \mathcal X_0 = \{0\}, +

    • +
+
+
+
+ +
+
+Note +
+
+
+

Obviously this is not a finite state automation – unless the queue is bounded – and whether the queue’s length is bounded is a modelling assumption.

+
+
+
    +
  • state transition: +f(x,e) = +\begin{cases} +x+1, & \text{if}\; x\leq 0 \land e = \mathrm{arrival}\\ +x-1, & \text{if}\; x > 0 \land e = \mathrm{departure}. +\end{cases} +
    • +
+
+
+

Queueing system as an automaton

+
+
+
+ +
+
+Figure 19: Queueing system as an automaton +
+
+
+
+
+
+ +
+
+Note +
+
+
+

Note how the states correspond to the value of the state variable.

+
+
+
+

Example 16 (Example of a queueing system: jobs processing by a CPU) …

+
+
+
+

Stochastic queueing systems

+

An important extension of the basic concept of a queueing system is the introduction of randomness. In particular, the arrivals can be modelled using random processes. Similarly, the departures given by the delays (the processing time) of the server can be modelled as random.

+

Obviously, the time needs to be included in the automaton, and so far we do not have it there. It is then high time to introduce it.

+
+
+
+

Timed automaton

+

So far, even if the automaton corresponded to a physical system (and did not just represent a generator of a language), the time was not included. The transitions were triggered by the events, but we did not specify the time at which the event occurred.

+

There are, however, many situations when it is useful or even crucial to incorporate time. We can then answer questions such as

+
    +
  • How many events of a certain type in a given interval?
    • +
  • Is the time interval between two events above a given threshold?
    • +
  • How long does the system spend in a given state?
    • +
  • …
    • +
+

There are several ways how to incorporate time into the automaton. We will follow the concept of a timed automaton with guards (introduced by Alur and Dill). Within their framework we have

+
    +
  • one or several resettable clocks: c_i,\, i=1,\ldots, k, driven by the ODE + \frac{\mathrm{d} c_i(t)}{\mathrm d t} = 1, \quad c_i(0) = 0; +
    • +
  • each transition labelled by the tripple {guard; event; reset}.
    • +
+
+
+
+ +
+
+Note +
+
+
+

Both satisfaction of the guard and arrival of the event constitute enabling conditions for the transition. They could be wrapped into a single compound condition.

+
+
+
+

Example 17 (Timed automaton with guards)  

+
+
+
+
+
+
+ + +G + + +init +init + + + +0 + +0 + + + +init->0 + + + + + +1 + +1 + + + +0->1 + + +-; msg; c₁ + + + +1->1 + + +c₁≥1; msg; c₁ + + + +2 + +2 + + + +1->2 + + +0<c₁<1; msg; c₁ + + + +3 + +3 + + + +2->3 + + +c₁<1; alarm; - + + + +
+
+
+Figure 20: Example of a timed automaton with guards +
+
+
+
+
+
+
+

Example 18 (Timed automaton with guards and invariant)  

+
+
+
+
+
+
+ + +G + + +init +init + + + +0 + +0 + + + +init->0 + + + + + +2 + +2 +c₁<1 + + + +3 + +3 + + + +2->3 + + +-; alarm; - + + + +1 + +1 + + + +0->1 + + +-; msg; c₁ + + + +1->2 + + +0<c₁<1; msg; c₁ + + + +1->1 + + +c₁≥1; msg; c₁ + + + +
+
+
+Figure 21: Example of a timed automaton with guards and invariant +
+
+
+
+
+
+
+

Invariant vs guard

+
    +
  • Invariant (of a location) gives an upper bound on the time the system can stay at the given location. It can leave earlier but not later.
    • +
  • Guard (of a given transition) gives an enabling condition on leaving the location through the given transition.
    • +
+
+

Example 19 (Several trains approaching a bridge) The example is taken from [1] and is included in the demos coming with the Uppaal tool.

+
+ + + +
+
+ + Back to top

References

+
+
[1]
G. Behrmann, A. David, and K. G. Larsen, “A Tutorial on Uppaal,” in Formal Methods for the Design of Real-Time Systems, M. Bernardo and F. Corradini, Eds., in Lecture Notes in Computer Science, no. 3185., Berlin, Heidelberg: Springer, 2004, pp. 200–236. doi: 10.1007/978-3-540-30080-9_7.
+
+
+ + +
+
+
+
+   +
+
+

Copyright 2024, Zdeněk Hurák

+
+
+   +
+
+
+ + + + + \ No newline at end of file diff --git a/des_automata.html b/des_automata.html index 9e5784f..7c50813 100644 --- a/des_automata.html +++ b/des_automata.html @@ -2131,16 +2131,16 @@

Mealy machine

@show update!(dtr), output(dtr)
-
x_initial = rand(0:k, n) = [4, 0, 3, 2]
+
x_initial = rand(0:k, n) = [2, 0, 2, 3]
 output(dtr) = [0, 1, 1, 1]
-(update!(dtr), output(dtr)) = ([4, 4, 0, 3], [0, 0, 1, 1])
-(update!(dtr), output(dtr)) = ([4, 4, 4, 0], [0, 0, 0, 1])
-(update!(dtr), output(dtr)) = ([4, 4, 4, 4], [1, 0, 0, 0])
-(update!(dtr), output(dtr)) = ([1, 4, 4, 4], [0, 1, 0, 0])
-(update!(dtr), output(dtr)) = ([1, 1, 4, 4], [0, 0, 1, 0])

+(update!(dtr), output(dtr)) = ([2, 2, 0, 2], [1, 0, 1, 1])
+(update!(dtr), output(dtr)) = ([3, 2, 2, 0], [0, 1, 0, 1])
+(update!(dtr), output(dtr)) = ([3, 3, 2, 2], [0, 0, 1, 0])
+(update!(dtr), output(dtr)) = ([3, 3, 3, 2], [0, 0, 0, 1])
+(update!(dtr), output(dtr)) = ([3, 3, 3, 3], [1, 0, 0, 0])
-
([1, 1, 4, 4], [0, 0, 1, 0])
+
([3, 3, 3, 3], [1, 0, 0, 0])

We can see that although initially the there can be more tokens, after a few iterations the algorithm achieves the goal of having just one token in the ring.

diff --git a/hybrid_automata_software 23.html b/hybrid_automata_software 23.html new file mode 100644 index 0000000..acc2cb0 --- /dev/null +++ b/hybrid_automata_software 23.html @@ -0,0 +1,1087 @@ + + + + + + + + + +Software – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + +
+ + + +
+ +
+
+

Software

+
+ + + +
+ + + + +
+ + + +
+ + +
+

Matlab

+ +
+
+

Julia

+ +
+
+

Python

+
    +
  • …
    • +
+ + +
+ + Back to top
+ + +
+
+
+
+   +
+
+

Copyright 2024, Zdeněk Hurák

+
+
+   +
+
+
+ + + + + \ No newline at end of file diff --git a/hybrid_equations.html b/hybrid_equations.html index 1e5056b..e6d2fde 100644 --- a/hybrid_equations.html +++ b/hybrid_equations.html @@ -1177,57 +1177,57 @@

- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Figure 7: Simulation of stabilization on a circle using a hybrid controller @@ -1251,54 +1251,54 @@

- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Figure 8: Simulation of stabilization on a circle using a hybrid controller diff --git a/hybrid_equations_references 24.html b/hybrid_equations_references 24.html new file mode 100644 index 0000000..a431e05 --- /dev/null +++ b/hybrid_equations_references 24.html @@ -0,0 +1,1093 @@ + + + + + + + + + +Literature – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + +
+ + + +
+ +
+
+

Literature

+
+ + + +
+ + + + +
+ + + +
+ + +

The theoretical and computational framework of hybrid equations has been mostly developed by a relatively small circle of researchers (Sanfelice, Goebel, Teel, …). The primary monograph is [1]. It is also supported by a freely available Matlab toolbox, see the section on software.

+

+

The book [2] can be regarded as a predecessor and/or complement of the just mentioned [1]. Although the book is not available online, a short version appears as an article [3] in the popular IEEE Control Systems magazine (the one with color figures :-).

+

+ + + + + Back to top

References

+
+
[1]
R. G. Sanfelice, Hybrid Feedback Control. Princeton University Press, 2021. Accessed: Sep. 23, 2020. [Online]. Available: https://press.princeton.edu/books/hardcover/9780691180229/hybrid-feedback-control
+
+
+
[2]
R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press, 2012. Available: https://press.princeton.edu/books/hardcover/9780691153896/hybrid-dynamical-systems
+
+
+
[3]
R. Goebel, R. G. Sanfelice, and A. R. Teel, “Hybrid dynamical systems,” IEEE Control Systems Magazine, vol. 29, no. 2, pp. 28–93, Apr. 2009, doi: 10.1109/MCS.2008.931718.
+
+
+ + +
+
+
+
+   +
+
+

Copyright 2024, Zdeněk Hurák

+
+
+   +
+
+
+ + + + + \ No newline at end of file diff --git a/intro_outline 25.html b/intro_outline 25.html new file mode 100644 index 0000000..27633d8 --- /dev/null +++ b/intro_outline 25.html @@ -0,0 +1,1097 @@ + + + + + + + + + +Course outline – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + +
+ + + +
+ +
+
+

Course outline

+
+ + + +
+ + + + +
+ + + +
+ + +

The course is structured into 14 topics, each of them corresponding to one lecture. The topics are as follows:

+
    +
  • Discrete-event systems +
      +
    1. (State) automata (state machines) (incl. timed variants)
      2. +
    2. Petri nets (and timed Petri nets),
      3. +
    3. Max-Plus algebra and Max-Plus Linear (MPL) systems
      4. +
    • +
  • Hybrid systems +
      +
    1. Hybrid automata
      2. +
    2. Hybrid equations
      3. +
    • +
  • Special classes of hybrid systems +
      +
    1. Reset (control) systems, Switched/switching systems, Piecewise affine systems (PWA)
      2. +
    2. Complementarity dynamical systems (and complementarity optimization constraints)
      +
      3. +
    • +
+
    +
  1. Solutions of hybrid systems
    2. +
+
    +
  • Stability of hybrid systems +
      +
    1. Common Lyapunov function +
        +
      • Quadratic Lyapunov function via linear matrix inequality (LMI) and semidefinite programming (SDP)
        • +
      • Polynomial Lyapunov function via sum-of-squares (SOS) programming
        • +
      2. +
    2. Piecewise quadratic/polynomial Lyapunov function via S-procedure
      3. +
    • +
+
    +
  1. Mixed-logical dynamical (MLD) description of hybrid systems
    2. +
  2. Model predictive control (MPC) for MLD systems
    3. +
  3. (Formal) verification of hybrid systems
    4. +
+ + + + Back to top
+ + +
+
+
+
+   +
+
+

Copyright 2024, Zdeněk Hurák

+
+
+   +
+
+
+ + + + + \ No newline at end of file diff --git a/max_plus_algebra 13.html b/max_plus_algebra 13.html new file mode 100644 index 0000000..ecd27b1 --- /dev/null +++ b/max_plus_algebra 13.html @@ -0,0 +1,8168 @@ + + + + + + + + + +Max-plus algebra – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + +
+ + + +
+ +
+
+

Max-plus algebra

+
+ + + +
+ + + + +
+ + + +
+ + +

Max-plus algebra, also written as (max,+) algebra (and also known as tropical algebra/geometry and dioid algebra), is an algebraic framework in which we can model and analyze a class of discrete-event systems, namely event graphs, which we have previously introduced as a subset of Petri nets. The framework is appealing in that the models than look like state equations \bm x_{k+1} = \mathbf A \bm x_k + \mathbf B \bm u_k for classical linear dynamical systems. We call these max-plus linear systems, or just MPL systems. Concepts such as poles, stability and observability can be defined, following closely the standard definitions. In fact, we can even formulate control problems for these models in a way that mimicks the conventional control theory for LTI systems, including MPC control.

+

But before we get to these applications, we first need to introduce the (max,+) algebra itself. And before we do that, we recapitulate the definition of a standard algebra.

+
+
+
+ +
+
+Algebra is not only a branch of mathematics +
+
+
+

Confusingly enough, algebra is used both as a name of both a branch of mathematics and a special mathematical structure. In what follows, we use the term algebra to refer to the latter.

+
+
+
+

Algebra

+

Algebra is a set of elements equipped with

+
    +
  • two operations: +
      +
    • addition (plus, +),
      • +
    • multiplication (times, ×),
      • +
    • +
  • neutral (identity) element with respect to addition: zero, 0, a+0=a,
    • +
  • neutral (identity) element with respect to multiplication: one, 1, a\times 1 = a.
    • +
+

Inverse elements can also be defined, namely

+
    +
  • Inverse element wrt addition: -a a+(-a) = 0.
    • +
  • Inverse element wrt multiplication (except for 0): a^{-1} a \times a^{-1} = 1.
    • +
+

If the inverse wrt multiplication exists for every (nonzero) element, the algebra is called a field, otherwise it is called a ring.

+

Prominent examples of a ring are integers and polynomials. For integers, it is only the numbers 1 and -1 that have integer inverses. For polynomials, it is only zero-th degree polynomials that have inverses qualifying as polynomials too. An example from the control theory is the ring of proper stable transfer functions, in which only the non-minimum phase transfer functions with zero relative degree have inverses, and thus qualify as units.

+

Prominent example of a field is the set of real numbers.

+
+
+

(max,+) algebra: redefining the addition and multiplication

+

Elements of the (max,+) algebra are real numbers, but it is still a ring and not a field since the two operations are defined differently.

+

The new operations of addition, which we denote by \oplus to distinguish it from the standard addition, is defined as \boxed{ + x\oplus y \equiv \max(x,y). + } +

+

The new operation of multiplication, which we denote by \otimes to distinguish it from the standard multiplication, is defined as \boxed{ + x\otimes y \equiv x+y}. +

+
+
+
+ +
+
+Important +
+
+
+

Indeed, there is no typo here, the standard addition is replaced by \otimes and not \oplus.

+
+
+
+
+
+ +
+
+(min,+) also possible +
+
+
+

Indeed, we can also define the (min,+) algebra. But for our later purposes in modelling we prefer the (max,+) algebra.

+
+
+
+

Reals must be extended with the negative infinity

+

Strictly speaking, the (max,+) algebra is a broader set than just \mathbb R. We need to extend the reals with the minus infinity. We denote the extended set by \mathbb R_\varepsilon \boxed{ +\mathbb R_\varepsilon \coloneqq \mathbb R \cup \{-\infty\}}. +

+

The reason for the notation is that a dedicated symbol \varepsilon is assigned to this minus infinity, that is, \boxed +{\varepsilon \coloneqq -\infty.} +

+

It may yield some later expressions less cluttered. Of course, at the cost of introducing one more symbol.

+

We are now going to see the reason for this extension.

+
+
+

Neutral elements

+
+

Neutral element with respect to \oplus

+

The neutral element with respect to \oplus, the zero, is -\infty. Indeed, for x \in \mathbb R_\varepsilon +x \oplus \varepsilon = x, + because \max(x,-\infty) = x.

+
+
+

Neutral element with respect to \otimes

+

The neutral element with respect to \otimes, the one, is 0. Indeed, for x \in \mathbb R_\varepsilon +x \otimes \varepsilon = x, + because x+0=x.

+
+
+
+ +
+
+Nonsymmetric notation, but who cares? +
+
+
+

The notation is rather nonsymmetric here. We now have a dedicated symbol \varepsilon for the zero element in the new algebra, but no dedicated symbol for the one element in the new algebra. It may be a bit confusing as “the old 0 is the new 1”. Perhaps similarly as we introduced dedicated symbols for the new operations of addition of multiplication, we should have introduced dedicated symbols such as ⓪ and ①, which would lead to expressions such as x⊕⓪=x and x ⊗ ① = x. In fact, in some software packages they do define something like mp-zero and mp-one to represent the two special elements. But this is not what we will mostly encounter in the literature. Perhaps the best attitude is to come to terms with this notational asymetry… After all, I myself was apparently not even able to figure out how to encircle numbers in LaTeX…

+
+
+
+
+
+

Inverse elements

+
+

Inverse with respect to \oplus

+

The inverse element with respect to \oplus in general does not exist! Think about it for a few moments, this is not necessarily intuitive. For which element(s) does it exist? Only for \varepsilon.

+

This has major consequences, for example, x\oplus x=x.

+

Can you verify this statement? How is is related to the fact that the inverse element with respect to \oplus does not exist in general?

+

This is the key difference with respect to a conventional algebra, wherein the inverse element of a wrt conventional addition is -a, while here we do not even define \ominus.

+

Formally speaking, the (max,+) algebra is only a semi-ring.

+
+
+

Inverse with respect to \otimes

+

The inverse element with respect to \otimes does not exist for all elements. The \varepsilon element does not have an inverse element with respect to \otimes. But in this aspect the (max,+) algebra just follows the conventional algebra, beucase 0 has no inverse there either.

+
+
+
+
+

Powers and the inverse with respect to \otimes

+

Having defined the fundamental operations and the fundamental elements, we can proceed with other operations. Namely, we consider powers. Fot an integer r\in\mathbb Z, the rth power of x, denoted by x^{\otimes^r}, is defined, unsurprisingly as x^{\otimes^r} \coloneqq x\otimes x \otimes \ldots \otimes x.

+

Observe that it corresponds to rx in the conventional algebra x^{\otimes^r} = rx.

+

But then the inverse element with respect to \otimes can also be determined using the (-1)th power as x^{\otimes^{-1}} = -x.

+

This is not actually surprising, is it?

+

There are few more implications. For example,
+x^{\otimes^0} = 0.

+

There is also no inverse element with respect to \otimes for \varepsilon, but it is expected as \varepsilon is a zero wrt \oplus. Furthermore, for r\neq -1, if r>0 , then \varepsilon^{\otimes^r} = \varepsilon, if r<0 , then \varepsilon^{\otimes^r} is undefined, which are both expected. Finally, \varepsilon^{\otimes^0} = 0 by convention.

+
+
+

Order of evaluation of (max,+) formulas

+

It is the same as that for the conventional algebra:

+
    +
  1. power,
    2. +
  2. multiplication,
    3. +
  3. addition.
    4. +
+
+
+

(max,+) polynomials (aka tropical polynomials)

+

Having covered addition, multiplication and powers, we can now define (max,+) polynomials. In order to get started, consider the the univariate polynomial p(x) = a_{n}\otimes x^{\otimes^{n}} \oplus a_{n-1}\otimes x^{\otimes^{n-1}} \oplus \ldots \oplus a_{1}\otimes x \oplus a_{0}, + where a_i\in \mathbb R_\varepsilon and n\in \mathbb N.

+

By interpreting the operations, this translates to the following function \boxed +{p(x) = \max\{nx + a_n, (n-1)x + a_{n-1}, \ldots, x+a_1, a_0\}.} +

+
+

Example 1 (1D polynomial) Consider the following (max,+) polynomial +p(x) = 2\otimes x^{\otimes^{2}} \oplus 3\otimes x \oplus 1. +

+

We can interpret it in the conventional algebra as +p(x) = \max\{2x+2,x+3,1\}, + which is a piecewise linear (actually affine) function.

+
+
+Show the code +
using Plots
+x = -5:3
+f(x) = max(2*x+2,x+3,1)
+plot(x,f.(x),label="",thickness_scaling = 2)
+xc = [-2,1]
+yc = f.(xc)
+scatter!(xc,yc,markercolor=[:red,:red],label="",thickness_scaling = 2)
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
+
+

Example 2 (Example of a 2D polynomial) Nothing prevents us from defining a polynomial in two (and more) variables. For example, consider the following (max,+) polynomial +p(x,y) = 0 \oplus x \oplus y. +

+
+
+Show the code +
using Plots
+x = -2:0.1:2;
+y = -2:0.1:2;
+f(x,y) = max(0,x,y)
+z = f.(x',y);
+wireframe(x,y,z,legend=false,camera=(5,30))
+xlabel!("x")
+ylabel!("y")
+zlabel!("f(x,y)")
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
+
+

Example 3 (Another 2D polynomial) Consider another 2D (max,+) polynomial +p(x,y) = 0 \oplus x \oplus y \oplus (-1)\otimes x^{\otimes^2} \oplus 1\otimes x\otimes y \oplus (-1)\otimes y^{\otimes^2}. +

+
+
+Show the code +
using Plots
+x = -2:0.1:2;
+y = -2:0.1:2;
+f(x,y) = max(0,x,y,2*x-1,x+y+1,2*y-1)
+z = f.(x',y);
+wireframe(x,y,z,legend=false,camera=(15,30))
+xlabel!("x")
+ylabel!("y")
+zlabel!("p(x,y)")
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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+Piecewise affine (PWA) functions +
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+

Piecewise affine (PWA) functions will turn out a frequent buddy in our course.

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+

Solution set (zero set)

+

…

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+

Matrix computations

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+

Addition and multiplication

+

What is attractive about the whole (max,+) framework is that it also extends nicely to matrices. For matrices, whose elements are in \mathbb R_\varepsilon, we define the operations of addition and multiplication identically as in the conventional case, we just use different definitions of the two basic scalar operations. (A\oplus B)_{ij} = a_{ij}\oplus b_{ij} = \max(a_{ij},b_{ij}) +\begin{aligned} +(A\otimes B)_{ij} &= \bigoplus_{k=1}^n a_{ik}\otimes b_{kj}\\ +&= \max_{k=1,\ldots, n}(a_{ik}+b_{kj}) +\end{aligned} +

+
+
+

Zero and identity matrices

+

(max,+) zero matrix \mathcal{E}_{m\times n} has all its elements equal to \varepsilon, that is, +\mathcal{E}_{m\times n} = +\begin{bmatrix} +\varepsilon & \varepsilon & \ldots & \varepsilon\\ +\varepsilon & \varepsilon & \ldots & \varepsilon\\ +\vdots & \vdots & \ddots & \vdots\\ +\varepsilon & \varepsilon & \ldots & \varepsilon +\end{bmatrix}. +

+

(max,+) identity matrix I_n has 0 on the diagonal and \varepsilon elsewhere, that is, +I_{n} = +\begin{bmatrix} +0 & \varepsilon & \ldots & \varepsilon\\ +\varepsilon & 0 & \ldots & \varepsilon\\ +\vdots & \vdots & \ddots & \vdots\\ +\varepsilon & \varepsilon & \ldots & 0 +\end{bmatrix}. +

+
+
+

Matrix powers

+

The zerothe power of a matrix is – unsurprisingly – the identity matrix, that is, A^{\otimes^0} = I_n.

+

The kth power of a matrix, for k\in \mathbb N\setminus\{0\}, is then defined using A^{\otimes^k} = A\otimes A^{\otimes^{k-1}}.

+
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+
+

Connection with graph theory – precedence graph

+

Consider A\in \mathbb R_\varepsilon^{n\times n}. For this matrix, we can define the precedence graph \mathcal{G}(A) as a weighted directed graph with the vertices 1, 2, …, n, and with the arcs (j,i) with the associated weights a_{ij} for all a_{ij}\neq \varepsilon. The kth power of the matrix is then

+

+(A)^{\otimes^k}_{ij} = \max_{i_1,\ldots,i_{k-1}\in \{1,2,\ldots,n\}} \{a_{ii_1} + a_{i_1i_2} + \ldots + a_{i_{k-1}j}\} + for all i,j and k\in \mathbb N\setminus 0.

+
+

Example 4 (Example) +A = +\begin{bmatrix} +2 & 3 & \varepsilon\\ +1 & \varepsilon & 0\\ +2 & -1 & 3 +\end{bmatrix} +\qquad +A^{\otimes^2} = +\begin{bmatrix} +4 & 5 & 3\\ +3 & 4 & 3\\ +5 & 5 & 6 +\end{bmatrix} +

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+Figure 1: An example of a precedence graph +
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Irreducibility of a matrix

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  • Matrix in \mathbb R_\varepsilon^{n\times n} is irreducible if its precedence graph is strongly connected.
    • +
  • Matrix is irreducible iff +(A \oplus A^{\otimes^2} \oplus \ldots A^{\otimes^{n-1}})_{ij} \neq \varepsilon \quad \forall i,j, i\neq j. +\tag{1}
    • +
+
+
+
+

Eigenvalues and eigenvectors

+

Eigenvalues and eigenvectors constitute another instance of a straightforward import of concepts from the conventional algebra into the (max,+) algebra – just take the standard definition of an eigenvalue-eigenvector pair and replace the conventional operations with the (max,+) alternatives +A\otimes v = \lambda \otimes v. +

+

A few comments:

+
    +
  • In general, total number of (max,+) eigenvalues <n.
    • +
  • An irreducible matrix has only one (max,+) eigenvalue.
    • +
  • Graph-theoretic interpretation: maximum average weight over all elementary circuits…
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+

Solving (max,+) linear equations

+

We can also define and solve linear equations within the (max,+) algebra. Considering A\in \mathbb R_\varepsilon^{n\times n},\, b\in \mathbb R_\varepsilon^n, we can formulate and solve the equation +A\otimes x = b. +

+

In general no solution even if A is square. However, often we can find some use for a subsolution defined as +A\otimes x \leq b. +

+

Typically we search for the maximal subsolution instead, or subsolutions optimal in some other sense.

+
+

Example 5 (Greatest subsolution) +A = +\begin{bmatrix} +2 & 3 & \varepsilon\\ +1 & \varepsilon & 0\\ +2 & -1 & 3 +\end{bmatrix}, +\qquad +b = +\begin{bmatrix} +1 \\ 2 \\ 3 +\end{bmatrix} +

+

+x = +\begin{bmatrix} +-1\\ -2 \\ 0 +\end{bmatrix} +

+

+A \otimes x = +\begin{bmatrix} +1\\ 0 \\ 1 +\end{bmatrix} +\leq b +

+
+

With this introduction to the (max,+) algebra, we are now ready to move on to the modeling of discrete-event systems using the max-plus linear (MPL) systems.

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Copyright 2024, Zdeněk Hurák

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Max-plus linear (MPL) systems

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We start with an example of a discrete-event systems modelled using (max,+) algebra.

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Example 1 (Production system)  

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  • There are 3 production units: \(P_1, P_2, P_3\).
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  • The unit \(P_3\) waits for the outputs from the units \(P_1\) and \(P_2\).
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  • Each unit introduces a processing delay: \(v_1 = 12, v_2 = 11, v_3 = 7\), respectively.
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  • There are also transportation delays: \(d_2 = 2\) from the entry to \(P_2\), and \(d_4 = 1\) from \(P_2\) to \(P_3\). All the other transportation delays are negligible.
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Timed Petri net (event graph) for the example is in Figure 1 below.

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+Figure 1: Example of a production system modelled by a timed Petri net (only the transitions are timed) +
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The Petri net can be made more compact by associating the delays also with the places, as in Figure 2 below.

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+Figure 2: Example of a production system modelled by a timed Petri net (both places and transitions are timed) +
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With the outputs from the three transitions (the rectangles) after the kth event labelled by \(x_{1,k}\), \(x_{2,k}\), and \(x_{3,k}\), the state equations are \[ +\begin{aligned} +x_{1,k} &= \max\{x_{1,k-1} + 12, u_k + 0\}\\ +x_{2,k} &= \max\{x_{2,k-1} + 11, u_k + 2\}\\ +x_{3,k} &= \max\{x_{3,k-1} + 7, x_{1,k} +12 + 0, x_{2,k} + 11 + 1\}\\ +&= \max\{x_{3,k-1} + 7, \max\{x_{1,k-1} + 12, u_k\} +12, \\ +&\qquad \max\{x_{2,k-1} + 11, u_k+2\} + 12\}\\ +&= \max\{x_{3,k-1} + 7, x_{1,k-1} + 24, x_{2,k-1} + 23, \\ +&\qquad\qquad u_k+14\}\\ +y_k &= x_{3,k} + 7 +\end{aligned} +\]

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The state equations can be rewritten in the (max,+) algebra \[ +\begin{aligned} +\begin{bmatrix} +x_{1,k} \\ x_{2,k} \\ x_{3,k} +\end{bmatrix} +&= +\begin{bmatrix} +12 & \varepsilon & \varepsilon\\ +\varepsilon & 11 & \varepsilon\\ +24 & 23 & 7 +\end{bmatrix} +\otimes +\begin{bmatrix} +x_{1,k-1} \\ x_{2,k-1} \\ x_{3,k-1} +\end{bmatrix} +\oplus +\begin{bmatrix} +0 \\ 2 \\ 14 +\end{bmatrix} +\otimes +u_k\\ +y_k &= +\begin{bmatrix} +\varepsilon & \varepsilon & 7 +\end{bmatrix} +\otimes +x_k +\end{aligned} +\]

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Model of an event graph as a Max-plus linear (MPL) state-space system

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Generalizing what we have seen in the previous example, we can write the MPL state-space system (actually a model) as \[\boxed{ + \begin{aligned} + x(k) &= A\otimes x(k-1) \oplus B\otimes u(k),\\ + y(k) &= C\otimes x(k), + \end{aligned}} +\tag{1}\]

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where \(A\), \(B\), and \(C\) are matrices of appropriate dimensions. or, equivalently (after relabelling) \[ + \begin{aligned} + x(k+1) &= A\otimes x(k) \oplus B\otimes u(k),\\ + y(k) &= C\otimes x(k), + \end{aligned} +\]
+which mimics the conventional state-space system \[ + \begin{aligned} + x(k+1) &= A x(k) + Bu(k),\\ + y(k) &= Cx(k). + \end{aligned} +\]

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We already know this from the example, but we need to emphasize it here again: the role of the variables \(u(k), x(k), y(k)\) is that they are event times. Namely the times of

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  • arrivals of inputs,
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  • beginning of processing
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respectively.

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The independent variable \(k\) is now a counter of the events.

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State response of an MPL system

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In order to simulate an MPL system, we can now find use of the definitions of the basic operations in (max,+) algebra that we studied previously. Note that \[ +\begin{aligned} +x_1 &= A\otimes x_0 \oplus B\otimes u_1\\ +x_2 &= A\otimes x_1 \oplus B\otimes u_2\\ + &= A\otimes (A\otimes x_0 \oplus B\otimes u_1) \oplus B\otimes u_2\\ + &= A^{\otimes^2}\otimes x_0 \oplus A\otimes B\otimes u_1 \oplus B\otimes u_2\\ + &\vdots +\end{aligned} +\] which can be generalized to \[\boxed{ +x_k = A^{\otimes^k}\otimes x_0 \oplus \bigoplus_{i=1}^k A^{\otimes^{k-i}} \otimes B\otimes u_i.} +\tag{2}\]

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+Response of an LTI state-space system +
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The response of a linear time-invariant (LTI) system described by a (vector) state equation \(x(k+1) = Ax(k) + Bu(k)\) is \[ +x_{k} = A^k x_0 + \sum_{i=0}^{k-1} A^{k-1-i}Bu_i. +\]

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+Lower and upper bounds for the summation shifted by 1 +
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Note how the lower and upper bounds for the summation are shifted by 1 compared to the traditional convolution.

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(max,+) linearity

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We should emphasize that the linearity exhibited by the state equation Equation 1 and the convolution Equation 2 must only be understood in the (max,+) sense.

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Indeed, if we consider two input sequences \(u_1= \{u_{1,1},u_{1,2},\ldots\}\) and \(u_2= \{u_{2,1},u_{2,2},\ldots\}\), a (max,+)-linear combination \(\alpha \otimes u_1 \oplus \beta \otimes u_2\) of the two inputs yields the same (max,+)-linear combination of the outputs \(y_1\) and \(y_2\).

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Input-output response of an MPL system

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We can also eliminate the state variables from the model and aim at finding the relation between the input and output sequences \[ +U = \begin{bmatrix}u_1 \\ u_2 \\ \vdots \\ u_p\end{bmatrix}, \qquad Y = \begin{bmatrix}y_1 \\ y_2 \\ \vdots \\ y_p\end{bmatrix} +\] in the form of \[\boxed +{Y = G\otimes x_0 \oplus H\otimes U,} +\] where \[ +H = +\begin{bmatrix} +C\otimes B & \varepsilon & \varepsilon & \ldots & \varepsilon\\ +C\otimes A\otimes B & C\otimes B & \varepsilon & \ldots & \varepsilon\\ +C\otimes A^{\otimes^2}\otimes B & C\otimes A\otimes B & C\otimes B & \ldots & \varepsilon\\ +\vdots & \vdots & \vdots & \ddots & \vdots\\ +C\otimes A^{\otimes^{p-1}}\otimes B & C\otimes A^{\otimes^{p-2}}\otimes B & C\otimes A^{\otimes^{p-3}}\otimes B & \ldots & C\otimes B +\end{bmatrix} +\] and \[ +G = +\begin{bmatrix} +C \\ C\otimes A \\ C\otimes A^{\otimes^2} \\ \vdots \\ C\otimes A^{\otimes^{p-1}} +\end{bmatrix}. +\]

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Example 2 (Production system) We consider again the production system in Example 1. On the time horizon of 4, and assuming zero initial state, the input-output model is paramaterized by \[ +Y = \begin{bmatrix}y_1 & y_2 & y_3 & y_4\end{bmatrix}^\top, \quad U = \begin{bmatrix}u_1 & u_2 & u_3 & u_4\end{bmatrix}^\top, +\]

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\[ +x_0 = \begin{bmatrix}\varepsilon & \varepsilon & \varepsilon\end{bmatrix}^\top, +\]

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\[ +H = +\begin{bmatrix} +21 & \varepsilon & \varepsilon & \varepsilon\\ +32 & 21 & \varepsilon & \varepsilon\\ +43 & 32 & 21 & \varepsilon\\ +55 & 43 & 32 & 21 +\end{bmatrix}. +\]

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Analysis of an irreducible MPL system

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We now consider an autonomous MPL system \[ +x_{k+c} = A^{\otimes^{k+c}}\otimes x_0, +\] for which we assume irreducibility of the matrix \(A\).

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We have learnt previously, that for large enough \(k\) and \(c\), \[ +x_{k+c} = \lambda^{\otimes^c}\otimes A^{\otimes^{k}}\otimes x_0 = \lambda^{\otimes^c}\otimes x_k. +\]

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This can be interpreted in the standard algabra as \[ +x_{k+c} = c\lambda + x_k, +\] from which it follows that \[ +x_{k+c}-x_k = c\lambda. +\]

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This is an insightful result. When the system under consideration is a production system, then once it reaches a cyclic behaviour, the average cycle is \(\lambda\). The average production rate is then \(1/\lambda\).

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Model Predictive Control (MPC) for MPL systems

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Now we are finally ready to consider control problems form MPL systems. We will consider the MPC approach.

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Cosf function for MPC

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We consider the const function composed of two parts \[ +J = J_\mathrm{output} + \lambda J_\mathrm{input}. +\]

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At “time” \(k\), with the prediction horizon \(N_\mathrm{p}\), and with the number of outputs \(n_\mathrm{y}\): \[ +J_\mathrm{output} = \sum_{j=0}^{N_\mathrm{p}-1}\sum_{i=1}^{n_\mathrm{y}} \max \{y_{i,k+j} - r_{i,k+j},0\} +\]

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This cost function penalizes tardiness (late delivery).

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Is the lower value for j correct?

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Alternative choice of the cost function is \[ +J_\mathrm{output} = \sum_{j=0}^{N_\mathrm{p}-1}\sum_{i=1}^{n_\mathrm{y}} \left|y_{i,k+j} - r_{i,k+j} \right |, +\] which penalizes difference between the due and actual dates, or \[ +J_\mathrm{output} = \sum_{j=1}^{N_\mathrm{p}-1}\sum_{i=1}^{n_\mathrm{y}} \left |\Delta^2 y_{i,k+j}\right |, +\] which balances the output rates.

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The input cost can be set to \[ +J_\mathrm{input} = -\sum_{j=0}^{N_\mathrm{p}-1}\sum_{l=1}^{n_\mathrm{u}} u_{l,k+j}, +\] which penalizes early feeding (favours just-in-time feeding). Note the minus sign.

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Control horizon vs. prediction horizon

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Assume constant feeding rate after the control horizon \(N_\mathrm{c}\) \[ +\Delta u_{k+j} = \Delta u_{k+N_\mathrm{c}-1},\qquad j=N_\mathrm{c},\ldots, N_\mathrm{p}-1 +\] where \(\Delta u_k = u_k - u_{k-1}\).

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Alternatively, \[ +\Delta^2 u_{k+j} = 0,\qquad j=N_\mathrm{c},\ldots, N_\mathrm{p}-1 +\] where \(\Delta^2 u_k = \Delta u_k - \Delta u_{k-1} = u_k - 2u_{k-1} + u_{k-2}\).

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Inequality constraints for MPC

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There are several possibilities for the constraints in the MPC for MPL systems. For example, we can constrain the minimum and maximum separation of input and output events \[ +a_{k+j} \leq \Delta u_{k+j} \leq b_{k+j},\qquad j=0,1,\ldots,N_\mathrm{c}-1, +\] \[ +c_{k+j} \leq \Delta y_{k+j} \leq d_{k+j},\qquad j=0,1,\ldots,N_\mathrm{p}-1. +\]

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We can also impose constraint on the maximum due dates for the output events \[ +y_{k+j} \leq r_{k+j},\qquad j=0,1,\ldots,N_\mathrm{p}-1. +\]

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We can also enforce the condition that the input and output events are consecutive \[ +\Delta u_{k+j} \geq 0, \qquad j=0,1,\ldots,N_\mathrm{c}-1. +\]

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MPC for MPL system leads to a nonlinear optimization problem

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Our motivation for formulating the problems within the (max,+) algebra was to fake the reality a bit and pretend that the problem is linear. This allowed us to invoke many concepts that we are familiar with from linear systems theory. However, at the end of the day, when it comes to actually solving the problem, we must reveal the nonlinear nature of the problem.

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When we consider the MPC for MPL systems, we are faced with a nonlinear optimization problem. We can use some general nonlinear solvers (fmincon, ipopt, …).

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Alternatively, there is a dedicated framework for solving these problem. It is called Extended Linear Complementarity Problem (ELCP) and was developed by [1]. We will introduce the complementarity problem(s) later in a chapter dedicated to complementarity.

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Yet another approach is through Mixed Integer (Linear) Programming (MILP).

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References

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[1]
B. De Schutter and B. De Moor, “The extended linear complementarity problem,” Mathematical Programming, vol. 71, no. 3, pp. 289–325, Dec. 1995, doi: 10.1007/BF01590958.
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Copyright 2024, Zdeněk Hurák

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Logic vs inequalities

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Our goal now is to turn the IF-THEN conditions in the model into linear inequalities. This will allow us to formulate the model as a mathematical program, actually a mixed-integer program (MIP).

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Propositional logic and connectives

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Propositions that are either true or false are composed of elementary propositions (Boolean variables) and connectives.

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Boolean variable (or elementary proposition)

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\(X\) evaluates to true or false. Oftentimes values 0 and 1 are used, but it should be clear that these are not numbers but logical values.

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Connectives

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  • Conjunction (logical and): \(X_1 \land X_2\)
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  • Disjunction (logical or): \(X_1 \lor X_2\)
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  • Negation: \(\neg X_2\) (or \(\overline{X_2}\) or \(\sim X_2\))
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  • Implication: \(X_1 \implies X_2\)
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  • Equivalence: \(X_1 \iff X_2\)
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Equivalences of logic propositions

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We will heavily used the following equivalences: \[ +\begin{aligned} +X_1 \implies X_2 \qquad &\equiv \qquad \neg X_2 \implies \neg X_1,\\ +X_1 \iff X_2 \qquad &\equiv \qquad (X_1 \implies X_2) \land (X_2 \implies X_1),\\ +X_1 \land X_2 \qquad &\equiv \qquad \neg (\neg X_1 \lor \neg X_2),\\ +X_1 \implies X_2 \qquad &\equiv \qquad \neg X_1 \lor X_2. +\end{aligned} +\]

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The last one can be seen as follows: it cannot happen that \(X1 \land \neg X2\), that is, it holds that \(\neg(X1 \land \neg X2)\). De Morgan gives \(\neg X1 \lor X2\).

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General transformation of Boolean expressions to integer inequalities

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From Conjunctive Normal Form (CNF) \[ +\bigwedge_{j=1}^m \left[\left(\lor_{i\in \mathcal{P}_j} X_i\right) \lor \left(\lor_{i\in \mathcal{N}_j} \neg X_i\right)\right] +\] to 0-1 integer inequalities defining a polyhedron \[ +\begin{aligned} +\sum_{i\in \mathcal{P}_1} \delta_i + \sum_{i\in \mathcal{N}_1} (1-\delta_i) &\geq 1,\\ +&\vdots\\ +\sum_{i\in \mathcal{P}_m} \delta_i + \sum_{i\in \mathcal{N}_m} (1-\delta_i) &\geq 1. +\end{aligned} +\]

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Finite state machine (FSM) using binary variables

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Encode the discrete state variables in binary \[ +x_b \in \{0,1\}^{n_b} +\]

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Similarly the discrete inputs \[ +u_b \in \{0,1\}^{m_b} +\]

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The logical state equation then \[ +x_b(k+1) = f_b(x_b(k),u_b(k),\delta_e(k)) +\]

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Example 1 (Example)  

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+Figure 1: Example of a FSM +
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The state update/transition equation is \[ +\begin{aligned} +x_d(k+1) = +\begin{cases} +\text{Red} & \text{if}\; ([x_d = \text{green}] \land \neg [\delta_3=1]) \lor ([x_d = \text{red}] \land \neg [\delta_3=1])\\ +\text{Green} & \text{if} \; \ldots\\ +\text{Blue} & \text{if} \; \ldots +\end{cases} +\end{aligned} +\]

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Binary encoding of the discrete states \[ +\text{Red}: x_b = \begin{bmatrix}0\\0 \end{bmatrix}, \; \text{Green}: x_b = \begin{bmatrix}0\\1 \end{bmatrix}, \; \text{Blue}: x_b = \begin{bmatrix}1\\0 \end{bmatrix} +\]

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Reformulating the state update equations for binary variables \[ +\begin{aligned} +x_{b1} &= (\neg [x_{b1} = 1] \land \neg [x_{b2} = 1] \land [\delta_1=1] \land \neg[u_{b2}=1]) \\ +&\quad \lor (\neg [x_{b1} = 1] \land [x_{b2} = 1] \land [\delta_3=1] \land [u_{b1}=1])\\ +&\quad \lor ([x_{b1} = 1]\land [x_{b2} = 1] \land \neg [\delta_2=1])\\ +x_{b2} &= \ldots +\end{aligned} +\]

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Finally, simplify, convert to CNF.

