93C55-DiscretizationOfContinuousSystems.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{DiscretizationOfContinuousSystems}
\pmcreated{2013-03-22 15:50:45}
\pmmodified{2013-03-22 15:50:45}
\pmowner{mathcam}{2727}
\pmmodifier{mathcam}{2727}
\pmtitle{discretization of continuous systems}
\pmrecord{7}{37827}
\pmprivacy{1}
\pmauthor{mathcam}{2727}
\pmtype{Topic}
\pmcomment{trigger rebuild}
\pmclassification{msc}{93C55}
\pmsynonym{Transform}{DiscretizationOfContinuousSystems}

\endmetadata

% this is the default PlanetMath preamble.  as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.

% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here
% Modified .tex file
%--------------------------------------------------------
%\makeindex
%\pagenumbering{roman}
%\tableofcontents
%\listoffigures
%\listoftables
%---------------------------------
%\input setwmf
%\input seteps
%\begin{figure}[!hbtp]
%\centerwmf{6in}{4in}{}
%\caption{Open-loop simulink structure}
%\label{yyy}
%\end{figure}
%See Fig.~\ref{yyy}
%=========================================================
%       Definitions                                 %
% Latex abbrevs ----------------------------------------
%\newcommand{\ba}{\begin{array}}
%\newcommand{\ea}{\end{array}}
%\newcommand{\bc}{\begin{center}}
%\newcommand{\ec}{\end{center}}

%\newcommand{\beqn}[1]{\begin{equation}\label{#1}}
%\newcommand{\eeqn}{\end{equation}}
%\newcommand{\be}{\begin{equation}}
%\newcommand{\ee}{\end{equation}}

%\newcommand{\beqnn}{\begin{eqnarray}}
%\newcommand{\eeqnn}{\end{eqnarray}}
\newcommand{\non}{\nonumber}
%-------------------------------------------------------
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{definition}{Definition}
\newtheorem{property}{Property}
\newtheorem{proposition}{Proposition}
%-------------------------------------------------------
\newcommand{\va}{\vspace*{5mm}}
\newcommand{\vb}{\vspace*{10mm}}
\newcommand{\vc}{\vspace*{15mm}}
%\newcommand{\ha}{\mbox{}\hspace*{5mm}}
%\newcommand{\hb}{\mbox{}\hspace*{10mm}}
%\newcommand{\hc}{\mbox{}\hspace*{15mm}}
\newcommand{\hsp}{\mbox{}\hspace{8mm}}
%-------------------------------------------------------
%\newcommand{\thetat}{{\theta}^{\T}}
%\newcommand{\btheta}{\bar{\theta}}
%\newcommand{\bthetat}{\bar{\theta}^{\T}}
%\newcommand{\htheta}{\hat{\theta}}
%\newcommand{\hthetat}{\hat{\theta}^{\T}}
%\newcommand{\ttheta}{\tilde{\theta}}
%\newcommand{\tthetat}{\tilde{\theta}^{\T}}

%\newcommand{\varphit}{\varphi^{\T}}
%\newcommand{\hvarphi}{\hat{\varphi}}
%\newcommand{\hvarphit}{\hat{\varphi}^{\T}}

%\newcommand{\hy}{\hat{y}}
%\newcommand{\ty}{\tilde{y}}
%\newcommand{\hu}{\hat{u}}
%\newcommand{\tu}{\tilde{u}}

\newcommand{\un}[1]{\underline{#1}\,}
%--------------------------------------------
% functions
%\newcommand{\ejt}{{\rm e}^{\jjj\theta}}
%\newcommand{\ejw}{{\rm e}^{\jjj\omega}}

%\newcommand{\ddd}[1]{{\rm d}{#1}}
%\newcommand{\eee}[1]{{\rm e}^{#1}}
%\newcommand{\e}[1]{{\rm e}^{#1}}

