Built site for gh-pages

hurak · Dec 6, 2024 · 335dafc · 335dafc
1 parent 81ffb3d
commit 335dafc
Show file tree

Hide file tree

Showing 49 changed files with 140,939 additions and 103,922 deletions.
diff --git a/.nojekyll b/.nojekyll
@@ -1 +1 @@
-9ce7609c
+78582dc2
diff --git a/classes_references 17.html b/classes_references 17.html
diff --git a/classes_reset.html b/classes_reset.html
diff --git a/classes_software 20.html b/classes_software 20.html
diff --git a/classes_switched.html b/classes_switched.html
diff --git a/classes_switched_files/figure-html/cell-2-output-1.svg b/classes_switched_files/figure-html/cell-2-output-1.svg
diff --git a/classes_switched_files/figure-html/cell-4-output-1.svg b/classes_switched_files/figure-html/cell-4-output-1.svg
diff --git a/classes_switched_files/figure-html/cell-5-output-1.svg b/classes_switched_files/figure-html/cell-5-output-1.svg
diff --git a/classes_switched_files/figure-html/cell-6-output-1.svg b/classes_switched_files/figure-html/cell-6-output-1.svg
diff --git a/complementarity_constraints 17.html b/complementarity_constraints 17.html
diff --git a/complementarity_references 21.html b/complementarity_references 21.html
diff --git a/complementarity_simulations.html b/complementarity_simulations.html
diff --git a/complementarity_simulations_files/figure-html/fig-state-portrait-output-1.svg b/complementarity_simulations_files/figure-html/fig-state-portrait-output-1.svg
diff --git a/complementarity_systems 13.html b/complementarity_systems 13.html
diff --git a/complementarity_systems.html b/complementarity_systems.html
@@ -661,9 +661,9 @@ <h2 class="anchored" data-anchor-id="linear-complementarity-system-lcs">Linear c
 <p>Having introduced the <em>complementarity constraints</em> and optimization problems with these constraints, we can now show how these constraints can be used to model a certain class of dynamical systems – <em>complementarity dynamical systems</em>. We start with linear ones, namely, linear complementarity systems (LCS). These are also called in the literature as Linear dynamical complementarity problems (LDCP).</p>
 <p>Linear complementarity system is modelled by <span id="eq-linear-complementarity-system"><span class="math display">\boxed{
 \begin{aligned}
-\dot x(t) &amp;= A x(t) + Bu(t)\\
-y(t) &amp;= C x(t) + Du(t)\\
-0&amp;\leq u(t) \perp y(t) \geq 0.
+\dot{\bm x}(t) &amp;= \mathbf A \bm x(t) + \mathbf B\bm u(t)\\
+\bm y(t) &amp;= \mathbf C \bm x(t) + \mathbf D\bm u(t)\\
+\mathbf 0&amp;\leq \bm u(t) \perp \bm y(t) \geq \mathbf 0.
 \end{aligned}}
 \tag{1}</span></span></p>
 <div id="exm-electrical-circuit-as-LCS" class="theorem example">
@@ -676,12 +676,12 @@ <h2 class="anchored" data-anchor-id="linear-complementarity-system-lcs">Linear c
 </div>
 <p>Note the upside-down orientation of the voltage and the current for the capacitor – we wanted the diode current identical to the capacitor current.</p>
 <p>Following the charge formalism within Lagrangian modelling, we can choose the generalized coordinates as <span class="math display">
-      q = \begin{bmatrix}
+      \bm q = \begin{bmatrix}
       q_L \\ q_C
       \end{bmatrix}.
 </span></p>
 <p>That this is indeed a sufficient number is obvious, but we can also check the classical formula <span class="math inline">B-N+1 = 4-3+1 = 2</span>. But we can also choose the state variables as <span class="math display">
-      x = \begin{bmatrix}
+      \bm x = \begin{bmatrix}
       i_L\\ q_c
       \end{bmatrix}.
 </span></p>
@@ -778,9 +778,9 @@ <h2 class="anchored" data-anchor-id="linear-complementarity-system-lcs">Linear c
 <p>The input <span class="math inline">u_1</span> corresponds to the reaction force of the hard stop applied to the left cart. Another input is <span class="math inline">u_2</span> that corresponds to the force exerted by the left cart onto the right cart.</p>
 <p>The state equations are <span class="math display">
 \begin{aligned}
-\dot x_1(t) &amp;= x_3\\
-\dot x_2(t) &amp;= x_4\\
-\dot x_3(t) &amp;= -\frac{k_1+k_2}{m_1}x_1(t) + \frac{k_2}{m_1}x_2(t) + \frac{1}{m_1}u_1(t) - \frac{1}{m_1}u_2(t)\\
+\dot x_1(t) &amp;= x_3,\\
+\dot x_2(t) &amp;= x_4,\\
+\dot x_3(t) &amp;= -\frac{k_1+k_2}{m_1}x_1(t) + \frac{k_2}{m_1}x_2(t) + \frac{1}{m_1}u_1(t) - \frac{1}{m_1}u_2(t),\\
 \dot x_4(t) &amp;= \frac{k_2}{m_2}x_1(t) - \frac{k_2}{m_2} x_2(t) + \frac{1}{m_2}u_2(t).
 \end{aligned}
 </span></p>
@@ -790,7 +790,7 @@ <h2 class="anchored" data-anchor-id="linear-complementarity-system-lcs">Linear c
 </span></p>
 <p>This motivates us to define two output variables as <span class="math display">
 \begin{aligned}
-y_1(t) &amp;= x_1(t)\\
+y_1(t) &amp;= x_1(t),\\
 y_2(t) &amp;= x_2(t)-x_1(t).
 \end{aligned}
 </span></p>