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Mixing logical and continuous through indicator variables

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Logical implies continuous

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\[X \implies [f(x)\leq 0]\]

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\[[\delta = 1] \implies [f(x)\leq 0]\]

+
    +

  • introduce \(M\) \[ +f(x) \leq (1-\delta) M +\]

    • +

  • that is large enough so that when \(\delta=0\), there is no practical restriction on \(f\).

    +
      +
    • Big-M technique.
      • +
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+
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Continuous implies logical

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\[[f(x)\leq 0] \implies X\]

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\[[f(x)\leq 0] \implies [\delta = 1]\]

+
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  • Equivalently \[\neg [\delta = 1] \implies \neg [f(x)\leq 0],\]

    • +

  • that is, \[[\delta = 0] \implies [f(x) > 0]\]

    • +

  • Introduce \(m\) such that \(f(x)>0\) is enforced when \(\delta=0\) \[ +f(x) > m\delta +\]

    • +

  • but small enough that there is no restriction on \(f\) when \(\delta=1\).

    • +

  • For numerical reasons, modify to nonstrict inequality \[ +f(x) \geq \epsilon + (m-\epsilon)\delta, +\] where \(\epsilon\approx 0\) (for example, machine epsilon).

    • +
+
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Equivalence between logical and continuous

+
    +
  • Combining the previous two implications.
    • +
+

\[ +\begin{aligned} +f(x) &\leq (1-\delta) M,\\ +f(x) &\geq \epsilon + (m-\epsilon)\delta. +\end{aligned} +\]

+
+
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IF-THEN-ELSE rule as an inequality

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    +
  • If \(X\) +
      +
    • then \(z = a^\top x + b^\top u + f\),
      • +
    • else \(z = 0\).
      • +
    • +
  • It can be expressed as a product \[ +z = \delta\,(a^\top x + b^\top u + f) +\]
    • +
+

\[ +\begin{aligned} +z &\leq M\delta,\\ +- z &\leq -m\delta,\\ +z &\leq a^\top x + b^\top u + f - m(1-\delta),\\ +-z &\leq -(a^\top x + b^\top u + f) + M(1-\delta). +\end{aligned} +\]

+
+

The reasoning is that if \(\delta=0\), then \(z\) is restricted, while \(a^\top x + b^\top u + f\) is not. And the other way around.

+
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+
+

Another IF-THEN-ELSE rule

+
    +
  • If \(X\) +
      +
    • then \(z = a_1^\top x + b_1^\top u + f_1\),
      • +
    • else \(z = a_2^\top x + b_2^\top u + f_2\).
      • +
    • +
  • It can be expressed as \[ +\begin{aligned} +z &= \delta\,(a_1^\top x + b_1^\top u + f_1) \\ +&\quad + (1-\delta)(a_2^\top x + b_2^\top u + f_2) +\end{aligned} +\]
    • +
+

\[ +\begin{aligned} +(m_2-M_1)\delta + z &\leq a_2^\top x + b_2^\top u + f_2,\\ +(m_1-M_2)\delta - z &\leq -a_2^\top x - b_2^\top u - f_2,\\ +(m_1-M_2)(1-\delta) + z &\leq a_1^\top x + b_1^\top u + f_1,\\ +(m_2-M_1)(1-\delta) - z &\leq -a_1^\top x - b_1^\top u - f_1. +\end{aligned} +\]

+
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Generation of events by mixing logical and continuous variables in inequalities

+

\[ +\begin{aligned} +h_i(x_c(k), u_c(k)) &\leq M_i (1-\delta_{e,i})\\ +h_i(x_c(k), u_c(k)) &\geq \epsilon + (m_i-\epsilon) \delta_{e,i} +\end{aligned} +\]

+
+
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Switched affine system

+
    +
  • We want to get rid of the IF-THEN and formulate the switching mechanism into the format of inequalities too.
    • +
+

\[ +x_c(k+1) = \sum_{i=1}^s z_i(k), +\]

+
    +
  • where \[ +z_1(k) = +\begin{cases} +A_1 x_c(k) + B_1 u_c(k) + f_1 & \text{if}\;i(k)=1\\ +0 & \text{otherwise} +\end{cases} +\]
    • +
+

\[\quad \vdots\]

+

\[ +z_s(k) = +\begin{cases} +A_s x_c(k) + B_s u_c(k) + f_s & \text{if}\;i(k)=s\\ +0 & \text{otherwise} +\end{cases} +\]

+
    +
  • For each \(i\in \{1, 2, \ldots, s\}\)
    • +
+

\[ +\begin{aligned} +z_i &\leq M_i\delta_i,\\ +- z_i &\leq -m_i\delta_i,\\ +z_i &\leq a_i^\top x + b_i^\top u + f_i - m_i(1-\delta_i),\\ +-z_i &\leq -(a_i^\top x + b_i^\top u + f_i) + M_i(1-\delta_i). +\end{aligned} +\]

+
+
+

Mixed logical dynamical (MLD) system

+

\[ +\begin{aligned} +x(k+1) &= Ax(k) + B_u u(k) + B_\delta\delta + B_zz(k) + B_0\\ +y(k) &= Cx(k) + D_u u(k) + D_\delta \delta + D_z z + D_0\\ +E_\delta \delta &+ E_z z(k) \leq E_u u(k) + E_x x(k) + E_0 +\end{aligned} +\]

+
+
+

Simple example

+
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HYSDEL language

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Piecewise affine systems

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\[ +\begin{aligned} +x(k+1) &= A_{i(k)}x(k) + B_{i(k)} u(k) + f_{i(k)}\\ +y(k) &= C_{i(k)}x(k) + D_{i(k)} u(k) + g_{i(k)}\\ +& \; H_{i(k)} x(k) + J_{i(k)} u(k) \leq K_{i(k)} +\end{aligned} +\]

+
    +
  • DHA, MLD, PWA are equivalent.
    • +
+ + +
+ + Back to top
+ + +
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+   +
+
+

Copyright 2024, Zdeněk Hurák

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+ + + + + \ No newline at end of file diff --git a/mpc_mld_software 24.html b/mpc_mld_software 24.html new file mode 100644 index 0000000..1c3af92 --- /dev/null +++ b/mpc_mld_software 24.html @@ -0,0 +1,1059 @@ + + + + + + + + + +Software – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Software

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Essentially the same as in the previous chapter.

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Copyright 2024, Zdeněk Hurák

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+ + + + + \ No newline at end of file diff --git a/petri_nets 11.html b/petri_nets 11.html new file mode 100644 index 0000000..486ad20 --- /dev/null +++ b/petri_nets 11.html @@ -0,0 +1,1679 @@ + + + + + + + + + +Petri nets – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Petri nets

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In this chapter we introduce another formalism for modelling discrete event systems (DES) – Petri nets. Petri nets offer an alternative perspective on discrete even systems compared to automata. And it is good to have alternatives, isn’t it? For some purposes, one framework can be more appropriate than the other.

+

Furthermore, the ideas behind Petri nets even made it into international standards. Either directly or through the derived GRAFCET language, which in turn served as the basis for the Sequential Function Chart (SFC) language for PLC programming. See the references.

+

Last but not least, an elegant algebraic framework based on the so-called (max,+) algebra has been developed for a subset of Petri nets (so-called event graphs) and it would be a shame not to mention it in our course (in the next chapter).

+
+

Definition of a Petri net

+

Similarly as in the case of automata, a Petri net (PN) can be defined as a tuple of sets and functions: \boxed{PN = \{\mathcal{P}, \mathcal{T}, \mathcal{A}, w\},} where

+
    +
  • \mathcal{P} = \{p_1, \dots, p_n\} is a finite set of places,
    • +
  • \mathcal{T} = \{t_1, \dots, t_m\} is a finite set of transitions,
    • +
  • \mathcal{A} \subseteq (\mathcal{P} \times \mathcal{T}) \cup (\mathcal{T} \times \mathcal{P}) is a finite set of arcs, and these arcs are directed and since there are two types of nodes, there are also two types of arcs: +
      +
    • (p_i, t_j) \in \mathcal{A} is from place p_i to transition t_j,
      • +
    • (t_j, p_i) \in \mathcal{A} is from transition t_j to place p_i,
      • +
    • +
  • w : \mathcal{A} \to \mathbb{N} is a weight function.
    • +
+

Similarly as in the case of automata, Petri nets can be visualized using graphs. But this time, we need to invoke the concept of a weighted bipartite graph. That is, a graph with two types of nodes:

+
    +
  • places = circles,
    • +
  • transitions = bars.
    • +
+

The nodes of different kinds are connected by arcs (arrowed curves). A integer weights are associated with arcs. Alternatively, for a smaller weight (2, 3, 4), the weithg can be graphically encoded by drawing multiple arcs.

+
+

Example 1 (Simple Petri net) We consider just two places, that is, \mathcal{P} = \{p_1, p_2\}, and one transition, that is, \mathcal{T} = \{t\}. The set of arcs is \mathcal{A} = \{\underbrace{(p_1, t)}_{a_1}, \underbrace{(t, p_2)}_{a_2}\}, and the associated weights are w(a_1) = w((p_1, t)) = 2 and w(a_2) = w((t, p_2)) = 1. The Petri net is depicted in Fig 1.

+
+
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+ +
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+Figure 1: Example of a simple Petri net +
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Additional definitions

+
    +
  • \mathcal{I}(t_j) … a set of input places of the transition t_j,
    • +
  • \mathcal{O}(t_j) … a set of output places of the transition t_j.
    • +
+
+

Example 2 (More complex Petri net)  

+
    +
  • \mathcal{P} = \{p_1, p_2, p_3, p_4\},
    • +
  • \mathcal{T} = \{t_1, t_2, t_3, t_4, t_5\},
    • +
  • \mathcal{A} = \{(p_1, t_1), (t_1, p_1), (p_1, t_2),\ldots\},
    • +
  • w((p_1, t_1)) = 2, \; w((t_1, p_1)) = 1, \; \ldots
    • +
+
+
+
+ +
+
+Figure 2: Example of a more complex Petri net +
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+
+

Marking and marked Petri nets

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An important concept that we must introduce now is that of marking. It is a function that assigns an integer to each place x: \mathcal{P} \rightarrow \mathbb{N}.

+

The vector composed of the values of the marking function for all places \bm x = \begin{bmatrix}x(p_1)\\ x(p_2)\\ \vdots \\ x(p_n) \end{bmatrix} can be viewed as the state vector (although the Petri nets community perhaps would not use this terminology and stick to just marking).

+

Marked Petri net is then a Petri net augmented with the marking

+

MPN = \{\mathcal{P}, \mathcal{T}, \mathcal{A}, w,x\}.

+
+

Visualization of marked Petri net using tokens

+

Marked Petri net can also be visualized by placing tokens (dots) into the places. The number of tokens in a place corresponds to the value of the marking function for that place.

+
+

Example 3 (Marked Petri net) Consider the Petri net from Example 1. The marking function is x(p_1) = 2 and x(p_2) = 1, which assembled into a vector gives \bm x = \begin{bmatrix}1\\ 0 \end{bmatrix}. The marked Petri net is depicted in Fig 3.

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+
+
+ +
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+Figure 3: Example of a marked Petri net +
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+
+

For another marking, namely \bm x = \begin{bmatrix}2\\ 1 \end{bmatrix}, the marked Petri net is depicted in Fig 4.

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+
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+ +
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+Figure 4: Example of a marked Petri net with different marking +
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Enabling and firing of a transition

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Finally, here comes the enabling (pun intended) component of the definition of a Petri net – enabled transition. A transition t_j does not just happen – we say fire – whenever it wants, it can only happen (fire) if it is enabled, and the marking is used to determined if it is enabled. Namely, the transition is enabled if the value of the marking function for each input place is greater than or equal to the weight of the arc from that place to the transition. That is, the transition t_j is enabled if +x(p_i) \geq w(p_i,t_j)\quad \forall p_i \in \mathcal{I}(t_j). +

+
+
+
+ +
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+Can but does not have to +
+
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The enabled transition can fire, but it doesn’t have to. We will exploit this in timed PN.

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Example 4 (Enabled transition) See the PN in Example 3: in the first marked PN the transition cannot fire, in the second it can.

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State transition function

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We now have a Petri net as a conceptual model with a graphical representation. But in order to use it for some quantitative analysis, it is useful to turn it into some computational form. Preferrably a familiar one. This is done by defining a state transition function. For a Petri net with n places, the state transition function is +f: \mathbb N^n \times \mathcal{T} \rightarrow \mathbb N^n, + which reads that the state transition fuction assignes a new marking (state) to the Petri net after a transition is fired at some given marking (state).

+

The function is only defined for a transition t_j iff the transition is enabled.

+

If the transition t_j is enabled and fired, the state evolves as +\bm x^+ = f(\bm x, t_j), + where the individual components of \bm x evolve according to \boxed{ + x^+(p_i) = x(p_i) - w(p_i,t_j) + w(t_j,p_i), \; i = 1,\ldots,n.} +

+

This has a visual interpretation – a fired transition moves tokens from the input to the output places.

+
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Example 5 (Moving tokens around) Consider the PN with the initial marking (state) \bm x_0 = \begin{bmatrix}2\\ 0\\ 0\\ 1 \end{bmatrix} (at discrete time 0), and the transition t_1 enabled

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+

+
Example PN in the initial state at time 0, the transition t_1 enabled, but not yet fired
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+
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+ +
+
+Conflict in notation. Again. Sorry. +
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+
+

We admit the notation here is confusing, because we use the lower index 0 in \bm x_0 to denote the discrete time, while the lower index 1 in t_1 to denote the transition and the lower indices 1 and 2 in p_1 and p_2 just number the transitions and places, respectively. We could have chosen something like \bm x(0) or \bm x[0], but we dare to hope that the context will make it clear.

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Now we assume that t_1 is fired

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+ +
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+Figure 5: Transition t_1 in the example PN fired at time 1 +
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The state vector changes to \bm x_1 = [1, 1, 1, 1]^\top, the discrete time is 1 now.

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As a result of this transition, note that t_1, t_2, t_3 are now enabled.

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+Number of tokens need not be preserved +
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In the example we can see for the first time, that the number of tokens need not be preserved.

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Now fire the t_2 transition

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Transition t_2 in the example PN fired at time 2
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The state vector changes to \bm x_2 = [1, 1, 0, 2]^\top, the discrete time is 2 now.

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Good, we can see the idea. But now we go back to time 1 (as in Fig 5) to explore the alternative evolution. With the state vector \bm x_1 = [1, 1, 1, 1]^\top and the transitions t_1, t_2, t_3 enabled, we fire t_3 this time.

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Transition t_3 in the example PN fired at time 1
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The state changes to \bm x_2 = [0, 1, 0, 0]^\top, the discrete time is 2. Apparently the PN evolved into at a different state. The lesson learnt with this example is that the order of firing of enabled transitions matters.

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+ +
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+The order in which the enabled transitions are fired does matter +
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The dependence of the state evolution upon the order of firing the transitions is not surprising. Wwe have already encountered it in automata when the active event set for a given state contains more then a single element.

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Reachability

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We have started talking about states and state transitions in Petri nets, which are all concepts that we are familiar with from dynamical systems. Another such concept is reachability. We explain it through an example.

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Example 6 (Not all states are reachable)  

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Example of an unreachability in a Petri net
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The Petri net is initial in the state [2,1]^\top. The only reachable state is [0,2]^\top.

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By the way, note that the weight of the arc from the place p_1 to the transition t is 2, so both tokens are removed from the place p_1 when the transition t fires. But then the arc to the place p_2 has weight 1, so only one token is added to the place p_2. The other token is “lost”.

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Reachability tree and graph

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Here we introduce two tools for analysis of reachability of a Petri net.

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Example 7 Consider the following example of a Petri net.

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Example of a Petri net for reachability analysis
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In Fig 6 we draw a reachability tree for this Petri net.

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+Figure 6: Reachability tree for an example Petri net +
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In Fig 7 we draw a reachability graph for this Petri net.

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+Figure 7: Reachability graph for an example Petri net +
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Number of tokens need not be preserved

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We have already commented on this before, but we emphasize it here. Indeed, it can be that +\sum_{p_i\in\mathcal{O}(t_j)}w(t_j,p_i) < \sum_{p_i\in\mathcal{I}(t_j)} w(p_i,t_j) +

+

or

+

+\sum_{p_i\in\mathcal{O}(t_j)}w(t_j,p_i) > \sum_{p_i\in\mathcal{I}(t_j)} w(p_i,t_j) +

+

With this reminder, we can now hightlight several patters that can be observed in Petri nets.

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AND-convergence, AND-divergence

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OR-convergence and OR-divergence

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Nondeterminism in a PN

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In the four patters just enumerated, we have seen that the last one – the OR-divergence – is not deterministic. Indeed, consider the following example.

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Example 8  

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Nondeterminism in a Petri net
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In other words, we can incorporate a nondeterminism in a model.

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+ +
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+Nondeterminism in automata +
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Recall that something similar can be encountered in automata, if the active event set for a given state contains more than one element (event,transition).

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Subclasses of Petri nets

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We can identify two subclasses of Petri nets:

+
    +
  • event graphs,
    • +
  • state machines.
    • +
+
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Event graph

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    +
  • Each place has just one input and one output transition (all ws equal to 1).
    • +
  • No OR-convergence, no OR-divergence.
    • +
  • Also known as Decision-free PN.
    • +
  • It can model synchronization.
    • +
+
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Example 9 (Event graph)  

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+
Example of an event graph
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State machine

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  • Each transition has just one input and one output place.
    • +
  • No AND-convergence, no AND-divergence.
    • +
  • Does not model synchronization.
    • +
  • It can model race conditions.
    • +
  • With no source (input) and sink (output) transitions, the number of tokens is preserved.
    • +
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Example 10 (State machine)  

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Example of a state machine
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Incidence matrix

+

We consider a Petri net with n places and m transitions. The incidence matrix is defined as +\bm A \in \mathbb{Z}^{n\times m}, + where +a_{ij} = w(t_j,p_i) - w(p_i,t_j). +

+
+
+
+ +
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+Transpose +
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Some define the incidence matrix as the transpose of our definition.

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State equation for a Petri net

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With the incidence matrix defined above, the state equation for a Petri net can be written as +\bm x^+ = \bm x + \bm A \bm u, + where \bm u is a firing vector for the enabled j-th transition +\bm u = \bm e_j = \begin{bmatrix}0 \\ \vdots \\ 0 \\ 1\\ 0\\ \vdots\\ 0\end{bmatrix} + with the 1 at the j-th position.

+
+
+
+ +
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+State vector as a column rather then a row +
+
+
+

Note that in [1] they define everything in terms of the transposed quantities, but we prefer sticking to the notion of a state vector as a column.

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Example 11 (State equation for a Petri net) Consider the Petri net from Example 5, which we show again below in Fig 8.

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+ +
+
+Figure 8: Example of a Petri net +
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+

The initial state is given by the vector +\bm x_0 += +\begin{bmatrix} +2\\ 0\\ 0\\ 1 +\end{bmatrix} +

+

The incidence matrix is +\bm A = \begin{bmatrix} +-1 & 0 & -1\\ +1 & 0 & 0\\ +1 & -1 & -1\\ +0 & 1 & -1 +\end{bmatrix} +

+

And the state vector evolves according to +\begin{aligned} +\bm x_1 &= \bm x_0 + \bm A \bm u_1\\ +\bm x_2 &= \bm x_1 + \bm A \bm u_2\\ +\vdots & +\end{aligned} +

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+
+
+
+ +
+
+Caution +
+
+
+

We repeat once again just to make sure: the lower index corresponds to the discrete time.

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Queueing systems modelled by PN

+

Although Petri nets can be used to model a vast variety of systems, below we single out one particular class of systems that can be modelled by Petri nets – queueing systems. The general symbol is shown in Fig 9.

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+
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+ +
+
+Figure 9: Queing systems and its components and transitions/events +
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We can associate the transitions with the events in the queing system:

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  • a is a spontaneous transition (no input places).
    • +
  • s needs a customer in the queue and the server being idle.
    • +
  • c needs the server being busy.
    • +
+

We can now start drawing the Petri net by drawin the bars corresponding to the transitions. Then in between every two bars, we draw a circle for a place. The places can be associated with three bold-face letters above, namely:

+

\quad \mathcal{P} = \{Q, I, B\}, that is, queue, idle, busy.

+
+
+

+
Petri net corresponding to the queing system
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+ +
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+The input transition adds tokens to the system +
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The transition a is an input transition – the tokens are added to the system through this transition.

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Note how we consider the token in the I place. This is not only to express that the server is initially idle, ready to serve as soon as a customer arrives to the queue, it also ensures that no serving of a new customer can start before the serving of the current customer is completed.

+

The initial state: [0,1,0]^\top. Consider now a particular trace (of transitions/events) \{a,s,a,a,c,s,a\}. Verify that this leads to the final state [2,0,1]^\top.

+
+

Some more extensions

+

We can keep adding features to the model of a queing system. In particular,

+
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  • the arrival transition always enabled,
    • +
  • the server can break down, and then be repaired,
    • +
  • completing the service \neq customer departure.
    • +
+

These are incorporated into the Petri net in Fig 10.

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+ +
+
+Figure 10: Extended model of a queueing system +
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In the Petri net, d is an output transition – the tokens are removed from the system.

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Example 12 (Beverage vending machine) Below we show a Petri net for a beverage vending machine. While building it, we find it useful to identify the events/transitions that can happen in the system.

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+

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Petri net for a beverage vending machine
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Some extensions of basic Petri nets

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    +
  • Coloured Petri nets (CPN): tokens can by of several types (colours), and the transitions can be enabled only if the tokens have the right colours.
    • +
  • …
    • +
+

We do not cover these extensions in our course. But there is one particular extension that we do want to cover, and this amounts to introducing time into Petri nets, leading to timed Petri nets, which we will discuss in the next chapter.

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References

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C. G. Cassandras and S. Lafortune, Introduction to Discrete Event Systems, 3rd ed. Cham: Springer, 2021. Available: https://doi.org/10.1007/978-3-030-72274-6
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Copyright 2024, Zdeněk Hurák

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Timed Petri nets

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Recall that when introducing enabled transitions, we emphasized that these can but do not have to fire immediately after having been enabled \boxed{\mathrm{ENABLING} \neq \text{FIRING}.}

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Delays associated with transitions

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Well then, the enabled transitions do not have to fire immediately, but they can possibly fire with some delay after enabling. This is for the first time that we are introducing the concept of time into the Petri nets, isn’t it?

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For the jth transition, the delay of the kth firing is v_{j,k}, and we collect the sequence of delayes into v_j = \{v_{j,1}, v_{j,2}, \ldots \}.

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But not all transitions have to be timed. Denote the timed transitions \mathcal{T}_\mathrm{D}\subseteq \mathcal{T}. We define the clock structure for a PN as \mathcal{V} = \{v_j\mid t_j\in\mathcal{T}_\mathrm{D}\}.

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The definition of a timed Petri net (TPN) is then obtained by augmenting the definition of a Petri net with the clock structure

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\boxed{TPN = \{\mathcal{P}, \mathcal{T}, \mathcal{A}, w, x, \mathcal{V}\}}.

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Example 1 (Timed Petri net) Model of processing multiple tasks: task 1 and task 2 are processed sequentially, and task 3 is processed in parallel with them; task 4 can only be processed after both tasks 2 and 3 have been finished. Finishing individual tasks corresponds to the individual transitions. The transition 4 is untimed, it only expresses the logical requirement.

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Example of a timed Petri net
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+Rectangles instead of bars +
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Sometimes instead of a bar, the untimed transitions are modelled using similarly thin rectangles as the timed transitions, but filled.

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Places can also be delayed

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With delays associated with just one type of a node in a Petri net, the situation is rather nonsymmetric. In some literature, delays can also associated with places. And yet in some other literature delays are only associated with places. Such delays associated with places are called holding time for a place It is the minimum duration the token must rest in the place. But the token can stay longer if the output transition is waiting for other places.

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+Delays associated with transitions and places +
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There is a major difference in delays associated with places compared to the delays associated with transitions. While the former tells the minimum duration the token has to dwell in the place, the latter tell the exact delay with which the transition fires after having been enabled.

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Timed Petri net dynamics

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With the introduction of time into the Petri nets, we can now study the dynamics of the system. For general Petri nets, alhough perfectly doable, it may quicly become too complex, and therefore here we only consider event graphs.

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Some notation:

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  • \{\tau_{j,1}, \tau_{j,2}, \ldots\} are the firing times of the jth transition,
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  • \{\pi_{i,1},\pi_{i,2},\ldots\} are the times when the ith place receives a token,
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  • x_i = x(p_i) is the number of tokens at the ith place,
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  • x_{i,k} = \left.x(p_i)\right|_k, is the number of tokens at the ith place after the kth firing.
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Now, assume first that x_{i,0} = 0. We can then relate the time of the arrival of the token to the place with the firing of the transition from which the token arrives \pi_{i,k} = \tau_{j,k},\quad p_i\in \mathcal{O}(t_j).

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But generally x_{i,0} \neq 0 and the above relation needs to be modified to \pi_{i,k+x_{i,0}} = \tau_{j,k},\quad p_i\in \mathcal{O}(t_j), or, equivalently \boxed{\pi_{i,k} = \tau_{j,k-x_{i,0}},\quad p_i\in \mathcal{O}(t_j).} \tag{1}

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This can be read in the following way. If there are initially, say, 3 tokens in the place, the time of the arrival of the 4th token is the time of the first firing of the transition from which the 4th token arrives.

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Good. Keep this result in mind. Now we go for another.

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For an untimed transition with a single input place, the firing time is the same as the time of the arrival of the token to the place +\tau_{j,k} = \pi_{i,k}. +

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Modifying this result for a timed transition with a single input place we get +\tau_{j,k} = \pi_{i,k} + v_{j,k}. +

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In words, the firing time is given by the time of the arrival of the token to the place, which enables the transition, and the delay associated with the transition.

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Finally, we extend this result to the case of a timed transition with multiple input places \boxed{ +\tau_{j,k} = \max_{p_i\in\mathcal{I}(t_j)}\{\pi_{i,k}\} + v_{j,k}.} +\tag{2}

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This is the other promised important result. Keep both boxed formulas Equation 1 and Equation 2 handy, they will be needed in what is coming.

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Example 2 (Timed Petri net dynamics) Consider the Petri net with three places and two transitions, one of which is timed, as in Fig 1.

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+Figure 1: Example of a Petri net for which the dynamics is analyzed +
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We first use Equation 2 to write down the firing times of the two transitions +\begin{aligned} +\tau_{1,k} &= \max\{\pi_{1,k},\pi_{3,k}\}\\ +\tau_{2,k} &= \pi_{2,k}+v_{2,k}. +\end{aligned} +

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Now we apply Equation 1 to write down the times of the arrival of the tokens to the places +\begin{aligned} +\pi_{1,k} &= \tau_{1,k-1}, \qquad k=2,\ldots, \qquad \pi_{1,0} = 0\\ +\pi_{2,k} &= \tau_{1,k-1}, \qquad k=2,\ldots, \qquad \pi_{2,0} = 0\\ +\pi_{3,k} &= \tau_{2,k}, \qquad k=1,\ldots +\end{aligned} +

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Substituting from the latter into the former we get +\begin{aligned} +\tau_{1,k} &= \max\{\tau_{1,k-1},\tau_{1,k-1}+v_{2,k}\}\\ +&= \tau_{1,k-1}+v_{2,k}, \quad \tau_{1,k} = 0\\ +\tau_{2,k} &= \tau_{1,k-1}+v_{2,k}. +\end{aligned} +

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This is the ultimate model for the dynamics of the Petri net. Should we need it, we can also get similar expressions for the times of the arrival of the tokens to the places.

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While with state equations we compute a sequences of values of the state vector (\bm x_0, \bm x_1, \bm x_2, \ldots), in other words, we compute the evolution of the state in time, here we compute the sequences of times when transitions fire (or token arrive to places). This update scheme for times resembles the state equations, but the interpretation is different.

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Queueing system using TPN

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We can also model a queueing system using a TPN. The Petri net is shown in Fig 2.

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+Figure 2: Timed Petri net modelling a queueing system +
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Of the three transitions \mathcal{T} = \{a,s,c\}, which we have already identified previously, we assume that only are times, namely \mathcal{T}_\mathrm{D} = \{a,c\}. The associated firing delays are \bm v = \begin{bmatrix}v_a \\ v_c\end{bmatrix}.

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For convenience we relabel the firing times of the transitions. Instead of t_{a,k} we will use a_k, and similarly s_k, and c_k. Application of Equation 2 and Equation 1 gives +\begin{aligned} +a_k &= a_{k-1} + v_{a,k},\quad k=1,2,\ldots,\quad a_0 = 0\\ +s_k &= \max\{\pi_{Q,k},\pi_{I,k}\}\\ +c_k &= \pi_{B,k} + v_{c,k}\\ +\pi_{Q,k} &= a_{k},\quad k=1,2,\ldots\\ +\pi_{I,k} &= c_{k-1},\quad k= 2, \ldots, \quad \pi_{I,0}=1\\ +\pi_{B,k} &= s_{k},\quad k=1,2,\ldots\\ +\end{aligned} +

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Combining gives the desired update equations +\begin{aligned} +a_k &= a_{k-1} + v_{a,k},\quad k=1,2,\ldots,\quad a_0 = 0\\ +s_k &= \max\{a_{k},c_{k-1}\}\\ +c_k &= s_{k} + v_{c,k}\\ +&= \max\{a_{k},c_{k-1}\} + v_{c,k},\quad k=1,\ldots, \quad c_0=0 +\end{aligned} +

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The time of completing the kth task is given by the time at which the previous task was completed and the time needed to complete the kth task itself, unless there is a gap in the queue after finishing the previous task, in which case the server must wait for the next task to arrive.

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Example 3 (Timed Petri net for synchronization of train lines) We consider three closed rail tracks and two stations as in Fig 3.

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+Figure 3: Example with three train lines +
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Departure of a train at a station must be synchronized with arrival of the other train so that passengers can change train. The timed Petri net for this system is shown in Fig 4.

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+Figure 4: Timed Petri net for the example of synchronization of three train lines +
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If time is associated with the places, the Petri net simplifies significantly to Fig 5.

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+Figure 5: Timed Petri net for the example of synchronization of three train lines +
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Example 4 (Manufacturing) tbd

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Extensions

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Stochastic Petri nets (SPN)

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Numerous extensions are possible, some of which we have already mentioned when discussing untimed Petri nets. But upon introducing time, stochastic Petr nets can be concived, in which delays are random variables.

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Copyright 2024, Zdeněk Hurák