%\newcommand{\aaa}{{\rm a}}
%\newcommand{\CCC}{{\rm C}}
%\newcommand{\ccc}{{\rm c}}
%\newcommand{\fff}{{\rm f}}
%\newcommand{\jjj}{{\rm j}}
%\newcommand{\HH}{{\rm H}}
%\newcommand{\ooo}{{\rm o}}
%--------------------------------------------
%¿Õ�Ä
\newcommand{\C}{{\mathbb C}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
%ÕýÌå
\newcommand{\E}{{\rm E}}
\newcommand{\T}{{\rm T}}

%
\newcommand{\adj}{{\rm\,adj}}
\newcommand{\AR}{{\rm AR}}
\newcommand{\arctg}{{\rm arctg}}
\newcommand{\asas}{{\rm a.s.}}

\newcommand{\co}{{\rm co}}
\newcommand{\col}{{\rm col}}
\newcommand{\coeff}{{\rm coeff}}
\newcommand{\const}{{\rm const}}
\newcommand{\cov}{{\rm cov}}
%\newcommand{\def}{{\rm def}}
\newcommand{\diag}{{\rm diag}}

\newcommand{\FFLS}{{\rm FFLS}}
\newcommand{\grad}{{\rm grad}}
\newcommand{\Ima}{{\rm Im}}
\newcommand{\IV}{{\rm IV}}
\newcommand{\LCM}{{\rm LCM}}
\newcommand{\LS}{{\rm LS}}
\newcommand{\msms}{{\rm m.s.}}
\newcommand{\ns}{{\rm ns}}
\newcommand{\ob}{{\rm ob}}
\newcommand{\rank}{{\rm rank}}
\newcommand{\Rea}{{\rm Re}}
\newcommand{\sgn}{{\rm sgn}}
%\newcommand{\th}{{\rm th}}
\newcommand{\tg}{{\rm tg}}
\newcommand{\tr}{{\rm tr}}
\newcommand{\var}{{\rm var}}

%\dim \det \exp \gcd
%-----------matrices--------------------------------
\newcommand{\vect}[2]{\left[\begin{array}{c} #1 \\  #2 \end{array} \right]}
\newcommand{\vectt}[3]{\left[\begin{array}{c} #1 \\  #2 \\  #3 \end{array}\right]}
\newcommand{\vectf}[4]{\left[\begin{array}{c} #1 \\  #2 \\  #3 \\ #4 \end{array}\right]}
\newcommand{\vectfd}[4]{\left[\begin{array}{c} #1 \\  #2 \\  #3 \\  \vdots \\ #4 \end{array}\right]}
\newcommand{\vectfive}[5]{\left[\begin{array}{c} #1 \\  #2 \\  #3 \\  #4 \\ #5 \end{array}\right]}
\newcommand{\vectsix}[6]{\left[\begin{array}{c} #1 \\  #2 \\  #3 \\  #4 \\ #5 \\ #6 \end{array}\right]}

\newcommand{\mtwo}[4]{\left[\begin{array}{cc}#1 & #2 \\  #3 & #4\end{array} \right]}

