pwa_structure.tex

由matlab开发的hybrid系统的描述语言

\chapter{PWA structure}
\section{Detailed Description}

An PWA system can be rewritten in this way:
\begin{eqnarray}
      x(t+1)&=&A_ix(t)+B_iu(t)+f_i\\
      y(t)&=&C_ix(t)+D_iu(t)+g_i\\
      &&\mbox{if} \PP_i=P_ix(t)+Q_iu(t)\le r_i+s_i\epsilon \label{eq:polyhedron}
\end{eqnarray}
and $\cup \PP_i=\{x(t),u(t): Tx(t)+Vu(t)\le w,\forall t\}$ {\tt
insert some comments on the role of $\epsilon$ $s_i(j)=0
\Rightarrow \le \mbox{constraint}$ and $s_i(j)=1 \Rightarrow <
\mbox{constrain}$}

The structure {\tt PWAsystem} contains the following fields:
\begin{itemize}
\item[{\tt A\{i\}($\cdot$,$\cdot$)}] $A_i$ \item[{\tt
B\{i\}($\cdot$,$\cdot$)}] $B_{i}$ \item[{\tt f\{i\}($\cdot$)}]
$f_{i}$ \item[{\tt C\{i\}($\cdot$,$\cdot$)}] $C_{i}$ \item[{\tt
D\{i\}($\cdot$,$\cdot$)}] $D_{i}$ \item[{\tt g\{i\}($\cdot$)}]
$g_{i}$ \item[{\tt P\{i\}($\cdot$,$\cdot$)}] $P_i$ \item[{\tt
Q\{i\}($\cdot$,$\cdot$)}] $Q_i$ \item[{\tt r\{i\}($\cdot$)}]
$r_{i}$ \item[{\tt s\{i\}($\cdot$)}] $s_{i}$ \item[{\tt
T($\cdot$,$\cdot$)}] $T_i$ \item[{\tt V($\cdot$,$\cdot$)}] $V_i$
\item[{\tt w($\cdot$)}] $w_{i}$ \item[{\tt nr}] number of regions
\item[{\tt nx}] size of $x$ \item[{\tt nu}] size of $u$ \item[{\tt
ny }] size of $y$ \item[{\tt PWAisvalid}] 1 if the model has
passed a validity check (sizes of the matrices, $P_i$, $Q_i$,
$r_i$ and $s_i$ define a partition \ldots). User functions will
trust this information \item[{\tt name}] string containing the
system name as specified in the \hysdel{} declaration \item[{\tt
PWAstructver}] integer containing the protocol version, needed for
future extensions. All the protocol versions should be backwards
compatible. User's function should check if the protocol version
is new enough.
\end{itemize}

This PWA structure is proposed as a standard for m-files developed
at IfA. This will ease file sharing between members of the hybrid
group.