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+ + + + + \ No newline at end of file diff --git a/search.json b/search.json index 565fd42..d5b9450 100644 --- a/search.json +++ b/search.json @@ -510,7 +510,7 @@ "href": "des_automata.html#extensions", "title": "State automata", "section": "Extensions", - "text": "Extensions\nThe concept of an automaton can be extended in several ways. In particular, the following two extensions introduce the concept of an output to an automaton.\n\nMoore machine\nOne extension of an automaton with outputs is Moore machine. The outputs assigned to the states by the output function y = g(x).\nThe output is produced (emitted) when the (new) state is entered.\nNote, in particular, that the output does not depend on the input. This has a major advantage when a feedback loop is closed around this system, since no algebraic loop is created.\nGraphically, we make a conventions that outputs are the labels of the states.\n\nExample 8 (Moore machine) The following automaton has just three states, but just two outputs (FLOW and NO FLOW).\n\n\n\n\n\n\n\n\nG\n\n\ninit\ninit\n\n\n\nclosed\n\nNO FLOW\nValve\nclosed\n\n\n\ninit->closed\n\n\n\n\n\npartial\n\nFLOW\nValve\npartially\nopen\n\n\n\nclosed->partial\n\n\nopen valve one turn\n\n\n\npartial->closed\n\n\nclose valve one turn\n\n\n\nfull\n\nFLOW\nValve\nfully open\n\n\n\npartial->full\n\n\nopen valve one turn\n\n\n\nfull->closed\n\n\nemergency shut off\n\n\n\nfull->partial\n\n\nclose valve one turn\n\n\n\n\n\n\nFigure 9: Example of a digraph representation of the Moore machine for a valve control\n\n\n\n\n\n\n\n\nMealy machine\nMealy machine is another extension of an automaton. Here the outputs are associated with the transitions rather than the states.\nSince the events already associated with the states can be viewed as the inputs, we now have input/output transition labels. The transition label e_\\mathrm{i}/e_\\mathrm{o} on the transion from x_1 to x_2 reads as “the input event e_\\mathrm{i} at state x_1 activates the transition to x_2, which outputs the event e_\\mathrm{o}” and can be written as x_1\\xrightarrow{e_\\mathrm{i}/e_\\mathrm{o}} x_2.\nIt can be viewed as if the output function also considers the input and not only the state y = e_\\mathrm{o} = g(x,e_\\mathrm{i}).\nIn contrast with the Moore machine, here the output is produced (emitted) during the transition (before the new state is entered).\n\nExample 9 (Mealy machine) Coffee machine: coffee for 30 CZK, machine accepting 10 and 20 CZK coins, no change.\n\n\n\n\n\n\n\n\nG\n\n\ninit\ninit\n\n\n\n0\n\nNo coin\n\n\n\ninit->0\n\n\n\n\n\n10\n\n10 CZK\n\n\n\n0->10\n\n\ninsert 10 CZK / no coffee\n\n\n\n20\n\n20 CZK\n\n\n\n0->20\n\n\ninsert 20 CZK / no coffee\n\n\n\n10->0\n\n\ninsert 20 CZK / coffee\n\n\n\n10->20\n\n\ninsert 10 CZK / no coffee\n\n\n\n20->0\n\n\ninsert 10 CZK / coffee\n\n\n\n20->10\n\n\ninsert 20 CZK / coffee\n\n\n\n\n\n\nFigure 10: Example of a digraph representation of the Mealy machine for a coffee machine\n\n\n\n\n\n\n\nExample 10 (Reformulate the previous example as a Moore machine) Two more states wrt Mealy\n\n\n\n\n\n\n\n\nG\n\n\ninit\ninit\n\n\n\n0\n\nNO COFFEE\nNo\ncoin\n\n\n\ninit->0\n\n\n\n\n\n10\n\nNO COFFEE\n10\nCZK\n\n\n\n0->10\n\n\ninsert 10 CZK\n\n\n\n20\n\nNO COFFEE\n20\nCZK\n\n\n\n0->20\n\n\ninsert 20 CZK\n\n\n\n10->20\n\n\ninsert 10 CZK\n\n\n\n30\n\nCOFFEE\n10+20\nCZK\n\n\n\n10->30\n\n\ninsert 20 CZK\n\n\n\n20->30\n\n\ninsert 10 CZK\n\n\n\n40\n\nCOFFEE\n20+20\nCZK\n\n\n\n20->40\n\n\ninsert 20 CZK\n\n\n\n30->0\n\n\n\n\n\n30->10\n\n\ninsert 10 CZK\n\n\n\n30->20\n\n\ninsert 20 CZK\n\n\n\n40->10\n\n\n\n\n\n40->20\n\n\ninsert 10 CZK\n\n\n\n40->30\n\n\ninsert 20 CZK\n\n\n\n\n\n\nFigure 11: Example of a digraph representation of the Moore machine for a coffee machine\n\n\n\n\n\n\n\n\n\n\n\n\nNote\n\n\n\nThere are transitions from 30 and 40 back to 0 that are not labelled by any event. This does not seem to follow the general rule that transitions are always triggered by events. Not what? It can be resolved upon introducing time as the timeout transitions.\n\n\n\nExample 11 (Dijkstra’s token passing) The motivation for this example is to show that it is perhaps not always productive to insist on visual description of the automaton using a graph. The four components of our formal definition of an automaton are just enough, and they translate directly to a code.\nThe example comes from the field of distributed computing systems. It considers several computers that are connected in ring topology, and the communication is just one-directional as Fig 12 shows. The task is to use the communication to determine in – a distributed way – which of the computers carries a (single) token at a given time. And to realize passing of the token to a neighbour. We assume a synchronous case, in which all the computers are sending simultaneously, say, with some fixed sending period.\n\n\n\n\n\n\n\n\nG\n\n\n0\n\n0\n\n\n\n1\n\n1\n\n\n\n0->1\n\n\n\n\n\n2\n\n2\n\n\n\n1->2\n\n\n\n\n\n3\n\n3\n\n\n\n2->3\n\n\n\n\n\n3->0\n\n\n\n\n\n\n\n\nFigure 12: Example of a ring topology for Dijkstra’s token passing in a distributed system\n\n\n\n\n\nOne popular method for this is called Dijkstra’s token passing. Each computer keeps a single integer value as its state variable. And it forwards this integer value to the neighbour (in the clockwise direction in our setting). Upon receiving the value from the other neighbour (in the counter-clockwise direction), it updates its own value according to the rule displayed in the code below. At every clock tick, the state vector (composed of the individual state variables) is updated according to the function update!() in the code. Based on the value of the state vector, an output is computed, which decodes the informovation about the location of the token from the state vector. Again, the details are in the output() function.\n\n\nShow the code\nstruct DijkstraTokenRing\n number_of_nodes::Int64\n max_value_of_state_variable::Int64\n state_vector::Vector{Int64}\nend\n\nfunction update!(dtr::DijkstraTokenRing) \n n = dtr.number_of_nodes\n k = dtr.max_value_of_state_variable\n x = dtr.state_vector\n xnext = copy(x)\n for i in eachindex(x) # Mind the +1 shift. x[2] corresponds to x₁ in the literature.\n if i == 1 \n xnext[i] = (x[i] == x[n]) ? mod(x[i] + 1,k) : x[i] # Increment if the left neighbour is identical.\n else \n xnext[i] = (x[i] != x[i-1]) ? x[i-1] : x[i] # Update by the differing left neighbour.\n end\n end\n dtr.state_vector .= xnext \nend\n\nfunction output(dtr::DijkstraTokenRing) # Token = 1, no token = 0 at the given position. \n x = dtr.state_vector\n y = similar(x)\n y[1] = iszero(x[1]-x[end])\n y[2:end] .= .!iszero.(diff(x))\n return y\nend\n\n\noutput (generic function with 1 method)\n\n\nWe now rund the code for a given number of computers and some initial state vector that does not necessarily comply with the requirement that there is only one token in the ring.\n\n\nShow the code\nn = 4 # Concrete number of nodes.\nk = n # Concrete max value of a state variable (>= n).\n@show x_initial = rand(0:k,n) # Initial state vector, not necessarily acceptable (>1 token in the ring).\ndtr = DijkstraTokenRing(n,k,x_initial)\n@show output(dtr) # Show where the token is (are).\n\n@show update!(dtr), output(dtr) # Perform the update, show the state vector and show where the token is.\n@show update!(dtr), output(dtr) # Repeat a few times to see the stabilization. \n@show update!(dtr), output(dtr)\n@show update!(dtr), output(dtr)\n@show update!(dtr), output(dtr)\n\n\nx_initial = rand(0:k, n) = [4, 0, 3, 2]\noutput(dtr) = [0, 1, 1, 1]\n(update!(dtr), output(dtr)) = ([4, 4, 0, 3], [0, 0, 1, 1])\n(update!(dtr), output(dtr)) = ([4, 4, 4, 0], [0, 0, 0, 1])\n(update!(dtr), output(dtr)) = ([4, 4, 4, 4], [1, 0, 0, 0])\n(update!(dtr), output(dtr)) = ([1, 4, 4, 4], [0, 1, 0, 0])\n(update!(dtr), output(dtr)) = ([1, 1, 4, 4], [0, 0, 1, 0])\n\n\n([1, 1, 4, 4], [0, 0, 1, 0])\n\n\nWe can see that although initially the there can be more tokens, after a few iterations the algorithm achieves the goal of having just one token in the ring.\n\n\n\nExtended-state automaton\nYet another extension of an automaton is the extended-state automaton. And indeed, the hyphen is there on purpose as we extend the state space.\nIn particular, we augment the state variable(s) that define the states/modes/locations (the nodes in the graph) by additional (typed) state variables: Int, Enum, Bool, …\nTransitions from one mode to another are then guarded by conditions on theses new extra state variables.\nBesides being guarded by a guard condition, a given transition can also be labelled by a reset function that resets the extended-state variables.\n\nExample 12 (Counting up to 10) In this example, there are two modes (on and off), which can be captured by a single binary state variable, say x. But then there is an additional integer variable k, and the two variables together characterize the extended state.\n\n\n\n\n\n\n\n\nG\n\n\ninit\ninit\n\n\n\nOFF\n\nOFF\n\n\n\ninit->OFF\n\n\nint k=0\n\n\n\nON\n\nON\n\n\n\nOFF->ON\n\n\npress\n\n\n\nON->OFF\n\n\n(press ⋁ k ≥ 10); k=0\n\n\n\nON->ON\n\n\n(press ∧ k < 10); k=k+1\n\n\n\n\n\n\nFigure 13: Example of a digraph representation of the extended-state automaton for counting up to ten", + "text": "Extensions\nThe concept of an automaton can be extended in several ways. In particular, the following two extensions introduce the concept of an output to an automaton.\n\nMoore machine\nOne extension of an automaton with outputs is Moore machine. The outputs assigned to the states by the output function y = g(x).\nThe output is produced (emitted) when the (new) state is entered.\nNote, in particular, that the output does not depend on the input. This has a major advantage when a feedback loop is closed around this system, since no algebraic loop is created.\nGraphically, we make a conventions that outputs are the labels of the states.\n\nExample 8 (Moore machine) The following automaton has just three states, but just two outputs (FLOW and NO FLOW).\n\n\n\n\n\n\n\n\nG\n\n\ninit\ninit\n\n\n\nclosed\n\nNO FLOW\nValve\nclosed\n\n\n\ninit->closed\n\n\n\n\n\npartial\n\nFLOW\nValve\npartially\nopen\n\n\n\nclosed->partial\n\n\nopen valve one turn\n\n\n\npartial->closed\n\n\nclose valve one turn\n\n\n\nfull\n\nFLOW\nValve\nfully open\n\n\n\npartial->full\n\n\nopen valve one turn\n\n\n\nfull->closed\n\n\nemergency shut off\n\n\n\nfull->partial\n\n\nclose valve one turn\n\n\n\n\n\n\nFigure 9: Example of a digraph representation of the Moore machine for a valve control\n\n\n\n\n\n\n\n\nMealy machine\nMealy machine is another extension of an automaton. Here the outputs are associated with the transitions rather than the states.\nSince the events already associated with the states can be viewed as the inputs, we now have input/output transition labels. The transition label e_\\mathrm{i}/e_\\mathrm{o} on the transion from x_1 to x_2 reads as “the input event e_\\mathrm{i} at state x_1 activates the transition to x_2, which outputs the event e_\\mathrm{o}” and can be written as x_1\\xrightarrow{e_\\mathrm{i}/e_\\mathrm{o}} x_2.\nIt can be viewed as if the output function also considers the input and not only the state y = e_\\mathrm{o} = g(x,e_\\mathrm{i}).\nIn contrast with the Moore machine, here the output is produced (emitted) during the transition (before the new state is entered).\n\nExample 9 (Mealy machine) Coffee machine: coffee for 30 CZK, machine accepting 10 and 20 CZK coins, no change.\n\n\n\n\n\n\n\n\nG\n\n\ninit\ninit\n\n\n\n0\n\nNo coin\n\n\n\ninit->0\n\n\n\n\n\n10\n\n10 CZK\n\n\n\n0->10\n\n\ninsert 10 CZK / no coffee\n\n\n\n20\n\n20 CZK\n\n\n\n0->20\n\n\ninsert 20 CZK / no coffee\n\n\n\n10->0\n\n\ninsert 20 CZK / coffee\n\n\n\n10->20\n\n\ninsert 10 CZK / no coffee\n\n\n\n20->0\n\n\ninsert 10 CZK / coffee\n\n\n\n20->10\n\n\ninsert 20 CZK / coffee\n\n\n\n\n\n\nFigure 10: Example of a digraph representation of the Mealy machine for a coffee machine\n\n\n\n\n\n\n\nExample 10 (Reformulate the previous example as a Moore machine) Two more states wrt Mealy\n\n\n\n\n\n\n\n\nG\n\n\ninit\ninit\n\n\n\n0\n\nNO COFFEE\nNo\ncoin\n\n\n\ninit->0\n\n\n\n\n\n10\n\nNO COFFEE\n10\nCZK\n\n\n\n0->10\n\n\ninsert 10 CZK\n\n\n\n20\n\nNO COFFEE\n20\nCZK\n\n\n\n0->20\n\n\ninsert 20 CZK\n\n\n\n10->20\n\n\ninsert 10 CZK\n\n\n\n30\n\nCOFFEE\n10+20\nCZK\n\n\n\n10->30\n\n\ninsert 20 CZK\n\n\n\n20->30\n\n\ninsert 10 CZK\n\n\n\n40\n\nCOFFEE\n20+20\nCZK\n\n\n\n20->40\n\n\ninsert 20 CZK\n\n\n\n30->0\n\n\n\n\n\n30->10\n\n\ninsert 10 CZK\n\n\n\n30->20\n\n\ninsert 20 CZK\n\n\n\n40->10\n\n\n\n\n\n40->20\n\n\ninsert 10 CZK\n\n\n\n40->30\n\n\ninsert 20 CZK\n\n\n\n\n\n\nFigure 11: Example of a digraph representation of the Moore machine for a coffee machine\n\n\n\n\n\n\n\n\n\n\n\n\nNote\n\n\n\nThere are transitions from 30 and 40 back to 0 that are not labelled by any event. This does not seem to follow the general rule that transitions are always triggered by events. Not what? It can be resolved upon introducing time as the timeout transitions.\n\n\n\nExample 11 (Dijkstra’s token passing) The motivation for this example is to show that it is perhaps not always productive to insist on visual description of the automaton using a graph. The four components of our formal definition of an automaton are just enough, and they translate directly to a code.\nThe example comes from the field of distributed computing systems. It considers several computers that are connected in ring topology, and the communication is just one-directional as Fig 12 shows. The task is to use the communication to determine in – a distributed way – which of the computers carries a (single) token at a given time. And to realize passing of the token to a neighbour. We assume a synchronous case, in which all the computers are sending simultaneously, say, with some fixed sending period.\n\n\n\n\n\n\n\n\nG\n\n\n0\n\n0\n\n\n\n1\n\n1\n\n\n\n0->1\n\n\n\n\n\n2\n\n2\n\n\n\n1->2\n\n\n\n\n\n3\n\n3\n\n\n\n2->3\n\n\n\n\n\n3->0\n\n\n\n\n\n\n\n\nFigure 12: Example of a ring topology for Dijkstra’s token passing in a distributed system\n\n\n\n\n\nOne popular method for this is called Dijkstra’s token passing. Each computer keeps a single integer value as its state variable. And it forwards this integer value to the neighbour (in the clockwise direction in our setting). Upon receiving the value from the other neighbour (in the counter-clockwise direction), it updates its own value according to the rule displayed in the code below. At every clock tick, the state vector (composed of the individual state variables) is updated according to the function update!() in the code. Based on the value of the state vector, an output is computed, which decodes the informovation about the location of the token from the state vector. Again, the details are in the output() function.\n\n\nShow the code\nstruct DijkstraTokenRing\n number_of_nodes::Int64\n max_value_of_state_variable::Int64\n state_vector::Vector{Int64}\nend\n\nfunction update!(dtr::DijkstraTokenRing) \n n = dtr.number_of_nodes\n k = dtr.max_value_of_state_variable\n x = dtr.state_vector\n xnext = copy(x)\n for i in eachindex(x) # Mind the +1 shift. x[2] corresponds to x₁ in the literature.\n if i == 1 \n xnext[i] = (x[i] == x[n]) ? mod(x[i] + 1,k) : x[i] # Increment if the left neighbour is identical.\n else \n xnext[i] = (x[i] != x[i-1]) ? x[i-1] : x[i] # Update by the differing left neighbour.\n end\n end\n dtr.state_vector .= xnext \nend\n\nfunction output(dtr::DijkstraTokenRing) # Token = 1, no token = 0 at the given position. \n x = dtr.state_vector\n y = similar(x)\n y[1] = iszero(x[1]-x[end])\n y[2:end] .= .!iszero.(diff(x))\n return y\nend\n\n\noutput (generic function with 1 method)\n\n\nWe now rund the code for a given number of computers and some initial state vector that does not necessarily comply with the requirement that there is only one token in the ring.\n\n\nShow the code\nn = 4 # Concrete number of nodes.\nk = n # Concrete max value of a state variable (>= n).\n@show x_initial = rand(0:k,n) # Initial state vector, not necessarily acceptable (>1 token in the ring).\ndtr = DijkstraTokenRing(n,k,x_initial)\n@show output(dtr) # Show where the token is (are).\n\n@show update!(dtr), output(dtr) # Perform the update, show the state vector and show where the token is.\n@show update!(dtr), output(dtr) # Repeat a few times to see the stabilization. \n@show update!(dtr), output(dtr)\n@show update!(dtr), output(dtr)\n@show update!(dtr), output(dtr)\n\n\nx_initial = rand(0:k, n) = [2, 0, 2, 3]\noutput(dtr) = [0, 1, 1, 1]\n(update!(dtr), output(dtr)) = ([2, 2, 0, 2], [1, 0, 1, 1])\n(update!(dtr), output(dtr)) = ([3, 2, 2, 0], [0, 1, 0, 1])\n(update!(dtr), output(dtr)) = ([3, 3, 2, 2], [0, 0, 1, 0])\n(update!(dtr), output(dtr)) = ([3, 3, 3, 2], [0, 0, 0, 1])\n(update!(dtr), output(dtr)) = ([3, 3, 3, 3], [1, 0, 0, 0])\n\n\n([3, 3, 3, 3], [1, 0, 0, 0])\n\n\nWe can see that although initially the there can be more tokens, after a few iterations the algorithm achieves the goal of having just one token in the ring.\n\n\n\nExtended-state automaton\nYet another extension of an automaton is the extended-state automaton. And indeed, the hyphen is there on purpose as we extend the state space.\nIn particular, we augment the state variable(s) that define the states/modes/locations (the nodes in the graph) by additional (typed) state variables: Int, Enum, Bool, …\nTransitions from one mode to another are then guarded by conditions on theses new extra state variables.\nBesides being guarded by a guard condition, a given transition can also be labelled by a reset function that resets the extended-state variables.\n\nExample 12 (Counting up to 10) In this example, there are two modes (on and off), which can be captured by a single binary state variable, say x. But then there is an additional integer variable k, and the two variables together characterize the extended state.\n\n\n\n\n\n\n\n\nG\n\n\ninit\ninit\n\n\n\nOFF\n\nOFF\n\n\n\ninit->OFF\n\n\nint k=0\n\n\n\nON\n\nON\n\n\n\nOFF->ON\n\n\npress\n\n\n\nON->OFF\n\n\n(press ⋁ k ≥ 10); k=0\n\n\n\nON->ON\n\n\n(press ∧ k < 10); k=k+1\n\n\n\n\n\n\nFigure 13: Example of a digraph representation of the extended-state automaton for counting up to ten", "crumbs": [ "1. Discrete-event systems: Automata", "State automata" @@ -873,7 +873,7 @@ "href": "verification_barrier.html#convex-relaxation-of-the-barrier-certificate-problem", "title": "Barrier certificates", "section": "Convex relaxation of the barrier certificate problem", - "text": "Convex relaxation of the barrier certificate problem\nWe relax the third condition so that it holds not only at B(\\bm x) = 0 but everywhere. The three conditions are then B(\\bm x) > 0,\\quad \\forall \\bm x \\in \\mathcal X_\\mathrm{u},\nB(\\bm x) \\leq 0,\\quad \\forall \\bm x \\in \\mathcal X_0,\n\\nabla B(\\bm x)^\\top \\mathbf f(\\bm x, \\bm d) \\leq 0,\\quad \\forall \\bm x\\in \\mathcal X, \\bm d \\in \\mathcal D.\nLet’s now demonstrate this by means of an example.\n\nExample 1 Consider the system modelled by \n\\begin{aligned}\n\\dot x_1 &= x_2\\\\\n\\dot x_2 &= -x_1 + \\frac{p}{3}x_1^3 - x_2,\n\\end{aligned}\n where the parameter p\\in [0.9,1.1].\nThe initial set is given by \n\\mathcal X_0 = \\{ \\bm x \\in \\mathbb R^2 \\mid (x_1-1.5)^2 + x_2^2 \\leq 0.25 \\}\n and the unsafe set is given by \n\\mathcal X_\\mathrm{u} = \\{ \\bm x \\in \\mathbb R^2 \\mid (x_1+1)^2 + (x_2+1)^2 \\leq 0.16 \\}.\n\nThe vector field \\mathbf f and the initial and unsafe sets are shown in the figure below.\n\n\nShow the code\nusing SumOfSquares\nusing DynamicPolynomials\n# using MosekTools \nusing CSDP\n\noptimizer = optimizer_with_attributes(CSDP.Optimizer, MOI.Silent() => true)\nmodel = SOSModel(optimizer)\n@polyvar x[1:2] \n\np = 1;\n\nf = [ x[2],\n -x[1] + (p/3)*x[1]^3 - x[2]]\n\ng₁ = -(x[1]+1)^2 - (x[2]+1)^2 + 0.16 # 𝒳ᵤ = {x ∈ R²: g₁(x) ≥ 0}\nh₁ = -(x[1]-1.5)^2 - x[2]^2 + 0.25 # 𝒳₀ = {x ∈ R²: h₁(x) ≥ 0}\n\nX = monomials(x, 0:4)\n@variable(model, B, Poly(X))\n\nε = 0.001\n@constraint(model, B >= ε, domain = @set(g₁ >= 0))\n\n@constraint(model, B <= 0, domain = @set(h₁ >= 0))\n\nusing LinearAlgebra # Needed for `dot`\ndBdt = dot(differentiate(B, x), f)\n@constraint(model, -dBdt >= 0)\n\nJuMP.optimize!(model)\n\nJuMP.primal_status(model)\n\nimport DifferentialEquations, Plots, ImplicitPlots\nfunction phase_plot(f, B, g₁, h₁, quiver_scaling, Δt, X0, solver = DifferentialEquations.Tsit5())\n X₀plot = ImplicitPlots.implicit_plot(h₁; xlims=(-2, 3), ylims=(-2.5, 2.5), resolution = 1000, label=\"X₀\", linecolor=:blue)\n Xᵤplot = ImplicitPlots.implicit_plot!(g₁; xlims=(-2, 3), ylims=(-2.5, 2.5), resolution = 1000, label=\"Xᵤ\", linecolor=:teal)\n Bplot = ImplicitPlots.implicit_plot!(B; xlims=(-2, 3), ylims=(-2.5, 2.5), resolution = 1000, label=\"B = 0\", linecolor=:red)\n Plots.plot(X₀plot)\n Plots.plot!(Xᵤplot)\n Plots.plot!(Bplot)\n ∇(vx, vy) = [fi(x[1] => vx, x[2] => vy) for fi in f]\n ∇pt(v, p, t) = ∇(v[1], v[2])\n function traj(v0)\n tspan = (0.0, Δt)\n prob = DifferentialEquations.ODEProblem(∇pt, v0, tspan)\n return DifferentialEquations.solve(prob, solver, reltol=1e-8, abstol=1e-8)\n end\n ticks = -5:0.5:5\n X = repeat(ticks, 1, length(ticks))\n Y = X'\n Plots.quiver!(X, Y, quiver = (x, y) -> ∇(x, y) / quiver_scaling, linewidth=0.5)\n for x0 in X0\n Plots.plot!(traj(x0), idxs=(1, 2), label = nothing)\n end\n Plots.plot!(xlims = (-2, 3), ylims = (-2.5, 2.5))\nend\n\nphase_plot(f, value(B), g₁, h₁, 10, 30.0, [[x1, x2] for x1 in 1.2:0.2:1.7, x2 in -0.35:0.1:0.35])\n\n\n┌ Warning: At t=4.423118290940107, dt was forced below floating point epsilon 8.881784197001252e-16, and step error estimate = 1.139033855908175. Aborting. There is either an error in your model specification or the true solution is unstable (or the true solution can not be represented in the precision of Float64).\n└ @ SciMLBase ~/.julia/packages/SciMLBase/rBKqA/src/integrator_interface.jl:623", + "text": "Convex relaxation of the barrier certificate problem\nWe relax the third condition so that it holds not only at B(\\bm x) = 0 but everywhere. The three conditions are then B(\\bm x) > 0,\\quad \\forall \\bm x \\in \\mathcal X_\\mathrm{u},\nB(\\bm x) \\leq 0,\\quad \\forall \\bm x \\in \\mathcal X_0,\n\\nabla B(\\bm x)^\\top \\mathbf f(\\bm x, \\bm d) \\leq 0,\\quad \\forall \\bm x\\in \\mathcal X, \\bm d \\in \\mathcal D.\nLet’s now demonstrate this by means of an example.\n\nExample 1 Consider the system modelled by \n\\begin{aligned}\n\\dot x_1 &= x_2\\\\\n\\dot x_2 &= -x_1 + \\frac{p}{3}x_1^3 - x_2,\n\\end{aligned}\n where the parameter p\\in [0.9,1.1].\nThe initial set is given by \n\\mathcal X_0 = \\{ \\bm x \\in \\mathbb R^2 \\mid (x_1-1.5)^2 + x_2^2 \\leq 0.25 \\}\n and the unsafe set is given by \n\\mathcal X_\\mathrm{u} = \\{ \\bm x \\in \\mathbb R^2 \\mid (x_1+1)^2 + (x_2+1)^2 \\leq 0.16 \\}.\n\nThe vector field \\mathbf f and the initial and unsafe sets are shown in the figure below.\n\n\nShow the code\nusing SumOfSquares\nusing DynamicPolynomials\n# using MosekTools \nusing CSDP\n\noptimizer = optimizer_with_attributes(CSDP.Optimizer, MOI.Silent() => true)\nmodel = SOSModel(optimizer)\n@polyvar x[1:2] \n\np = 1;\n\nf = [ x[2],\n -x[1] + (p/3)*x[1]^3 - x[2]]\n\ng₁ = -(x[1]+1)^2 - (x[2]+1)^2 + 0.16 # 𝒳ᵤ = {x ∈ R²: g₁(x) ≥ 0}\nh₁ = -(x[1]-1.5)^2 - x[2]^2 + 0.25 # 𝒳₀ = {x ∈ R²: h₁(x) ≥ 0}\n\nX = monomials(x, 0:4)\n@variable(model, B, Poly(X))\n\nε = 0.001\n@constraint(model, B >= ε, domain = @set(g₁ >= 0))\n\n@constraint(model, B <= 0, domain = @set(h₁ >= 0))\n\nusing LinearAlgebra # Needed for `dot`\ndBdt = dot(differentiate(B, x), f)\n@constraint(model, -dBdt >= 0)\n\nJuMP.optimize!(model)\n\nJuMP.primal_status(model)\n\nimport DifferentialEquations, Plots, ImplicitPlots\nfunction phase_plot(f, B, g₁, h₁, quiver_scaling, Δt, X0, solver = DifferentialEquations.Tsit5())\n X₀plot = ImplicitPlots.implicit_plot(h₁; xlims=(-2, 3), ylims=(-2.5, 2.5), resolution = 1000, label=\"X₀\", linecolor=:blue)\n Xᵤplot = ImplicitPlots.implicit_plot!(g₁; xlims=(-2, 3), ylims=(-2.5, 2.5), resolution = 1000, label=\"Xᵤ\", linecolor=:teal)\n Bplot = ImplicitPlots.implicit_plot!(B; xlims=(-2, 3), ylims=(-2.5, 2.5), resolution = 1000, label=\"B = 0\", linecolor=:red)\n Plots.plot(X₀plot)\n Plots.plot!(Xᵤplot)\n Plots.plot!(Bplot)\n ∇(vx, vy) = [fi(x[1] => vx, x[2] => vy) for fi in f]\n ∇pt(v, p, t) = ∇(v[1], v[2])\n function traj(v0)\n tspan = (0.0, Δt)\n prob = DifferentialEquations.ODEProblem(∇pt, v0, tspan)\n return DifferentialEquations.solve(prob, solver, reltol=1e-8, abstol=1e-8)\n end\n ticks = -5:0.5:5\n X = repeat(ticks, 1, length(ticks))\n Y = X'\n Plots.quiver!(X, Y, quiver = (x, y) -> ∇(x, y) / quiver_scaling, linewidth=0.5)\n for x0 in X0\n Plots.plot!(traj(x0), idxs=(1, 2), label = nothing)\n end\n Plots.plot!(xlims = (-2, 3), ylims = (-2.5, 2.5))\nend\n\nphase_plot(f, value(B), g₁, h₁, 10, 30.0, [[x1, x2] for x1 in 1.2:0.2:1.7, x2 in -0.35:0.1:0.35])\n\n\n┌ Warning: At t=4.423118290940107, dt was forced below floating point epsilon 8.881784197001252e-16, and step error estimate = 1.139033855908175. Aborting. There is either an error in your model specification or the true solution is unstable (or the true solution can not be represented in the precision of Float64).\n└ @ SciMLBase ~/.julia/packages/SciMLBase/BaHpR/src/integrator_interface.jl:623", "crumbs": [ "12. Formal verification", "Barrier certificates" @@ -1720,7 +1720,7 @@ "href": "complementarity_simulations.html", "title": "Simulations of complementarity systems using time-stepping", "section": "", - "text": "One of the useful outcomes of the theory of complementarity systems is a new family of methods for numerical simulation of discontinuous systems. Here we will demonstrate the essence by introducing the method of time-stepping. And we do it by means of an example.\n\nExample 1 (Simulation using time-stepping) Consider the following discontinuous dynamical system in \\mathbb R^2: \n\\begin{aligned}\n\\dot x_1 &= -\\operatorname{sign} x_1 + 2 \\operatorname{sign} x_2\\\\\n\\dot x_2 &= -2\\operatorname{sign} x_1 -\\operatorname{sign} x_2.\n\\end{aligned}\n\nThe state portrait is in Fig. 1.\n\n\nShow the code\nf₁(x) = -sign(x[1]) + 2*sign(x[2])\nf₂(x) = -2*sign(x[1]) - sign(x[2])\nf(x) = [f₁(x), f₂(x)]\n\nusing CairoMakie\nfig = Figure(size = (800, 800),fontsize=20)\nax = Axis(fig[1, 1], xlabel = \"x₁\", ylabel = \"x₂\")\nstreamplot!(ax,(x₁,x₂)->Point2f(f([x₁,x₂])), -2.0..2.0, -2.0..2.0, colormap = :magma)\nfig\n\n\n\n\n\n\n\n\nFigure 1: State portrait of the discontinuous system\n\n\n\n\n\nOne particular (vector) state trajectory obtained by some default ODE solver is in Fig. 2.\n\n\nShow the code\nusing DifferentialEquations\n\nfunction f!(dx,x,p,t)\n dx[1] = -sign(x[1]) + 2*sign(x[2])\n dx[2] = -2*sign(x[1]) - sign(x[2])\nend\n\nx0 = [-1.0, 1.0]\ntfinal = 2.0\ntspan = (0.0,tfinal)\nprob = ODEProblem(f!,x0,tspan)\nsol = solve(prob)\n\nusing Plots\nPlots.plot(sol,xlabel=\"t\",ylabel=\"x\",label=false,lw=3)\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 2: Trajectory of the discontinuous system\n\n\n\n\nWe can also plot the trajectory in the state space, as in Fig. 3.\n\n\nShow the code\nPlots.plot(sol[1,:],sol[2,:],xlabel=\"x₁\",ylabel=\"x₂\",label=false,aspect_ratio=:equal,lw=3,xlims=(-1.2,0.5))\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 3: Trajectory of the discontinuous system in the state space\n\n\n\n\nNote that for the default setting of absolute and relative tolerances, the adaptive-step ODE solver actually needed a huge number of steps:\n\n\nShow the code\nlength(sol.t)\n\n\n288594\n\n\nThis is indeed quite a lot? Can we do better?\nIn this particular example, we have the benefit of being able to analyze the solution analytically. In particular, we can ask: how fast does the solution approach the origin? We turn this into a question of how fast the distance from the origin decreases. With some anticipation we use the 1-norm \\|\\bm x\\|_1 = |x_1| + |x_2| to the distance of the state from the origin. We then ask: \n\\frac{\\mathrm d}{\\mathrm dt}\\|\\bm x\\|_1 = ?\n\nWe avoid the troubles with nonsmoothness of the absolute value by considering each quadrant separately. Let’s start in the first (upper right) quadrant, that is, x_1>0 and x_2>0, from which it follows that |x_1| = x_1, \\;|x_2| = x_2, and therefore \n\\frac{\\mathrm d}{\\mathrm dt}\\|\\bm x\\|_1 = \\dot x_1 + \\dot x_2 = 1 - 3 = -2.\n\nThe situation is identical in the other quadrants. We do not worry that the norm is undefined on the axes, because the trajectory obviously just crosses them.\nThe conclusion is that the trajectory will hit the origin in finite time (!): with, say, x_1(0) = 1 and x_2(0) = 1, the trajectory hits the origin at t=(|x_1(0)|+|x_2(0)|)/2 = 1. Surprisingly (or not), this will happen after an infinite number of revolutions around the origin…\nWe have seen above in Fig. 2 that a default solver for ODE can handle the situation in a decent way. But we have also seen that this was at the cost of a huge number of steps. To get some insight, we implement our own solver.\n\nForward Euler with fixed step size\nWe start with the simplest of all methods, the forward Euler method with a fixed step length. The computation of the next state is done just by the assignment \n\\begin{aligned}\n{\\color{blue}x_{1,k+1}} &= x_{1,k} + h (-\\operatorname{sign} x_{1,k} + 2 \\operatorname{sign} x_{2,k})\\\\\n{\\color{blue}x_{2,k+1}} &= x_{1,k} + h (-2\\operatorname{sign} x_{1,k} - \\operatorname{sign} x_{2,k}).\n\\end{aligned}\n\n\n\n\n\n\n\nBlue color in our text is used to denote the unknown\n\n\n\nWe use the blue color here (and in the next few paragraphs) to highlight what is uknown.\n\n\n\n\nShow the code\nf(x) = [-sign(x[1]) + 2*sign(x[2]), -2*sign(x[1]) - sign(x[2])]\n\nusing LinearAlgebra\nN = 1000\nx₀ = [-1.0, 1.0] \nx = [x₀]\ntfinal = norm(x₀,1)/2\ntfinal = 5.0\nh = tfinal/N \nt = range(0.0, step=h, stop=tfinal)\n\nfor i=1:N\n xnext = x[i] + h*f(x[i]) \n push!(x,xnext)\nend\n\nX = [getindex.(x, i) for i in 1:length(x[1])]\n\nPlots.plot(t,X,lw=3,label=false,xlabel=\"t\",ylabel=\"x\")\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 4: Solution trajectory for a discontinuous system obtained by the forward Euler method\n\n\n\n\nThen number of steps is\n\n\nShow the code\nlength(t)\n\n\n1001\n\n\nwhich is significantly less than before, but the solution is not perfect – notice the chattering. It could be reduced by using a smaller step size, but again, at the cost of a longer simulation time.\n\n\nBackward Euler\nAs an alternative to the forward Euler method, we can use the backward Euler method. The computation of the next state is done by solving the nonlinear equations \n\\begin{aligned}\n{\\color{blue} x_{1,k+1}} &= x_{1,k} + h (-\\operatorname{sign} {\\color{blue}x_{1,k+1}} + 2 \\operatorname{sign} {\\color{blue}x_{2,k+1}})\\\\\n{\\color{blue} x_{2,k+1}} &= x_{1,k} + h (-2\\operatorname{sign} {\\color{blue}x_{1,k+1}} - \\operatorname{sign} {\\color{blue}x_{2,k+1}}).\n\\end{aligned}\n\nThe discontinuities on the right-hand sides constitute a challenge for solvers of nonlinear equations – recall that the Newton’s method requires the first derivatives (assembled into the Jacobian) or their approximation. Instead of this struggle, we can use the linear complementarity problem (LCP) formulation.\n\n\nFormulation of the backward Euler using LCP\nInstead solving the above nonlinear equations with discontinuities, we introduce new variables u_1 and u_2 as the outputs of the \\operatorname{sign} functions: \n\\begin{aligned}\n{\\color{blue} x_{1,k+1}} &= x_{1,k} + h (-{\\color{blue}u_{1}} + 2 {\\color{blue}u_{2}})\\\\\n{\\color{blue} x_{2,k+1}} &= x_{1,k} + h (-2{\\color{blue}u_{1}} - {\\color{blue}u_{2}}).\n\\end{aligned}\n\nBut now we have to enforce the relationship between \\bm u and \\bm x_{k+1}. Recall the standard definition of the \\operatorname{sign} function is \n\\operatorname{sign}(x) = \\begin{cases}\n1 & x>0\\\\\n0 & x=0\\\\\n-1 & x<0,\n\\end{cases}\n but we change the definition to a set-valued function \n\\begin{cases}\n\\operatorname{sign}(x) = 1 & x>0\\\\\n\\operatorname{sign}(x) \\in [-1,1] & x=0\\\\\n\\operatorname{sign}(x) = -1 & x<0.\n\\end{cases}\n\nAccordingly, we set the relationship between \\bm u and \\bm x to \n\\begin{cases}\nu_1 = 1 & x_1>0\\\\\nu_1 \\in [-1,1] & x_1=0\\\\\nu_1 = -1 & x_1<0,\n\\end{cases}\n and \n\\begin{cases}\nu_2 = 1 & x_2>0\\\\\nu_2 \\in [-1,1] & x_2=0\\\\\nu_2 = -1 & x_2<0.\n\\end{cases}\n\nBut these are mixed complementarity contraints we have defined previously! We can thus write the single step of the simulation algorithm as the MCP \n\\boxed{\n\\begin{aligned}\n\\begin{bmatrix}\n{\\color{blue} x_{1,k+1}}\\\\\n{\\color{blue} x_{1,k+1}}\n\\end{bmatrix}\n&=\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix} + h\n\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}\n\\begin{bmatrix}\n{\\color{blue}u_{1}}\\\\\n{\\color{blue}u_{2}}\n\\end{bmatrix}\\\\\n-1 &\\leq {\\color{blue} u_1} \\leq 1 \\quad \\bot \\quad -{\\color{blue}x_{1,k+1}}\\\\\n-1 &\\leq {\\color{blue} u_2} \\leq 1 \\quad \\bot \\quad -{\\color{blue}x_{2,k+1}}.\n\\end{aligned}\n}\n\\tag{1}\nWe now have enough to start implementing a solver for this system, provided we have an access to a solver the MCP (see the page dedicated to software for complementarity problems). But before we do it, we analyze the problem a bit deeper. In this particular small and simple case, it is still doable with just a pencil and paper.\n\n\n9 possible combinations\nThere are now 9 possible combinations of the values of u_1 and u_2, each of them strictly inside their respective intervals, or at the either end. Let’s explore some combinations. We start with x_{1,k+1} = x_{2,k+1} = 0, while u_1 \\in [-1,1] and u_2 \\in [-1,1]:\n\n\\begin{aligned}\n\\begin{bmatrix}\n0\\\\\n0\n\\end{bmatrix}\n&=\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix} + h\n\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}\n\\begin{bmatrix}\n{\\color{blue}u_{1}}\\\\\n{\\color{blue}u_{2}}\n\\end{bmatrix}\\\\\n& -1 \\leq {\\color{blue} u_1} \\leq 1, \\quad -1 \\leq {\\color{blue} u_2} \\leq 1\n\\end{aligned}\n\nHow does the set of states from which the next state is zero look like? We isolate the vector \\bm u from the equation and impose the constraints on it: \n\\begin{aligned}\n-\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix}\n&= h\n\\begin{bmatrix}\n{\\color{blue}u_{1}}\\\\\n{\\color{blue}u_{2}}\n\\end{bmatrix}\\\\\n-1 \\leq {\\color{blue} u_1} \\leq 1, \\quad -1 &\\leq {\\color{blue} u_2} \\leq 1\n\\end{aligned}\n\nWe then get \n\\begin{bmatrix}\n-h\\\\-h\n\\end{bmatrix}\n\\leq\n\\begin{bmatrix}\n0.2 & 0.4 \\\\\n-0.4 & 0.2\n\\end{bmatrix}\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix}\n\\leq\n\\begin{bmatrix}\nh\\\\ h\n\\end{bmatrix}\n\nFor h=0.2, the resulting set is a rotated square, as shown in Fig. 5.\n\n\nShow the code\nusing LazySets\nh = 0.2\nH1u = HalfSpace([0.2, 0.4], h)\nH2u = HalfSpace([-0.4, 0.2], h)\nH1l = HalfSpace(-[0.2, 0.4], h)\nH2l = HalfSpace(-[-0.4, 0.2], h)\n\nHa = H1u ∩ H2u ∩ H1l ∩ H2l\n\nusing Plots\nPlots.plot(Ha, aspect_ratio=:equal,xlabel=\"x₁\",ylabel=\"x₂\",label=false,xlims=(-1.5,1.5),ylims=(-1.5,1.5))\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 5: The set of states from which the next state is zero (the origin)\n\n\n\n\nIndeed, if the current state is in this rotated square, then the next state will be zero. No more infite spiralling around the origin! No more chattering! Perfect zero.\n\n\nAnother combination\nWe now consider u_1 = 1, u_2 = 1, which we substitute into Eq. 1: \n\\begin{aligned}\n\\begin{bmatrix}\n{\\color{blue} x_{1,k+1}}\\\\\n{\\color{blue} x_{1,k+1}}\n\\end{bmatrix}\n&=\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix} + h\n\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}\n\\begin{bmatrix}\n{1}\\\\\n{1}\n\\end{bmatrix}\\\\\n\\color{blue}x_{1,k+1} &\\geq 0\\\\\n\\color{blue}x_{2,k+1} &\\geq 0,\n\\end{aligned}\n which can be reformatted to \n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix} + h\n\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}\n\\begin{bmatrix}\n1\\\\\n1\n\\end{bmatrix}\\geq \\bm 0,\n and further to \n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix}\n\\geq h\n\\begin{bmatrix}\n-1\\\\\n3\n\\end{bmatrix}.\n\nThe set of states for which the next state is such that both its state variables are positive (hence u_1 = 1, u_2 = 1) is shown in Fig. 6.\n\n\nShow the code\nusing LazySets\nh = 0.2\nA = [-1.0 2.0; -2.0 -1.0]\nu = [1.0, 1.0]\nb = h*A*u\n\nH1 = HalfSpace([-1.0, 0.0], b[1])\nH2 = HalfSpace([0.0, -1.0], b[2])\nHb = H1 ∩ H2\n\nusing Plots\nPlots.plot(Ha, aspect_ratio=:equal,xlabel=\"x₁\",ylabel=\"x₂\",label=false,xlims=(-1.5,1.5),ylims=(-1.5,1.5))\nPlots.plot!(Hb)\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 6: The set of states from which the next state has both state variables positive\n\n\n\n\n\n\nAll nine regions\nWe plot all nine regions in Fig. 7. Note that we do not color them all – the white regions are to be counted as well.\n\n\nShow the code\nusing LazySets\nh = 0.2\nA = [-1.0 2.0; -2.0 -1.0]\n\nu = [1, -1]\nb = h*A*u\n\nH1 = HalfSpace(-[1.0, 0.0], b[1])\nH2 = HalfSpace([0.0, 1.0], -b[2])\nHc = H1 ∩ H2\n\nu = [-1, 1]\nb = h*A*u\n\nH1 = HalfSpace([1.0, 0.0], -b[1])\nH2 = HalfSpace(-[0.0, 1.0], b[2])\nHd = H1 ∩ H2\n\nu = [-1, -1]\nb = h*A*u\n\nH1 = HalfSpace([1.0, 0.0], -b[1])\nH2 = HalfSpace([0.0, 1.0], -b[2])\nHe = H1 ∩ H2\n\nusing Plots\nPlots.plot(Ha, aspect_ratio=:equal,xlabel=\"x₁\",ylabel=\"x₂\",label=false,xlims=(-1.5,1.5),ylims=(-1.5,1.5))\nPlots.plot!(Hb)\nPlots.plot!(Hc)\nPlots.plot!(Hd)\nPlots.plot!(He)\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 7: The nine regions of the state space for the spiralling system\n\n\n\n\n\n\nSolutions using a MCP solver\nNow that we have got some insight into the algorithm, we can implement it by wrapping a for-loop around the MCP solver.\n\n\nShow the code\nM = [-1 2; -2 -1]\nh = 2e-1\ntfinal = 2.0\nN = tfinal/h\n\nx0 = [-1.0, 1.0]\nx = [x0]\n\nusing JuMP\nusing PATHSolver\n\nfor i = 1:N\n model = Model(PATHSolver.Optimizer)\n set_optimizer_attribute(model, \"output\", \"no\")\n set_silent(model)\n @variable(model, -1 <= u[1:2] <= 1)\n @constraint(model, -h*M * u - x[end] ⟂ u)\n optimize!(model)\n push!(x, x[end]+h*M*value.(u))\nend\n\n\nThe code is admittedly unoptimized (for example, it is not wise to start building the optimization model from the scratch in every iteration), but it was our intention to keep it simple to be read conveniently.\nThe code outcome can be plotted as in Fig. 8. A solution obtained by a classical (default) ODE solver with default setting is plotted for comparison.\n\n\nShow the code\nt = range(0.0, step=h, stop=tfinal)\nX = [getindex.(x, i) for i in 1:length(x[1])]\n\nusing Plots\nPlots.plot(Ha, aspect_ratio=:equal,xlabel=\"x₁\",ylabel=\"x₂\",label=false,xlims=(-1.5,1.5),ylims=(-1.5,1.5))\nPlots.plot!(Hb)\nPlots.plot!(Hc)\nPlots.plot!(Hd)\nPlots.plot!(He)\nPlots.plot!(X[1],X[2],xlabel=\"x₁\",ylabel=\"x₂\",label=\"Time-stepping\",aspect_ratio=:equal,lw=3,markershape=:circle)\nPlots.plot!(sol[1,:],sol[2,:],label=false,lw=3)\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 8: Solution trajectory for a discontinuous system obtained by the MCP solver\n\n\n\n\nIt is worth emphasizing that once the solution trajectory hits the blue square, it is just one step from the origin. And once the solution trajectory reaches the origin, it stays there forever, no more spiralling, no more chattering.\nApparently, the chosen step length was too large. If we reduce it, the resulting trajectory resembles the one obtained by a default ODE solver, as shown in Fig. 9.\n\n\nShow the code\nM = [-1 2; -2 -1]\nh = 1e-2\ntfinal = 2.0\nN = tfinal/h\n\nx0 = [-1.0, 1.0]\nx = [x0]\n\nusing JuMP\nusing PATHSolver\n\nfor i = 1:N\n model = Model(PATHSolver.Optimizer)\n set_optimizer_attribute(model, \"output\", \"no\")\n set_silent(model)\n @variable(model, -1 <= u[1:2] <= 1)\n @constraint(model, -h*M * u - x[end] ⟂ u)\n optimize!(model)\n push!(x, x[end]+h*M*value.(u))\nend\n\nt = range(0.0, step=h, stop=tfinal)\nX = [getindex.(x, i) for i in 1:length(x[1])]\n\nPlots.plot(X[1],X[2],xlabel=\"x₁\",ylabel=\"x₂\",label=\"Time-stepping\",aspect_ratio=:equal,lw=3,markershape=:circle)\nPlots.plot!(sol[1,:],sol[2,:],lw=3,label=\"Default ODE solver with adaptive step\")\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 9: Solution trajectory for a discontinuous system obtained by the MCP solver with a smaller step size\n\n\n\n\nAnd yet the total number of steps is still much smaller:\n\n\nShow the code\nlength(t)\n\n\n201\n\n\n\n\n\nExample 2 (Time-stepping for a mechanical system with a hard stop) Let’s now apply the time-stepping method for a mechanical system with a hard stop as described in the Example 2 in the section on complementarity systems. We start by formulating the semi-explicit Euler scheme\n\n\\begin{aligned}\n\\bm x_{k+1} &= \\bm x_k + h \\bm v_{k+1}\\\\\n\\bm v_{k+1} &= \\bm v_k + h \\mathbf A \\bm x_{k} + h \\mathbf B \\bm u_k,\n\\end{aligned}\n where \n\\mathbf A = \\begin{bmatrix} -\\frac{k_1+k2}{m_1} & \\frac{k_2}{m_1}\\\\ \\frac{k_2}{m_2} & -\\frac{k_2}{m_2} \\end{bmatrix}, \\quad\n\\mathbf B = \\begin{bmatrix} \\frac{1}{m_1} & -\\frac{1}{m_1}\\\\ 0 & \\frac{1}{m_2} \\end{bmatrix}.\n\nSubstituting from the second to the first equation, we get \n{\\color{blue}\\bm x_{k+1}} = \\bm x_k + h \\bm v_k + h^2 \\mathbf A \\bm x_k + h^2 \\mathbf B {\\color{blue}\\bm u_k},\n in which we again highlight the unknown terms by blue color for convenience. These two vector unknowns must satisfy the following complementarity condition \n0 \\leq x_{1,k+1} \\perp u_{1,k} \\geq 0,\n and \n0 \\leq (x_{2,k+1} - x_{1,k+1}) \\perp u_{2,k} \\geq 0.\n\nThis can be written compactly as \n\\bm 0 \\leq \\mathbf C \\bm x_{k+1} \\perp \\bm u_{k} \\geq 0,\n where \\mathbf C is known from the previous section as \n\\mathbf C = \\begin{bmatrix}1 & 0\\\\ -1 & 1\\end{bmatrix}.\n\nThe constraint can be expanded into \n\\bm 0 \\leq \\mathbf C\\left(\\bm x_k + h \\bm v_k + h^2 \\mathbf A \\bm x_k + h^2 \\mathbf B {\\color{blue}\\bm u_k}\\right) \\perp \\bm u_{k} \\geq 0,\n which is an LCP problem with \\bm q = \\mathbf C\\left(\\bm x_k + h \\bm v_k + h^2 \\mathbf A \\bm x_k\\right) and \\mathbf M = h^2 \\mathbf C \\mathbf B.\nAn implementation of the time-stepping for this system is shown below.\n\n\nShow the code\nm1 = 1.0\nm2 = 1.0\nk1 = 1.0\nk2 = 1.0\n\nA = [-(k1+k2)/m1 k2/m1; k2/m2 -k2/m2]\nB = [1/m1 -1/m1; 0 1/m2]\nh = 1e-1\n\nx10 = 1.0\nx20 = 3.5\nv10 = 0.0\nv20 = 0.0\n\nx0 = [x10, x20] \nv0 = [v10, v20]\n\nx = [x0]\nv = [v0]\n\ntfinal = 10.0\nN = tfinal/h\n\nfor i = 1:N\n model = Model(PATHSolver.Optimizer)\n set_optimizer_attribute(model, \"output\", \"no\")\n set_silent(model)\n @variable(model, u[1:2] >= 0)\n @constraint(model, [1 0; -1 1]*(h^2*B*u + x[end] + h*v[end] + h^2*A*x[end]) ⟂ u)\n optimize!(model) \n push!(x, h^2*B*value.(u) + (x[end] + h*v[end] + h^2*A*x[end]))\n push!(v, v[end] + h*A*x[end] + h*B*value.(u))\nend\n\nt = range(0.0, step=h, stop=tfinal)\nX = [getindex.(x, i) for i in 1:length(x[1])]\n\nPlots.plot(t,X[1],label=\"x₁\",lw=3,xlabel=\"t\",ylabel=\"Positions\")\nPlots.plot!(t,X[2],label=\"x₂\",lw=3)\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nNote, once again, the amazingly small number of steps\n\n\nShow the code\nlength(t)\n\n\n101\n\n\nwhile the shape of the trajectory is already quite accurate (you can try reducing the step lenght by yourself)and whatever chattering is absent in the solution.\n\n\n\n\n\n\n\nImportant\n\n\n\nWe have not certainly presented the most advanced and state-or-the-art methods. But the essence of using complementarity concepts in numerical simulation is a bit clearer now, hopefully.\n\n\n\n\n\n Back to top", + "text": "One of the useful outcomes of the theory of complementarity systems is a new family of methods for numerical simulation of discontinuous systems. Here we will demonstrate the essence by introducing the method of time-stepping. And we do it by means of an example.\n\nExample 1 (Simulation using time-stepping) Consider the following discontinuous dynamical system in \\mathbb R^2: \n\\begin{aligned}\n\\dot x_1 &= -\\operatorname{sign} x_1 + 2 \\operatorname{sign} x_2\\\\\n\\dot x_2 &= -2\\operatorname{sign} x_1 -\\operatorname{sign} x_2.\n\\end{aligned}\n\nThe state portrait is in Fig. 1.\n\n\nShow the code\nf₁(x) = -sign(x[1]) + 2*sign(x[2])\nf₂(x) = -2*sign(x[1]) - sign(x[2])\nf(x) = [f₁(x), f₂(x)]\n\nusing CairoMakie\nfig = Figure(size = (800, 800),fontsize=20)\nax = Axis(fig[1, 1], xlabel = \"x₁\", ylabel = \"x₂\")\nstreamplot!(ax,(x₁,x₂)->Point2f(f([x₁,x₂])), -2.0..2.0, -2.0..2.0, colormap = :magma)\nfig\n\n\n\n\n\n\n\n\nFigure 1: State portrait of the discontinuous system\n\n\n\n\n\nOne particular (vector) state trajectory obtained by some default ODE solver is in Fig. 2.\n\n\nShow the code\nusing DifferentialEquations\n\nfunction f!(dx,x,p,t)\n dx[1] = -sign(x[1]) + 2*sign(x[2])\n dx[2] = -2*sign(x[1]) - sign(x[2])\nend\n\nx0 = [-1.0, 1.0]\ntfinal = 2.0\ntspan = (0.0,tfinal)\nprob = ODEProblem(f!,x0,tspan)\nsol = solve(prob)\n\nusing Plots\nPlots.plot(sol,xlabel=\"t\",ylabel=\"x\",label=false,lw=3)\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 2: Trajectory of the discontinuous system\n\n\n\n\nWe can also plot the trajectory in the state space, as in Fig. 3.\n\n\nShow the code\nPlots.plot(sol[1,:],sol[2,:],xlabel=\"x₁\",ylabel=\"x₂\",label=false,aspect_ratio=:equal,lw=3,xlims=(-1.2,0.5))\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 3: Trajectory of the discontinuous system in the state space\n\n\n\n\nNote that for the default setting of absolute and relative tolerances, the adaptive-step ODE solver actually needed a huge number of steps:\n\n\nShow the code\nlength(sol.t)\n\n\n288594\n\n\nThis is indeed quite a lot? Can we do better?\nIn this particular example, we have the benefit of being able to analyze the solution analytically. In particular, we can ask: how fast does the solution approach the origin? We turn this into a question of how fast the distance from the origin decreases. With some anticipation we use the 1-norm \\|\\bm x\\|_1 = |x_1| + |x_2| to the distance of the state from the origin. We then ask: \n\\frac{\\mathrm d}{\\mathrm dt}\\|\\bm x\\|_1 = ?