\newcommand{\tfmat}[4]{\left[ \begin{array}{c|c} #1 & #2 \\  \hline
                                                 #3 & #4 \end{array} \right] }
%»¨Ìå´ó�´-------------------------------------------------------------------------------------------------------
\def\CalA{{\mathcal A}}
\def\CalB{{\mathcal B}}
\def\CalC{{\mathcal C}}
\def\CalD{{\mathcal D}}
\def\CalE{{\mathcal E}}
\def\CalF{{\mathcal F}}
\def\CalG{{\mathcal G}}
\def\CalH{{\mathcal H}}
\def\CalI{{\mathcal I}}
\def\CalJ{{\mathcal J}}
\def\CalK{{\mathcal K}}
\def\CalL{{\mathcal L}}
\def\CalM{{\mathcal M}}
\def\CalN{{\mathcal N}}
\def\CalO{{\mathcal O}}
\def\CalP{{\mathcal P}}
\def\CalQ{{\mathcal Q}}
\def\CalR{{\mathcal R}}
\def\CalS{{\mathcal S}}
\def\CalT{{\mathcal T}}
\def\CalU{{\mathcal U}}
\def\CalV{{\mathcal V}}
\def\CalW{{\mathcal W}}
\def\CalX{{\mathcal X}}
\def\CalY{{\mathcal Y}}
\def\CalZ{{\mathcal Z}}
%ÕýÌå�¡�´-------------------------------------------------------
\newcommand{\rma}{{\rm a}}
\newcommand{\rmb}{{\rm b}}
\newcommand{\rmc}{{\rm c}}
\newcommand{\rmd}{{\rm d}}
\newcommand{\rme}{{\rm e}}
\newcommand{\rmf}{{\rm f}}
\newcommand{\rmg}{{\rm g}}
\newcommand{\rmh}{{\rm h}}
\newcommand{\rmi}{{\rm i}}
\newcommand{\rmj}{{\rm j}}
\newcommand{\rmk}{{\rm k}}
\newcommand{\rml}{{\rm l}}
\newcommand{\rmm}{{\rm m}}
\newcommand{\rmn}{{\rm n}}
\newcommand{\rmo}{{\rm o}}
\newcommand{\rmp}{{\rm p}}
\newcommand{\rmq}{{\rm q}}
\newcommand{\rmr}{{\rm r}}
\newcommand{\rms}{{\rm s}}
\newcommand{\rmt}{{\rm t}}
\newcommand{\rmu}{{\rm u}}
\newcommand{\rmv}{{\rm v}}
\newcommand{\rmw}{{\rm w}}
\newcommand{\rmx}{{\rm x}}
\newcommand{\rmy}{{\rm y}}
\newcommand{\rmz}{{\rm z}}
%ÕýÌå´ó�´
\newcommand{\rmA}{{\rm A}}
\newcommand{\rmB}{{\rm B}}
\newcommand{\rmC}{{\rm C}}
\newcommand{\rmD}{{\rm D}}
\newcommand{\rmE}{{\rm E}}
\newcommand{\rmF}{{\rm F}}
\newcommand{\rmG}{{\rm G}}
\newcommand{\rmH}{{\rm H}}
\newcommand{\rmI}{{\rm I}}
\newcommand{\rmJ}{{\rm J}}
\newcommand{\rmK}{{\rm K}}
\newcommand{\rmL}{{\rm L}}
\newcommand{\rmM}{{\rm M}}