\n\nWe avoid the troubles with nonsmoothness of the absolute value by considering each quadrant separately. Let’s start in the first (upper right) quadrant, that is, x_1>0 and x_2>0, from which it follows that |x_1| = x_1, \\;|x_2| = x_2, and therefore \n\\frac{\\mathrm d}{\\mathrm dt}\\|\\bm x\\|_1 = \\dot x_1 + \\dot x_2 = 1 - 3 = -2.\n\nThe situation is identical in the other quadrants. We do not worry that the norm is undefined on the axes, because the trajectory obviously just crosses them.\nThe conclusion is that the trajectory will hit the origin in finite time (!): with, say, x_1(0) = 1 and x_2(0) = 1, the trajectory hits the origin at t=(|x_1(0)|+|x_2(0)|)/2 = 1. Surprisingly (or not), this will happen after an infinite number of revolutions around the origin…\nWe have seen above in Fig. 2 that a default solver for ODE can handle the situation in a decent way. But we have also seen that this was at the cost of a huge number of steps. To get some insight, we implement our own solver.\n\nForward Euler with fixed step size\nWe start with the simplest of all methods, the forward Euler method with a fixed step length. The computation of the next state is done just by the assignment \n\\begin{aligned}\n{\\color{blue}x_{1,k+1}} &= x_{1,k} + h (-\\operatorname{sign} x_{1,k} + 2 \\operatorname{sign} x_{2,k})\\\\\n{\\color{blue}x_{2,k+1}} &= x_{1,k} + h (-2\\operatorname{sign} x_{1,k} - \\operatorname{sign} x_{2,k}).\n\\end{aligned}\n\n\n\n\n\n\n\nBlue color in our text is used to denote the unknown\n\n\n\nWe use the blue color here (and in the next few paragraphs) to highlight what is uknown.\n\n\n\n\nShow the code\nf(x) = [-sign(x[1]) + 2*sign(x[2]), -2*sign(x[1]) - sign(x[2])]\n\nusing LinearAlgebra\nN = 1000\nx₀ = [-1.0, 1.0] \nx = [x₀]\ntfinal = norm(x₀,1)/2\ntfinal = 5.0\nh = tfinal/N \nt = range(0.0, step=h, stop=tfinal)\n\nfor i=1:N\n xnext = x[i] + h*f(x[i]) \n push!(x,xnext)\nend\n\nX = [getindex.(x, i) for i in 1:length(x[1])]\n\nPlots.plot(t,X,lw=3,label=false,xlabel=\"t\",ylabel=\"x\")\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 4: Solution trajectory for a discontinuous system obtained by the forward Euler method\n\n\n\n\nThen number of steps is\n\n\nShow the code\nlength(t)\n\n\n1001\n\n\nwhich is significantly less than before, but the solution is not perfect – notice the chattering. It could be reduced by using a smaller step size, but again, at the cost of a longer simulation time.\n\n\nBackward Euler\nAs an alternative to the forward Euler method, we can use the backward Euler method. The computation of the next state is done by solving the nonlinear equations \n\\begin{aligned}\n{\\color{blue} x_{1,k+1}} &= x_{1,k} + h (-\\operatorname{sign} {\\color{blue}x_{1,k+1}} + 2 \\operatorname{sign} {\\color{blue}x_{2,k+1}})\\\\\n{\\color{blue} x_{2,k+1}} &= x_{1,k} + h (-2\\operatorname{sign} {\\color{blue}x_{1,k+1}} - \\operatorname{sign} {\\color{blue}x_{2,k+1}}).\n\\end{aligned}\n\nThe discontinuities on the right-hand sides constitute a challenge for solvers of nonlinear equations – recall that the Newton’s method requires the first derivatives (assembled into the Jacobian) or their approximation. Instead of this struggle, we can use the linear complementarity problem (LCP) formulation.\n\n\nFormulation of the backward Euler using LCP\nInstead solving the above nonlinear equations with discontinuities, we introduce new variables u_1 and u_2 as the outputs of the \\operatorname{sign} functions: \n\\begin{aligned}\n{\\color{blue} x_{1,k+1}} &= x_{1,k} + h (-{\\color{blue}u_{1}} + 2 {\\color{blue}u_{2}})\\\\\n{\\color{blue} x_{2,k+1}} &= x_{1,k} + h (-2{\\color{blue}u_{1}} - {\\color{blue}u_{2}}).\n\\end{aligned}\n\nBut now we have to enforce the relationship between \\bm u and \\bm x_{k+1}. Recall the standard definition of the \\operatorname{sign} function is \n\\operatorname{sign}(x) = \\begin{cases}\n1 & x>0\\\\\n0 & x=0\\\\\n-1 & x<0,\n\\end{cases}\n but we change the definition to a set-valued function \n\\begin{cases}\n\\operatorname{sign}(x) = 1 & x>0\\\\\n\\operatorname{sign}(x) \\in [-1,1] & x=0\\\\\n\\operatorname{sign}(x) = -1 & x<0.\n\\end{cases}\n\nAccordingly, we set the relationship between \\bm u and \\bm x to \n\\begin{cases}\nu_1 = 1 & x_1>0\\\\\nu_1 \\in [-1,1] & x_1=0\\\\\nu_1 = -1 & x_1<0,\n\\end{cases}\n and \n\\begin{cases}\nu_2 = 1 & x_2>0\\\\\nu_2 \\in [-1,1] & x_2=0\\\\\nu_2 = -1 & x_2<0.\n\\end{cases}\n\nBut these are mixed complementarity contraints we have defined previously! We can thus write the single step of the simulation algorithm as the MCP \n\\boxed{\n\\begin{aligned}\n\\begin{bmatrix}\n{\\color{blue} x_{1,k+1}}\\\\\n{\\color{blue} x_{1,k+1}}\n\\end{bmatrix}\n&=\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix} + h\n\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}\n\\begin{bmatrix}\n{\\color{blue}u_{1}}\\\\\n{\\color{blue}u_{2}}\n\\end{bmatrix}\\\\\n-1 &\\leq {\\color{blue} u_1} \\leq 1 \\quad \\bot \\quad -{\\color{blue}x_{1,k+1}}\\\\\n-1 &\\leq {\\color{blue} u_2} \\leq 1 \\quad \\bot \\quad -{\\color{blue}x_{2,k+1}}.\n\\end{aligned}\n}\n\\tag{1}\nWe now have enough to start implementing a solver for this system, provided we have an access to a solver the MCP (see the page dedicated to software for complementarity problems). But before we do it, we analyze the problem a bit deeper. In this particular small and simple case, it is still doable with just a pencil and paper.\n\n\n9 possible combinations\nThere are now 9 possible combinations of the values of u_1 and u_2, each of them strictly inside their respective intervals, or at the either end. Let’s explore some combinations. We start with x_{1,k+1} = x_{2,k+1} = 0, while u_1 \\in [-1,1] and u_2 \\in [-1,1]:\n\n\\begin{aligned}\n\\begin{bmatrix}\n0\\\\\n0\n\\end{bmatrix}\n&=\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix} + h\n\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}\n\\begin{bmatrix}\n{\\color{blue}u_{1}}\\\\\n{\\color{blue}u_{2}}\n\\end{bmatrix}\\\\\n& -1 \\leq {\\color{blue} u_1} \\leq 1, \\quad -1 \\leq {\\color{blue} u_2} \\leq 1\n\\end{aligned}\n\nHow does the set of states from which the next state is zero look like? We isolate the vector \\bm u from the equation and impose the constraints on it: \n\\begin{aligned}\n-\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix}\n&= h\n\\begin{bmatrix}\n{\\color{blue}u_{1}}\\\\\n{\\color{blue}u_{2}}\n\\end{bmatrix}\\\\\n-1 \\leq {\\color{blue} u_1} \\leq 1, \\quad -1 &\\leq {\\color{blue} u_2} \\leq 1\n\\end{aligned}\n\nWe then get \n\\begin{bmatrix}\n-h\\\\-h\n\\end{bmatrix}\n\\leq\n\\begin{bmatrix}\n0.2 & 0.4 \\\\\n-0.4 & 0.2\n\\end{bmatrix}\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix}\n\\leq\n\\begin{bmatrix}\nh\\\\ h\n\\end{bmatrix}\n\nFor h=0.2, the resulting set is a rotated square, as shown in Fig. 5.\n\n\nShow the code\nusing LazySets\nh = 0.2\nH1u = HalfSpace([0.2, 0.4], h)\nH2u = HalfSpace([-0.4, 0.2], h)\nH1l = HalfSpace(-[0.2, 0.4], h)\nH2l = HalfSpace(-[-0.4, 0.2], h)\n\nHa = H1u ∩ H2u ∩ H1l ∩ H2l\n\nusing Plots\nPlots.plot(Ha, aspect_ratio=:equal,xlabel=\"x₁\",ylabel=\"x₂\",label=false,xlims=(-1.5,1.5),ylims=(-1.5,1.5))\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 5: The set of states from which the next state is zero (the origin)\n\n\n\n\nIndeed, if the current state is in this rotated square, then the next state will be zero. No more infite spiralling around the origin! No more chattering! Perfect zero.\n\n\nAnother combination\nWe now consider u_1 = 1, u_2 = 1, which we substitute into Eq. 1: \n\\begin{aligned}\n\\begin{bmatrix}\n{\\color{blue} x_{1,k+1}}\\\\\n{\\color{blue} x_{1,k+1}}\n\\end{bmatrix}\n&=\n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix} + h\n\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}\n\\begin{bmatrix}\n{1}\\\\\n{1}\n\\end{bmatrix}\\\\\n\\color{blue}x_{1,k+1} &\\geq 0\\\\\n\\color{blue}x_{2,k+1} &\\geq 0,\n\\end{aligned}\n which can be reformatted to \n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix} + h\n\\begin{bmatrix}\n-1 & 2 \\\\\n-2 & -1\n\\end{bmatrix}\n\\begin{bmatrix}\n1\\\\\n1\n\\end{bmatrix}\\geq \\bm 0,\n and further to \n\\begin{bmatrix}\nx_{1,k}\\\\\nx_{2,k}\n\\end{bmatrix}\n\\geq h\n\\begin{bmatrix}\n-1\\\\\n3\n\\end{bmatrix}.\n\nThe set of states for which the next state is such that both its state variables are positive (hence u_1 = 1, u_2 = 1) is shown in Fig. 6.\n\n\nShow the code\nusing LazySets\nh = 0.2\nA = [-1.0 2.0; -2.0 -1.0]\nu = [1.0, 1.0]\nb = h*A*u\n\nH1 = HalfSpace([-1.0, 0.0], b[1])\nH2 = HalfSpace([0.0, -1.0], b[2])\nHb = H1 ∩ H2\n\nusing Plots\nPlots.plot(Ha, aspect_ratio=:equal,xlabel=\"x₁\",ylabel=\"x₂\",label=false,xlims=(-1.5,1.5),ylims=(-1.5,1.5))\nPlots.plot!(Hb)\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 6: The set of states from which the next state has both state variables positive\n\n\n\n\n\n\nAll nine regions\nWe plot all nine regions in Fig. 7. Note that we do not color them all – the white regions are to be counted as well.\n\n\nShow the code\nusing LazySets\nh = 0.2\nA = [-1.0 2.0; -2.0 -1.0]\n\nu = [1, -1]\nb = h*A*u\n\nH1 = HalfSpace(-[1.0, 0.0], b[1])\nH2 = HalfSpace([0.0, 1.0], -b[2])\nHc = H1 ∩ H2\n\nu = [-1, 1]\nb = h*A*u\n\nH1 = HalfSpace([1.0, 0.0], -b[1])\nH2 = HalfSpace(-[0.0, 1.0], b[2])\nHd = H1 ∩ H2\n\nu = [-1, -1]\nb = h*A*u\n\nH1 = HalfSpace([1.0, 0.0], -b[1])\nH2 = HalfSpace([0.0, 1.0], -b[2])\nHe = H1 ∩ H2\n\nusing Plots\nPlots.plot(Ha, aspect_ratio=:equal,xlabel=\"x₁\",ylabel=\"x₂\",label=false,xlims=(-1.5,1.5),ylims=(-1.5,1.5))\nPlots.plot!(Hb)\nPlots.plot!(Hc)\nPlots.plot!(Hd)\nPlots.plot!(He)\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 7: The nine regions of the state space for the spiralling system\n\n\n\n\n\n\nSolutions using a MCP solver\nNow that we have got some insight into the algorithm, we can implement it by wrapping a for-loop around the MCP solver.\n\n\nShow the code\nM = [-1 2; -2 -1]\nh = 2e-1\ntfinal = 2.0\nN = tfinal/h\n\nx0 = [-1.0, 1.0]\nx = [x0]\n\nusing JuMP\nusing PATHSolver\n\nfor i = 1:N\n model = Model(PATHSolver.Optimizer)\n set_optimizer_attribute(model, \"output\", \"no\")\n set_silent(model)\n @variable(model, -1 <= u[1:2] <= 1)\n @constraint(model, -h*M * u - x[end] ⟂ u)\n optimize!(model)\n push!(x, x[end]+h*M*value.(u))\nend\n\n\nThe code is admittedly unoptimized (for example, it is not wise to start building the optimization model from the scratch in every iteration), but it was our intention to keep it simple to be read conveniently.\nThe code outcome can be plotted as in Fig. 8. A solution obtained by a classical (default) ODE solver with default setting is plotted for comparison.\n\n\nShow the code\nt = range(0.0, step=h, stop=tfinal)\nX = [getindex.(x, i) for i in 1:length(x[1])]\n\nusing Plots\nPlots.plot(Ha, aspect_ratio=:equal,xlabel=\"x₁\",ylabel=\"x₂\",label=false,xlims=(-1.5,1.5),ylims=(-1.5,1.5))\nPlots.plot!(Hb)\nPlots.plot!(Hc)\nPlots.plot!(Hd)\nPlots.plot!(He)\nPlots.plot!(X[1],X[2],xlabel=\"x₁\",ylabel=\"x₂\",label=\"Time-stepping\",aspect_ratio=:equal,lw=3,markershape=:circle)\nPlots.plot!(sol[1,:],sol[2,:],label=false,lw=3)\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 8: Solution trajectory for a discontinuous system obtained by the MCP solver\n\n\n\n\nIt is worth emphasizing that once the solution trajectory hits the blue square, it is just one step from the origin. And once the solution trajectory reaches the origin, it stays there forever, no more spiralling, no more chattering.\nApparently, the chosen step length was too large. If we reduce it, the resulting trajectory resembles the one obtained by a default ODE solver, as shown in Fig. 9.\n\n\nShow the code\nM = [-1 2; -2 -1]\nh = 1e-2\ntfinal = 2.0\nN = tfinal/h\n\nx0 = [-1.0, 1.0]\nx = [x0]\n\nusing JuMP\nusing PATHSolver\n\nfor i = 1:N\n model = Model(PATHSolver.Optimizer)\n set_optimizer_attribute(model, \"output\", \"no\")\n set_silent(model)\n @variable(model, -1 <= u[1:2] <= 1)\n @constraint(model, -h*M * u - x[end] ⟂ u)\n optimize!(model)\n push!(x, x[end]+h*M*value.(u))\nend\n\nt = range(0.0, step=h, stop=tfinal)\nX = [getindex.(x, i) for i in 1:length(x[1])]\n\nPlots.plot(X[1],X[2],xlabel=\"x₁\",ylabel=\"x₂\",label=\"Time-stepping\",aspect_ratio=:equal,lw=3,markershape=:circle)\nPlots.plot!(sol[1,:],sol[2,:],lw=3,label=\"Default ODE solver with adaptive step\")\n\n\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 9: Solution trajectory for a discontinuous system obtained by the MCP solver with a smaller step size\n\n\n\n\nAnd yet the total number of steps is still much smaller:\n\n\nShow the code\nlength(t)\n\n\n201\n\n\n\n\n\nExample 2 (Time-stepping for a mechanical system with a hard stop) Let’s now apply the time-stepping method for a mechanical system with a hard stop as described in the Example 2 in the section on complementarity systems. We start by formulating the semi-explicit Euler scheme\n\n\\begin{aligned}\n\\bm x_{k+1} &= \\bm x_k + h \\bm v_{k+1}\\\\\n\\bm v_{k+1} &= \\bm v_k + h \\mathbf A \\bm x_{k} + h \\mathbf B \\bm u_k,\n\\end{aligned}\n where \n\\mathbf A = \\begin{bmatrix} -\\frac{k_1+k2}{m_1} & \\frac{k_2}{m_1}\\\\ \\frac{k_2}{m_2} & -\\frac{k_2}{m_2} \\end{bmatrix}, \\quad\n\\mathbf B = \\begin{bmatrix} \\frac{1}{m_1} & -\\frac{1}{m_1}\\\\ 0 & \\frac{1}{m_2} \\end{bmatrix}.\n\nSubstituting from the second to the first equation, we get \n{\\color{blue}\\bm x_{k+1}} = \\bm x_k + h \\bm v_k + h^2 \\mathbf A \\bm x_k + h^2 \\mathbf B {\\color{blue}\\bm u_k},\n in which we again highlight the unknown terms by blue color for convenience. These two vector unknowns must satisfy the following complementarity condition \n0 \\leq x_{1,k+1} \\perp u_{1,k} \\geq 0,\n and \n0 \\leq (x_{2,k+1} - x_{1,k+1}) \\perp u_{2,k} \\geq 0.\n\nThis can be written compactly as \n\\bm 0 \\leq \\mathbf C \\bm x_{k+1} \\perp \\bm u_{k} \\geq 0,\n where \\mathbf C is known from the previous section as \n\\mathbf C = \\begin{bmatrix}1 & 0\\\\ -1 & 1\\end{bmatrix}.\n\nThe constraint can be expanded into \n\\bm 0 \\leq \\mathbf C\\left(\\bm x_k + h \\bm v_k + h^2 \\mathbf A \\bm x_k + h^2 \\mathbf B {\\color{blue}\\bm u_k}\\right) \\perp \\bm u_{k} \\geq 0,\n which is an LCP problem with \\bm q = \\mathbf C\\left(\\bm x_k + h \\bm v_k + h^2 \\mathbf A \\bm x_k\\right) and \\mathbf M = h^2 \\mathbf C \\mathbf B.\nAn implementation of the time-stepping for this system is shown below.\n\n\nShow the code\nm1 = 1.0\nm2 = 1.0\nk1 = 1.0\nk2 = 1.0\n\nA = [-(k1+k2)/m1 k2/m1; k2/m2 -k2/m2]\nB = [1/m1 -1/m1; 0 1/m2]\nh = 1e-1\n\nx10 = 1.0\nx20 = 3.5\nv10 = 0.0\nv20 = 0.0\n\nx0 = [x10, x20] \nv0 = [v10, v20]\n\nx = [x0]\nv = [v0]\n\ntfinal = 10.0\nN = tfinal/h\n\nfor i = 1:N\n model = Model(PATHSolver.Optimizer)\n set_optimizer_attribute(model, \"output\", \"no\")\n set_silent(model)\n @variable(model, u[1:2] >= 0)\n @constraint(model, [1 0; -1 1]*(h^2*B*u + x[end] + h*v[end] + h^2*A*x[end]) ⟂ u)\n optimize!(model) \n push!(x, h^2*B*value.(u) + (x[end] + h*v[end] + h^2*A*x[end]))\n push!(v, v[end] + h*A*x[end] + h*B*value.(u))\nend\n\nt = range(0.0, step=h, stop=tfinal)\nX = [getindex.(x, i) for i in 1:length(x[1])]\n\nPlots.plot(t,X[1],label=\"x₁\",lw=3,xlabel=\"t\",ylabel=\"Positions\",markershape=:circle)\nPlots.plot!(t,X[2],label=\"x₂\",lw=3,markershape=:circle)\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nNote, once again, the amazingly small number of steps\n\n\nShow the code\nlength(t)\n\n\n101\n\n\nwhile the shape of the trajectory is already quite accurate (you can try reducing the step lenght by yourself)and whatever chattering is absent in the solution.\n\n\nExample 3 (Time stepping for a mechanical system with Coulomb friction modelled as complementarity constraints) We consider an object of mass m moving on a horizontal surface. The object is subject to an applied force F_\\mathrm{a} and a friction force F_\\mathrm{f}, both oriented in the positive direction of position and velocity, as in Fig. 10.\n\n\n\n\n\n\nFigure 10: Coulomb friction to be modelled using complementarity constraints\n\n\n\nThe friction force is modelled as a Coulomb friction model \nF_\\mathrm{f}(t) = -m g \\mu\\, \\mathrm{sgn}(v(t)),\n where \\mu is a coefficient that relates the frictional force to the normal force mg.\nThe motion equations are \n\\begin{aligned}\n\\dot x(t) &= v(t)\\\\\n\\dot v(t) &= \\frac{1}{m}(F_\\mathrm{a}(t) + F_\\mathrm{f}(t)).\n\\end{aligned}\n\nWe now express the friction force as a difference of two nonnegative variables \nF_\\mathrm{f}(t) = F^+_\\mathrm{f}(t) - F^-_\\mathrm{f}(t),\\quad F^+_\\mathrm{f}(t), \\,F^-_\\mathrm{f}(t) \\geq 0.\n\nAnd we also introduce another auxiliary variable \\nu(t) that is just the absolute value of the velocity \n\\nu(t) = |v(t)|.\n\nWith the three new variables we formulate the Coulomb friction model as \n\\begin{aligned}\n0 \\leq v + \\nu &\\perp F^+_\\mathrm{f} \\geq 0,\\\\\n0 \\leq -v + \\nu &\\perp F^-_\\mathrm{f} \\geq 0, \\\\\n0 \\leq \\mu m g - F^+_\\mathrm{f} - F^-_\\mathrm{f} &\\perp \\nu \\geq 0.\n\\end{aligned}\n\nIf F^-_\\mathrm{f} > 0, that is, if the friction force is negative, the second complementarity constraint implies that the velocity is equal to its absolute value, in other words, it is nonnegative, the object moves to the right (or stays still). If the velocity is positive, the first constraint implies that F^+_\\mathrm{f} = 0, and the third constraint then implies that F^-_\\mathrm{f} = mg\\mu. Feel free to explore other cases.\nWe now proceed to implement the time-stepping method for this system. Once again, we use the semi-explicit Euler scheme \n\\begin{aligned}\nx_{k+1} &= x_k + h v_{k+1},\\\\\nv_{k+1} &= v_k + \\frac{h}{m} (F^+_{\\mathrm{f},k} - F^-_{\\mathrm{f},k}),\n\\end{aligned}\n where the velocity is subject to the complementarity constraints \n\\begin{aligned}\n0 \\leq v_{k+1} + \\nu_{k+1} &\\perp F^+_{\\mathrm{f},k} \\geq 0,\\\\\n0 \\leq -v_{k+1} + \\nu_{k+1} &\\perp F^-_{\\mathrm{f},k} \\geq 0, \\\\\n0 \\leq \\mu m g - F^+_{\\mathrm{f},k} - F^-_{\\mathrm{f},k} &\\perp \\nu_{k+1} \\geq 0,\n\\end{aligned}\n into which we substitute for v_{k+1} \\boxed{\n\\begin{aligned}\n0 \\leq v_k + \\frac{h}{m} ({\\color{blue}F^+_{\\mathrm{f},k}} - {\\color{blue}F^-_{\\mathrm{f},k}}) + {\\color{blue}\\nu_{k+1}}\\, &\\perp\\, {\\color{blue}F^+_{\\mathrm{f},k}} \\geq 0,\\\\\n0 \\leq -\\left(v_k + \\frac{h}{m} ({\\color{blue}F^+_{\\mathrm{f},k}} - {\\color{blue}F^-_{\\mathrm{f},k}})\\right) + {\\color{blue}\\nu_{k+1}}\\, &\\perp\\, {\\color{blue}F^-_{\\mathrm{f},k}} \\geq 0, \\\\\n0 \\leq \\mu m g - {\\color{blue}F^+_{\\mathrm{f},k}} - {\\color{blue}F^-_{\\mathrm{f},k}} \\,&\\perp\\, {\\color{blue}\\nu_{k+1}} \\geq 0,\n\\end{aligned}\n}\n in which we highlighted the unknowns by blue color for convenience.\n\n\nShow the code\nusing JuMP\nusing PATHSolver\nusing Plots\n\nm = 100.0\ng = 9.81\nμ = 10.0\n\nh = 2e-1\n\nx0 = 0.0\nv0 = 100.0\n\nx = [x0]\nv = [v0]\n\ntfinal = 5.0\nN = tfinal/h\n\nfor i = 1:N\n model = Model(PATHSolver.Optimizer)\n set_optimizer_attribute(model, \"output\", \"no\")\n set_silent(model)\n @variable(model, vabs >= 0)\n @variable(model, F⁺ >= 0)\n @variable(model, F⁻ >= 0)\n @constraint(model, (v[end] + h/m*(F⁺-F⁻) + vabs) ⟂ F⁺)\n @constraint(model, (-(v[end] + h/m*(F⁺-F⁻)) + vabs) ⟂ F⁻)\n @constraint(model, (μ*m*g - (F⁺+F⁻)) ⟂ vabs)\n optimize!(model) \n push!(v, v[end] + h/m*(value(F⁺)-value(F⁻)))\n push!(x, x[end] + h*v[end]) # Here v[end] on the left already equals v_k+1\nend\n\nt = range(0.0, step=h, stop=tfinal)\nPlots.plot(t,v,label=\"v\",lw=3,markershape=:circle)\nPlots.plot!(t,x,label=\"x\",lw=3,markershape=:circle,xlabel=\"t\")\n\n\n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nOnce again, we can see that the method correctly simulates the system coming to a complete stop due to friction.\n\n\n\n\n\n\n\nImportant\n\n\n\nWhile concluding this section about time-stepping methods, we have to emphasize that this was really just an introduction. But the essence of using complementarity concepts in numerical simulation is perhaps a bit clearer now.\n\n\n\n\n\n Back to top", "crumbs": [ "9. 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992px){.navbar-expand-lg{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-lg .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-lg .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-lg .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-lg .navbar-nav-scroll{overflow:visible}.navbar-expand-lg .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-lg .navbar-toggler{display:none}.navbar-expand-lg .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-lg .offcanvas .offcanvas-header{display:none}.navbar-expand-lg .offcanvas 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!important;transform:none !important;transition:none}.navbar-expand-xl .offcanvas .offcanvas-header{display:none}.navbar-expand-xl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 1400px){.navbar-expand-xxl{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-xxl .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-xxl .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-xxl .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-xxl .navbar-nav-scroll{overflow:visible}.navbar-expand-xxl .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-xxl .navbar-toggler{display:none}.navbar-expand-xxl .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: #545555;--bs-navbar-hover-color: rgba(31, 78, 182, 0.8);--bs-navbar-disabled-color: rgba(84, 85, 85, 0.75);--bs-navbar-active-color: #1f4eb6;--bs-navbar-brand-color: #545555;--bs-navbar-brand-hover-color: #1f4eb6;--bs-navbar-toggler-border-color: rgba(84, 85, 85, 0);--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23545555' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23545555' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: 1px;--bs-card-border-color: rgba(0, 0, 0, 0.175);--bs-card-border-radius: 0.25rem;--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(0.25rem - 1px);--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(52, 58, 64, 0.25);--bs-card-cap-color: ;--bs-card-height: ;--bs-card-color: ;--bs-card-bg: #fff;--bs-card-img-overlay-padding: 1rem;--bs-card-group-margin: 0.75rem;position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0}.card>.list-group:last-child{border-bottom-width:0}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 576px){.card-group{display:flex;display:-webkit-flex;flex-flow:row wrap;-webkit-flex-flow:row wrap}.card-group>.card{flex:1 0 0%;-webkit-flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}}.accordion{--bs-accordion-color: #343a40;--bs-accordion-bg: #fff;--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: #dee2e6;--bs-accordion-border-width: 1px;--bs-accordion-border-radius: 0.25rem;--bs-accordion-inner-border-radius: calc(0.25rem - 1px);--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: #343a40;--bs-accordion-btn-bg: #fff;--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23343a40'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%2310335b'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-focus-border-color: #93c0f1;--bs-accordion-btn-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: #10335b;--bs-accordion-active-bg: #d4e6f9}.accordion-button{position:relative;display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:not(:first-of-type){border-top:0}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(52, 58, 64, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(52, 58, 64, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #2761e3;--bs-pagination-bg: #fff;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #1f4eb6;--bs-pagination-hover-bg: #f8f9fa;--bs-pagination-hover-border-color: #dee2e6;--bs-pagination-focus-color: #1f4eb6;--bs-pagination-focus-bg: #e9ecef;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #2780e3;--bs-pagination-active-border-color: #2780e3;--bs-pagination-disabled-color: rgba(52, 58, 64, 0.75);--bs-pagination-disabled-bg: #e9ecef;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 0 solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:.5rem}}.progress,.progress-stacked{--bs-progress-height: 0.5rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #e9ecef;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #2780e3;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #343a40;--bs-list-group-bg: #fff;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(52, 58, 64, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #343a40;--bs-list-group-action-active-bg: #e9ecef;--bs-list-group-disabled-color: rgba(52, 58, 64, 0.75);--bs-list-group-disabled-bg: #fff;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #2780e3;--bs-list-group-active-border-color: #2780e3;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(255, 255, 255, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(52, 58, 64, 0.75);--bs-toast-header-bg: rgba(255, 255, 255, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color)}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #fff;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;min-height:calc(100% - var(--bs-modal-margin)*2)}.modal-content{position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;width:100%;color:var(--bs-modal-color);pointer-events:auto;background-color:var(--bs-modal-bg);background-clip:padding-box;border:var(--bs-modal-border-width) solid var(--bs-modal-border-color);outline:0}.modal-backdrop{--bs-backdrop-zindex: 1050;--bs-backdrop-bg: #000;--bs-backdrop-opacity: 0.5;position:fixed;top:0;left:0;z-index:var(--bs-backdrop-zindex);width:100vw;height:100vh;background-color:var(--bs-backdrop-bg)}.modal-backdrop.fade{opacity:0}.modal-backdrop.show{opacity:var(--bs-backdrop-opacity)}.modal-header{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;align-items:center;-webkit-align-items:center;justify-content:space-between;-webkit-justify-content:space-between;padding:var(--bs-modal-header-padding);border-bottom:var(--bs-modal-header-border-width) solid var(--bs-modal-header-border-color)}.modal-header .btn-close{padding:calc(var(--bs-modal-header-padding-y)*.5) calc(var(--bs-modal-header-padding-x)*.5);margin:calc(-0.5*var(--bs-modal-header-padding-y)) calc(-0.5*var(--bs-modal-header-padding-x)) calc(-0.5*var(--bs-modal-header-padding-y)) auto}.modal-title{margin-bottom:0;line-height:var(--bs-modal-title-line-height)}.modal-body{position:relative;flex:1 1 auto;-webkit-flex:1 1 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auto);grid-template-areas:"top" "bottom";overflow:hidden}.bslib-value-box.showcase-bottom .value-box-grid .value-box-showcase{grid-area:bottom;padding:0;margin:0}.bslib-value-box.showcase-bottom .value-box-grid .value-box-area{grid-area:top}.bslib-value-box.showcase-bottom[data-full-screen=true] .value-box-grid{grid-template-rows:1fr var(---bslib-value-box-showcase-h-fs, 2fr)}.bslib-value-box.showcase-bottom[data-full-screen=true] .value-box-grid .value-box-showcase{padding:1rem}[data-bs-theme=dark] .bslib-value-box{--bslib-value-box-shadow: 0 0.5rem 1rem rgb(0 0 0 / 50%)}@media(min-width: 576px){.nav:not(.nav-hidden){display:flex !important;display:-webkit-flex !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column){float:none !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.bslib-nav-spacer{margin-left:auto 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.accordion-header{font-size:1.65rem}}.accordion .accordion-icon:not(:empty){margin-right:.75rem;display:flex}.accordion .accordion-button:not(.collapsed){box-shadow:none}.accordion .accordion-button:not(.collapsed):focus{box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.navbar+.container-fluid:has(>.tab-content>.tab-pane.active.html-fill-container),.navbar+.container-sm:has(>.tab-content>.tab-pane.active.html-fill-container),.navbar+.container-md:has(>.tab-content>.tab-pane.active.html-fill-container),.navbar+.container-lg:has(>.tab-content>.tab-pane.active.html-fill-container),.navbar+.container-xl:has(>.tab-content>.tab-pane.active.html-fill-container),.navbar+.container-xxl:has(>.tab-content>.tab-pane.active.html-fill-container){padding-left:0;padding-right:0}.navbar+.container-fluid>.tab-content>.tab-pane.active.html-fill-container,.navbar+.container-sm>.tab-content>.tab-pane.active.html-fill-container,.navbar+.container-md>.tab-content>.tab-pane.active.html-fill-container,.navbar+.container-lg>.tab-content>.tab-pane.active.html-fill-container,.navbar+.container-xl>.tab-content>.tab-pane.active.html-fill-container,.navbar+.container-xxl>.tab-content>.tab-pane.active.html-fill-container{padding:var(--bslib-spacer, 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1rem);gap:var(--bslib-spacer, 1rem)}@media(max-width: 575.98px){.bslib-page-fill{height:var(--bslib-page-fill-mobile-height, auto)}}:root{--bslib-page-sidebar-title-bg: #f8f9fa;--bslib-page-sidebar-title-color: #000}.bslib-page-title{background-color:var(--bslib-page-sidebar-title-bg);color:var(--bslib-page-sidebar-title-color);font-size:1.25rem;font-weight:300;padding:var(--bslib-spacer, 1rem);padding-left:1.5rem;margin-bottom:0;border-bottom:1px solid #dee2e6}.bslib-sidebar-layout{--bslib-sidebar-transition-duration: 500ms;--bslib-sidebar-transition-easing-x: cubic-bezier(0.8, 0.78, 0.22, 1.07);--bslib-sidebar-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-border-radius: var(--bs-border-radius);--bslib-sidebar-vert-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.05);--bslib-sidebar-fg: 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#dfd8e5;--quarto-scss-export-purple-200: #c0b1cb;--quarto-scss-export-purple-300: #a08bb0;--quarto-scss-export-purple-400: #816496;--quarto-scss-export-purple-500: #613d7c;--quarto-scss-export-purple-600: #4e3163;--quarto-scss-export-purple-700: #3a254a;--quarto-scss-export-purple-800: #271832;--quarto-scss-export-purple-900: #130c19;--quarto-scss-export-pink-100: #fad8e8;--quarto-scss-export-pink-200: #f6b2d1;--quarto-scss-export-pink-300: #f18bba;--quarto-scss-export-pink-400: #ed65a3;--quarto-scss-export-pink-500: #e83e8c;--quarto-scss-export-pink-600: #ba3270;--quarto-scss-export-pink-700: #8b2554;--quarto-scss-export-pink-800: #5d1938;--quarto-scss-export-pink-900: #2e0c1c;--quarto-scss-export-red-100: #ffccd7;--quarto-scss-export-red-200: #ff99b0;--quarto-scss-export-red-300: #ff6688;--quarto-scss-export-red-400: #ff3361;--quarto-scss-export-red-500: #ff0039;--quarto-scss-export-red-600: #cc002e;--quarto-scss-export-red-700: #990022;--quarto-scss-export-red-800: #660017;--quarto-scss-export-red-900: #33000b;--quarto-scss-export-orange-100: #fcefdc;--quarto-scss-export-orange-200: #f9deb8;--quarto-scss-export-orange-300: #f6ce95;--quarto-scss-export-orange-400: #f3bd71;--quarto-scss-export-orange-500: #f0ad4e;--quarto-scss-export-orange-600: #c08a3e;--quarto-scss-export-orange-700: #90682f;--quarto-scss-export-orange-800: #60451f;--quarto-scss-export-orange-900: #302310;--quarto-scss-export-yellow-100: #ffe3d1;--quarto-scss-export-yellow-200: #ffc8a3;--quarto-scss-export-yellow-300: #ffac74;--quarto-scss-export-yellow-400: #ff9146;--quarto-scss-export-yellow-500: #ff7518;--quarto-scss-export-yellow-600: #cc5e13;--quarto-scss-export-yellow-700: #99460e;--quarto-scss-export-yellow-800: #662f0a;--quarto-scss-export-yellow-900: #331705;--quarto-scss-export-green-100: #d9f0d1;--quarto-scss-export-green-200: #b2e2a3;--quarto-scss-export-green-300: #8cd374;--quarto-scss-export-green-400: #65c546;--quarto-scss-export-green-500: #3fb618;--quarto-scss-export-green-600: #329213;--quarto-scss-export-green-700: #266d0e;--quarto-scss-export-green-800: #19490a;--quarto-scss-export-green-900: #0d2405;--quarto-scss-export-teal-100: #d2f4ea;--quarto-scss-export-teal-200: #a6e9d5;--quarto-scss-export-teal-300: #79dfc1;--quarto-scss-export-teal-400: #4dd4ac;--quarto-scss-export-teal-500: #20c997;--quarto-scss-export-teal-600: #1aa179;--quarto-scss-export-teal-700: #13795b;--quarto-scss-export-teal-800: #0d503c;--quarto-scss-export-teal-900: #06281e;--quarto-scss-export-cyan-100: #ebddf1;--quarto-scss-export-cyan-200: #d6bbe4;--quarto-scss-export-cyan-300: #c298d6;--quarto-scss-export-cyan-400: #ad76c9;--quarto-scss-export-cyan-500: #9954bb;--quarto-scss-export-cyan-600: #7a4396;--quarto-scss-export-cyan-700: #5c3270;--quarto-scss-export-cyan-800: #3d224b;--quarto-scss-export-cyan-900: #1f1125;--quarto-scss-export-default: #343a40;--quarto-scss-export-primary-text-emphasis: #10335b;--quarto-scss-export-secondary-text-emphasis: #15171a;--quarto-scss-export-success-text-emphasis: #19490a;--quarto-scss-export-info-text-emphasis: #3d224b;--quarto-scss-export-warning-text-emphasis: #662f0a;--quarto-scss-export-danger-text-emphasis: #660017;--quarto-scss-export-light-text-emphasis: #495057;--quarto-scss-export-dark-text-emphasis: #495057;--quarto-scss-export-primary-bg-subtle: #d4e6f9;--quarto-scss-export-secondary-bg-subtle: #d6d8d9;--quarto-scss-export-success-bg-subtle: #d9f0d1;--quarto-scss-export-info-bg-subtle: #ebddf1;--quarto-scss-export-warning-bg-subtle: #ffe3d1;--quarto-scss-export-danger-bg-subtle: #ffccd7;--quarto-scss-export-light-bg-subtle: #fcfcfd;--quarto-scss-export-dark-bg-subtle: #ced4da;--quarto-scss-export-primary-border-subtle: #a9ccf4;--quarto-scss-export-secondary-border-subtle: #aeb0b3;--quarto-scss-export-success-border-subtle: #b2e2a3;--quarto-scss-export-info-border-subtle: #d6bbe4;--quarto-scss-export-warning-border-subtle: #ffc8a3;--quarto-scss-export-danger-border-subtle: #ff99b0;--quarto-scss-export-light-border-subtle: #e9ecef;--quarto-scss-export-dark-border-subtle: #adb5bd;--quarto-scss-export-body-text-align: ;--quarto-scss-export-body-bg: #fff;--quarto-scss-export-body-secondary-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-body-secondary-bg: #e9ecef;--quarto-scss-export-body-tertiary-color: rgba(52, 58, 64, 0.5);--quarto-scss-export-body-tertiary-bg: #f8f9fa;--quarto-scss-export-body-emphasis-color: #000;--quarto-scss-export-link-hover-color: #1f4eb6;--quarto-scss-export-link-hover-decoration: ;--quarto-scss-export-border-color-translucent: rgba(0, 0, 0, 0.175);--quarto-scss-export-component-active-bg: #2780e3;--quarto-scss-export-component-active-color: #fff;--quarto-scss-export-focus-ring-color: rgba(39, 128, 227, 0.25);--quarto-scss-export-headings-font-family: ;--quarto-scss-export-headings-font-style: ;--quarto-scss-export-display-font-family: ;--quarto-scss-export-display-font-style: ;--quarto-scss-export-blockquote-footer-color: #6c757d;--quarto-scss-export-blockquote-border-color: #e9ecef;--quarto-scss-export-hr-bg-color: ;--quarto-scss-export-hr-height: ;--quarto-scss-export-hr-border-color: ;--quarto-scss-export-legend-font-weight: ;--quarto-scss-export-mark-bg: #ffe3d1;--quarto-scss-export-table-color: #343a40;--quarto-scss-export-table-bg: #fff;--quarto-scss-export-table-accent-bg: transparent;--quarto-scss-export-table-th-font-weight: ;--quarto-scss-export-table-striped-color: #343a40;--quarto-scss-export-table-striped-bg: rgba(0, 0, 0, 0.05);--quarto-scss-export-table-active-color: #343a40;--quarto-scss-export-table-active-bg: rgba(0, 0, 0, 0.1);--quarto-scss-export-table-hover-color: #343a40;--quarto-scss-export-table-hover-bg: rgba(0, 0, 0, 0.075);--quarto-scss-export-table-caption-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-input-btn-font-family: ;--quarto-scss-export-input-btn-focus-color: rgba(39, 128, 227, 0.25);--quarto-scss-export-btn-color: #343a40;--quarto-scss-export-btn-font-family: ;--quarto-scss-export-btn-white-space: ;--quarto-scss-export-btn-link-color: #2761e3;--quarto-scss-export-btn-link-hover-color: #1f4eb6;--quarto-scss-export-btn-link-disabled-color: #6c757d;--quarto-scss-export-form-text-font-style: ;--quarto-scss-export-form-text-font-weight: ;--quarto-scss-export-form-text-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-form-label-font-size: ;--quarto-scss-export-form-label-font-style: ;--quarto-scss-export-form-label-font-weight: ;--quarto-scss-export-form-label-color: ;--quarto-scss-export-input-font-family: ;--quarto-scss-export-input-disabled-color: ;--quarto-scss-export-input-disabled-bg: #e9ecef;--quarto-scss-export-input-disabled-border-color: ;--quarto-scss-export-input-color: #343a40;--quarto-scss-export-input-focus-bg: #fff;--quarto-scss-export-input-focus-border-color: #93c0f1;--quarto-scss-export-input-focus-color: #343a40;--quarto-scss-export-input-placeholder-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-input-plaintext-color: #343a40;--quarto-scss-export-form-check-label-color: ;--quarto-scss-export-form-check-transition: ;--quarto-scss-export-form-check-input-bg: #fff;--quarto-scss-export-form-check-input-focus-border: #93c0f1;--quarto-scss-export-form-check-input-checked-color: #fff;--quarto-scss-export-form-check-input-checked-bg-color: #2780e3;--quarto-scss-export-form-check-input-checked-border-color: #2780e3;--quarto-scss-export-form-check-input-indeterminate-color: #fff;--quarto-scss-export-form-check-input-indeterminate-bg-color: #2780e3;--quarto-scss-export-form-check-input-indeterminate-border-color: #2780e3;--quarto-scss-export-form-switch-color: rgba(0, 0, 0, 0.25);--quarto-scss-export-form-switch-focus-color: #93c0f1;--quarto-scss-export-form-switch-checked-color: #fff;--quarto-scss-export-input-group-addon-color: #343a40;--quarto-scss-export-input-group-addon-bg: #f8f9fa;--quarto-scss-export-input-group-addon-border-color: #dee2e6;--quarto-scss-export-form-select-font-family: ;--quarto-scss-export-form-select-color: #343a40;--quarto-scss-export-form-select-bg: #fff;--quarto-scss-export-form-select-disabled-color: ;--quarto-scss-export-form-select-disabled-bg: #e9ecef;--quarto-scss-export-form-select-disabled-border-color: ;--quarto-scss-export-form-select-indicator-color: #343a40;--quarto-scss-export-form-select-border-color: #dee2e6;--quarto-scss-export-form-select-focus-border-color: #93c0f1;--quarto-scss-export-form-range-track-bg: #f8f9fa;--quarto-scss-export-form-range-thumb-bg: #2780e3;--quarto-scss-export-form-range-thumb-active-bg: #bed9f7;--quarto-scss-export-form-range-thumb-disabled-bg: rgba(52, 58, 64, 0.75);--quarto-scss-export-form-file-button-color: #343a40;--quarto-scss-export-form-file-button-bg: #f8f9fa;--quarto-scss-export-form-file-button-hover-bg: #e9ecef;--quarto-scss-export-form-floating-label-disabled-color: #6c757d;--quarto-scss-export-form-feedback-font-style: ;--quarto-scss-export-form-feedback-valid-color: #3fb618;--quarto-scss-export-form-feedback-invalid-color: #ff0039;--quarto-scss-export-form-feedback-icon-valid-color: #3fb618;--quarto-scss-export-form-feedback-icon-invalid-color: #ff0039;--quarto-scss-export-form-valid-color: #3fb618;--quarto-scss-export-form-valid-border-color: #3fb618;--quarto-scss-export-form-invalid-color: #ff0039;--quarto-scss-export-form-invalid-border-color: #ff0039;--quarto-scss-export-nav-link-font-size: ;--quarto-scss-export-nav-link-font-weight: ;--quarto-scss-export-nav-link-color: #2761e3;--quarto-scss-export-nav-link-hover-color: #1f4eb6;--quarto-scss-export-nav-link-disabled-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-nav-tabs-border-color: #dee2e6;--quarto-scss-export-nav-tabs-link-hover-border-color: #e9ecef #e9ecef #dee2e6;--quarto-scss-export-nav-tabs-link-active-color: #000;--quarto-scss-export-nav-tabs-link-active-bg: #fff;--quarto-scss-export-nav-pills-link-active-bg: #2780e3;--quarto-scss-export-nav-pills-link-active-color: #fff;--quarto-scss-export-nav-underline-link-active-color: #000;--quarto-scss-export-navbar-padding-x: ;--quarto-scss-export-navbar-light-contrast: #000;--quarto-scss-export-navbar-dark-contrast: #000;--quarto-scss-export-navbar-light-icon-color: rgba(0, 0, 0, 0.75);--quarto-scss-export-navbar-dark-icon-color: rgba(0, 0, 0, 0.75);--quarto-scss-export-dropdown-color: #343a40;--quarto-scss-export-dropdown-bg: #fff;--quarto-scss-export-dropdown-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-dropdown-divider-bg: rgba(0, 0, 0, 0.175);--quarto-scss-export-dropdown-link-color: #343a40;--quarto-scss-export-dropdown-link-hover-color: #343a40;--quarto-scss-export-dropdown-link-hover-bg: #f8f9fa;--quarto-scss-export-dropdown-link-active-bg: #2780e3;--quarto-scss-export-dropdown-link-active-color: #fff;--quarto-scss-export-dropdown-link-disabled-color: rgba(52, 58, 64, 0.5);--quarto-scss-export-dropdown-header-color: #6c757d;--quarto-scss-export-dropdown-dark-color: #dee2e6;--quarto-scss-export-dropdown-dark-bg: #343a40;--quarto-scss-export-dropdown-dark-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-dropdown-dark-divider-bg: rgba(0, 0, 0, 0.175);--quarto-scss-export-dropdown-dark-box-shadow: ;--quarto-scss-export-dropdown-dark-link-color: #dee2e6;--quarto-scss-export-dropdown-dark-link-hover-color: #fff;--quarto-scss-export-dropdown-dark-link-hover-bg: rgba(255, 255, 255, 0.15);--quarto-scss-export-dropdown-dark-link-active-color: #fff;--quarto-scss-export-dropdown-dark-link-active-bg: #2780e3;--quarto-scss-export-dropdown-dark-link-disabled-color: #adb5bd;--quarto-scss-export-dropdown-dark-header-color: #adb5bd;--quarto-scss-export-pagination-color: #2761e3;--quarto-scss-export-pagination-bg: #fff;--quarto-scss-export-pagination-border-color: #dee2e6;--quarto-scss-export-pagination-focus-color: #1f4eb6;--quarto-scss-export-pagination-focus-bg: #e9ecef;--quarto-scss-export-pagination-hover-color: #1f4eb6;--quarto-scss-export-pagination-hover-bg: #f8f9fa;--quarto-scss-export-pagination-hover-border-color: #dee2e6;--quarto-scss-export-pagination-active-color: #fff;--quarto-scss-export-pagination-active-bg: #2780e3;--quarto-scss-export-pagination-active-border-color: #2780e3;--quarto-scss-export-pagination-disabled-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-pagination-disabled-bg: #e9ecef;--quarto-scss-export-pagination-disabled-border-color: #dee2e6;--quarto-scss-export-card-title-color: ;--quarto-scss-export-card-subtitle-color: ;--quarto-scss-export-card-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-card-box-shadow: ;--quarto-scss-export-card-cap-color: ;--quarto-scss-export-card-height: ;--quarto-scss-export-card-color: ;--quarto-scss-export-card-bg: #fff;--quarto-scss-export-accordion-color: #343a40;--quarto-scss-export-accordion-bg: #fff;--quarto-scss-export-accordion-border-color: #dee2e6;--quarto-scss-export-accordion-button-color: #343a40;--quarto-scss-export-accordion-button-bg: #fff;--quarto-scss-export-accordion-button-active-bg: #d4e6f9;--quarto-scss-export-accordion-button-active-color: #10335b;--quarto-scss-export-accordion-button-focus-border-color: #93c0f1;--quarto-scss-export-accordion-icon-color: #343a40;--quarto-scss-export-accordion-icon-active-color: #10335b;--quarto-scss-export-tooltip-color: #fff;--quarto-scss-export-tooltip-bg: #000;--quarto-scss-export-tooltip-margin: ;--quarto-scss-export-tooltip-arrow-color: ;--quarto-scss-export-form-feedback-tooltip-line-height: ;--quarto-scss-export-popover-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-popover-header-bg: #e9ecef;--quarto-scss-export-popover-body-color: #343a40;--quarto-scss-export-popover-arrow-color: #fff;--quarto-scss-export-popover-arrow-outer-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-toast-color: ;--quarto-scss-export-toast-background-color: rgba(255, 255, 255, 0.85);--quarto-scss-export-toast-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-toast-header-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-toast-header-background-color: rgba(255, 255, 255, 0.85);--quarto-scss-export-toast-header-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-badge-color: #fff;--quarto-scss-export-modal-content-color: ;--quarto-scss-export-modal-content-bg: #fff;--quarto-scss-export-modal-content-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-modal-backdrop-bg: #000;--quarto-scss-export-modal-header-border-color: #dee2e6;--quarto-scss-export-modal-footer-bg: ;--quarto-scss-export-modal-footer-border-color: #dee2e6;--quarto-scss-export-progress-bg: #e9ecef;--quarto-scss-export-progress-bar-color: #fff;--quarto-scss-export-progress-bar-bg: #2780e3;--quarto-scss-export-list-group-color: #343a40;--quarto-scss-export-list-group-bg: #fff;--quarto-scss-export-list-group-border-color: #dee2e6;--quarto-scss-export-list-group-hover-bg: #f8f9fa;--quarto-scss-export-list-group-active-bg: #2780e3;--quarto-scss-export-list-group-active-color: #fff;--quarto-scss-export-list-group-active-border-color: #2780e3;--quarto-scss-export-list-group-disabled-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-list-group-disabled-bg: #fff;--quarto-scss-export-list-group-action-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-list-group-action-hover-color: #000;--quarto-scss-export-list-group-action-active-color: #343a40;--quarto-scss-export-list-group-action-active-bg: #e9ecef;--quarto-scss-export-thumbnail-bg: #fff;--quarto-scss-export-thumbnail-border-color: #dee2e6;--quarto-scss-export-figure-caption-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-breadcrumb-font-size: ;--quarto-scss-export-breadcrumb-bg: ;--quarto-scss-export-breadcrumb-divider-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-breadcrumb-active-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-breadcrumb-border-radius: ;--quarto-scss-export-carousel-control-color: #fff;--quarto-scss-export-carousel-indicator-active-bg: #fff;--quarto-scss-export-carousel-caption-color: #fff;--quarto-scss-export-carousel-dark-indicator-active-bg: #000;--quarto-scss-export-carousel-dark-caption-color: #000;--quarto-scss-export-btn-close-color: #000;--quarto-scss-export-offcanvas-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-offcanvas-bg-color: #fff;--quarto-scss-export-offcanvas-color: #343a40;--quarto-scss-export-offcanvas-backdrop-bg: #000;--quarto-scss-export-code-color-dark: white;--quarto-scss-export-kbd-color: #fff;--quarto-scss-export-kbd-bg: #343a40;--quarto-scss-export-nested-kbd-font-weight: ;--quarto-scss-export-pre-bg: #f8f9fa;--quarto-scss-export-pre-color: #000;--quarto-scss-export-bslib-page-sidebar-title-bg: #f8f9fa;--quarto-scss-export-bslib-page-sidebar-title-color: #000;--quarto-scss-export-bslib-sidebar-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 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var(--bs-navbar-hover-color);--bs-nav-link-disabled-color: var(--bs-navbar-disabled-color);display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0;list-style:none}.navbar-nav .nav-link.active,.navbar-nav .nav-link.show{color:var(--bs-navbar-active-color)}.navbar-nav .dropdown-menu{position:static}.navbar-text{padding-top:.5rem;padding-bottom:.5rem;color:var(--bs-navbar-color)}.navbar-text a,.navbar-text a:hover,.navbar-text a:focus{color:var(--bs-navbar-active-color)}.navbar-collapse{flex-basis:100%;-webkit-flex-basis:100%;flex-grow:1;-webkit-flex-grow:1;align-items:center;-webkit-align-items:center}.navbar-toggler{padding:var(--bs-navbar-toggler-padding-y) var(--bs-navbar-toggler-padding-x);font-size:var(--bs-navbar-toggler-font-size);line-height:1;color:var(--bs-navbar-color);background-color:rgba(0,0,0,0);border:var(--bs-border-width) solid 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.navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-md .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-md .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-md .navbar-nav-scroll{overflow:visible}.navbar-expand-md .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-md .navbar-toggler{display:none}.navbar-expand-md .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-md .offcanvas .offcanvas-header{display:none}.navbar-expand-md .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 992px){.navbar-expand-lg{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-lg .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-lg .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-lg .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-lg .navbar-nav-scroll{overflow:visible}.navbar-expand-lg .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-lg .navbar-toggler{display:none}.navbar-expand-lg .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-lg .offcanvas .offcanvas-header{display:none}.navbar-expand-lg .offcanvas 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.offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: #545555;--bs-navbar-hover-color: rgba(31, 78, 182, 0.8);--bs-navbar-disabled-color: rgba(84, 85, 85, 0.75);--bs-navbar-active-color: #1f4eb6;--bs-navbar-brand-color: #545555;--bs-navbar-brand-hover-color: #1f4eb6;--bs-navbar-toggler-border-color: rgba(84, 85, 85, 0);--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 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0.75rem;position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0}.card>.list-group:last-child{border-bottom-width:0}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 576px){.card-group{display:flex;display:-webkit-flex;flex-flow:row wrap;-webkit-flex-flow:row wrap}.card-group>.card{flex:1 0 0%;-webkit-flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}}.accordion{--bs-accordion-color: #343a40;--bs-accordion-bg: #fff;--bs-accordion-transition: 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reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:not(:first-of-type){border-top:0}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(52, 58, 64, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(52, 58, 64, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #2761e3;--bs-pagination-bg: #fff;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #1f4eb6;--bs-pagination-hover-bg: #f8f9fa;--bs-pagination-hover-border-color: #dee2e6;--bs-pagination-focus-color: #1f4eb6;--bs-pagination-focus-bg: #e9ecef;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #2780e3;--bs-pagination-active-border-color: #2780e3;--bs-pagination-disabled-color: rgba(52, 58, 64, 0.75);--bs-pagination-disabled-bg: #e9ecef;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 0 solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:.5rem}}.progress,.progress-stacked{--bs-progress-height: 0.5rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #e9ecef;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #2780e3;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #343a40;--bs-list-group-bg: #fff;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(52, 58, 64, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #343a40;--bs-list-group-action-active-bg: #e9ecef;--bs-list-group-disabled-color: rgba(52, 58, 64, 0.75);--bs-list-group-disabled-bg: #fff;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #2780e3;--bs-list-group-active-border-color: #2780e3;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(255, 255, 255, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(52, 58, 64, 0.75);--bs-toast-header-bg: rgba(255, 255, 255, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 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0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static 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0.5;position:fixed;top:0;left:0;z-index:var(--bs-backdrop-zindex);width:100vw;height:100vh;background-color:var(--bs-backdrop-bg)}.modal-backdrop.fade{opacity:0}.modal-backdrop.show{opacity:var(--bs-backdrop-opacity)}.modal-header{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;align-items:center;-webkit-align-items:center;justify-content:space-between;-webkit-justify-content:space-between;padding:var(--bs-modal-header-padding);border-bottom:var(--bs-modal-header-border-width) solid var(--bs-modal-header-border-color)}.modal-header .btn-close{padding:calc(var(--bs-modal-header-padding-y)*.5) calc(var(--bs-modal-header-padding-x)*.5);margin:calc(-0.5*var(--bs-modal-header-padding-y)) calc(-0.5*var(--bs-modal-header-padding-x)) calc(-0.5*var(--bs-modal-header-padding-y)) auto}.modal-title{margin-bottom:0;line-height:var(--bs-modal-title-line-height)}.modal-body{position:relative;flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-modal-padding)}.modal-footer{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;flex-wrap:wrap;-webkit-flex-wrap:wrap;align-items:center;-webkit-align-items:center;justify-content:flex-end;-webkit-justify-content:flex-end;padding:calc(var(--bs-modal-padding) - var(--bs-modal-footer-gap)*.5);background-color:var(--bs-modal-footer-bg);border-top:var(--bs-modal-footer-border-width) solid var(--bs-modal-footer-border-color)}.modal-footer>*{margin:calc(var(--bs-modal-footer-gap)*.5)}@media(min-width: 576px){.modal{--bs-modal-margin: 1.75rem;--bs-modal-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15)}.modal-dialog{max-width:var(--bs-modal-width);margin-right:auto;margin-left:auto}.modal-sm{--bs-modal-width: 300px}}@media(min-width: 992px){.modal-lg,.modal-xl{--bs-modal-width: 800px}}@media(min-width: 1200px){.modal-xl{--bs-modal-width: 1140px}}.modal-fullscreen{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen .modal-content{height:100%;border:0}.modal-fullscreen .modal-body{overflow-y:auto}@media(max-width: 575.98px){.modal-fullscreen-sm-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-sm-down .modal-content{height:100%;border:0}.modal-fullscreen-sm-down .modal-body{overflow-y:auto}}@media(max-width: 767.98px){.modal-fullscreen-md-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-md-down .modal-content{height:100%;border:0}.modal-fullscreen-md-down .modal-body{overflow-y:auto}}@media(max-width: 991.98px){.modal-fullscreen-lg-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-lg-down .modal-content{height:100%;border:0}.modal-fullscreen-lg-down .modal-body{overflow-y:auto}}@media(max-width: 1199.98px){.modal-fullscreen-xl-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-xl-down .modal-content{height:100%;border:0}.modal-fullscreen-xl-down .modal-body{overflow-y:auto}}@media(max-width: 1399.98px){.modal-fullscreen-xxl-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-xxl-down .modal-content{height:100%;border:0}.modal-fullscreen-xxl-down .modal-body{overflow-y:auto}}.tooltip{--bs-tooltip-zindex: 1080;--bs-tooltip-max-width: 200px;--bs-tooltip-padding-x: 0.5rem;--bs-tooltip-padding-y: 0.25rem;--bs-tooltip-margin: ;--bs-tooltip-font-size:0.875rem;--bs-tooltip-color: #fff;--bs-tooltip-bg: #000;--bs-tooltip-border-radius: 0.25rem;--bs-tooltip-opacity: 0.9;--bs-tooltip-arrow-width: 0.8rem;--bs-tooltip-arrow-height: 0.4rem;z-index:var(--bs-tooltip-zindex);display:block;margin:var(--bs-tooltip-margin);font-family:"Source Sans Pro",-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,"Helvetica Neue",Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol";font-style:normal;font-weight:400;line-height:1.5;text-align:left;text-align:start;text-decoration:none;text-shadow:none;text-transform:none;letter-spacing:normal;word-break:normal;white-space:normal;word-spacing:normal;line-break:auto;font-size:var(--bs-tooltip-font-size);word-wrap:break-word;opacity:0}.tooltip.show{opacity:var(--bs-tooltip-opacity)}.tooltip .tooltip-arrow{display:block;width:var(--bs-tooltip-arrow-width);height:var(--bs-tooltip-arrow-height)}.tooltip .tooltip-arrow::before{position:absolute;content:"";border-color:rgba(0,0,0,0);border-style:solid}.bs-tooltip-top .tooltip-arrow,.bs-tooltip-auto[data-popper-placement^=top] .tooltip-arrow{bottom:calc(-1*var(--bs-tooltip-arrow-height))}.bs-tooltip-top .tooltip-arrow::before,.bs-tooltip-auto[data-popper-placement^=top] .tooltip-arrow::before{top:-1px;border-width:var(--bs-tooltip-arrow-height) calc(var(--bs-tooltip-arrow-width)*.5) 0;border-top-color:var(--bs-tooltip-bg)}.bs-tooltip-end 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var(--bs-popover-border-width))}.bs-popover-top>.popover-arrow::before,.bs-popover-auto[data-popper-placement^=top]>.popover-arrow::before,.bs-popover-top>.popover-arrow::after,.bs-popover-auto[data-popper-placement^=top]>.popover-arrow::after{border-width:var(--bs-popover-arrow-height) calc(var(--bs-popover-arrow-width)*.5) 0}.bs-popover-top>.popover-arrow::before,.bs-popover-auto[data-popper-placement^=top]>.popover-arrow::before{bottom:0;border-top-color:var(--bs-popover-arrow-border)}.bs-popover-top>.popover-arrow::after,.bs-popover-auto[data-popper-placement^=top]>.popover-arrow::after{bottom:var(--bs-popover-border-width);border-top-color:var(--bs-popover-bg)}.bs-popover-end>.popover-arrow,.bs-popover-auto[data-popper-placement^=right]>.popover-arrow{left:calc(-1*(var(--bs-popover-arrow-height)) - 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#fff);--bslib-value-box-border-color-default: var(--bs-card-border-color, rgba(0, 0, 0, 0.175));color:var(--bslib-value-box-color);background-color:var(--bslib-value-box-bg, var(--bslib-value-box-bg-default));border-color:var(--bslib-value-box-border-color, var(--bslib-value-box-border-color-default))}.bslib-value-box .value-box-grid{display:grid;grid-template-areas:"left right";align-items:center;overflow:hidden}.bslib-value-box .value-box-showcase{height:100%;max-height:var(---bslib-value-box-showcase-max-h, 100%)}.bslib-value-box .value-box-showcase,.bslib-value-box .value-box-showcase>.html-fill-item{width:100%}.bslib-value-box[data-full-screen=true] .value-box-showcase{max-height:var(---bslib-value-box-showcase-max-h-fs, 100%)}@media screen and (min-width: 575.98px){@container bslib-value-box (max-width: 300px){.bslib-value-box:not(.showcase-bottom) .value-box-grid{grid-template-columns:1fr !important;grid-template-rows:auto auto;grid-template-areas:"top" "bottom"}.bslib-value-box:not(.showcase-bottom) .value-box-grid .value-box-showcase{grid-area:top !important}.bslib-value-box:not(.showcase-bottom) .value-box-grid .value-box-area{grid-area:bottom !important;justify-content:end}}}.bslib-value-box .value-box-area{justify-content:center;padding:1.5rem 1rem;font-size:.9rem;font-weight:500}.bslib-value-box .value-box-area *{margin-bottom:0;margin-top:0}.bslib-value-box .value-box-title{font-size:1rem;margin-top:0;margin-bottom:.5rem;font-weight:400;line-height:1.2}.bslib-value-box .value-box-title:empty::after{content:" "}.bslib-value-box .value-box-value{font-size:calc(1.29rem + 0.48vw);margin-top:0;margin-bottom:.5rem;font-weight:400;line-height:1.2}@media(min-width: 1200px){.bslib-value-box .value-box-value{font-size:1.65rem}}.bslib-value-box .value-box-value:empty::after{content:" "}.bslib-value-box .value-box-showcase{align-items:center;justify-content:center;margin-top:auto;margin-bottom:auto;padding:1rem}.bslib-value-box 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1fr)}.bslib-value-box.showcase-top-right[data-full-screen=true] .value-box-grid>div{align-self:center}.bslib-value-box.showcase-top-right:not([data-full-screen=true]) .value-box-showcase{margin-top:0}@container bslib-value-box (max-width: 300px){.bslib-value-box.showcase-top-right:not([data-full-screen=true]) .value-box-grid .value-box-showcase{padding-left:1rem}}.bslib-value-box.showcase-left-center .value-box-grid{grid-template-columns:var(---bslib-value-box-showcase-w, 30%) auto}.bslib-value-box.showcase-left-center[data-full-screen=true] .value-box-grid{grid-template-columns:var(---bslib-value-box-showcase-w-fs, 1fr) auto}.bslib-value-box.showcase-left-center:not([data-fill-screen=true]) .value-box-grid .value-box-showcase{grid-area:left}.bslib-value-box.showcase-left-center:not([data-fill-screen=true]) .value-box-grid .value-box-area{grid-area:right}.bslib-value-box.showcase-bottom .value-box-grid{grid-template-columns:1fr;grid-template-rows:1fr var(---bslib-value-box-showcase-h, 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#dee2e6;--quarto-scss-export-form-select-font-family: ;--quarto-scss-export-form-select-color: #343a40;--quarto-scss-export-form-select-bg: #fff;--quarto-scss-export-form-select-disabled-color: ;--quarto-scss-export-form-select-disabled-bg: #e9ecef;--quarto-scss-export-form-select-disabled-border-color: ;--quarto-scss-export-form-select-indicator-color: #343a40;--quarto-scss-export-form-select-border-color: #dee2e6;--quarto-scss-export-form-select-focus-border-color: #93c0f1;--quarto-scss-export-form-range-track-bg: #f8f9fa;--quarto-scss-export-form-range-thumb-bg: #2780e3;--quarto-scss-export-form-range-thumb-active-bg: #bed9f7;--quarto-scss-export-form-range-thumb-disabled-bg: rgba(52, 58, 64, 0.75);--quarto-scss-export-form-file-button-color: #343a40;--quarto-scss-export-form-file-button-bg: #f8f9fa;--quarto-scss-export-form-file-button-hover-bg: #e9ecef;--quarto-scss-export-form-floating-label-disabled-color: #6c757d;--quarto-scss-export-form-feedback-font-style: ;--quarto-scss-export-form-feedback-valid-color: #3fb618;--quarto-scss-export-form-feedback-invalid-color: #ff0039;--quarto-scss-export-form-feedback-icon-valid-color: #3fb618;--quarto-scss-export-form-feedback-icon-invalid-color: #ff0039;--quarto-scss-export-form-valid-color: #3fb618;--quarto-scss-export-form-valid-border-color: #3fb618;--quarto-scss-export-form-invalid-color: #ff0039;--quarto-scss-export-form-invalid-border-color: #ff0039;--quarto-scss-export-nav-link-font-size: ;--quarto-scss-export-nav-link-font-weight: ;--quarto-scss-export-nav-link-color: #2761e3;--quarto-scss-export-nav-link-hover-color: #1f4eb6;--quarto-scss-export-nav-link-disabled-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-nav-tabs-border-color: #dee2e6;--quarto-scss-export-nav-tabs-link-hover-border-color: #e9ecef #e9ecef #dee2e6;--quarto-scss-export-nav-tabs-link-active-color: #000;--quarto-scss-export-nav-tabs-link-active-bg: #fff;--quarto-scss-export-nav-pills-link-active-bg: #2780e3;--quarto-scss-export-nav-pills-link-active-color: #fff;--quarto-scss-export-nav-underline-link-active-color: #000;--quarto-scss-export-navbar-padding-x: ;--quarto-scss-export-navbar-light-contrast: #000;--quarto-scss-export-navbar-dark-contrast: #000;--quarto-scss-export-navbar-light-icon-color: rgba(0, 0, 0, 0.75);--quarto-scss-export-navbar-dark-icon-color: rgba(0, 0, 0, 0.75);--quarto-scss-export-dropdown-color: #343a40;--quarto-scss-export-dropdown-bg: #fff;--quarto-scss-export-dropdown-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-dropdown-divider-bg: rgba(0, 0, 0, 0.175);--quarto-scss-export-dropdown-link-color: #343a40;--quarto-scss-export-dropdown-link-hover-color: #343a40;--quarto-scss-export-dropdown-link-hover-bg: #f8f9fa;--quarto-scss-export-dropdown-link-active-bg: #2780e3;--quarto-scss-export-dropdown-link-active-color: #fff;--quarto-scss-export-dropdown-link-disabled-color: rgba(52, 58, 64, 0.5);--quarto-scss-export-dropdown-header-color: #6c757d;--quarto-scss-export-dropdown-dark-color: #dee2e6;--quarto-scss-export-dropdown-dark-bg: #343a40;--quarto-scss-export-dropdown-dark-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-dropdown-dark-divider-bg: rgba(0, 0, 0, 0.175);--quarto-scss-export-dropdown-dark-box-shadow: ;--quarto-scss-export-dropdown-dark-link-color: #dee2e6;--quarto-scss-export-dropdown-dark-link-hover-color: #fff;--quarto-scss-export-dropdown-dark-link-hover-bg: rgba(255, 255, 255, 0.15);--quarto-scss-export-dropdown-dark-link-active-color: #fff;--quarto-scss-export-dropdown-dark-link-active-bg: #2780e3;--quarto-scss-export-dropdown-dark-link-disabled-color: #adb5bd;--quarto-scss-export-dropdown-dark-header-color: #adb5bd;--quarto-scss-export-pagination-color: #2761e3;--quarto-scss-export-pagination-bg: #fff;--quarto-scss-export-pagination-border-color: #dee2e6;--quarto-scss-export-pagination-focus-color: #1f4eb6;--quarto-scss-export-pagination-focus-bg: #e9ecef;--quarto-scss-export-pagination-hover-color: #1f4eb6;--quarto-scss-export-pagination-hover-bg: #f8f9fa;--quarto-scss-export-pagination-hover-border-color: #dee2e6;--quarto-scss-export-pagination-active-color: #fff;--quarto-scss-export-pagination-active-bg: #2780e3;--quarto-scss-export-pagination-active-border-color: #2780e3;--quarto-scss-export-pagination-disabled-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-pagination-disabled-bg: #e9ecef;--quarto-scss-export-pagination-disabled-border-color: #dee2e6;--quarto-scss-export-card-title-color: ;--quarto-scss-export-card-subtitle-color: ;--quarto-scss-export-card-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-card-box-shadow: ;--quarto-scss-export-card-cap-color: ;--quarto-scss-export-card-height: ;--quarto-scss-export-card-color: ;--quarto-scss-export-card-bg: #fff;--quarto-scss-export-accordion-color: #343a40;--quarto-scss-export-accordion-bg: #fff;--quarto-scss-export-accordion-border-color: #dee2e6;--quarto-scss-export-accordion-button-color: #343a40;--quarto-scss-export-accordion-button-bg: #fff;--quarto-scss-export-accordion-button-active-bg: #d4e6f9;--quarto-scss-export-accordion-button-active-color: #10335b;--quarto-scss-export-accordion-button-focus-border-color: #93c0f1;--quarto-scss-export-accordion-icon-color: #343a40;--quarto-scss-export-accordion-icon-active-color: #10335b;--quarto-scss-export-tooltip-color: #fff;--quarto-scss-export-tooltip-bg: #000;--quarto-scss-export-tooltip-margin: ;--quarto-scss-export-tooltip-arrow-color: ;--quarto-scss-export-form-feedback-tooltip-line-height: ;--quarto-scss-export-popover-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-popover-header-bg: #e9ecef;--quarto-scss-export-popover-body-color: #343a40;--quarto-scss-export-popover-arrow-color: #fff;--quarto-scss-export-popover-arrow-outer-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-toast-color: ;--quarto-scss-export-toast-background-color: rgba(255, 255, 255, 0.85);--quarto-scss-export-toast-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-toast-header-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-toast-header-background-color: rgba(255, 255, 255, 0.85);--quarto-scss-export-toast-header-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-badge-color: #fff;--quarto-scss-export-modal-content-color: ;--quarto-scss-export-modal-content-bg: #fff;--quarto-scss-export-modal-content-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-modal-backdrop-bg: #000;--quarto-scss-export-modal-header-border-color: #dee2e6;--quarto-scss-export-modal-footer-bg: ;--quarto-scss-export-modal-footer-border-color: #dee2e6;--quarto-scss-export-progress-bg: #e9ecef;--quarto-scss-export-progress-bar-color: #fff;--quarto-scss-export-progress-bar-bg: #2780e3;--quarto-scss-export-list-group-color: #343a40;--quarto-scss-export-list-group-bg: #fff;--quarto-scss-export-list-group-border-color: #dee2e6;--quarto-scss-export-list-group-hover-bg: #f8f9fa;--quarto-scss-export-list-group-active-bg: #2780e3;--quarto-scss-export-list-group-active-color: #fff;--quarto-scss-export-list-group-active-border-color: #2780e3;--quarto-scss-export-list-group-disabled-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-list-group-disabled-bg: #fff;--quarto-scss-export-list-group-action-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-list-group-action-hover-color: #000;--quarto-scss-export-list-group-action-active-color: #343a40;--quarto-scss-export-list-group-action-active-bg: #e9ecef;--quarto-scss-export-thumbnail-bg: #fff;--quarto-scss-export-thumbnail-border-color: #dee2e6;--quarto-scss-export-figure-caption-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-breadcrumb-font-size: ;--quarto-scss-export-breadcrumb-bg: ;--quarto-scss-export-breadcrumb-divider-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-breadcrumb-active-color: rgba(52, 58, 64, 0.75);--quarto-scss-export-breadcrumb-border-radius: ;--quarto-scss-export-carousel-control-color: #fff;--quarto-scss-export-carousel-indicator-active-bg: #fff;--quarto-scss-export-carousel-caption-color: #fff;--quarto-scss-export-carousel-dark-indicator-active-bg: #000;--quarto-scss-export-carousel-dark-caption-color: #000;--quarto-scss-export-btn-close-color: #000;--quarto-scss-export-offcanvas-border-color: rgba(0, 0, 0, 0.175);--quarto-scss-export-offcanvas-bg-color: #fff;--quarto-scss-export-offcanvas-color: #343a40;--quarto-scss-export-offcanvas-backdrop-bg: #000;--quarto-scss-export-code-color-dark: white;--quarto-scss-export-kbd-color: #fff;--quarto-scss-export-kbd-bg: #343a40;--quarto-scss-export-nested-kbd-font-weight: ;--quarto-scss-export-pre-bg: #f8f9fa;--quarto-scss-export-pre-color: #000;--quarto-scss-export-bslib-page-sidebar-title-bg: #f8f9fa;--quarto-scss-export-bslib-page-sidebar-title-color: #000;--quarto-scss-export-bslib-sidebar-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.05);--quarto-scss-export-bslib-sidebar-toggle-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.1);--quarto-scss-export-sidebar-color: #545555;--quarto-scss-export-sidebar-hover-color: rgba(31, 78, 182, 0.8);--quarto-scss-export-sidebar-disabled-color: rgba(84, 85, 85, 0.75);--quarto-scss-export-valuebox-bg-primary: #5397e9;--quarto-scss-export-valuebox-bg-secondary: #343a40;--quarto-scss-export-valuebox-bg-success: #3aa716;--quarto-scss-export-valuebox-bg-info: rgba(153, 84, 187, 0.7019607843);--quarto-scss-export-valuebox-bg-warning: #fa6400;--quarto-scss-export-valuebox-bg-danger: rgba(255, 0, 57, 0.7019607843);--quarto-scss-export-valuebox-bg-light: #f8f9fa;--quarto-scss-export-valuebox-bg-dark: #343a40;--quarto-scss-export-mermaid-bg-color: #fff;--quarto-scss-export-mermaid-edge-color: #343a40;--quarto-scss-export-mermaid-node-fg-color: #343a40;--quarto-scss-export-mermaid-fg-color: #343a40;--quarto-scss-export-mermaid-fg-color--lighter: #4b545c;--quarto-scss-export-mermaid-fg-color--lightest: #626d78;--quarto-scss-export-mermaid-label-bg-color: #fff;--quarto-scss-export-mermaid-label-fg-color: #2780e3;--quarto-scss-export-mermaid-node-bg-color: rgba(39, 128, 227, 0.1);--quarto-scss-export-code-block-border-left-color: #dee2e6;--quarto-scss-export-callout-color-note: #2780e3;--quarto-scss-export-callout-color-tip: #3fb618;--quarto-scss-export-callout-color-important: #ff0039;--quarto-scss-export-callout-color-caution: #f0ad4e;--quarto-scss-export-callout-color-warning: #ff7518} \ No newline at end of file diff --git a/site_libs 27/bootstrap/bootstrap-icons.css b/site_libs 27/bootstrap/bootstrap-icons.css new file mode 100644 index 0000000..285e444 --- /dev/null +++ b/site_libs 27/bootstrap/bootstrap-icons.css @@ -0,0 +1,2078 @@ +/*! + * Bootstrap Icons v1.11.1 (https://icons.getbootstrap.com/) + * Copyright 2019-2023 The Bootstrap Authors + * Licensed under MIT (https://github.com/twbs/icons/blob/main/LICENSE) + */ + +@font-face { + font-display: block; + font-family: "bootstrap-icons"; + src: +url("./bootstrap-icons.woff?2820a3852bdb9a5832199cc61cec4e65") format("woff"); +} + +.bi::before, +[class^="bi-"]::before, +[class*=" bi-"]::before { + display: inline-block; + font-family: bootstrap-icons !important; + font-style: normal; + font-weight: normal !important; + font-variant: normal; + text-transform: none; + line-height: 1; + vertical-align: -.125em; + -webkit-font-smoothing: antialiased; + -moz-osx-font-smoothing: grayscale; +} + +.bi-123::before { content: "\f67f"; } +.bi-alarm-fill::before { content: "\f101"; } +.bi-alarm::before { content: "\f102"; } +.bi-align-bottom::before { content: "\f103"; } +.bi-align-center::before { content: "\f104"; } +.bi-align-end::before { content: "\f105"; } +.bi-align-middle::before { content: "\f106"; } +.bi-align-start::before { content: "\f107"; } +.bi-align-top::before { content: "\f108"; } +.bi-alt::before { content: "\f109"; } +.bi-app-indicator::before { content: "\f10a"; } +.bi-app::before { content: "\f10b"; } +.bi-archive-fill::before { content: "\f10c"; } +.bi-archive::before { content: "\f10d"; } +.bi-arrow-90deg-down::before { content: "\f10e"; } +.bi-arrow-90deg-left::before { content: "\f10f"; } +.bi-arrow-90deg-right::before { content: "\f110"; } +.bi-arrow-90deg-up::before { content: "\f111"; } +.bi-arrow-bar-down::before { content: "\f112"; } +.bi-arrow-bar-left::before { content: "\f113"; } +.bi-arrow-bar-right::before { content: "\f114"; } +.bi-arrow-bar-up::before { content: "\f115"; } +.bi-arrow-clockwise::before { content: "\f116"; } +.bi-arrow-counterclockwise::before { content: "\f117"; } +.bi-arrow-down-circle-fill::before { content: "\f118"; } +.bi-arrow-down-circle::before { content: "\f119"; } +.bi-arrow-down-left-circle-fill::before { content: "\f11a"; } +.bi-arrow-down-left-circle::before { content: "\f11b"; } +.bi-arrow-down-left-square-fill::before { content: "\f11c"; } +.bi-arrow-down-left-square::before { content: "\f11d"; } +.bi-arrow-down-left::before { content: "\f11e"; } +.bi-arrow-down-right-circle-fill::before { content: "\f11f"; } +.bi-arrow-down-right-circle::before { content: "\f120"; } +.bi-arrow-down-right-square-fill::before { content: "\f121"; } +.bi-arrow-down-right-square::before { content: "\f122"; } +.bi-arrow-down-right::before { content: "\f123"; } +.bi-arrow-down-short::before { content: "\f124"; } +.bi-arrow-down-square-fill::before { content: "\f125"; } +.bi-arrow-down-square::before { content: "\f126"; } +.bi-arrow-down-up::before { content: "\f127"; } +.bi-arrow-down::before { content: "\f128"; } +.bi-arrow-left-circle-fill::before { content: "\f129"; } +.bi-arrow-left-circle::before { content: "\f12a"; } +.bi-arrow-left-right::before { content: "\f12b"; } +.bi-arrow-left-short::before { content: "\f12c"; } +.bi-arrow-left-square-fill::before { content: "\f12d"; } +.bi-arrow-left-square::before { content: "\f12e"; } +.bi-arrow-left::before { content: "\f12f"; } +.bi-arrow-repeat::before { content: "\f130"; } +.bi-arrow-return-left::before { content: "\f131"; } +.bi-arrow-return-right::before { content: "\f132"; } +.bi-arrow-right-circle-fill::before { content: "\f133"; } +.bi-arrow-right-circle::before { content: "\f134"; } +.bi-arrow-right-short::before { content: "\f135"; } +.bi-arrow-right-square-fill::before { content: "\f136"; } +.bi-arrow-right-square::before { content: "\f137"; } +.bi-arrow-right::before { content: "\f138"; } +.bi-arrow-up-circle-fill::before { content: "\f139"; } +.bi-arrow-up-circle::before { content: "\f13a"; } +.bi-arrow-up-left-circle-fill::before { content: "\f13b"; } +.bi-arrow-up-left-circle::before { content: "\f13c"; } +.bi-arrow-up-left-square-fill::before { content: "\f13d"; } +.bi-arrow-up-left-square::before { content: "\f13e"; } +.bi-arrow-up-left::before { content: "\f13f"; } +.bi-arrow-up-right-circle-fill::before { content: "\f140"; } +.bi-arrow-up-right-circle::before { content: "\f141"; } 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+.bi-type-h6::before { content: "\f8de"; } +.bi-backpack-fill::before { content: "\f8df"; } +.bi-backpack::before { content: "\f8e0"; } +.bi-backpack2-fill::before { content: "\f8e1"; } +.bi-backpack2::before { content: "\f8e2"; } +.bi-backpack3-fill::before { content: "\f8e3"; } +.bi-backpack3::before { content: "\f8e4"; } +.bi-backpack4-fill::before { content: "\f8e5"; } +.bi-backpack4::before { content: "\f8e6"; } +.bi-brilliance::before { content: "\f8e7"; } +.bi-cake-fill::before { content: "\f8e8"; } +.bi-cake2-fill::before { content: "\f8e9"; } +.bi-duffle-fill::before { content: "\f8ea"; } +.bi-duffle::before { content: "\f8eb"; } +.bi-exposure::before { content: "\f8ec"; } +.bi-gender-neuter::before { content: "\f8ed"; } +.bi-highlights::before { content: "\f8ee"; } +.bi-luggage-fill::before { content: "\f8ef"; } +.bi-luggage::before { content: "\f8f0"; } +.bi-mailbox-flag::before { content: "\f8f1"; } +.bi-mailbox2-flag::before { content: "\f8f2"; } +.bi-noise-reduction::before { content: "\f8f3"; } +.bi-passport-fill::before { content: "\f8f4"; } +.bi-passport::before { content: "\f8f5"; } +.bi-person-arms-up::before { content: "\f8f6"; } +.bi-person-raised-hand::before { content: "\f8f7"; } +.bi-person-standing-dress::before { content: "\f8f8"; } +.bi-person-standing::before { content: "\f8f9"; } +.bi-person-walking::before { content: "\f8fa"; } +.bi-person-wheelchair::before { content: "\f8fb"; } +.bi-shadows::before { content: "\f8fc"; } +.bi-suitcase-fill::before { content: "\f8fd"; } +.bi-suitcase-lg-fill::before { content: "\f8fe"; } +.bi-suitcase-lg::before { content: "\f8ff"; } +.bi-suitcase::before { content: "\f900"; } +.bi-suitcase2-fill::before { content: "\f901"; } +.bi-suitcase2::before { content: "\f902"; } +.bi-vignette::before { content: "\f903"; } diff --git a/site_libs 27/bootstrap/bootstrap-icons.woff b/site_libs 27/bootstrap/bootstrap-icons.woff new file mode 100644 index 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t.defaultInterval=t.interval,t}_addEventListeners(){this._config.keyboard&&N.on(this._element,ft,(t=>this._keydown(t))),"hover"===this._config.pause&&(N.on(this._element,pt,(()=>this.pause())),N.on(this._element,mt,(()=>this._maybeEnableCycle()))),this._config.touch&&st.isSupported()&&this._addTouchEventListeners()}_addTouchEventListeners(){for(const t of z.find(".carousel-item img",this._element))N.on(t,gt,(t=>t.preventDefault()));const t={leftCallback:()=>this._slide(this._directionToOrder(ct)),rightCallback:()=>this._slide(this._directionToOrder(ht)),endCallback:()=>{"hover"===this._config.pause&&(this.pause(),this.touchTimeout&&clearTimeout(this.touchTimeout),this.touchTimeout=setTimeout((()=>this._maybeEnableCycle()),500+this._config.interval))}};this._swipeHelper=new st(this._element,t)}_keydown(t){if(/input|textarea/i.test(t.target.tagName))return;const e=Tt[t.key];e&&(t.preventDefault(),this._slide(this._directionToOrder(e)))}_getItemIndex(t){return this._getItems().indexOf(t)}_setActiveIndicatorElement(t){if(!this._indicatorsElement)return;const e=z.findOne(wt,this._indicatorsElement);e.classList.remove(yt),e.removeAttribute("aria-current");const i=z.findOne(`[data-bs-slide-to="${t}"]`,this._indicatorsElement);i&&(i.classList.add(yt),i.setAttribute("aria-current","true"))}_updateInterval(){const t=this._activeElement||this._getActive();if(!t)return;const e=Number.parseInt(t.getAttribute("data-bs-interval"),10);this._config.interval=e||this._config.defaultInterval}_slide(t,e=null){if(this._isSliding)return;const i=this._getActive(),n=t===at,s=e||b(this._getItems(),i,n,this._config.wrap);if(s===i)return;const o=this._getItemIndex(s),r=e=>N.trigger(this._element,e,{relatedTarget:s,direction:this._orderToDirection(t),from:this._getItemIndex(i),to:o});if(r(dt).defaultPrevented)return;if(!i||!s)return;const a=Boolean(this._interval);this.pause(),this._isSliding=!0,this._setActiveIndicatorElement(o),this._activeElement=s;const l=n?"carousel-item-start":"carousel-item-end",c=n?"carousel-item-next":"carousel-item-prev";s.classList.add(c),d(s),i.classList.add(l),s.classList.add(l),this._queueCallback((()=>{s.classList.remove(l,c),s.classList.add(yt),i.classList.remove(yt,c,l),this._isSliding=!1,r(ut)}),i,this._isAnimated()),a&&this.cycle()}_isAnimated(){return this._element.classList.contains("slide")}_getActive(){return z.findOne(Et,this._element)}_getItems(){return z.find(At,this._element)}_clearInterval(){this._interval&&(clearInterval(this._interval),this._interval=null)}_directionToOrder(t){return p()?t===ct?lt:at:t===ct?at:lt}_orderToDirection(t){return p()?t===lt?ct:ht:t===lt?ht:ct}static jQueryInterface(t){return this.each((function(){const e=xt.getOrCreateInstance(this,t);if("number"!=typeof t){if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}else e.to(t)}))}}N.on(document,bt,"[data-bs-slide], 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i=`scroll${e[0].toUpperCase()+e.slice(1)}`;this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(Mt),this._element.classList.add(Pt,Nt),this._element.style[e]="",N.trigger(this._element,St)}),this._element,!0),this._element.style[e]=`${this._element[i]}px`}hide(){if(this._isTransitioning||!this._isShown())return;if(N.trigger(this._element,Dt).defaultPrevented)return;const t=this._getDimension();this._element.style[t]=`${this._element.getBoundingClientRect()[t]}px`,d(this._element),this._element.classList.add(Mt),this._element.classList.remove(Pt,Nt);for(const t of this._triggerArray){const e=z.getElementFromSelector(t);e&&!this._isShown(e)&&this._addAriaAndCollapsedClass([t],!1)}this._isTransitioning=!0,this._element.style[t]="",this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(Mt),this._element.classList.add(Pt),N.trigger(this._element,$t)}),this._element,!0)}_isShown(t=this._element){return 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J,Z="x"===y?zt:Vt,tt="x"===y?Rt:qt,et=A[w],it="y"===w?"height":"width",nt=et+g[Z],st=et-g[tt],ot=-1!==[zt,Vt].indexOf(_),rt=null!=(J=null==x?void 0:x[w])?J:0,at=ot?nt:et-E[it]-T[it]-rt+O.altAxis,lt=ot?et+E[it]+T[it]-rt-O.altAxis:st,ct=f&&ot?function(t,e,i){var n=Ne(t,e,i);return n>i?i:n}(at,et,lt):Ne(f?at:nt,et,f?lt:st);A[w]=ct,k[w]=ct-et}e.modifiersData[n]=k}},requiresIfExists:["offset"]};function di(t,e,i){void 0===i&&(i=!1);var n,s,o=me(e),r=me(e)&&function(t){var e=t.getBoundingClientRect(),i=we(e.width)/t.offsetWidth||1,n=we(e.height)/t.offsetHeight||1;return 1!==i||1!==n}(e),a=Le(e),l=Te(t,r,i),c={scrollLeft:0,scrollTop:0},h={x:0,y:0};return(o||!o&&!i)&&(("body"!==ue(e)||Ue(a))&&(c=(n=e)!==fe(n)&&me(n)?{scrollLeft:(s=n).scrollLeft,scrollTop:s.scrollTop}:Xe(n)),me(e)?((h=Te(e,!0)).x+=e.clientLeft,h.y+=e.clientTop):a&&(h.x=Ye(a))),{x:l.left+c.scrollLeft-h.x,y:l.top+c.scrollTop-h.y,width:l.width,height:l.height}}function ui(t){var e=new Map,i=new Set,n=[];function s(t){i.add(t.name),[].concat(t.requires||[],t.requiresIfExists||[]).forEach((function(t){if(!i.has(t)){var n=e.get(t);n&&s(n)}})),n.push(t)}return t.forEach((function(t){e.set(t.name,t)})),t.forEach((function(t){i.has(t.name)||s(t)})),n}var fi={placement:"bottom",modifiers:[],strategy:"absolute"};function pi(){for(var t=arguments.length,e=new Array(t),i=0;iNumber.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_getPopperConfig(){const t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return(this._inNavbar||"static"===this._config.display)&&(F.setDataAttribute(this._menu,"popper","static"),t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,...g(this._config.popperConfig,[t])}}_selectMenuItem({key:t,target:e}){const i=z.find(".dropdown-menu 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e=/input|textarea/i.test(t.target.tagName),i="Escape"===t.key,n=[Ei,Ti].includes(t.key);if(!n&&!i)return;if(e&&!i)return;t.preventDefault();const s=this.matches(Ii)?this:z.prev(this,Ii)[0]||z.next(this,Ii)[0]||z.findOne(Ii,t.delegateTarget.parentNode),o=qi.getOrCreateInstance(s);if(n)return t.stopPropagation(),o.show(),void o._selectMenuItem(t);o._isShown()&&(t.stopPropagation(),o.hide(),s.focus())}}N.on(document,Si,Ii,qi.dataApiKeydownHandler),N.on(document,Si,Pi,qi.dataApiKeydownHandler),N.on(document,Li,qi.clearMenus),N.on(document,Di,qi.clearMenus),N.on(document,Li,Ii,(function(t){t.preventDefault(),qi.getOrCreateInstance(this).toggle()})),m(qi);const Vi="backdrop",Ki="show",Qi=`mousedown.bs.${Vi}`,Xi={className:"modal-backdrop",clickCallback:null,isAnimated:!1,isVisible:!0,rootElement:"body"},Yi={className:"string",clickCallback:"(function|null)",isAnimated:"boolean",isVisible:"boolean",rootElement:"(element|string)"};class Ui extends H{constructor(t){super(),this._config=this._getConfig(t),this._isAppended=!1,this._element=null}static get Default(){return Xi}static get DefaultType(){return Yi}static get NAME(){return Vi}show(t){if(!this._config.isVisible)return void g(t);this._append();const e=this._getElement();this._config.isAnimated&&d(e),e.classList.add(Ki),this._emulateAnimation((()=>{g(t)}))}hide(t){this._config.isVisible?(this._getElement().classList.remove(Ki),this._emulateAnimation((()=>{this.dispose(),g(t)}))):g(t)}dispose(){this._isAppended&&(N.off(this._element,Qi),this._element.remove(),this._isAppended=!1)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_configAfterMerge(t){return t.rootElement=r(t.rootElement),t}_append(){if(this._isAppended)return;const t=this._getElement();this._config.rootElement.append(t),N.on(t,Qi,(()=>{g(this._config.clickCallback)})),this._isAppended=!0}_emulateAnimation(t){_(t,this._getElement(),this._config.isAnimated)}}const Gi=".bs.focustrap",Ji=`focusin${Gi}`,Zi=`keydown.tab${Gi}`,tn="backward",en={autofocus:!0,trapElement:null},nn={autofocus:"boolean",trapElement:"element"};class sn extends H{constructor(t){super(),this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}static get Default(){return en}static get DefaultType(){return nn}static get NAME(){return"focustrap"}activate(){this._isActive||(this._config.autofocus&&this._config.trapElement.focus(),N.off(document,Gi),N.on(document,Ji,(t=>this._handleFocusin(t))),N.on(document,Zi,(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,N.off(document,Gi))}_handleFocusin(t){const{trapElement:e}=this._config;if(t.target===document||t.target===e||e.contains(t.target))return;const i=z.focusableChildren(e);0===i.length?e.focus():this._lastTabNavDirection===tn?i[i.length-1].focus():i[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?tn:"forward")}}const on=".fixed-top, .fixed-bottom, .is-fixed, .sticky-top",rn=".sticky-top",an="padding-right",ln="margin-right";class cn{constructor(){this._element=document.body}getWidth(){const t=document.documentElement.clientWidth;return Math.abs(window.innerWidth-t)}hide(){const t=this.getWidth();this._disableOverFlow(),this._setElementAttributes(this._element,an,(e=>e+t)),this._setElementAttributes(on,an,(e=>e+t)),this._setElementAttributes(rn,ln,(e=>e-t))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,an),this._resetElementAttributes(on,an),this._resetElementAttributes(rn,ln)}isOverflowing(){return this.getWidth()>0}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t).getPropertyValue(e);t.style.setProperty(e,`${i(Number.parseFloat(s))}px`)}))}_saveInitialAttribute(t,e){const i=t.style.getPropertyValue(e);i&&F.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=F.getDataAttribute(t,e);null!==i?(F.removeDataAttribute(t,e),t.style.setProperty(e,i)):t.style.removeProperty(e)}))}_applyManipulationCallback(t,e){if(o(t))e(t);else for(const i of z.find(t,this._element))e(i)}}const hn=".bs.modal",dn=`hide${hn}`,un=`hidePrevented${hn}`,fn=`hidden${hn}`,pn=`show${hn}`,mn=`shown${hn}`,gn=`resize${hn}`,_n=`click.dismiss${hn}`,bn=`mousedown.dismiss${hn}`,vn=`keydown.dismiss${hn}`,yn=`click${hn}.data-api`,wn="modal-open",An="show",En="modal-static",Tn={backdrop:!0,focus:!0,keyboard:!0},Cn={backdrop:"(boolean|string)",focus:"boolean",keyboard:"boolean"};class On extends W{constructor(t,e){super(t,e),this._dialog=z.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._isTransitioning=!1,this._scrollBar=new cn,this._addEventListeners()}static get Default(){return Tn}static get DefaultType(){return Cn}static get NAME(){return"modal"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||N.trigger(this._element,pn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isTransitioning=!0,this._scrollBar.hide(),document.body.classList.add(wn),this._adjustDialog(),this._backdrop.show((()=>this._showElement(t))))}hide(){this._isShown&&!this._isTransitioning&&(N.trigger(this._element,dn).defaultPrevented||(this._isShown=!1,this._isTransitioning=!0,this._focustrap.deactivate(),this._element.classList.remove(An),this._queueCallback((()=>this._hideModal()),this._element,this._isAnimated())))}dispose(){N.off(window,hn),N.off(this._dialog,hn),this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new Ui({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new sn({trapElement:this._element})}_showElement(t){document.body.contains(this._element)||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0;const e=z.findOne(".modal-body",this._dialog);e&&(e.scrollTop=0),d(this._element),this._element.classList.add(An),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,N.trigger(this._element,mn,{relatedTarget:t})}),this._dialog,this._isAnimated())}_addEventListeners(){N.on(this._element,vn,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():this._triggerBackdropTransition())})),N.on(window,gn,(()=>{this._isShown&&!this._isTransitioning&&this._adjustDialog()})),N.on(this._element,bn,(t=>{N.one(this._element,_n,(e=>{this._element===t.target&&this._element===e.target&&("static"!==this._config.backdrop?this._config.backdrop&&this.hide():this._triggerBackdropTransition())}))}))}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(wn),this._resetAdjustments(),this._scrollBar.reset(),N.trigger(this._element,fn)}))}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(N.trigger(this._element,un).defaultPrevented)return;const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._element.style.overflowY;"hidden"===e||this._element.classList.contains(En)||(t||(this._element.style.overflowY="hidden"),this._element.classList.add(En),this._queueCallback((()=>{this._element.classList.remove(En),this._queueCallback((()=>{this._element.style.overflowY=e}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;if(i&&!t){const t=p()?"paddingLeft":"paddingRight";this._element.style[t]=`${e}px`}if(!i&&t){const t=p()?"paddingRight":"paddingLeft";this._element.style[t]=`${e}px`}}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=On.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}N.on(document,yn,'[data-bs-toggle="modal"]',(function(t){const e=z.getElementFromSelector(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),N.one(e,pn,(t=>{t.defaultPrevented||N.one(e,fn,(()=>{a(this)&&this.focus()}))}));const i=z.findOne(".modal.show");i&&On.getInstance(i).hide(),On.getOrCreateInstance(e).toggle(this)})),R(On),m(On);const xn=".bs.offcanvas",kn=".data-api",Ln=`load${xn}${kn}`,Sn="show",Dn="showing",$n="hiding",In=".offcanvas.show",Nn=`show${xn}`,Pn=`shown${xn}`,Mn=`hide${xn}`,jn=`hidePrevented${xn}`,Fn=`hidden${xn}`,Hn=`resize${xn}`,Wn=`click${xn}${kn}`,Bn=`keydown.dismiss${xn}`,zn={backdrop:!0,keyboard:!0,scroll:!1},Rn={backdrop:"(boolean|string)",keyboard:"boolean",scroll:"boolean"};class qn extends W{constructor(t,e){super(t,e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get Default(){return zn}static get DefaultType(){return Rn}static get NAME(){return"offcanvas"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||N.trigger(this._element,Nn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._backdrop.show(),this._config.scroll||(new cn).hide(),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add(Dn),this._queueCallback((()=>{this._config.scroll&&!this._config.backdrop||this._focustrap.activate(),this._element.classList.add(Sn),this._element.classList.remove(Dn),N.trigger(this._element,Pn,{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(N.trigger(this._element,Mn).defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.add($n),this._backdrop.hide(),this._queueCallback((()=>{this._element.classList.remove(Sn,$n),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._config.scroll||(new cn).reset(),N.trigger(this._element,Fn)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_initializeBackDrop(){const t=Boolean(this._config.backdrop);return new Ui({className:"offcanvas-backdrop",isVisible:t,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:t?()=>{"static"!==this._config.backdrop?this.hide():N.trigger(this._element,jn)}:null})}_initializeFocusTrap(){return new sn({trapElement:this._element})}_addEventListeners(){N.on(this._element,Bn,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():N.trigger(this._element,jn))}))}static jQueryInterface(t){return this.each((function(){const e=qn.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}N.on(document,Wn,'[data-bs-toggle="offcanvas"]',(function(t){const e=z.getElementFromSelector(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),l(this))return;N.one(e,Fn,(()=>{a(this)&&this.focus()}));const i=z.findOne(In);i&&i!==e&&qn.getInstance(i).hide(),qn.getOrCreateInstance(e).toggle(this)})),N.on(window,Ln,(()=>{for(const t of z.find(In))qn.getOrCreateInstance(t).show()})),N.on(window,Hn,(()=>{for(const t of z.find("[aria-modal][class*=show][class*=offcanvas-]"))"fixed"!==getComputedStyle(t).position&&qn.getOrCreateInstance(t).hide()})),R(qn),m(qn);const Vn={"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],div:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},Kn=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Qn=/^(?!javascript:)(?:[a-z0-9+.-]+:|[^&:/?#]*(?:[/?#]|$))/i,Xn=(t,e)=>{const i=t.nodeName.toLowerCase();return e.includes(i)?!Kn.has(i)||Boolean(Qn.test(t.nodeValue)):e.filter((t=>t instanceof RegExp)).some((t=>t.test(i)))},Yn={allowList:Vn,content:{},extraClass:"",html:!1,sanitize:!0,sanitizeFn:null,template:"