\newcommand{\rmN}{{\rm N}}
\newcommand{\rmO}{{\rm O}}
\newcommand{\rmP}{{\rm P}}
\newcommand{\rmQ}{{\rm Q}}
\newcommand{\rmR}{{\rm R}}
\newcommand{\rmS}{{\rm S}}
\newcommand{\rmT}{{\rm T}}
\newcommand{\rmU}{{\rm U}}
\newcommand{\rmV}{{\rm V}}
\newcommand{\rmW}{{\rm W}}
\newcommand{\rmX}{{\rm X}}
\newcommand{\rmY}{{\rm Y}}
\newcommand{\rmZ}{{\rm Z}}
%--ºÚ�±Ìå�¡�´-----------------------------------------------------------------
\def\bfa{\mbox{\boldmath$a$}}
\def\bfb{\mbox{\boldmath$b$}}
\def\bfc{\mbox{\boldmath$c$}}
\def\bfd{\mbox{\boldmath$d$}}
\def\bfe{\mbox{\boldmath$e$}}
\def\bff{\mbox{\boldmath$f$}}
\def\bfg{\mbox{\boldmath$g$}}
\def\bfh{\mbox{\boldmath$h$}}
\def\bfi{\mbox{\boldmath$i$}}
\def\bfj{\mbox{\boldmath$j$}}
\def\bfk{\mbox{\boldmath$k$}}
\def\bfl{\mbox{\boldmath$l$}}
\def\bfm{\mbox{\boldmath$m$}}
\def\bfn{\mbox{\boldmath$n$}}
\def\bfo{\mbox{\boldmath$o$}}
\def\bfp{\mbox{\boldmath$p$}}
\def\bfq{\mbox{\boldmath$q$}}
\def\bfr{\mbox{\boldmath$r$}}
\def\bfs{\mbox{\boldmath$s$}}
\def\bft{\mbox{\boldmath$t$}}
\def\bfu{\mbox{\boldmath$u$}}
\def\bfv{\mbox{\boldmath$v$}}
\def\bfw{\mbox{\boldmath$w$}}
\def\bfx{\mbox{\boldmath$x$}}
\def\bfy{\mbox{\boldmath$y$}}
\def\bfz{\mbox{\boldmath$z$}}
%ºÚ�±Ìå´ó�´
\def\bfA{\mbox{\boldmath$A$}}
\def\bfB{\mbox{\boldmath$B$}}
\def\bfC{\mbox{\boldmath$C$}}
\def\bfD{\mbox{\boldmath$D$}}
\def\bfE{\mbox{\boldmath$E$}}
\def\bfF{\mbox{\boldmath$F$}}
\def\bfG{\mbox{\boldmath$G$}}
\def\bfH{\mbox{\boldmath$H$}}
\def\bfI{\mbox{\boldmath$I$}}
\def\bfJ{\mbox{\boldmath$J$}}
\def\bfK{\mbox{\boldmath$K$}}
\def\bfL{\mbox{\boldmath$L$}}
\def\bfM{\mbox{\boldmath$M$}}
\def\bfN{\mbox{\boldmath$N$}}
\def\bfO{\mbox{\boldmath$O$}}
\def\bfP{\mbox{\boldmath$P$}}
\def\bfQ{\mbox{\boldmath$Q$}}
\def\bfR{\mbox{\boldmath$R$}}
\def\bfS{\mbox{\boldmath$S$}}
\def\bfT{\mbox{\boldmath$T$}}
\def\bfU{\mbox{\boldmath$U$}}
\def\bfV{\mbox{\boldmath$V$}}
\def\bfW{\mbox{\boldmath$W$}}
\def\bfX{\mbox{\boldmath$X$}}
\def\bfY{\mbox{\boldmath$Y$}}
\def\bfZ{\mbox{\boldmath$Z$}}