"},Un={allowList:"object",content:"object",extraClass:"(string|function)",html:"boolean",sanitize:"boolean",sanitizeFn:"(null|function)",template:"string"},Gn={entry:"(string|element|function|null)",selector:"(string|element)"};class Jn extends H{constructor(t){super(),this._config=this._getConfig(t)}static get Default(){return Yn}static get DefaultType(){return Un}static get NAME(){return"TemplateFactory"}getContent(){return Object.values(this._config.content).map((t=>this._resolvePossibleFunction(t))).filter(Boolean)}hasContent(){return this.getContent().length>0}changeContent(t){return this._checkContent(t),this._config.content={...this._config.content,...t},this}toHtml(){const t=document.createElement("div");t.innerHTML=this._maybeSanitize(this._config.template);for(const[e,i]of Object.entries(this._config.content))this._setContent(t,i,e);const e=t.children[0],i=this._resolvePossibleFunction(this._config.extraClass);return i&&e.classList.add(...i.split(" ")),e}_typeCheckConfig(t){super._typeCheckConfig(t),this._checkContent(t.content)}_checkContent(t){for(const[e,i]of Object.entries(t))super._typeCheckConfig({selector:e,entry:i},Gn)}_setContent(t,e,i){const n=z.findOne(i,t);n&&((e=this._resolvePossibleFunction(e))?o(e)?this._putElementInTemplate(r(e),n):this._config.html?n.innerHTML=this._maybeSanitize(e):n.textContent=e:n.remove())}_maybeSanitize(t){return this._config.sanitize?function(t,e,i){if(!t.length)return t;if(i&&"function"==typeof i)return i(t);const n=(new window.DOMParser).parseFromString(t,"text/html"),s=[].concat(...n.body.querySelectorAll("*"));for(const t of s){const i=t.nodeName.toLowerCase();if(!Object.keys(e).includes(i)){t.remove();continue}const n=[].concat(...t.attributes),s=[].concat(e["*"]||[],e[i]||[]);for(const e of n)Xn(e,s)||t.removeAttribute(e.nodeName)}return n.body.innerHTML}(t,this._config.allowList,this._config.sanitizeFn):t}_resolvePossibleFunction(t){return g(t,[this])}_putElementInTemplate(t,e){if(this._config.html)return e.innerHTML="",void e.append(t);e.textContent=t.textContent}}const Zn=new Set(["sanitize","allowList","sanitizeFn"]),ts="fade",es="show",is=".modal",ns="hide.bs.modal",ss="hover",os="focus",rs={AUTO:"auto",TOP:"top",RIGHT:p()?"left":"right",BOTTOM:"bottom",LEFT:p()?"right":"left"},as={allowList:Vn,animation:!0,boundary:"clippingParents",container:!1,customClass:"",delay:0,fallbackPlacements:["top","right","bottom","left"],html:!1,offset:[0,6],placement:"top",popperConfig:null,sanitize:!0,sanitizeFn:null,selector:!1,template:'',title:"",trigger:"hover focus"},ls={allowList:"object",animation:"boolean",boundary:"(string|element)",container:"(string|element|boolean)",customClass:"(string|function)",delay:"(number|object)",fallbackPlacements:"array",html:"boolean",offset:"(array|string|function)",placement:"(string|function)",popperConfig:"(null|object|function)",sanitize:"boolean",sanitizeFn:"(null|function)",selector:"(string|boolean)",template:"string",title:"(string|element|function)",trigger:"string"};class cs extends W{constructor(t,e){if(void 0===vi)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t,e),this._isEnabled=!0,this._timeout=0,this._isHovered=null,this._activeTrigger={},this._popper=null,this._templateFactory=null,this._newContent=null,this.tip=null,this._setListeners(),this._config.selector||this._fixTitle()}static get Default(){return as}static get DefaultType(){return ls}static get NAME(){return"tooltip"}enable(){this._isEnabled=!0}disable(){this._isEnabled=!1}toggleEnabled(){this._isEnabled=!this._isEnabled}toggle(){this._isEnabled&&(this._activeTrigger.click=!this._activeTrigger.click,this._isShown()?this._leave():this._enter())}dispose(){clearTimeout(this._timeout),N.off(this._element.closest(is),ns,this._hideModalHandler),this._element.getAttribute("data-bs-original-title")&&this._element.setAttribute("title",this._element.getAttribute("data-bs-original-title")),this._disposePopper(),super.dispose()}show(){if("none"===this._element.style.display)throw new Error("Please use show on visible elements");if(!this._isWithContent()||!this._isEnabled)return;const t=N.trigger(this._element,this.constructor.eventName("show")),e=(c(this._element)||this._element.ownerDocument.documentElement).contains(this._element);if(t.defaultPrevented||!e)return;this._disposePopper();const i=this._getTipElement();this._element.setAttribute("aria-describedby",i.getAttribute("id"));const{container:n}=this._config;if(this._element.ownerDocument.documentElement.contains(this.tip)||(n.append(i),N.trigger(this._element,this.constructor.eventName("inserted"))),this._popper=this._createPopper(i),i.classList.add(es),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))N.on(t,"mouseover",h);this._queueCallback((()=>{N.trigger(this._element,this.constructor.eventName("shown")),!1===this._isHovered&&this._leave(),this._isHovered=!1}),this.tip,this._isAnimated())}hide(){if(this._isShown()&&!N.trigger(this._element,this.constructor.eventName("hide")).defaultPrevented){if(this._getTipElement().classList.remove(es),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))N.off(t,"mouseover",h);this._activeTrigger.click=!1,this._activeTrigger[os]=!1,this._activeTrigger[ss]=!1,this._isHovered=null,this._queueCallback((()=>{this._isWithActiveTrigger()||(this._isHovered||this._disposePopper(),this._element.removeAttribute("aria-describedby"),N.trigger(this._element,this.constructor.eventName("hidden")))}),this.tip,this._isAnimated())}}update(){this._popper&&this._popper.update()}_isWithContent(){return Boolean(this._getTitle())}_getTipElement(){return this.tip||(this.tip=this._createTipElement(this._newContent||this._getContentForTemplate())),this.tip}_createTipElement(t){const e=this._getTemplateFactory(t).toHtml();if(!e)return null;e.classList.remove(ts,es),e.classList.add(`bs-${this.constructor.NAME}-auto`);const i=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME).toString();return e.setAttribute("id",i),this._isAnimated()&&e.classList.add(ts),e}setContent(t){this._newContent=t,this._isShown()&&(this._disposePopper(),this.show())}_getTemplateFactory(t){return this._templateFactory?this._templateFactory.changeContent(t):this._templateFactory=new Jn({...this._config,content:t,extraClass:this._resolvePossibleFunction(this._config.customClass)}),this._templateFactory}_getContentForTemplate(){return{".tooltip-inner":this._getTitle()}}_getTitle(){return this._resolvePossibleFunction(this._config.title)||this._element.getAttribute("data-bs-original-title")}_initializeOnDelegatedTarget(t){return this.constructor.getOrCreateInstance(t.delegateTarget,this._getDelegateConfig())}_isAnimated(){return this._config.animation||this.tip&&this.tip.classList.contains(ts)}_isShown(){return this.tip&&this.tip.classList.contains(es)}_createPopper(t){const e=g(this._config.placement,[this,t,this._element]),i=rs[e.toUpperCase()];return bi(this._element,t,this._getPopperConfig(i))}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_resolvePossibleFunction(t){return g(t,[this._element])}_getPopperConfig(t){const e={placement:t,modifiers:[{name:"flip",options:{fallbackPlacements:this._config.fallbackPlacements}},{name:"offset",options:{offset:this._getOffset()}},{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"arrow",options:{element:`.${this.constructor.NAME}-arrow`}},{name:"preSetPlacement",enabled:!0,phase:"beforeMain",fn:t=>{this._getTipElement().setAttribute("data-popper-placement",t.state.placement)}}]};return{...e,...g(this._config.popperConfig,[e])}}_setListeners(){const t=this._config.trigger.split(" ");for(const e of t)if("click"===e)N.on(this._element,this.constructor.eventName("click"),this._config.selector,(t=>{this._initializeOnDelegatedTarget(t).toggle()}));else if("manual"!==e){const t=e===ss?this.constructor.eventName("mouseenter"):this.constructor.eventName("focusin"),i=e===ss?this.constructor.eventName("mouseleave"):this.constructor.eventName("focusout");N.on(this._element,t,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusin"===t.type?os:ss]=!0,e._enter()})),N.on(this._element,i,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusout"===t.type?os:ss]=e._element.contains(t.relatedTarget),e._leave()}))}this._hideModalHandler=()=>{this._element&&this.hide()},N.on(this._element.closest(is),ns,this._hideModalHandler)}_fixTitle(){const t=this._element.getAttribute("title");t&&(this._element.getAttribute("aria-label")||this._element.textContent.trim()||this._element.setAttribute("aria-label",t),this._element.setAttribute("data-bs-original-title",t),this._element.removeAttribute("title"))}_enter(){this._isShown()||this._isHovered?this._isHovered=!0:(this._isHovered=!0,this._setTimeout((()=>{this._isHovered&&this.show()}),this._config.delay.show))}_leave(){this._isWithActiveTrigger()||(this._isHovered=!1,this._setTimeout((()=>{this._isHovered||this.hide()}),this._config.delay.hide))}_setTimeout(t,e){clearTimeout(this._timeout),this._timeout=setTimeout(t,e)}_isWithActiveTrigger(){return Object.values(this._activeTrigger).includes(!0)}_getConfig(t){const e=F.getDataAttributes(this._element);for(const t of Object.keys(e))Zn.has(t)&&delete e[t];return t={...e,..."object"==typeof t&&t?t:{}},t=this._mergeConfigObj(t),t=this._configAfterMerge(t),this._typeCheckConfig(t),t}_configAfterMerge(t){return t.container=!1===t.container?document.body:r(t.container),"number"==typeof t.delay&&(t.delay={show:t.delay,hide:t.delay}),"number"==typeof t.title&&(t.title=t.title.toString()),"number"==typeof t.content&&(t.content=t.content.toString()),t}_getDelegateConfig(){const t={};for(const[e,i]of Object.entries(this._config))this.constructor.Default[e]!==i&&(t[e]=i);return t.selector=!1,t.trigger="manual",t}_disposePopper(){this._popper&&(this._popper.destroy(),this._popper=null),this.tip&&(this.tip.remove(),this.tip=null)}static jQueryInterface(t){return this.each((function(){const e=cs.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}m(cs);const hs={...cs.Default,content:"",offset:[0,8],placement:"right",template:'',trigger:"click"},ds={...cs.DefaultType,content:"(null|string|element|function)"};class us extends cs{static get Default(){return hs}static get DefaultType(){return ds}static get NAME(){return"popover"}_isWithContent(){return this._getTitle()||this._getContent()}_getContentForTemplate(){return{".popover-header":this._getTitle(),".popover-body":this._getContent()}}_getContent(){return this._resolvePossibleFunction(this._config.content)}static jQueryInterface(t){return this.each((function(){const e=us.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}m(us);const fs=".bs.scrollspy",ps=`activate${fs}`,ms=`click${fs}`,gs=`load${fs}.data-api`,_s="active",bs="[href]",vs=".nav-link",ys=`${vs}, .nav-item > ${vs}, .list-group-item`,ws={offset:null,rootMargin:"0px 0px -25%",smoothScroll:!1,target:null,threshold:[.1,.5,1]},As={offset:"(number|null)",rootMargin:"string",smoothScroll:"boolean",target:"element",threshold:"array"};class Es extends W{constructor(t,e){super(t,e),this._targetLinks=new Map,this._observableSections=new 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e=this._observableSections.get(t.target.hash);if(e){t.preventDefault();const i=this._rootElement||window,n=e.offsetTop-this._element.offsetTop;if(i.scrollTo)return void i.scrollTo({top:n,behavior:"smooth"});i.scrollTop=n}})))}_getNewObserver(){const t={root:this._rootElement,threshold:this._config.threshold,rootMargin:this._config.rootMargin};return new IntersectionObserver((t=>this._observerCallback(t)),t)}_observerCallback(t){const e=t=>this._targetLinks.get(`#${t.target.id}`),i=t=>{this._previousScrollData.visibleEntryTop=t.target.offsetTop,this._process(e(t))},n=(this._rootElement||document.documentElement).scrollTop,s=n>=this._previousScrollData.parentScrollTop;this._previousScrollData.parentScrollTop=n;for(const o of t){if(!o.isIntersecting){this._activeTarget=null,this._clearActiveClass(e(o));continue}const t=o.target.offsetTop>=this._previousScrollData.visibleEntryTop;if(s&&t){if(i(o),!n)return}else s||t||i(o)}}_initializeTargetsAndObservables(){this._targetLinks=new Map,this._observableSections=new Map;const t=z.find(bs,this._config.target);for(const e of t){if(!e.hash||l(e))continue;const t=z.findOne(decodeURI(e.hash),this._element);a(t)&&(this._targetLinks.set(decodeURI(e.hash),e),this._observableSections.set(e.hash,t))}}_process(t){this._activeTarget!==t&&(this._clearActiveClass(this._config.target),this._activeTarget=t,t.classList.add(_s),this._activateParents(t),N.trigger(this._element,ps,{relatedTarget:t}))}_activateParents(t){if(t.classList.contains("dropdown-item"))z.findOne(".dropdown-toggle",t.closest(".dropdown")).classList.add(_s);else for(const e of z.parents(t,".nav, .list-group"))for(const t of z.prev(e,ys))t.classList.add(_s)}_clearActiveClass(t){t.classList.remove(_s);const e=z.find(`${bs}.${_s}`,t);for(const t of e)t.classList.remove(_s)}static jQueryInterface(t){return this.each((function(){const e=Es.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}))}}N.on(window,gs,(()=>{for(const t of z.find('[data-bs-spy="scroll"]'))Es.getOrCreateInstance(t)})),m(Es);const Ts=".bs.tab",Cs=`hide${Ts}`,Os=`hidden${Ts}`,xs=`show${Ts}`,ks=`shown${Ts}`,Ls=`click${Ts}`,Ss=`keydown${Ts}`,Ds=`load${Ts}`,$s="ArrowLeft",Is="ArrowRight",Ns="ArrowUp",Ps="ArrowDown",Ms="Home",js="End",Fs="active",Hs="fade",Ws="show",Bs=":not(.dropdown-toggle)",zs='[data-bs-toggle="tab"], [data-bs-toggle="pill"], [data-bs-toggle="list"]',Rs=`.nav-link${Bs}, .list-group-item${Bs}, [role="tab"]${Bs}, ${zs}`,qs=`.${Fs}[data-bs-toggle="tab"], .${Fs}[data-bs-toggle="pill"], .${Fs}[data-bs-toggle="list"]`;class Vs extends W{constructor(t){super(t),this._parent=this._element.closest('.list-group, .nav, [role="tablist"]'),this._parent&&(this._setInitialAttributes(this._parent,this._getChildren()),N.on(this._element,Ss,(t=>this._keydown(t))))}static get NAME(){return"tab"}show(){const t=this._element;if(this._elemIsActive(t))return;const 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t=e.state,n=e.name,r=t.rects.reference,o=t.rects.popper,i=t.modifiersData.preventOverflow,a=J(t,{elementContext:"reference"}),s=J(t,{altBoundary:!0}),f=ve(a,r),c=ve(s,o,i),p=ye(f),u=ye(c);t.modifiersData[n]={referenceClippingOffsets:f,popperEscapeOffsets:c,isReferenceHidden:p,hasPopperEscaped:u},t.attributes.popper=Object.assign({},t.attributes.popper,{"data-popper-reference-hidden":p,"data-popper-escaped":u})}},be=Z({defaultModifiers:[ee,te,oe,ie]}),xe=[ee,te,oe,ie,ae,le,he,me,ge],we=Z({defaultModifiers:xe});e.applyStyles=ie,e.arrow=me,e.computeStyles=oe,e.createPopper=we,e.createPopperLite=be,e.defaultModifiers=xe,e.detectOverflow=J,e.eventListeners=ee,e.flip=le,e.hide=ge,e.offset=ae,e.popperGenerator=Z,e.popperOffsets=te,e.preventOverflow=he,Object.defineProperty(e,"__esModule",{value:!0})})); + diff --git a/site_libs 27/quarto-html/quarto-syntax-highlighting-e26003cea8cd680ca0c55a263523d882.css b/site_libs 27/quarto-html/quarto-syntax-highlighting-e26003cea8cd680ca0c55a263523d882.css new file mode 100644 index 0000000..74e04ef --- /dev/null +++ b/site_libs 27/quarto-html/quarto-syntax-highlighting-e26003cea8cd680ca0c55a263523d882.css @@ -0,0 +1,205 @@ +/* quarto syntax highlight colors */ +:root { + --quarto-hl-ot-color: #003B4F; + --quarto-hl-at-color: #657422; + --quarto-hl-ss-color: #20794D; + --quarto-hl-an-color: #5E5E5E; + --quarto-hl-fu-color: #4758AB; + --quarto-hl-st-color: #20794D; + --quarto-hl-cf-color: #003B4F; + --quarto-hl-op-color: #5E5E5E; + --quarto-hl-er-color: #AD0000; + --quarto-hl-bn-color: #AD0000; + --quarto-hl-al-color: #AD0000; + --quarto-hl-va-color: #111111; + --quarto-hl-bu-color: inherit; + --quarto-hl-ex-color: inherit; + --quarto-hl-pp-color: #AD0000; + --quarto-hl-in-color: #5E5E5E; + --quarto-hl-vs-color: #20794D; + --quarto-hl-wa-color: #5E5E5E; + --quarto-hl-do-color: #5E5E5E; + --quarto-hl-im-color: #00769E; + --quarto-hl-ch-color: #20794D; + --quarto-hl-dt-color: #AD0000; + --quarto-hl-fl-color: #AD0000; + --quarto-hl-co-color: #5E5E5E; + --quarto-hl-cv-color: #5E5E5E; + --quarto-hl-cn-color: #8f5902; + --quarto-hl-sc-color: #5E5E5E; + --quarto-hl-dv-color: #AD0000; + --quarto-hl-kw-color: #003B4F; +} + +/* other quarto variables */ +:root { + --quarto-font-monospace: SFMono-Regular, Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace; +} + +pre > code.sourceCode > span { + color: #003B4F; +} + +code span { + color: #003B4F; +} + +code.sourceCode > span { + color: #003B4F; +} + +div.sourceCode, +div.sourceCode pre.sourceCode { + color: #003B4F; +} + +code span.ot { + color: #003B4F; + font-style: inherit; +} + +code span.at { + color: #657422; + font-style: inherit; +} + +code span.ss { + color: #20794D; + font-style: inherit; +} + +code span.an { + color: #5E5E5E; + font-style: inherit; +} + +code span.fu { + color: #4758AB; + font-style: inherit; +} + +code span.st { + color: #20794D; + font-style: inherit; +} + +code span.cf { + color: #003B4F; + font-weight: bold; + font-style: inherit; +} + +code span.op { + color: #5E5E5E; + font-style: inherit; +} + +code span.er { + color: #AD0000; + font-style: inherit; +} + +code span.bn { + color: #AD0000; + font-style: inherit; +} + +code span.al { + color: #AD0000; + font-style: inherit; +} + +code span.va { + color: #111111; + font-style: inherit; +} + +code span.bu { + font-style: inherit; +} + +code span.ex { + font-style: inherit; +} + +code span.pp { + color: #AD0000; + font-style: inherit; +} + +code span.in { + color: #5E5E5E; + font-style: inherit; +} + +code span.vs { + color: #20794D; + font-style: inherit; +} + +code span.wa { + color: #5E5E5E; + font-style: italic; +} + +code span.do { + color: #5E5E5E; + font-style: italic; +} + +code span.im { + color: #00769E; + font-style: inherit; +} + +code span.ch { + color: #20794D; + font-style: inherit; +} + +code span.dt { + color: #AD0000; + font-style: inherit; +} + +code span.fl { + color: #AD0000; + font-style: inherit; +} + +code span.co { + color: #5E5E5E; + font-style: inherit; +} + +code span.cv { + color: #5E5E5E; + font-style: italic; +} + +code span.cn { + color: #8f5902; + font-style: inherit; +} + +code span.sc { + color: #5E5E5E; + font-style: inherit; +} + +code span.dv { + color: #AD0000; + font-style: inherit; +} + +code span.kw { + color: #003B4F; + font-weight: bold; + font-style: inherit; +} + +.prevent-inlining { + content: " { + // Find any conflicting margin elements and add margins to the + // top to prevent overlap + const marginChildren = window.document.querySelectorAll( + ".column-margin.column-container > *, .margin-caption, .aside" + ); + + let lastBottom = 0; + for (const marginChild of marginChildren) { + if (marginChild.offsetParent !== null) { + // clear the top margin so we recompute it + marginChild.style.marginTop = null; + const top = marginChild.getBoundingClientRect().top + window.scrollY; + if (top < lastBottom) { + const marginChildStyle = window.getComputedStyle(marginChild); + const marginBottom = parseFloat(marginChildStyle["marginBottom"]); + const margin = lastBottom - top + marginBottom; + marginChild.style.marginTop = `${margin}px`; + } + const styles = window.getComputedStyle(marginChild); + const marginTop = parseFloat(styles["marginTop"]); + lastBottom = top + marginChild.getBoundingClientRect().height + marginTop; + } + } +}; + +window.document.addEventListener("DOMContentLoaded", function (_event) { + // Recompute the position of margin elements anytime the body size changes + if (window.ResizeObserver) { + const resizeObserver = new window.ResizeObserver( + throttle(() => { + layoutMarginEls(); + if ( + window.document.body.getBoundingClientRect().width < 990 && + isReaderMode() + ) { + quartoToggleReader(); + } + }, 50) + ); + resizeObserver.observe(window.document.body); + } + + const tocEl = window.document.querySelector('nav.toc-active[role="doc-toc"]'); + const sidebarEl = window.document.getElementById("quarto-sidebar"); + const leftTocEl = window.document.getElementById("quarto-sidebar-toc-left"); + const marginSidebarEl = window.document.getElementById( + "quarto-margin-sidebar" + ); + // function to determine whether the element has a previous sibling that is active + const prevSiblingIsActiveLink = (el) => { + const sibling = el.previousElementSibling; + if (sibling && sibling.tagName === "A") { + return sibling.classList.contains("active"); + } else { + return false; + } + }; + + // fire slideEnter for bootstrap tab activations (for htmlwidget resize behavior) + function fireSlideEnter(e) { + const event = window.document.createEvent("Event"); + event.initEvent("slideenter", true, true); + window.document.dispatchEvent(event); + } + const tabs = window.document.querySelectorAll('a[data-bs-toggle="tab"]'); + tabs.forEach((tab) => { + tab.addEventListener("shown.bs.tab", fireSlideEnter); + }); + + // fire slideEnter for tabby tab activations (for htmlwidget resize behavior) + document.addEventListener("tabby", fireSlideEnter, false); + + // Track scrolling and mark TOC links as active + // get table of contents and sidebar (bail if we don't have at least one) + const tocLinks = tocEl + ? [...tocEl.querySelectorAll("a[data-scroll-target]")] + : []; + const makeActive = (link) => tocLinks[link].classList.add("active"); + const removeActive = (link) => tocLinks[link].classList.remove("active"); + const removeAllActive = () => + [...Array(tocLinks.length).keys()].forEach((link) => removeActive(link)); + + // activate the anchor for a section associated with this TOC entry + tocLinks.forEach((link) => { + link.addEventListener("click", () => { + if (link.href.indexOf("#") !== -1) { + const anchor = link.href.split("#")[1]; + const heading = window.document.querySelector( + `[data-anchor-id="${anchor}"]` + ); + if (heading) { + // Add the class + heading.classList.add("reveal-anchorjs-link"); + + // function to show the anchor + const handleMouseout = () => { + heading.classList.remove("reveal-anchorjs-link"); + heading.removeEventListener("mouseout", handleMouseout); + }; + + // add a function to clear the anchor when the user mouses out of it + heading.addEventListener("mouseout", handleMouseout); + } + } + }); + }); + + const sections = tocLinks.map((link) => { + const target = link.getAttribute("data-scroll-target"); + if (target.startsWith("#")) { + return window.document.getElementById(decodeURI(`${target.slice(1)}`)); + } else { + return window.document.querySelector(decodeURI(`${target}`)); + } + }); + + const sectionMargin = 200; + let currentActive = 0; + // track whether we've initialized state the first time + let init = false; + + const updateActiveLink = () => { + // The index from bottom to top (e.g. reversed list) + let sectionIndex = -1; + if ( + window.innerHeight + window.pageYOffset >= + window.document.body.offsetHeight + ) { + // This is the no-scroll case where last section should be the active one + sectionIndex = 0; + } else { + // This finds the last section visible on screen that should be made active + sectionIndex = [...sections].reverse().findIndex((section) => { + if (section) { + return window.pageYOffset >= section.offsetTop - sectionMargin; + } else { + return false; + } + }); + } + if (sectionIndex > -1) { + const current = sections.length - sectionIndex - 1; + if (current !== currentActive) { + removeAllActive(); + currentActive = current; + makeActive(current); + if (init) { + window.dispatchEvent(sectionChanged); + } + init = true; + } + } + }; + + const inHiddenRegion = (top, bottom, hiddenRegions) => { + for (const region of hiddenRegions) { + if (top <= region.bottom && bottom >= region.top) { + return true; + } + } + return false; + }; + + const categorySelector = "header.quarto-title-block .quarto-category"; + const activateCategories = (href) => { + // Find any categories + // Surround them with a link pointing back to: + // #category=Authoring + try { + const categoryEls = window.document.querySelectorAll(categorySelector); + for (const categoryEl of categoryEls) { + const categoryText = categoryEl.textContent; + if (categoryText) { + const link = `${href}#category=${encodeURIComponent(categoryText)}`; + const linkEl = window.document.createElement("a"); + linkEl.setAttribute("href", link); + for (const child of categoryEl.childNodes) { + linkEl.append(child); + } + categoryEl.appendChild(linkEl); + } + } + } catch { + // Ignore errors + } + }; + function hasTitleCategories() { + return window.document.querySelector(categorySelector) !== null; + } + + function offsetRelativeUrl(url) { + const offset = getMeta("quarto:offset"); + return offset ? offset + url : url; + } + + function offsetAbsoluteUrl(url) { + const offset = getMeta("quarto:offset"); + const baseUrl = new URL(offset, window.location); + + const projRelativeUrl = url.replace(baseUrl, ""); + if (projRelativeUrl.startsWith("/")) { + return projRelativeUrl; + } else { + return "/" + projRelativeUrl; + } + } + + // read a meta tag value + function getMeta(metaName) { + const metas = window.document.getElementsByTagName("meta"); + for (let i = 0; i < metas.length; i++) { + if (metas[i].getAttribute("name") === metaName) { + return metas[i].getAttribute("content"); + } + } + return ""; + } + + async function findAndActivateCategories() { + // Categories search with listing only use path without query + const currentPagePath = offsetAbsoluteUrl( + window.location.origin + window.location.pathname + ); + const response = await fetch(offsetRelativeUrl("listings.json")); + if (response.status == 200) { + return response.json().then(function (listingPaths) { + const listingHrefs = []; + for (const listingPath of listingPaths) { + const pathWithoutLeadingSlash = listingPath.listing.substring(1); + for (const item of listingPath.items) { + if ( + item === currentPagePath || + item === currentPagePath + "index.html" + ) { + // Resolve this path against the offset to be sure + // we already are using the correct path to the listing + // (this adjusts the listing urls to be rooted against + // whatever root the page is actually running against) + const relative = offsetRelativeUrl(pathWithoutLeadingSlash); + const baseUrl = window.location; + const resolvedPath = new URL(relative, baseUrl); + listingHrefs.push(resolvedPath.pathname); + break; + } + } + } + + // Look up the tree for a nearby linting and use that if we find one + const nearestListing = findNearestParentListing( + offsetAbsoluteUrl(window.location.pathname), + listingHrefs + ); + if (nearestListing) { + activateCategories(nearestListing); + } else { + // See if the referrer is a listing page for this item + const referredRelativePath = offsetAbsoluteUrl(document.referrer); + const referrerListing = listingHrefs.find((listingHref) => { + const isListingReferrer = + listingHref === referredRelativePath || + listingHref === referredRelativePath + "index.html"; + return isListingReferrer; + }); + + if (referrerListing) { + // Try to use the referrer if possible + activateCategories(referrerListing); + } else if (listingHrefs.length > 0) { + // Otherwise, just fall back to the first listing + activateCategories(listingHrefs[0]); + } + } + }); + } + } + if (hasTitleCategories()) { + findAndActivateCategories(); + } + + const findNearestParentListing = (href, listingHrefs) => { + if (!href || !listingHrefs) { + return undefined; + } + // Look up the tree for a nearby linting and use that if we find one + const relativeParts = href.substring(1).split("/"); + while (relativeParts.length > 0) { + const path = relativeParts.join("/"); + for (const listingHref of listingHrefs) { + if (listingHref.startsWith(path)) { + return listingHref; + } + } + relativeParts.pop(); + } + + return undefined; + }; + + const manageSidebarVisiblity = (el, placeholderDescriptor) => { + let isVisible = true; + let elRect; + + return (hiddenRegions) => { + if (el === null) { + return; + } + + // Find the last element of the TOC + const lastChildEl = el.lastElementChild; + + if (lastChildEl) { + // Converts the sidebar to a menu + const convertToMenu = () => { + for (const child of el.children) { + child.style.opacity = 0; + child.style.overflow = "hidden"; + child.style.pointerEvents = "none"; + } + + nexttick(() => { + const toggleContainer = window.document.createElement("div"); + toggleContainer.style.width = "100%"; + toggleContainer.classList.add("zindex-over-content"); + toggleContainer.classList.add("quarto-sidebar-toggle"); + toggleContainer.classList.add("headroom-target"); // Marks this to be managed by headeroom + toggleContainer.id = placeholderDescriptor.id; + toggleContainer.style.position = "fixed"; + + const toggleIcon = window.document.createElement("i"); + toggleIcon.classList.add("quarto-sidebar-toggle-icon"); + toggleIcon.classList.add("bi"); + toggleIcon.classList.add("bi-caret-down-fill"); + + const toggleTitle = window.document.createElement("div"); + const titleEl = window.document.body.querySelector( + placeholderDescriptor.titleSelector + ); + if (titleEl) { + toggleTitle.append( + titleEl.textContent || titleEl.innerText, + toggleIcon + ); + } + toggleTitle.classList.add("zindex-over-content"); + toggleTitle.classList.add("quarto-sidebar-toggle-title"); + toggleContainer.append(toggleTitle); + + const toggleContents = window.document.createElement("div"); + toggleContents.classList = el.classList; + toggleContents.classList.add("zindex-over-content"); + toggleContents.classList.add("quarto-sidebar-toggle-contents"); + for (const child of el.children) { + if (child.id === "toc-title") { + continue; + } + + const clone = child.cloneNode(true); + clone.style.opacity = 1; + clone.style.pointerEvents = null; + clone.style.display = null; + toggleContents.append(clone); + } + toggleContents.style.height = "0px"; + const positionToggle = () => { + // position the element (top left of parent, same width as parent) + if (!elRect) { + elRect = el.getBoundingClientRect(); + } + toggleContainer.style.left = `${elRect.left}px`; + toggleContainer.style.top = `${elRect.top}px`; + toggleContainer.style.width = `${elRect.width}px`; + }; + positionToggle(); + + toggleContainer.append(toggleContents); + el.parentElement.prepend(toggleContainer); + + // Process clicks + let tocShowing = false; + // Allow the caller to control whether this is dismissed + // when it is clicked (e.g. sidebar navigation supports + // opening and closing the nav tree, so don't dismiss on click) + const clickEl = placeholderDescriptor.dismissOnClick + ? toggleContainer + : toggleTitle; + + const closeToggle = () => { + if (tocShowing) { + toggleContainer.classList.remove("expanded"); + toggleContents.style.height = "0px"; + tocShowing = false; + } + }; + + // Get rid of any expanded toggle if the user scrolls + window.document.addEventListener( + "scroll", + throttle(() => { + closeToggle(); + }, 50) + ); + + // Handle positioning of the toggle + window.addEventListener( + "resize", + throttle(() => { + elRect = undefined; + positionToggle(); + }, 50) + ); + + window.addEventListener("quarto-hrChanged", () => { + elRect = undefined; + }); + + // Process the click + clickEl.onclick = () => { + if (!tocShowing) { + toggleContainer.classList.add("expanded"); + toggleContents.style.height = null; + tocShowing = true; + } else { + closeToggle(); + } + }; + }); + }; + + // Converts a sidebar from a menu back to a sidebar + const convertToSidebar = () => { + for (const child of el.children) { + child.style.opacity = 1; + child.style.overflow = null; + child.style.pointerEvents = null; + } + + const placeholderEl = window.document.getElementById( + placeholderDescriptor.id + ); + if (placeholderEl) { + placeholderEl.remove(); + } + + el.classList.remove("rollup"); + }; + + if (isReaderMode()) { + convertToMenu(); + isVisible = false; + } else { + // Find the top and bottom o the element that is being managed + const elTop = el.offsetTop; + const elBottom = + elTop + lastChildEl.offsetTop + lastChildEl.offsetHeight; + + if (!isVisible) { + // If the element is current not visible reveal if there are + // no conflicts with overlay regions + if (!inHiddenRegion(elTop, elBottom, hiddenRegions)) { + convertToSidebar(); + isVisible = true; + } + } else { + // If the element is visible, hide it if it conflicts with overlay regions + // and insert a placeholder toggle (or if we're in reader mode) + if (inHiddenRegion(elTop, elBottom, hiddenRegions)) { + convertToMenu(); + isVisible = false; + } + } + } + } + }; + }; + + const tabEls = document.querySelectorAll('a[data-bs-toggle="tab"]'); + for (const tabEl of tabEls) { + const id = tabEl.getAttribute("data-bs-target"); + if (id) { + const columnEl = document.querySelector( + `${id} .column-margin, .tabset-margin-content` + ); + if (columnEl) + tabEl.addEventListener("shown.bs.tab", function (event) { + const el = event.srcElement; + if (el) { + const visibleCls = `${el.id}-margin-content`; + // walk up until we find a parent tabset + let panelTabsetEl = el.parentElement; + while (panelTabsetEl) { + if (panelTabsetEl.classList.contains("panel-tabset")) { + break; + } + panelTabsetEl = panelTabsetEl.parentElement; + } + + if (panelTabsetEl) { + const prevSib = panelTabsetEl.previousElementSibling; + if ( + prevSib && + prevSib.classList.contains("tabset-margin-container") + ) { + const childNodes = prevSib.querySelectorAll( + ".tabset-margin-content" + ); + for (const childEl of childNodes) { + if (childEl.classList.contains(visibleCls)) { + childEl.classList.remove("collapse"); + } else { + childEl.classList.add("collapse"); + } + } + } + } + } + + layoutMarginEls(); + }); + } + } + + // Manage the visibility of the toc and the sidebar + const marginScrollVisibility = manageSidebarVisiblity(marginSidebarEl, { + id: "quarto-toc-toggle", + titleSelector: "#toc-title", + dismissOnClick: true, + }); + const sidebarScrollVisiblity = manageSidebarVisiblity(sidebarEl, { + id: "quarto-sidebarnav-toggle", + titleSelector: ".title", + dismissOnClick: false, + }); + let tocLeftScrollVisibility; + if (leftTocEl) { + tocLeftScrollVisibility = manageSidebarVisiblity(leftTocEl, { + id: "quarto-lefttoc-toggle", + titleSelector: "#toc-title", + dismissOnClick: true, + }); + } + + // Find the first element that uses formatting in special columns + const conflictingEls = window.document.body.querySelectorAll( + '[class^="column-"], [class*=" column-"], aside, [class*="margin-caption"], [class*=" margin-caption"], [class*="margin-ref"], [class*=" margin-ref"]' + ); + + // Filter all the possibly conflicting elements into ones + // the do conflict on the left or ride side + const arrConflictingEls = Array.from(conflictingEls); + const leftSideConflictEls = arrConflictingEls.filter((el) => { + if (el.tagName === "ASIDE") { + return false; + } + return Array.from(el.classList).find((className) => { + return ( + className !== "column-body" && + className.startsWith("column-") && + !className.endsWith("right") && + !className.endsWith("container") && + className !== "column-margin" + ); + }); + }); + const rightSideConflictEls = arrConflictingEls.filter((el) => { + if (el.tagName === "ASIDE") { + return true; + } + + const hasMarginCaption = Array.from(el.classList).find((className) => { + return className == "margin-caption"; + }); + if (hasMarginCaption) { + return true; + } + + return Array.from(el.classList).find((className) => { + return ( + className !== "column-body" && + !className.endsWith("container") && + className.startsWith("column-") && + !className.endsWith("left") + ); + }); + }); + + const kOverlapPaddingSize = 10; + function toRegions(els) { + return els.map((el) => { + const boundRect = el.getBoundingClientRect(); + const top = + boundRect.top + + document.documentElement.scrollTop - + kOverlapPaddingSize; + return { + top, + bottom: top + el.scrollHeight + 2 * kOverlapPaddingSize, + }; + }); + } + + let hasObserved = false; + const visibleItemObserver = (els) => { + let visibleElements = [...els]; + const intersectionObserver = new IntersectionObserver( + (entries, _observer) => { + entries.forEach((entry) => { + if (entry.isIntersecting) { + if (visibleElements.indexOf(entry.target) === -1) { + visibleElements.push(entry.target); + } + } else { + visibleElements = visibleElements.filter((visibleEntry) => { + return visibleEntry !== entry; + }); + } + }); + + if (!hasObserved) { + hideOverlappedSidebars(); + } + hasObserved = true; + }, + {} + ); + els.forEach((el) => { + intersectionObserver.observe(el); + }); + + return { + getVisibleEntries: () => { + return visibleElements; + }, + }; + }; + + const rightElementObserver = visibleItemObserver(rightSideConflictEls); + const leftElementObserver = visibleItemObserver(leftSideConflictEls); + + const hideOverlappedSidebars = () => { + marginScrollVisibility(toRegions(rightElementObserver.getVisibleEntries())); + sidebarScrollVisiblity(toRegions(leftElementObserver.getVisibleEntries())); + if (tocLeftScrollVisibility) { + tocLeftScrollVisibility( + toRegions(leftElementObserver.getVisibleEntries()) + ); + } + }; + + window.quartoToggleReader = () => { + // Applies a slow class (or removes it) + // to update the transition speed + const slowTransition = (slow) => { + const manageTransition = (id, slow) => { + const el = document.getElementById(id); + if (el) { + if (slow) { + el.classList.add("slow"); + } else { + el.classList.remove("slow"); + } + } + }; + + manageTransition("TOC", slow); + manageTransition("quarto-sidebar", slow); + }; + const readerMode = !isReaderMode(); + setReaderModeValue(readerMode); + + // If we're entering reader mode, slow the transition + if (readerMode) { + slowTransition(readerMode); + } + highlightReaderToggle(readerMode); + hideOverlappedSidebars(); + + // If we're exiting reader mode, restore the non-slow transition + if (!readerMode) { + slowTransition(!readerMode); + } + }; + + const highlightReaderToggle = (readerMode) => { + const els = document.querySelectorAll(".quarto-reader-toggle"); + if (els) { + els.forEach((el) => { + if (readerMode) { + el.classList.add("reader"); + } else { + el.classList.remove("reader"); + } + }); + } + }; + + const setReaderModeValue = (val) => { + if (window.location.protocol !== "file:") { + window.localStorage.setItem("quarto-reader-mode", val); + } else { + localReaderMode = val; + } + }; + + const isReaderMode = () => { + if (window.location.protocol !== "file:") { + return window.localStorage.getItem("quarto-reader-mode") === "true"; + } else { + return localReaderMode; + } + }; + let localReaderMode = null; + + const tocOpenDepthStr = tocEl?.getAttribute("data-toc-expanded"); + const tocOpenDepth = tocOpenDepthStr ? Number(tocOpenDepthStr) : 1; + + // Walk the TOC and collapse/expand nodes + // Nodes are expanded if: + // - they are top level + // - they have children that are 'active' links + // - they are directly below an link that is 'active' + const walk = (el, depth) => { + // Tick depth when we enter a UL + if (el.tagName === "UL") { + depth = depth + 1; + } + + // It this is active link + let isActiveNode = false; + if (el.tagName === "A" && el.classList.contains("active")) { + isActiveNode = true; + } + + // See if there is an active child to this element + let hasActiveChild = false; + for (child of el.children) { + hasActiveChild = walk(child, depth) || hasActiveChild; + } + + // Process the collapse state if this is an UL + if (el.tagName === "UL") { + if (tocOpenDepth === -1 && depth > 1) { + // toc-expand: false + el.classList.add("collapse"); + } else if ( + depth <= tocOpenDepth || + hasActiveChild || + prevSiblingIsActiveLink(el) + ) { + el.classList.remove("collapse"); + } else { + el.classList.add("collapse"); + } + + // untick depth when we leave a UL + depth = depth - 1; + } + return hasActiveChild || isActiveNode; + }; + + // walk the TOC and expand / collapse any items that should be shown + if (tocEl) { + updateActiveLink(); + walk(tocEl, 0); + } + + // Throttle the scroll event and walk peridiocally + window.document.addEventListener( + "scroll", + throttle(() => { + if (tocEl) { + updateActiveLink(); + walk(tocEl, 0); + } + if (!isReaderMode()) { + hideOverlappedSidebars(); + } + }, 5) + ); + window.addEventListener( + "resize", + throttle(() => { + if (tocEl) { + updateActiveLink(); + walk(tocEl, 0); + } + if (!isReaderMode()) { + hideOverlappedSidebars(); + } + }, 10) + ); + hideOverlappedSidebars(); + highlightReaderToggle(isReaderMode()); +}); + +// grouped tabsets +window.addEventListener("pageshow", (_event) => { + function getTabSettings() { + const data = localStorage.getItem("quarto-persistent-tabsets-data"); + if (!data) { + localStorage.setItem("quarto-persistent-tabsets-data", "{}"); + return {}; + } + if (data) { + return JSON.parse(data); + } + } + + function setTabSettings(data) { + localStorage.setItem( + "quarto-persistent-tabsets-data", + JSON.stringify(data) + ); + } + + function setTabState(groupName, groupValue) { + const data = getTabSettings(); + data[groupName] = groupValue; + setTabSettings(data); + } + + function toggleTab(tab, active) { + const tabPanelId = tab.getAttribute("aria-controls"); + const tabPanel = document.getElementById(tabPanelId); + if (active) { + tab.classList.add("active"); + tabPanel.classList.add("active"); + } else { + tab.classList.remove("active"); + tabPanel.classList.remove("active"); + } + } + + function toggleAll(selectedGroup, selectorsToSync) { + for (const [thisGroup, tabs] of Object.entries(selectorsToSync)) { + const active = selectedGroup === thisGroup; + for (const tab of tabs) { + toggleTab(tab, active); + } + } + } + + function findSelectorsToSyncByLanguage() { + const result = {}; + const tabs = Array.from( + document.querySelectorAll(`div[data-group] a[id^='tabset-']`) + ); + for (const item of tabs) { + const div = item.parentElement.parentElement.parentElement; + const group = div.getAttribute("data-group"); + if (!result[group]) { + result[group] = {}; + } + const selectorsToSync = result[group]; + const value = item.innerHTML; + if (!selectorsToSync[value]) { + selectorsToSync[value] = []; + } + selectorsToSync[value].push(item); + } + return result; + } + + function setupSelectorSync() { + const selectorsToSync = findSelectorsToSyncByLanguage(); + Object.entries(selectorsToSync).forEach(([group, tabSetsByValue]) => { + Object.entries(tabSetsByValue).forEach(([value, items]) => { + items.forEach((item) => { + item.addEventListener("click", (_event) => { + setTabState(group, value); + toggleAll(value, selectorsToSync[group]); + }); + }); + }); + }); + return selectorsToSync; + } + + const selectorsToSync = setupSelectorSync(); + for (const [group, selectedName] of Object.entries(getTabSettings())) { + const selectors = selectorsToSync[group]; + // it's possible that stale state gives us empty selections, so we explicitly check here. + if (selectors) { + toggleAll(selectedName, selectors); + } + } +}); + +function throttle(func, wait) { + let waiting = false; + return function () { + if (!waiting) { + func.apply(this, arguments); + waiting = true; + setTimeout(function () { + waiting = false; + }, wait); + } + }; +} + +function nexttick(func) { + return setTimeout(func, 0); +} diff --git a/site_libs 27/quarto-html/tippy.css b/site_libs 27/quarto-html/tippy.css new file mode 100644 index 0000000..e6ae635 --- /dev/null +++ b/site_libs 27/quarto-html/tippy.css @@ -0,0 +1 @@ +.tippy-box[data-animation=fade][data-state=hidden]{opacity:0}[data-tippy-root]{max-width:calc(100vw - 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s.cutsTheMustard&&!this.initialised&&(this.addClass("initial"),this.initialised=!0,setTimeout(function(t){t.scrollTracker=n(t.scroller,{offset:t.offset,tolerance:t.tolerance},t.update.bind(t))},100,this)),this},destroy:function(){this.initialised=!1,Object.keys(this.classes).forEach(this.removeClass,this),this.scrollTracker.destroy()},unpin:function(){!this.hasClass("pinned")&&this.hasClass("unpinned")||(this.addClass("unpinned"),this.removeClass("pinned"),this.onUnpin&&this.onUnpin.call(this))},pin:function(){this.hasClass("unpinned")&&(this.addClass("pinned"),this.removeClass("unpinned"),this.onPin&&this.onPin.call(this))},freeze:function(){this.frozen=!0,this.addClass("frozen")},unfreeze:function(){this.frozen=!1,this.removeClass("frozen")},top:function(){this.hasClass("top")||(this.addClass("top"),this.removeClass("notTop"),this.onTop&&this.onTop.call(this))},notTop:function(){this.hasClass("notTop")||(this.addClass("notTop"),this.removeClass("top"),this.onNotTop&&this.onNotTop.call(this))},bottom:function(){this.hasClass("bottom")||(this.addClass("bottom"),this.removeClass("notBottom"),this.onBottom&&this.onBottom.call(this))},notBottom:function(){this.hasClass("notBottom")||(this.addClass("notBottom"),this.removeClass("bottom"),this.onNotBottom&&this.onNotBottom.call(this))},shouldUnpin:function(t){return"down"===t.direction&&!t.top&&t.toleranceExceeded},shouldPin:function(t){return"up"===t.direction&&t.toleranceExceeded||t.top},addClass:function(t){this.elem.classList.add.apply(this.elem.classList,this.classes[t].split(" "))},removeClass:function(t){this.elem.classList.remove.apply(this.elem.classList,this.classes[t].split(" "))},hasClass:function(t){return this.classes[t].split(" ").every(function(t){return this.classList.contains(t)},this.elem)},update:function(t){t.isOutOfBounds||!0!==this.frozen&&(t.top?this.top():this.notTop(),t.bottom?this.bottom():this.notBottom(),this.shouldUnpin(t)?this.unpin():this.shouldPin(t)&&this.pin())}},s.options={tolerance:{up:0,down:0},offset:0,scroller:t()?window:null,classes:{frozen:"headroom--frozen",pinned:"headroom--pinned",unpinned:"headroom--unpinned",top:"headroom--top",notTop:"headroom--not-top",bottom:"headroom--bottom",notBottom:"headroom--not-bottom",initial:"headroom"}},s.cutsTheMustard=!!(t()&&function(){}.bind&&"classList"in document.documentElement&&Object.assign&&Object.keys&&requestAnimationFrame),s}); diff --git a/site_libs 27/quarto-nav/quarto-nav.js b/site_libs 27/quarto-nav/quarto-nav.js new file mode 100644 index 0000000..38cc430 --- /dev/null +++ b/site_libs 27/quarto-nav/quarto-nav.js @@ -0,0 +1,325 @@ +const headroomChanged = new CustomEvent("quarto-hrChanged", { + detail: {}, + bubbles: true, + cancelable: false, + composed: false, +}); + +const announceDismiss = () => { + const annEl = window.document.getElementById("quarto-announcement"); + if (annEl) { + annEl.remove(); + + const annId = annEl.getAttribute("data-announcement-id"); + window.localStorage.setItem(`quarto-announce-${annId}`, "true"); + } +}; + +const announceRegister = () => { + const annEl = window.document.getElementById("quarto-announcement"); + if (annEl) { + const annId = annEl.getAttribute("data-announcement-id"); + const isDismissed = + window.localStorage.getItem(`quarto-announce-${annId}`) || false; + if (isDismissed) { + announceDismiss(); + return; + } else { + annEl.classList.remove("hidden"); + } + + const actionEl = annEl.querySelector(".quarto-announcement-action"); + if (actionEl) { + actionEl.addEventListener("click", function (e) { + e.preventDefault(); + // Hide the bar immediately + announceDismiss(); + }); + } + } +}; + +window.document.addEventListener("DOMContentLoaded", function () { + let init = false; + + announceRegister(); + + // Manage the back to top button, if one is present. + let lastScrollTop = window.pageYOffset || document.documentElement.scrollTop; + const scrollDownBuffer = 5; + const scrollUpBuffer = 35; + const btn = document.getElementById("quarto-back-to-top"); + const hideBackToTop = () => { + btn.style.display = "none"; + }; + const showBackToTop = () => { + btn.style.display = "inline-block"; + }; + if (btn) { + window.document.addEventListener( + "scroll", + function () { + const currentScrollTop = + window.pageYOffset || document.documentElement.scrollTop; + + // Shows and hides the button 'intelligently' as the user scrolls + if (currentScrollTop - scrollDownBuffer > lastScrollTop) { + hideBackToTop(); + lastScrollTop = currentScrollTop <= 0 ? 0 : currentScrollTop; + } else if (currentScrollTop < lastScrollTop - scrollUpBuffer) { + showBackToTop(); + lastScrollTop = currentScrollTop <= 0 ? 0 : currentScrollTop; + } + + // Show the button at the bottom, hides it at the top + if (currentScrollTop <= 0) { + hideBackToTop(); + } else if ( + window.innerHeight + currentScrollTop >= + document.body.offsetHeight + ) { + showBackToTop(); + } + }, + false + ); + } + + function throttle(func, wait) { + var timeout; + return function () { + const context = this; + const args = arguments; + const later = function () { + clearTimeout(timeout); + timeout = null; + func.apply(context, args); + }; + + if (!timeout) { + timeout = setTimeout(later, wait); + } + }; + } + + function headerOffset() { + // Set an offset if there is are fixed top navbar + const headerEl = window.document.querySelector("header.fixed-top"); + if (headerEl) { + return headerEl.clientHeight; + } else { + return 0; + } + } + + function footerOffset() { + const footerEl = window.document.querySelector("footer.footer"); + if (footerEl) { + return footerEl.clientHeight; + } else { + return 0; + } + } + + function dashboardOffset() { + const dashboardNavEl = window.document.getElementById( + "quarto-dashboard-header" + ); + if (dashboardNavEl !== null) { + return dashboardNavEl.clientHeight; + } else { + return 0; + } + } + + function updateDocumentOffsetWithoutAnimation() { + updateDocumentOffset(false); + } + + function updateDocumentOffset(animated) { + // set body offset + const topOffset = headerOffset(); + const bodyOffset = topOffset + footerOffset() + dashboardOffset(); + const bodyEl = window.document.body; + bodyEl.setAttribute("data-bs-offset", topOffset); + bodyEl.style.paddingTop = topOffset + "px"; + + // deal with sidebar offsets + const sidebars = window.document.querySelectorAll( + ".sidebar, .headroom-target" + ); + sidebars.forEach((sidebar) => { + if (!animated) { + sidebar.classList.add("notransition"); + // Remove the no transition class after the animation has time to complete + setTimeout(function () { + sidebar.classList.remove("notransition"); + }, 201); + } + + if (window.Headroom && sidebar.classList.contains("sidebar-unpinned")) { + sidebar.style.top = "0"; + sidebar.style.maxHeight = "100vh"; + } else { + sidebar.style.top = topOffset + "px"; + sidebar.style.maxHeight = "calc(100vh - " + topOffset + "px)"; + } + }); + + // allow space for footer + const mainContainer = window.document.querySelector(".quarto-container"); + if (mainContainer) { + mainContainer.style.minHeight = "calc(100vh - " + bodyOffset + "px)"; + } + + // link offset + let linkStyle = window.document.querySelector("#quarto-target-style"); + if (!linkStyle) { + linkStyle = window.document.createElement("style"); + linkStyle.setAttribute("id", "quarto-target-style"); + window.document.head.appendChild(linkStyle); + } + while (linkStyle.firstChild) { + linkStyle.removeChild(linkStyle.firstChild); + } + if (topOffset > 0) { + linkStyle.appendChild( + window.document.createTextNode(` + section:target::before { + content: ""; + display: block; + height: ${topOffset}px; + margin: -${topOffset}px 0 0; + }`) + ); + } + if (init) { + window.dispatchEvent(headroomChanged); + } + init = true; + } + + // initialize headroom + var header = window.document.querySelector("#quarto-header"); + if (header && window.Headroom) { + const headroom = new window.Headroom(header, { + tolerance: 5, + onPin: function () { + const sidebars = window.document.querySelectorAll( + ".sidebar, .headroom-target" + ); + sidebars.forEach((sidebar) => { + sidebar.classList.remove("sidebar-unpinned"); + }); + updateDocumentOffset(); + }, + onUnpin: function () { + const sidebars = window.document.querySelectorAll( + ".sidebar, .headroom-target" + ); + sidebars.forEach((sidebar) => { + sidebar.classList.add("sidebar-unpinned"); + }); + updateDocumentOffset(); + }, + }); + headroom.init(); + + let frozen = false; + window.quartoToggleHeadroom = function () { + if (frozen) { + headroom.unfreeze(); + frozen = false; + } else { + headroom.freeze(); + frozen = true; + } + }; + } + + window.addEventListener( + "hashchange", + function (e) { + if ( + getComputedStyle(document.documentElement).scrollBehavior !== "smooth" + ) { + window.scrollTo(0, window.pageYOffset - headerOffset()); + } + }, + false + ); + + // Observe size changed for the header + const headerEl = window.document.querySelector("header.fixed-top"); + if (headerEl && window.ResizeObserver) { + const observer = new window.ResizeObserver(() => { + setTimeout(updateDocumentOffsetWithoutAnimation, 0); + }); + observer.observe(headerEl, { + attributes: true, + childList: true, + characterData: true, + }); + } else { + window.addEventListener( + "resize", + throttle(updateDocumentOffsetWithoutAnimation, 50) + ); + } + setTimeout(updateDocumentOffsetWithoutAnimation, 250); + + // fixup index.html links if we aren't on the filesystem + if (window.location.protocol !== "file:") { + const links = window.document.querySelectorAll("a"); + for (let i = 0; i < links.length; i++) { + if (links[i].href) { + links[i].dataset.originalHref = links[i].href; + links[i].href = links[i].href.replace(/\/index\.html/, "/"); + } + } + + // Fixup any sharing links that require urls + // Append url to any sharing urls + const sharingLinks = window.document.querySelectorAll( + "a.sidebar-tools-main-item, a.quarto-navigation-tool, a.quarto-navbar-tools, a.quarto-navbar-tools-item" + ); + for (let i = 0; i < sharingLinks.length; i++) { + const sharingLink = sharingLinks[i]; + const href = sharingLink.getAttribute("href"); + if (href) { + sharingLink.setAttribute( + "href", + href.replace("|url|", window.location.href) + ); + } + } + + // Scroll the active navigation item into view, if necessary + const navSidebar = window.document.querySelector("nav#quarto-sidebar"); + if (navSidebar) { + // Find the active item + const activeItem = navSidebar.querySelector("li.sidebar-item a.active"); + if (activeItem) { + // Wait for the scroll height and height to resolve by observing size changes on the + // nav element that is scrollable + const resizeObserver = new ResizeObserver((_entries) => { + // The bottom of the element + const elBottom = activeItem.offsetTop; + const viewBottom = navSidebar.scrollTop + navSidebar.clientHeight; + + // The element height and scroll height are the same, then we are still loading + if (viewBottom !== navSidebar.scrollHeight) { + // Determine if the item isn't visible and scroll to it + if (elBottom >= viewBottom) { + navSidebar.scrollTop = elBottom; + } + + // stop observing now since we've completed the scroll + resizeObserver.unobserve(navSidebar); + } + }); + resizeObserver.observe(navSidebar); + } + } + } +}); diff --git a/site_libs 27/quarto-search/autocomplete.umd.js b/site_libs 27/quarto-search/autocomplete.umd.js new file mode 100644 index 0000000..ae0063a --- /dev/null +++ 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0!==arguments[1]?arguments[1]:{},n=t.limit,r=void 0===n?-1:n,i=this.options,o=i.includeMatches,c=i.includeScore,a=i.shouldSort,s=i.sortFn,u=i.ignoreFieldNorm,h=g(e)?g(this._docs[0])?this._searchStringList(e):this._searchObjectList(e):this._searchLogical(e);return fe(h,{ignoreFieldNorm:u}),a&&h.sort(s),y(r)&&r>-1&&(h=h.slice(0,r)),ge(h,this._docs,{includeMatches:o,includeScore:c})}},{key:"_searchStringList",value:function(e){var t=re(e,this.options),n=this._myIndex.records,r=[];return n.forEach((function(e){var n=e.v,i=e.i,o=e.n;if(k(n)){var c=t.searchIn(n),a=c.isMatch,s=c.score,u=c.indices;a&&r.push({item:n,idx:i,matches:[{score:s,value:n,norm:o,indices:u}]})}})),r}},{key:"_searchLogical",value:function(e){var t=this,n=function(e,t){var n=(arguments.length>2&&void 0!==arguments[2]?arguments[2]:{}).auto,r=void 0===n||n,i=function e(n){var i=Object.keys(n),o=ue(n);if(!o&&i.length>1&&!se(n))return e(le(n));if(he(n)){var c=o?n[ce]:i[0],a=o?n[ae]:n[c];if(!g(a))throw new Error(x(c));var s={keyId:j(c),pattern:a};return r&&(s.searcher=re(a,t)),s}var u={children:[],operator:i[0]};return i.forEach((function(t){var r=n[t];v(r)&&r.forEach((function(t){u.children.push(e(t))}))})),u};return se(e)||(e=le(e)),i(e)}(e,this.options),r=function e(n,r,i){if(!n.children){var o=n.keyId,c=n.searcher,a=t._findMatches({key:t._keyStore.get(o),value:t._myIndex.getValueForItemAtKeyId(r,o),searcher:c});return a&&a.length?[{idx:i,item:r,matches:a}]:[]}for(var s=[],u=0,h=n.children.length;u1&&void 0!==arguments[1]?arguments[1]:{},n=t.getFn,r=void 0===n?I.getFn:n,i=t.fieldNormWeight,o=void 0===i?I.fieldNormWeight:i,c=e.keys,a=e.records,s=new $({getFn:r,fieldNormWeight:o});return s.setKeys(c),s.setIndexRecords(a),s},ye.config=I,function(){ne.push.apply(ne,arguments)}(te),ye},"object"==typeof exports&&"undefined"!=typeof module?module.exports=t():"function"==typeof define&&define.amd?define(t):(e="undefined"!=typeof globalThis?globalThis:e||self).Fuse=t(); \ No newline at end of file diff --git a/site_libs 27/quarto-search/quarto-search.js b/site_libs 27/quarto-search/quarto-search.js new file mode 100644 index 0000000..d788a95 --- /dev/null +++ b/site_libs 27/quarto-search/quarto-search.js @@ -0,0 +1,1290 @@ +const kQueryArg = "q"; +const kResultsArg = "show-results"; + +// If items don't provide a URL, then both the navigator and the onSelect +// function aren't called (and therefore, the default implementation is used) +// +// We're using this sentinel URL to signal to those handlers that this +// item is a more item (along with the type) and can be handled appropriately +const kItemTypeMoreHref = "0767FDFD-0422-4E5A-BC8A-3BE11E5BBA05"; + +window.document.addEventListener("DOMContentLoaded", function (_event) { + // Ensure that search is available on this page. If it isn't, + // should return early and not do anything + var searchEl = window.document.getElementById("quarto-search"); + if (!searchEl) return; + + const { autocomplete } = window["@algolia/autocomplete-js"]; + + let quartoSearchOptions = {}; + let language = {}; + const searchOptionEl = window.document.getElementById( + "quarto-search-options" + ); + if (searchOptionEl) { + const jsonStr = searchOptionEl.textContent; + quartoSearchOptions = JSON.parse(jsonStr); + language = quartoSearchOptions.language; + } + + // note the search mode + if (quartoSearchOptions.type === "overlay") { + searchEl.classList.add("type-overlay"); + } else { + searchEl.classList.add("type-textbox"); + } + + // Used to determine highlighting behavior for this page + // A `q` query param is expected when the user follows a search + // to this page + const currentUrl = new URL(window.location); + const query = currentUrl.searchParams.get(kQueryArg); + const showSearchResults = currentUrl.searchParams.get(kResultsArg); + const mainEl = window.document.querySelector("main"); + + // highlight matches on the page + if (query && mainEl) { + // perform any highlighting + highlight(escapeRegExp(query), mainEl); + + // fix up the URL to remove the q query param + const replacementUrl = new URL(window.location); + replacementUrl.searchParams.delete(kQueryArg); + window.history.replaceState({}, "", replacementUrl); + } + + // function to clear highlighting on the page when the search query changes + // (e.g. if the user edits the query or clears it) + let highlighting = true; + const resetHighlighting = (searchTerm) => { + if (mainEl && highlighting && query && searchTerm !== query) { + clearHighlight(query, mainEl); + highlighting = false; + } + }; + + // Clear search highlighting when the user scrolls sufficiently + const resetFn = () => { + resetHighlighting(""); + window.removeEventListener("quarto-hrChanged", resetFn); + window.removeEventListener("quarto-sectionChanged", resetFn); + }; + + // Register this event after the initial scrolling and settling of events + // on the page + window.addEventListener("quarto-hrChanged", resetFn); + window.addEventListener("quarto-sectionChanged", resetFn); + + // Responsively switch to overlay mode if the search is present on the navbar + // Note that switching the sidebar to overlay mode requires more coordinate (not just + // the media query since we generate different HTML for sidebar overlays than we do + // for sidebar input UI) + const detachedMediaQuery = + quartoSearchOptions.type === "overlay" ? "all" : "(max-width: 991px)"; + + // If configured, include the analytics client to send insights + const plugins = configurePlugins(quartoSearchOptions); + + let lastState = null; + const { setIsOpen, setQuery, setCollections } = autocomplete({ + container: searchEl, + detachedMediaQuery: detachedMediaQuery, + defaultActiveItemId: 0, + panelContainer: "#quarto-search-results", + panelPlacement: quartoSearchOptions["panel-placement"], + debug: false, + openOnFocus: true, + plugins, + classNames: { + form: "d-flex", + }, + placeholder: language["search-text-placeholder"], + translations: { + clearButtonTitle: language["search-clear-button-title"], + detachedCancelButtonText: language["search-detached-cancel-button-title"], + submitButtonTitle: language["search-submit-button-title"], + }, + initialState: { + query, + }, + getItemUrl({ item }) { + return item.href; + }, + onStateChange({ state }) { + // If this is a file URL, note that + + // Perhaps reset highlighting + resetHighlighting(state.query); + + // If the panel just opened, ensure the panel is positioned properly + if (state.isOpen) { + if (lastState && !lastState.isOpen) { + setTimeout(() => { + positionPanel(quartoSearchOptions["panel-placement"]); + }, 150); + } + } + + // Perhaps show the copy link + showCopyLink(state.query, quartoSearchOptions); + + lastState = state; + }, + reshape({ sources, state }) { + return sources.map((source) => { + try { + const items = source.getItems(); + + // Validate the items + validateItems(items); + + // group the items by document + const groupedItems = new Map(); + items.forEach((item) => { + const hrefParts = item.href.split("#"); + const baseHref = hrefParts[0]; + const isDocumentItem = hrefParts.length === 1; + + const items = groupedItems.get(baseHref); + if (!items) { + groupedItems.set(baseHref, [item]); + } else { + // If the href for this item matches the document + // exactly, place this item first as it is the item that represents + // the document itself + if (isDocumentItem) { + items.unshift(item); + } else { + items.push(item); + } + groupedItems.set(baseHref, items); + } + }); + + const reshapedItems = []; + let count = 1; + for (const [_key, value] of groupedItems) { + const firstItem = value[0]; + reshapedItems.push({ + ...firstItem, + type: kItemTypeDoc, + }); + + const collapseMatches = quartoSearchOptions["collapse-after"]; + const collapseCount = + typeof collapseMatches === "number" ? collapseMatches : 1; + + if (value.length > 1) { + const target = `search-more-${count}`; + const isExpanded = + state.context.expanded && + state.context.expanded.includes(target); + + const remainingCount = value.length - collapseCount; + + for (let i = 1; i < value.length; i++) { + if (collapseMatches && i === collapseCount) { + reshapedItems.push({ + target, + title: isExpanded + ? language["search-hide-matches-text"] + : remainingCount === 1 + ? `${remainingCount} ${language["search-more-match-text"]}` + : `${remainingCount} ${language["search-more-matches-text"]}`, + type: kItemTypeMore, + href: kItemTypeMoreHref, + }); + } + + if (isExpanded || !collapseMatches || i < collapseCount) { + reshapedItems.push({ + ...value[i], + type: kItemTypeItem, + target, + }); + } + } + } + count += 1; + } + + return { + ...source, + getItems() { + return reshapedItems; + }, + }; + } catch (error) { + // Some form of error occurred + return { + ...source, + getItems() { + return [ + { + title: error.name || "An Error Occurred While Searching", + text: + error.message || + "An unknown error occurred while attempting to perform the requested search.", + type: kItemTypeError, + }, + ]; + }, + }; + } + }); + }, + navigator: { + navigate({ itemUrl }) { + if (itemUrl !== offsetURL(kItemTypeMoreHref)) { + window.location.assign(itemUrl); + } + }, + navigateNewTab({ itemUrl }) { + if (itemUrl !== offsetURL(kItemTypeMoreHref)) { + const windowReference = window.open(itemUrl, "_blank", "noopener"); + if (windowReference) { + windowReference.focus(); + } + } + }, + navigateNewWindow({ itemUrl }) { + if (itemUrl !== offsetURL(kItemTypeMoreHref)) { + window.open(itemUrl, "_blank", "noopener"); + } + }, + }, + getSources({ state, setContext, setActiveItemId, refresh }) { + return [ + { + sourceId: "documents", + getItemUrl({ item }) { + if (item.href) { + return offsetURL(item.href); + } else { + return undefined; + } + }, + onSelect({ + item, + state, + setContext, + setIsOpen, + setActiveItemId, + refresh, + }) { + if (item.type === kItemTypeMore) { + toggleExpanded(item, state, setContext, setActiveItemId, refresh); + + // Toggle more + setIsOpen(true); + } + }, + getItems({ query }) { + if (query === null || query === "") { + return []; + } + + const limit = quartoSearchOptions.limit; + if (quartoSearchOptions.algolia) { + return algoliaSearch(query, limit, quartoSearchOptions.algolia); + } else { + // Fuse search options + const fuseSearchOptions = { + isCaseSensitive: false, + shouldSort: true, + minMatchCharLength: 2, + limit: limit, + }; + + return readSearchData().then(function (fuse) { + return fuseSearch(query, fuse, fuseSearchOptions); + }); + } + }, + templates: { + noResults({ createElement }) { + const hasQuery = lastState.query; + + return createElement( + "div", + { + class: `quarto-search-no-results${ + hasQuery ? "" : " no-query" + }`, + }, + language["search-no-results-text"] + ); + }, + header({ items, createElement }) { + // count the documents + const count = items.filter((item) => { + return item.type === kItemTypeDoc; + }).length; + + if (count > 0) { + return createElement( + "div", + { class: "search-result-header" }, + `${count} ${language["search-matching-documents-text"]}` + ); + } else { + return createElement( + "div", + { class: "search-result-header-no-results" }, + `` + ); + } + }, + footer({ _items, createElement }) { + if ( + quartoSearchOptions.algolia && + quartoSearchOptions.algolia["show-logo"] + ) { + const libDir = quartoSearchOptions.algolia["libDir"]; + const logo = createElement("img", { + src: offsetURL( + `${libDir}/quarto-search/search-by-algolia.svg` + ), + class: "algolia-search-logo", + }); + return createElement( + "a", + { href: "http://www.algolia.com/" }, + logo + ); + } + }, + + item({ item, createElement }) { + return renderItem( + item, + createElement, + state, + setActiveItemId, + setContext, + refresh, + quartoSearchOptions + ); + }, + }, + }, + ]; + }, + }); + + window.quartoOpenSearch = () => { + setIsOpen(false); + setIsOpen(true); + focusSearchInput(); + }; + + document.addEventListener("keyup", (event) => { + const { key } = event; + const kbds = quartoSearchOptions["keyboard-shortcut"]; + const focusedEl = document.activeElement; + + const isFormElFocused = [ + "input", + "select", + "textarea", + "button", + "option", + ].find((tag) => { + return focusedEl.tagName.toLowerCase() === tag; + }); + + if ( + kbds && + kbds.includes(key) && + !isFormElFocused && + !document.activeElement.isContentEditable + ) { + event.preventDefault(); + window.quartoOpenSearch(); + } + }); + + // Remove the labeleledby attribute since it is pointing + // to a non-existent label + if (quartoSearchOptions.type === "overlay") { + const inputEl = window.document.querySelector( + "#quarto-search .aa-Autocomplete" + ); + if (inputEl) { + inputEl.removeAttribute("aria-labelledby"); + } + } + + function throttle(func, wait) { + let waiting = false; + return function () { + if (!waiting) { + func.apply(this, arguments); + waiting = true; + setTimeout(function () { + waiting = false; + }, wait); + } + }; + } + + // If the main document scrolls dismiss the search results + // (otherwise, since they're floating in the document they can scroll with the document) + window.document.body.onscroll = throttle(() => { + // Only do this if we're not detached + // Bug #7117 + // This will happen when the keyboard is shown on ios (resulting in a scroll) + // which then closed the search UI + if (!window.matchMedia(detachedMediaQuery).matches) { + setIsOpen(false); + } + }, 50); + + if (showSearchResults) { + setIsOpen(true); + focusSearchInput(); + } +}); + +function configurePlugins(quartoSearchOptions) { + const autocompletePlugins = []; + const algoliaOptions = quartoSearchOptions.algolia; + if ( + algoliaOptions && + algoliaOptions["analytics-events"] && + algoliaOptions["search-only-api-key"] && + algoliaOptions["application-id"] + ) { + const apiKey = algoliaOptions["search-only-api-key"]; + const appId = algoliaOptions["application-id"]; + + // Aloglia insights may not be loaded because they require cookie consent + // Use deferred loading so events will start being recorded when/if consent + // is granted. + const algoliaInsightsDeferredPlugin = deferredLoadPlugin(() => { + if ( + window.aa && + window["@algolia/autocomplete-plugin-algolia-insights"] + ) { + window.aa("init", { + appId, + apiKey, + useCookie: true, + }); + + const { createAlgoliaInsightsPlugin } = + window["@algolia/autocomplete-plugin-algolia-insights"]; + // Register the insights client + const algoliaInsightsPlugin = createAlgoliaInsightsPlugin({ + insightsClient: window.aa, + onItemsChange({ insights, insightsEvents }) { + const events = insightsEvents.flatMap((event) => { + // This API limits the number of items per event to 20 + const chunkSize = 20; + const itemChunks = []; + const eventItems = event.items; + for (let i = 0; i < eventItems.length; i += chunkSize) { + itemChunks.push(eventItems.slice(i, i + chunkSize)); + } + // Split the items into multiple events that can be sent + const events = itemChunks.map((items) => { + return { + ...event, + items, + }; + }); + return events; + }); + + for (const event of events) { + insights.viewedObjectIDs(event); + } + }, + }); + return algoliaInsightsPlugin; + } + }); + + // Add the plugin + autocompletePlugins.push(algoliaInsightsDeferredPlugin); + return autocompletePlugins; + } +} + +// For plugins that may not load immediately, create a wrapper +// plugin and forward events and plugin data once the plugin +// is initialized. This is useful for cases like cookie consent +// which may prevent the analytics insights event plugin from initializing +// immediately. +function deferredLoadPlugin(createPlugin) { + let plugin = undefined; + let subscribeObj = undefined; + const wrappedPlugin = () => { + if (!plugin && subscribeObj) { + plugin = createPlugin(); + if (plugin && plugin.subscribe) { + plugin.subscribe(subscribeObj); + } + } + return plugin; + }; + + return { + subscribe: (obj) => { + subscribeObj = obj; + }, + onStateChange: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.onStateChange) { + plugin.onStateChange(obj); + } + }, + onSubmit: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.onSubmit) { + plugin.onSubmit(obj); + } + }, + onReset: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.onReset) { + plugin.onReset(obj); + } + }, + getSources: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.getSources) { + return plugin.getSources(obj); + } else { + return Promise.resolve([]); + } + }, + data: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.data) { + plugin.data(obj); + } + }, + }; +} + +function validateItems(items) { + // Validate the first item + if (items.length > 0) { + const item = items[0]; + const missingFields = []; + if (item.href == undefined) { + missingFields.push("href"); + } + if (!item.title == undefined) { + missingFields.push("title"); + } + if (!item.text == undefined) { + missingFields.push("text"); + } + + if (missingFields.length === 1) { + throw { + name: `Error: Search index is missing the ${missingFields[0]} field.`, + message: `The items being returned for this search do not include all the required fields. Please ensure that your index items include the ${missingFields[0]} field or use index-fields in your _quarto.yml file to specify the field names.`, + }; + } else if (missingFields.length > 1) { + const missingFieldList = missingFields + .map((field) => { + return `${field}`; + }) + .join(", "); + + throw { + name: `Error: Search index is missing the following fields: ${missingFieldList}.`, + message: `The items being returned for this search do not include all the required fields. Please ensure that your index items includes the following fields: ${missingFieldList}, or use index-fields in your _quarto.yml file to specify the field names.`, + }; + } + } +} + +let lastQuery = null; +function showCopyLink(query, options) { + const language = options.language; + lastQuery = query; + // Insert share icon + const inputSuffixEl = window.document.body.querySelector( + ".aa-Form .aa-InputWrapperSuffix" + ); + + if (inputSuffixEl) { + let copyButtonEl = window.document.body.querySelector( + ".aa-Form .aa-InputWrapperSuffix .aa-CopyButton" + ); + + if (copyButtonEl === null) { + copyButtonEl = window.document.createElement("button"); + copyButtonEl.setAttribute("class", "aa-CopyButton"); + copyButtonEl.setAttribute("type", "button"); + copyButtonEl.setAttribute("title", language["search-copy-link-title"]); + copyButtonEl.onmousedown = (e) => { + e.preventDefault(); + e.stopPropagation(); + }; + + const linkIcon = "bi-clipboard"; + const checkIcon = "bi-check2"; + + const shareIconEl = window.document.createElement("i"); + shareIconEl.setAttribute("class", `bi ${linkIcon}`); + copyButtonEl.appendChild(shareIconEl); + inputSuffixEl.prepend(copyButtonEl); + + const clipboard = new window.ClipboardJS(".aa-CopyButton", { + text: function (_trigger) { + const copyUrl = new URL(window.location); + copyUrl.searchParams.set(kQueryArg, lastQuery); + copyUrl.searchParams.set(kResultsArg, "1"); + return copyUrl.toString(); + }, + }); + clipboard.on("success", function (e) { + // Focus the input + + // button target + const button = e.trigger; + const icon = button.querySelector("i.bi"); + + // flash "checked" + icon.classList.add(checkIcon); + icon.classList.remove(linkIcon); + setTimeout(function () { + icon.classList.remove(checkIcon); + icon.classList.add(linkIcon); + }, 1000); + }); + } + + // If there is a query, show the link icon + if (copyButtonEl) { + if (lastQuery && options["copy-button"]) { + copyButtonEl.style.display = "flex"; + } else { + copyButtonEl.style.display = "none"; + } + } + } +} + +/* Search Index Handling */ +// create the index +var fuseIndex = undefined; +var shownWarning = false; + +// fuse index options +const kFuseIndexOptions = { + keys: [ + { name: "title", weight: 20 }, + { name: "section", weight: 20 }, + { name: "text", weight: 10 }, + ], + ignoreLocation: true, + threshold: 0.1, +}; + +async function readSearchData() { + // Initialize the search index on demand + if (fuseIndex === undefined) { + if (window.location.protocol === "file:" && !shownWarning) { + window.alert( + "Search requires JavaScript features disabled when running in file://... URLs. In order to use search, please run this document in a web server." + ); + shownWarning = true; + return; + } + const fuse = new window.Fuse([], kFuseIndexOptions); + + // fetch the main search.json + const response = await fetch(offsetURL("search.json")); + if (response.status == 200) { + return response.json().then(function (searchDocs) { + searchDocs.forEach(function (searchDoc) { + fuse.add(searchDoc); + }); + fuseIndex = fuse; + return fuseIndex; + }); + } else { + return Promise.reject( + new Error( + "Unexpected status from search index request: " + response.status + ) + ); + } + } + + return fuseIndex; +} + +function inputElement() { + return window.document.body.querySelector(".aa-Form .aa-Input"); +} + +function focusSearchInput() { + setTimeout(() => { + const inputEl = inputElement(); + if (inputEl) { + inputEl.focus(); + } + }, 50); +} + +/* Panels */ +const kItemTypeDoc = "document"; +const kItemTypeMore = "document-more"; +const kItemTypeItem = "document-item"; +const kItemTypeError = "error"; + +function renderItem( + item, + createElement, + state, + setActiveItemId, + setContext, + refresh, + quartoSearchOptions +) { + switch (item.type) { + case kItemTypeDoc: + return createDocumentCard( + createElement, + "file-richtext", + item.title, + item.section, + item.text, + item.href, + item.crumbs, + quartoSearchOptions + ); + case kItemTypeMore: + return createMoreCard( + createElement, + item, + state, + setActiveItemId, + setContext, + refresh + ); + case kItemTypeItem: + return createSectionCard( + createElement, + item.section, + item.text, + item.href + ); + case kItemTypeError: + return createErrorCard(createElement, item.title, item.text); + default: + return undefined; + } +} + +function createDocumentCard( + createElement, + icon, + title, + section, + text, + href, + crumbs, + quartoSearchOptions +) { + const iconEl = createElement("i", { + class: `bi bi-${icon} search-result-icon`, + }); + const titleEl = createElement("p", { class: "search-result-title" }, title); + const titleContents = [iconEl, titleEl]; + const showParent = quartoSearchOptions["show-item-context"]; + if (crumbs && showParent) { + let crumbsOut = undefined; + const crumbClz = ["search-result-crumbs"]; + if (showParent === "root") { + crumbsOut = crumbs.length > 1 ? crumbs[0] : undefined; + } else if (showParent === "parent") { + crumbsOut = crumbs.length > 1 ? crumbs[crumbs.length - 2] : undefined; + } else { + crumbsOut = crumbs.length > 1 ? crumbs.join(" > ") : undefined; + crumbClz.push("search-result-crumbs-wrap"); + } + + const crumbEl = createElement( + "p", + { class: crumbClz.join(" ") }, + crumbsOut + ); + titleContents.push(crumbEl); + } + + const titleContainerEl = createElement( + "div", + { class: "search-result-title-container" }, + titleContents + ); + + const textEls = []; + if (section) { + const sectionEl = createElement( + "p", + { class: "search-result-section" }, + section + ); + textEls.push(sectionEl); + } + const descEl = createElement("p", { + class: "search-result-text", + dangerouslySetInnerHTML: { + __html: text, + }, + }); + textEls.push(descEl); + + const textContainerEl = createElement( + "div", + { class: "search-result-text-container" }, + textEls + ); + + const containerEl = createElement( + "div", + { + class: "search-result-container", + }, + [titleContainerEl, textContainerEl] + ); + + const linkEl = createElement( + "a", + { + href: offsetURL(href), + class: "search-result-link", + }, + containerEl + ); + + const classes = ["search-result-doc", "search-item"]; + if (!section) { + classes.push("document-selectable"); + } + + return createElement( + "div", + { + class: classes.join(" "), + }, + linkEl + ); +} + +function createMoreCard( + createElement, + item, + state, + setActiveItemId, + setContext, + refresh +) { + const moreCardEl = createElement( + "div", + { + class: "search-result-more search-item", + onClick: (e) => { + // Handle expanding the sections by adding the expanded + // section to the list of expanded sections + toggleExpanded(item, state, setContext, setActiveItemId, refresh); 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"&" : "?"; + anchorParts[0] = baseUrl + sep + name + "=" + value; + return anchorParts.join("#"); + }; + + return { + title: result.item.title, + section: result.item.section, + href: addParam(result.item.href, kQueryArg, query), + text: highlightMatch(query, result.item.text), + crumbs: result.item.crumbs, + }; + }); + + // If we don't have a subfuse and the query is long enough, go ahead + // and create a subfuse to use for subsequent queries + if ( + now - then > kFuseMaxWait && + subSearchFuse === undefined && + resultsRaw.length < fuseOptions.limit + ) { + subSearchTerm = query; + subSearchFuse = new window.Fuse([], kFuseIndexOptions); + resultsRaw.forEach((rr) => { + subSearchFuse.add(rr.item); + }); + } + return results; +} diff --git a/sitemap.xml b/sitemap.xml index 5281abd..84f5e05 100644 --- a/sitemap.xml +++ b/sitemap.xml @@ -126,7 +126,7 @@ https://hurak.github.io/hys/complementarity_simulations.html - 2024-12-09T23:16:08.120Z + 2024-12-14T20:43:13.990Z https://hurak.github.io/hys/classes_PWA.html @@ -162,7 +162,7 @@ https://hurak.github.io/hys/complementarity_systems.html - 2024-12-09T22:53:12.747Z + 2024-12-14T16:40:41.900Z https://hurak.github.io/hys/mpc_mld_online.html @@ -178,7 +178,7 @@ https://hurak.github.io/hys/complementarity_constraints.html - 2024-11-26T20:54:28.053Z + 2024-12-13T20:39:20.740Z https://hurak.github.io/hys/verification_intro.html diff --git a/stability_via_multiple_lyapunov_function 25.html b/stability_via_multiple_lyapunov_function 25.html new file mode 100644 index 0000000..ed52d2c --- /dev/null +++ b/stability_via_multiple_lyapunov_function 25.html @@ -0,0 +1,1550 @@ + + + + + + + + + +Stability via multiple Lyapunov functions – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ + +
+ + + +
+ +
+
+