%\def\bfE{\mbox{\boldmath$E$}}
%\def\bfG{\mbox{\boldmath$G$}}

%--ºÚ�±Ìå�£À°´ó�´----------------------------------
\def\bfDelta{\mbox{\boldmath$\Delta$}}
\def\bfGamma{\mbox{\boldmath$\Gamma$}}
\def\bfTheta{\mbox{\boldmath$\Theta$}}
\def\bfLambda{\mbox{\boldmath$\Lambda$}}
\def\bfXi{\mbox{\boldmath$\Xi$}}
\def\bfPi{\mbox{\boldmath$\Pi$}}
\def\bfSigma{\mbox{\boldmath$\Sigma$}}
\def\bfUpsilon{\mbox{\boldmath$\Upsilon$}}
\def\bfPhi{\mbox{\boldmath$\Phi$}}
\def\bfPsi{\mbox{\boldmath$\Psi$}}
\def\bfOmega{\mbox{\boldmath$\Omega$}}
%ºÚ�±Ìå�£À°�¡�´
\def\bfalpha{\mbox{\boldmath${\alpha}$}}
\def\bfbeta{\mbox{\boldmath${\beta}$}}
\def\bfgamma{\mbox{\boldmath${\gamma}$}}
\def\bfdelta{\mbox{\boldmath${\delta}$}}
\def\bfepsilon{\mbox{\boldmath${\epsilon}$}}
\def\bfvarepsilon{\mbox{\boldmath${\varepsilon}$}}
\def\bfzeta{\mbox{\boldmath${\zeta}$}}
\def\bfeta{\mbox{\boldmath${\eta}$}}         % note the variation
\def\bftheta{\mbox{\boldmath${\theta}$}}
\def\bfiota{\mbox{\boldmath${\iota}$}}
\def\bfkappa{\mbox{\boldmath${\kappa}$}}
\def\bflambda{\mbox{\boldmath${\lambda}$}}
\def\bfmu{\mbox{\boldmath${\mu}$}}
\def\bfnu{ {\bf\nu}}
\def\bfxi{\mbox{\boldmath${\xi}$}}
\def\bfomicron{\mbox{\boldmath ${\omicron}$}}
\def\bfpi{\mbox{\boldmath${\pi}$}}
\def\bfrho{\mbox{\boldmath${\rho}$}}
\def\bfsigma{\mbox{\boldmath${\sigma}$}}
\def\bftau{\mbox{\boldmath${\tau}$}}
\def\bfphi{\mbox{\boldmath${\phi}$}}
\def\bfchi{\mbox{\boldmath${\chi}$}}
\def\bfupsilon{\mbox{\boldmath${\upsilon}$}}
\def\bfpsi{\mbox{\boldmath${\psi}$}}
\def\bfomega{\mbox{\boldmath${\omega}$}}
%
\def\bfvaralpha{\mbox{\boldmath${\varalpha}$}}
\def\bfvarbeta{\mbox{\boldmath${\varbeta}$}}
\def\bfvargamma{\mbox{\boldmath${\vargamma}$}}
\def\bfvardelta{\mbox{\boldmath${\vardelta}$}}
\def\bfvarepsilon{\mbox{\boldmath${\varepsilon}$}}
\def\bfvarzeta{\mbox{\boldmath${\varzeta}$}}
\def\bfvareta{\mbox{\boldmath${\vareta}$}}
\def\bfvartheta{\mbox{\boldmath${\vartheta}$}}
\def\bfvariota{\mbox{\boldmath${\variota}$}}
\def\bfvarkappa{\mbox{\boldmath${\varkappa}$}}
\def\bfvarlambda{\mbox{\boldmath${\varlambda}$}}
\def\bfvarmu{\mbox{\boldmath${\varmu}$}}
\def\bfvarnu{\mbox{\boldmath${\varnu}$}}
\def\bfvarxi{\mbox{\boldmath${\varxi}$}}
\def\bfvaromicron{\mbox{\boldmath ${\varomicron}$}}
\def\bfvarpi{\mbox{\boldmath${\pi}$}}
\def\bfvarrho{\mbox{\boldmath${\varrho}$}}
\def\bfvarsigma{\mbox{\boldmath${\varsigma}$}}
\def\bfvartau{\mbox{\boldmath${\vartau}$}}
\def\bfvarphi{\mbox{\boldmath${\varphi}$}}
\def\bfvarchi{\mbox{\boldmath${\varchi}$}}
\def\bfvarupsilon{\mbox{\boldmath${\varupsilon}$}}
\def\bfvarpsi{\mbox{\boldmath${\varpsi}$}}
\def\bfvaromega{\mbox{\boldmath${\varomega}$}}