Stability via multiple Lyapunov functions

+
+ + + +
+ + + + +
+ + + +
+ + +

We start with an example in which we show it can happen that even if the hybrid (switched) system is stable, there is no common quadratic Lyapunov function.

+
+

Example 1 (No common quadratic Lyapunov function can be found) We consider a switched system with two modes, linear models of which are parameterized by the matrices \mathbf A_1 and \mathbf A_2:

+
+
A₁ = [-0.1 -1; 2 -0.1]
+A₂ = [-0.1 -2; 1 -0.1]
+
+

Switching curve is given by x_1=0, that is, the vertical axis. State portrait is in Fig. 1.

+
+
+Show the code +
f₁(x) = A₁*x
+f₂(x) = A₂*x
+f(x) = x[1] <= 0.0 ? f₁(x) : f₂(x)
+
+using CairoMakie
+fig = Figure(size = (800, 800),fontsize=20)
+ax = Axis(fig[1, 1], xlabel = "x₁", ylabel = "x₂")
+streamplot!(ax,(x₁,x₂)->Point2f(f([x₁,x₂])), -2.0..2.0, -2.0..2.0, colormap = :magma)
+vlines!(ax,0.0,ymin=-2.0,ymax=2.0, color = :red, linewidth=3)
+fig
+
+
+
+
+
+ +
+
+Figure 1: State portrait for the switched system with no common quadratic Lyapunov function +
+
+
+
+
+

The equilibrium (the origin) of this switched system appears stable.

+

The individual systems are stable, which we can immediately see by computing the eigenvalues of the matrices A_1 and A_2:

+
+
using LinearAlgebra
+eigvals(A₁), eigvals(A₂)
+
+
(ComplexF64[-0.1 - 1.4142135623730951im, -0.1 + 1.4142135623730951im], ComplexF64[-0.1 - 1.4142135623730951im, -0.1 + 1.4142135623730951im])
+
+
+

We now try to find a common quadratic Lyapunov function for both subsystems. We will formulate the problem as an LMI feasibility problem.

+
+
using Convex, SCS
+X = Semidefinite(2)
+constraint₁ = A₁'*X + X*A₁ ⪯ -Matrix{Float64}(I, 2, 2)
+constraint₂ = A₂'*X + X*A₂ ⪯ -Matrix{Float64}(I, 2, 2)
+constraints = [constraint₁, constraint₂]
+problem = satisfy(constraints)
+solve!(problem,SCS.Optimizer,silent=true)
+
+
Problem statistics
+  problem is DCP         : true
+  number of variables    : 1 (4 scalar elements)
+  number of constraints  : 3 (12 scalar elements)
+  number of coefficients : 24
+  number of atoms        : 10
+
+Solution summary
+  termination status : INFEASIBLE
+  primal status      : INFEASIBLE_POINT
+  dual status        : INFEASIBILITY_CERTIFICATE
+
+Expression graph
+  satisfy
+   └─ nothing
+  subject to
+   ├─ PSD constraint (convex)
+   │  └─ + (affine; real)
+   │     ├─ 2×2 Matrix{Float64}
+   │     └─ Convex.NegateAtom (affine; real)
+   │        └─ …
+   ├─ PSD constraint (convex)
+   │  └─ + (affine; real)
+   │     ├─ 2×2 Matrix{Float64}
+   │     └─ Convex.NegateAtom (affine; real)
+   │        └─ …
+   ├─ PSD constraint (convex)
+   │  └─ 2×2 real variable (id: 126…042)
+   ⋮
+
+
+

The solver does not find a solution. Well, perhaps trying another solver or two would make our conclusion more robust (MosekTools is another recommendable alternative to SCS). But we are now be tempted to conclude that there is no common quadratic Lyapunov function for both subsystems.

+
+
+
+ +
+
+Dual LMI problem confirms the infeasibility of the primal one +
+
+
+

It is possible to formulate a dual LMI problem. Its feasibility certifies infeasibility of the primal one. Namely, the problem is, if there are matrices \mathbf R_1\succ 0 and \mathbf R_2\succ 0 such that the following LMI holds: +\mathbf R_1\mathbf A_1^\top+\mathbf A_1\mathbf R_1 + \mathbf R_2\mathbf A_2^\top+\mathbf A_2\mathbf R_2 \prec 0. +

+
+
+

The conclusion about the imposibility to find a single quadratic Lyapunov function for both subsystems is also suppported by plotting the invariant sets for the two subsystems. First, we need to compute Lyapunov functions for the two subsystems.

+
+
X₁ = Semidefinite(2)
+constraint₁ = A₁'*X₁ + X₁*A₁ ⪯ -Matrix{Float64}(I, 2, 2)
+problem₁ = satisfy(constraint₁)
+solve!(problem₁,SCS.Optimizer,silent=true)
+X₁.value
+
+
2×2 Matrix{Float64}:
+ 423.71     12.5739
+  12.5739  212.123
+
+
+
+
X₂ = Semidefinite(2)
+constraint₂ = A₂'*X₂ + X₂*A₂ ⪯ -Matrix{Float64}(I, 2, 2)
+problem₂ = satisfy(constraint₂)
+solve!(problem₂,SCS.Optimizer,silent=true)
+X₂.value
+
+
2×2 Matrix{Float64}:
+ 212.123   -12.5739
+ -12.5739  423.71
+
+
+

Generally, a Lyapunov function has the property that its sublevel set \{\bm x \mid V(\bm x) \leq \alpha\} is forward (also positive) invariant. We plot invariant sets corresponding to some particular value for both subsystems superposed on the state portrait in Fig. 2.

+
+
+Show the code +
x1s = LinRange(-2, 2, 100)
+x2s = LinRange(-2, 2, 100)
+V₁(x) = x'*X₁.value*x
+V₂(x) = x'*X₂.value*x
+V1s = [V₁([x₁,x₂]) for x₁ in x1s, x₂ in x2s]
+V2s = [V₂([x₁,x₂]) for x₁ in x1s, x₂ in x2s]
+contour!(x1s, x2s, V1s, levels=[300.0], linewidth=3, color=:blue)
+contour!(x1s, x2s, V2s, levels=[300.0], linewidth=3, color=:green)
+fig
+
+
+
+
+
+ +
+
+Figure 2: Invariant ellipses for the two subsystems superposed on the state portrait +
+
+
+
+
+

We can see in Fig. 2 that none of the two ellipses works as an invariant set for the switched system – a state trajectory entering the set leaves it afterwards. No way to to come up with a single ellipse (hence a single quadratic Lyapunov function) that would work here.

+
+

Good, we have seen in the example that it is not always possible to find a common quadratic Lyapunov function for a switched system, even it it is stable. We need to expand the set of functions in which we search for a Lyapunov function. We have proposed one way to do it in the previouse chapter wherein we considered higher degree polynomials on which we imposed the nonnegativity constraint through in the form of SOS constraint. Here we are going to consider another approach. We are goint to stitch together several Lyapunov-like functions, each of which is a Lyapunov function only on some subset of the state space (that is why we call them just Lyapunov-like and not Lyapunov). This approach is sometimes called the Multiple Lyapunov Function (MLF) approach, or Piecewise Lyapunov Function approach.

+
+

Multiple Lyapunov Function (MLF) approach to analysis of stability

+

Instead of just a single common Lyapunov function V(\bm x), we are now going to consider several Lyapunov-like functions V_i(\bm x),\; i=1,\ldots,r, that qualify as Lyapunov function only on some subsets of the state space \Omega_i. And we “stitch” them together to form a piecewise Lyapunov function V(\bm x): +V(x) = +\begin{cases} +V_1(\bm x) & \text{if } \bm x\in \Omega_1, \\ +\vdots\\ +V_r(\bm x) & \text{if } \bm x\in \Omega_r. +\end{cases} +

+
+
+

S-procedure

+

In order to restrict the requirement of positive definiteness of the Lyapunov function to some region in the state space, and similarly for the regurement of negative definiteness of its time derivative, we need to introduce the the S-procedure. This is a result about solvability of two or more quadratic inequalities, not necessarily convex ones (for convex problems we have nonlinear Farkas’ lemma). Origins of this result can be found in the control theory (analysis of stability of nonlinear systems, hence the S letter) with the first rigorous result provided by Yakubovich in 1970s.

+

It gives conditions under which (satisfaction of) one quadratic inequality follows from (satisfaction of) another one (or more). Namely, it gives a condition under which the following implication holds:

+

\boxed +{\text{Quadratic inequality \#1 satisfied by some}\; \bm x \Rightarrow \text{Quadratic inequality \#0 satisfied by the same}\; \bm x.} +

+

In other words, it gives a condition under which the solution set of the inequality #1 denoted as \mathcal X_1 is included in the solution set \mathcal X_0 of the inequality #0.

+
+

S-procedure with nonstrict inequalities

+

Consider the general quadratic functions F_i(\bm x) = \bm x^\top \mathbf A_i \bm x + 2\mathbf b_i^\top \bm x + c_i, \; i=0,\ldots, p.

+

The question is: under which conditions it holds that F_0(\bm x) \geq 0 for all \bm x satisfying F_i(\bm x)\geq 0,\; i=1,\ldots,p ?

+

In other words, we are looking for conditions under which the implication +F_i(\bm x) \geq 0,\; i=1,\ldots,p \quad \Rightarrow \quad F_0(\bm x) \geq 0 +
+holds.

+

In the simplest (yet relevant) case p=1 we search for conditions under which F_0(\bm x) \geq 0 for all \bm x satisfying F_1(\bm x)\geq 0, that is, conditions under which the implication +F_1(\bm x) \geq 0 \quad \Rightarrow \quad F_0(\bm x) \geq 0 +
+holds.

+
+
+

Sufficient conditions

+

The existence of \alpha_i\geq 0,\; i=1,\ldots,p such that +F_0(\bm x)-\sum_{i=1}^p \alpha_i F_i(\bm x) \geq 0 + is sufficient for the original implication to hold. Generally, it is not necessary; the condition is conservative.

+

It can be formulated as an LMI +\begin{bmatrix} +\mathbf A_0 & \mathbf b_0 \\ +\mathbf b_0^\top & c_0 +\end{bmatrix} - +\sum_{i=1}^p +\alpha_i +\begin{bmatrix} +\mathbf A_i & \mathbf b_i \\ +\mathbf b_i^\top & c_i +\end{bmatrix} +\succeq 0 +
+where \alpha_i \geq 0,\; i=1,\ldots,p.

+
+
+

Sufficient and also necessary

+

It is nontrivial that for p=1 it is also necessary, provided that there is some \bm x_0 such that F_1(\bm x_0)>0. Then we have the following equivalence between the two constraints: +\begin{aligned} +F_0(\bm x) &\geq 0 \; \forall \bm x \;\mathrm{satisfying}\; F_1(\bm x)\geq 0 \\ +&\Longleftrightarrow \\ +F_0(\bm x)-\alpha F_1(\bm x) &\geq 0,\;\text{for some}\; \alpha\in\mathbb{R}, \; \alpha\geq 0, +\end{aligned} +
+which again can be formulated as an LMI, namely
+ +\begin{bmatrix} +\mathbf A_0 & \mathbf b_0 \\ +\mathbf b_0^\top & c_0 +\end{bmatrix} - \alpha +\begin{bmatrix} +\mathbf A_1 & \mathbf b_1 \\ +\mathbf b_1^\top & c_1 +\end{bmatrix} +\succeq 0,\quad \alpha\geq 0. +

+
+
+

More on S-procedure

+

There are several variants

+
    +
  • strict, nonstrict or mixed inequalities,
    • +
  • just two or more,
    • +
  • some of the constraints can be equations.
    • +
+
+
+
+

Piecewise quadratic Lyapunov function

+

We now restrict ourselves to quadratic Lyapunov-like functions, that is, quadratic functions V_i(\bm x) = \bm x^\top \mathbf P_i \bm x that qualify as Lyapunov functions only on respective subsets \Omega_i\sub \mathbb R^n:

+

+V_i(\bm x) = \bm x^\top \mathbf P_i \bm x > 0\quad \forall \;\bm x\in \Omega_i, +

+

+\dot V_i(\bm x) = \bm x^\top \left( \mathbf A_i^\top \mathbf P_i + \mathbf P_i \mathbf A_i \right) \bm x < 0\quad \forall \;\bm x\in \Omega_i. +

+
+
+

Using comparison functions and nonstrict inequalities

+

We can use our good old comparison functions to formulate the conditions of positive definiteness and negative definiteness. +\alpha_1 \bm x^\top \mathbf I \bm x \leq \bm x^\top \mathbf P_i \bm x \leq \alpha_2 \bm x^\top \mathbf I \bm x \quad \forall \;\bm x\in \Omega_i, +

+

+\bm x^\top \left( \mathbf A_i^\top \mathbf P_i + \mathbf P_i \mathbf A_i \right) \bm x \leq -\alpha_3 \bm x^\top \mathbf I \bm x\quad \forall \;\bm x\in \Omega_i. +

+

The difference now is that these conditions are only reuired to hold on some state regions, some subsets of the state space. It is now time to discuss how to characterize those regions.

+
+
+

Characterization of subsets of state space using LMI

+

Some subsets \Omega_i\sub \mathbb R^n characterized using linear and quadratic inequalities can be formulated within the LMI framework as +\bm x^\top \mathbf Q_i \bm x \geq 0. +

+

In particular, centered ellipsoids and cones.

+

For example,
+ +\begin{aligned} +\Omega_i &= \{\bm x \in \mathbb R^n \mid (\mathbf c^\top \bm x \geq 0 \land \mathbf d^\top \bm x \geq 0) \\ +& \qquad \qquad \qquad \lor (\mathbf c^\top \bm x \leq 0 \land \mathbf d^\top \bm x \leq 0)\}. +\end{aligned} +

+

This constraint can be reformulated as +(\mathbf c^\top \bm x) (\mathbf d^\top \bm x) \geq 0, +
+which can be reformatted to +\bm x^\top \mathbf c \mathbf d^\top \bm x \geq 0, +
+which can further be symmetrized to +\bm x^\top \left(\frac{\mathbf c \mathbf d^\top + \mathbf d \mathbf c^\top}{2}\right) \bm x \geq 0. +

+

More general sets (general polyhedra, noncentered ellipsoids) can also be modelled using LMI too… We are going to have a look at them, but first we hurry to show how to combine the subset characterization and Lyapunov-ness using the S-procedure.

+
+
+

Combining the subset characterization and Lyapunov-ness using the S-procedure

+

We want to learn if the the following hold +\alpha_i \bm x^\top \mathbf I \bm x \leq \bm x^\top \mathbf P_i \bm x, +

+

+\bm x^\top \left( \mathbf A_i^\top \mathbf P_i + \mathbf P_i \mathbf A_i \right) \bm x \leq -\gamma_i \bm x^\top \mathbf I \bm x, +

+

but not for all \bm x, but only for \bm x\in \Omega_i, that is, all \bm x satisfying \bm x^\top \mathbf Q_i \bm x \geq 0. But this is now a perfect opportunity for application of the S-procedure:

+

+\mathbf P_i - \alpha_i \mathbf I - \mu_i \mathbf Q_i \succeq 0,\quad \mu_i \geq 0,\; \alpha_i > 0, +

+

+\mathbf A_i^\top \mathbf P_i + \mathbf P_i \mathbf A_i + \gamma_i \mathbf I + \xi_i \mathbf Q \preceq 0,\quad \mu_i \geq 0,\; \gamma_i > 0. +

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+
+

More general sets using LMI

+

How can we model more general sets using LMI?

+

The inequality +\bm x^\top \mathbf Q \bm x + 2\mathbf r^\top \bm x + s \geq 0, +

+

can be reformulated as +\begin{bmatrix} +\bm x^\top & 1 +\end{bmatrix} +\underbrace{ +\begin{bmatrix} +\mathbf Q & \mathbf r \\ \mathbf r^\top & s +\end{bmatrix}}_{\bar{\mathbf{Q}}} +\underbrace{ +\begin{bmatrix} +\bm x \\ 1 +\end{bmatrix}}_{\bar{\bm x}} +\geq 0, +

+

that is, as an LMI +\begin{bmatrix} +\mathbf Q & \mathbf r \\ \mathbf r^\top & s +\end{bmatrix} +\succeq 0. +

+
+

Affine subspace

+

+\mathbf c^\top \bm x + d \geq 0, +

+

+\begin{bmatrix} +\bm x^\top & 1 +\end{bmatrix} +\begin{bmatrix} +\mathbf 0 & \mathbf c \\ \mathbf c^\top & 2d +\end{bmatrix} +\begin{bmatrix} +\bm x \\ 1 +\end{bmatrix} +\geq 0, +

+

+\begin{bmatrix} +\mathbf 0 & \mathbf c \\ \mathbf c^\top & 2d +\end{bmatrix} +\succeq 0. +

+

But then the Lyapunov-like functions and system matrices must also be extended +V(\bm x) = +\begin{bmatrix} +\bm x^\top & 1 +\end{bmatrix} +\underbrace{ +\begin{bmatrix} +\mathbf P & \mathbf P_{12} \\ \mathbf P_{12} & P_{22} +\end{bmatrix}}_{\bar{\mathbf P}} +\begin{bmatrix} +\bm x \\ 1 +\end{bmatrix}, +

+

+\bar{\mathbf{A}} = +\begin{bmatrix} +\mathbf A & \mathbf 0 \\ \mathbf 0 & 0 +\end{bmatrix}. +

+
+
+
+

Continuity conditions

+

The boundary between the regions \Omega_i and \Omega_j described by +\Omega_{ij} = \{\bm x \in \mathbb R^n \mid \mathbf F_{ij} \bm z + \mathbf{l}_{ij}\}, + where \bm z\in \mathbb R^p, \mathbf F_{ij}\in \mathbb R^{n\times p}, and \mathbf l_{ij}\in \mathbb R^{n}.

+

The continuity conditions are +V_i(\bm x) = V_j(\bm x) \quad \forall \bm x \in \Omega_{ij}, + which can be reformulated as +\begin{aligned} +&\left(\mathbf F_{ij} \bm z + \mathbf{l}_{ij}\right)^\top \mathbf P_i \left(\mathbf F_{ij} \bm z + \mathbf{l}_{ij}\right) \\ +& \qquad + 2\left(\mathbf F_{ij} \bm z + \mathbf{l}_{ij}\right)^\top \bm q_i + r_i \\ +&= \left(\mathbf F_{ij} \bm z + \mathbf{l}_{ij}\right)^\top \mathbf P_j \left(\mathbf F_{ij} \bm z + \mathbf{l}_{ij}\right) \\ +& \qquad + 2\left(\mathbf F_{ij} \bm z + \mathbf{l}_{ij}\right)^\top \bm q_j + r_j +\end{aligned}. +

+

Collecting the terms with equal powers of \bm z. +\begin{aligned} +\mathbf F_{ij}^\top (\mathbf P_1 - \mathbf P_2) \mathbf F_{ij} &= 0, \\ +\mathbf F_{ij}^\top (\mathbf P_1 - \mathbf P_2) \mathbf l_{ij} + (\mathbf q_1-\mathbf q_2) &= 0, \\ +\mathbf l_{ij}^\top (\mathbf P_1 - \mathbf P_2)\mathbf l_{ij} + 2\mathbf l_{ij}^\top (\mathbf q_1-\mathbf q_2) + r_1-r_2 &= 0. +\end{aligned} +

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Copyright 2024, Zdeněk Hurák

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+ + + + + \ No newline at end of file diff --git a/stability_via_multiple_lyapunov_function.html b/stability_via_multiple_lyapunov_function.html index d18c81e..14b709f 100644 --- a/stability_via_multiple_lyapunov_function.html +++ b/stability_via_multiple_lyapunov_function.html @@ -784,7 +784,7 @@

Stability via multiple Lyapunov functions

│ └─ Convex.NegateAtom (affine; real) │ └─ … ├─ PSD constraint (convex) - │ └─ 2×2 real variable (id: 440…606) + │ └─ 2×2 real variable (id: 113…477) ⋮ diff --git a/verification_barrier.html b/verification_barrier.html index 58376f4..87754fa 100644 --- a/verification_barrier.html +++ b/verification_barrier.html @@ -810,1409 +810,1409 @@ 

┌ Warning: At t=4.423118290940107, dt was forced below floating point epsilon 8.881784197001252e-16, and step error estimate = 1.139033855908175. Aborting. There is either an error in your model specification or the true solution is unstable (or the true solution can not be represented in the precision of Float64).
-└ @ SciMLBase ~/.julia/packages/SciMLBase/rBKqA/src/integrator_interface.jl:623
+└ @ SciMLBase ~/.julia/packages/SciMLBase/BaHpR/src/integrator_interface.jl:623
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diff --git a/verification_references 24.html b/verification_references 24.html new file mode 100644 index 0000000..c946582 --- /dev/null +++ b/verification_references 24.html @@ -0,0 +1,1126 @@ + + + + + + + + + +Literature – B(E)3M35HYS – Hybrid systems + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Literature

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The topic of verification of hybrid systems is vast. While we only reserved a single week/chapter/block for it, it would easily fill a dedicated course, supported by a couple of books. Having a smaller time budget, we have still found some encouragement that modest introduction is feasible in the Chapter 3 of [1]. Although we are not following the book closely, we are covering some of their topics.

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Among general references for hybrid system verification, we can recommend [2]. Although the book is not freely available for download, its web page contains quite some additional material such as slides and codes.

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Reachability analysis

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[3] [4]

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Barier certificates

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[5]

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Temporal logics

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[6], [7]

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[8], [9].

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References

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[1]
H. Lin and P. J. Antsaklis, Hybrid Dynamical Systems: Fundamentals and Methods. in Advanced Textbooks in Control and Signal Processing. Cham: Springer, 2022. Accessed: Jul. 09, 2022. [Online]. Available: https://doi.org/10.1007/978-3-030-78731-8
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[2]
S. Mitra, Verifying Cyber-Physical Systems: A Path to Safe Autonomy. in Cyber Physical Systems Series. Cambridge, MA, USA: MIT Press, 2021. Available: https://sayanmitracode.github.io/cpsbooksite/about.html
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[3]
M. Althoff, G. Frehse, and A. Girard, “Set Propagation Techniques for Reachability Analysis,” Annual Review of Control, Robotics, and Autonomous Systems, vol. 4, no. 1, pp. 369–395, 2021, doi: 10.1146/annurev-control-071420-081941.
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[4]
M. Althoff, N. Kochdumper, M. Wetzlinger, and T. Ladner, “CORA 2024 Manual.” 2023. Accessed: Jan. 10, 2023. [Online]. Available: https://tumcps.github.io/CORA/data/archive/manual/Cora2024Manual.pdf
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[5]
S. Prajna and A. Jadbabaie, “Safety Verification of Hybrid Systems Using Barrier Certificates,” in Hybrid Systems: Computation and Control, R. Alur and G. J. Pappas, Eds., in Lecture Notes in Computer Science. Berlin, Heidelberg: Springer, 2004, pp. 477–492. doi: 10.1007/978-3-540-24743-2_32.
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[6]
C. Baier and J.-P. Katoen, Principles of Model Checking. Cambridge, MA, USA: MIT Press, 2008. Available: https://mitpress.mit.edu/books/principles-model-checking
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[7]
E. M. Clarke, Jr, O. Grumberg, D. Kroening, D. Peled, and H. Veith, Model Checking, 2nd ed. in Cyber Physical Systems Series. Cambridge, MA, USA: MIT Press, 2018. Available: https://mitpress.mit.edu/9780262038836/model-checking/
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[8]
R. M. Murray, U. Topcu, and N. Wongpiromsarn, “Lecture 3 Linear Temporal Logic (LTL).” Belgrade (Serbia), Mar. 2020. Available: http://www.cds.caltech.edu/~murray/courses/eeci-sp2020/L3_ltl-09Mar2020.pdf
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[9]
N. Wongpiromsarn, R. M. Murray, and U. Topcu, “Lecture 4 Model Checking and Logic Synthesis.” Belgrade (Serbia), Mar. 2020. Available: http://www.cds.caltech.edu/~murray/courses/eeci-sp2020//L4_model_checking-09Mar2020.pdf
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Copyright 2024, Zdeněk Hurák

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