%--ÕýºÚÌå�¡�´------------------------------------------------------------
\def\rbfa{{\bf a}}
\def\rbfb{{\bf b}}
\def\rbfc{{\bf c}}
\def\rbfd{{\bf d}}
\def\rbfe{{\bf e}}
\def\rbff{{\bf f}}
\def\rbfg{{\bf g}}
\def\rbfh{{\bf h}}
\def\rbfi{{\bf i}}
\def\rbfj{{\bf j}}
\def\rbfk{{\bf k}}
\def\rbfl{{\bf l}}
\def\rbfm{{\bf m}}
\def\rbfn{{\bf n}}
\def\rbfo{{\bf o}}
\def\rbfp{{\bf p}}
\def\rbfq{{\bf q}}
\def\rbfr{{\bf r}}
\def\rbfs{{\bf s}}
\def\rbft{{\bf t}}
\def\rbfu{{\bf u}}
\def\rbfv{{\bf v}}
\def\rbfw{{\bf w}}
\def\rbfx{{\bf x}}
\def\rbfy{{\bf y}}
\def\rbfz{{\bf z}}
%ÕýºÚÌå´ó�´
\def\rbfA{{\bf A}}
\def\rbfB{{\bf B}}
\def\rbfC{{\bf C}}
\def\rbfD{{\bf D}}
\def\rbfE{{\bf E}}
\def\rbfF{{\bf F}}
\def\rbfG{{\bf G}}
\def\rbfH{{\bf H}}
\def\rbfI{{\bf I}}
\def\rbfJ{{\bf J}}
\def\rbfK{{\bf K}}
\def\rbfL{{\bf L}}
\def\rbfM{{\bf M}}
\def\rbfN{{\bf N}}
\def\rbfO{{\bf O}}
\def\rbfP{{\bf P}}
\def\rbfQ{{\bf Q}}
\def\rbfR{{\bf R}}
\def\rbfS{{\bf S}}
\def\rbfT{{\bf T}}
\def\rbfU{{\bf U}}
\def\rbfV{{\bf V}}
\def\rbfW{{\bf W}}
\def\rbfX{{\bf X}}
\def\rbfY{{\bf Y}}
\def\rbfZ{{\bf Z}}
%ÕýºÚÌå�£À°´ó�´
\def\rbfDelta{{\bf \Delta}}
\def\rbfGamma{{\bf \Gamma}}
\def\rbfTheta{{\bf \Theta}}
\def\rbfLambda{{\bf \Lambda}}
\def\rbfXi{{\bf \Xi}}
\def\rbfPi{{\bf \Pi}}
\def\rbfSigma{{\bf \Sigma}}
\def\rbfUpsilon{{\bf \Upsilon}}
\def\rbfPhi{{\bf \Phi}}
\def\rbfPsi{{\bf \Psi}}
\def\rbfOmega{{\bf \Omega}}
%ÕýºÚÌå�£À°�¡�´
\def\rbfalpha{\mbox{\boldmath${\alpha}$}}
\def\rbfbeta{\mbox{\boldmath${\beta}$}}
\def\rbfgamma{\mbox{\boldmath${\gamma}$}}
\def\rbfdelta{\mbox{\boldmath${\delta}$}}
\def\rbfepsilon{\mbox{\boldmath${\epsilon}$}}
\def\rbfvarepsilon{\mbox{\boldmath${\varepsilon}$}}
\def\rbfzeta{\mbox{\boldmath${\zeta}$}}
\def\rbfeta{\mbox{\boldmath${\eta}$}}         % note the variation
\def\rbftheta{\mbox{\boldmath${\theta}$}}
\def\rbfiota{\mbox{\boldmath${\iota}$}}
\def\rbfkappa{\mbox{\boldmath${\kappa}$}}
\def\rbflambda{\mbox{\boldmath${\lambda}$}}
\def\rbfmu{\mbox{\boldmath${\mu}$}}
\def\rbfnu{ {\bf\nu}}
\def\rbfxi{\mbox{\boldmath${\xi}$}}
\def\rbfomicron{\mbox{\boldmath ${\omicron}$}}
\def\rbfpi{\mbox{\boldmath${\pi}$}}
\def\rbfrho{\mbox{\boldmath${\rho}$}}
\def\rbfsigma{\mbox{\boldmath${\sigma}$}}
\def\rbftau{\mbox{\boldmath${\tau}$}}
\def\rbfphi{\mbox{\boldmath${\phi}$}}
\def\rbfchi{\mbox{\boldmath${\chi}$}}
\def\rbfupsilon{\mbox{\boldmath${\upsilon}$}}
\def\rbfpsi{\mbox{\boldmath${\psi}$}}
\def\rbfomega{\mbox{\boldmath${\omega}$}}
%ÕýºÚÌå�£À°�¡�´ var
\def\rbfvaralpha{\mbox{\boldmath${\varalpha}$}}
\def\rbfvarbeta{\mbox{\boldmath${\varbeta}$}}
\def\rbfvargamma{\mbox{\boldmath${\vargamma}$}}
\def\rbfvardelta{\mbox{\boldmath${\vardelta}$}}
\def\rbfvarepsilon{\mbox{\boldmath${\varepsilon}$}}
\def\rbfvarzeta{\mbox{\boldmath${\varzeta}$}}
\def\rbfvareta{\mbox{\boldmath${\vareta}$}}
\def\rbfvartheta{\mbox{\boldmath${\vartheta}$}}
\def\rbfvariota{\mbox{\boldmath${\variota}$}}
\def\rbfvarkappa{\mbox{\boldmath${\varkappa}$}}
\def\rbfvarlambda{\mbox{\boldmath${\varlambda}$}}
\def\rbfvarmu{\mbox{\boldmath${\varmu}$}}
\def\rbfvarnu{\mbox{\boldmath${\varnu}$}}
\def\rbfvarxi{\mbox{\boldmath${\varxi}$}}
\def\rbfvaromicron{\mbox{\boldmath ${\varomicron}$}}
\def\rbfvarpi{\mbox{\boldmath${\pi}$}}
\def\rbfvarrho{\mbox{\boldmath${\varrho}$}}
\def\rbfvarsigma{\mbox{\boldmath${\varsigma}$}}
\def\rbfvartau{\mbox{\boldmath${\vartau}$}}
\def\rbfvarphi{\mbox{\boldmath${\varphi}$}}
\def\rbfvarchi{\mbox{\boldmath${\varchi}$}}
\def\rbfvarupsilon{\mbox{\boldmath${\varupsilon}$}}
\def\rbfvarpsi{\mbox{\boldmath${\varpsi}$}}
\def\rbfvaromega{\mbox{\boldmath${\varomega}$}}
%----------------------------------------------




% 5. shortened caligraphic symbolics
\begin{document}
Consider a continuous-time system with the following state space
representation
\begin{equation}
\label{re_01a}
 P:\ \ \ \  \left\{\begin{array}{ccl} \dot{x}(t)&=&A x(t)+Bu(t),\\
    y(t) &=& Cx(t)+D u(t),\end{array}\right.
\end{equation}
where $x(t)\in\R^n$, $u(t)\in\R^r$ and $y(t)\in\R^m$ are the state
vector, input vector and output vector of the system,
respectively; $A\in\R^{n\times n}$, $B\in\R^{n\times r}$,
$C\in\R^{m\times n}$ and $D\in\R^{m\times r}$ are the constant
real or complex matrices.

Suppose that the sampling interval is $\tau$. By using the step
invariance transform or the zero-order hold (ZOH), i.e.,
$u(t)=u(k\tau),\ k\tau\leq t<(k+1)\tau$, discretizing the system
in (\ref{re_01a}) gives a discrete-time model,
\begin{equation}\label{re_01c}
 P_\tau:\ \ \ \ \left\{\begin{array}{ccl}x(k\tau+\tau)&=&G_\tau x(k\tau) +F_\tau u(k\tau),\\
              y(k\tau) &=& C x(k\tau)+Du(k\tau),\ k=0, 1, 2, \cdots\end{array}\right.
\end{equation}
where $x(k\tau)=x(t)\left|_{t=k\tau}\right.$,
$y(k\tau)=y(t)\left|_{t=k\tau}\right.$, and
\begin{equation}\label{re_01d}
 G_\tau:=\rme^{A \tau}, \ F_\tau:=\int^{\tau}_{0}\rme^{A t}\rmd t\ B.
\end{equation}
%%%%%
%%%%%
\end{document}