models.tex

由matlab开发的hybrid系统的描述语言

        f^i_{\rm H}(x_r(k),u_r(k),k)&>&m^i\delta^i_e{\rm ,} ~~~~~~~~~~~~i=1,\ldots,n_e {\rm ,}\label{eq:ti2}
\eeqar  where $M^i$, $m^i$ are upper and lower bounds,
respectively, on $f^i_{\rm H}(x_r(k),u_r(k),k)$. As we will point
out in Section~\ref{sect.mld}, sometimes from a computational
point of view, it may be convenient to have a system of
inequalities without strict inequalities. In this case we will
follow the common practice~\cite{Will93} to replace the strict
Inequality ~(\ref{eq:ti2}) as
\begin{equation}
       f^i_{\rm H}(x_r(k),u_r(k),k) \geq\epsilon+(m-\epsilon)\delta\label{eq:ti3}
\end{equation}
where $\epsilon$ is a small positive scalar, e.g. the machine
precision, although the equivalence does not hold for $ 0 \le
f^i_{\rm H}(x_r(k),u_r(k),k)<\epsilon$. \label{eq:threshold-ineq}
\end{subequations}

The most common {\em logic to continuous} interface is the
if-then-else construct
\begin{equation}
  \mbox{IF}\ \delta\ \mbox{THEN}\ z=a_1^T x+b_1^Tu+f_1\ \mbox{ELSE}\
  z=a_2^T x -b_2^T u +f_2
    \label{eq:if-then-else}
\end{equation}
which can be translated into~\cite{BTM01a}
\begin{subequations}
    \beqar
          (m_2-M_1)\delta+z&\leq &a_2x+b_2u+f_2, \\
          (m_1-M_2)\delta-z&\leq &-a_2x-b_2u-f_2, \\
          (m_1-M_2)(1-\delta)+z&\leq &a_1x+b_1u+f_1, \\
          (m_2-M_1)(1-\delta)-z&\leq &-a_1x-b_1u-f_1{\rm ,}
    \eeqar
    \label{eq:if-ineq}
\end{subequations}

\noindent where $M_i$, $m_i$ are upper and lower bounds on
$a_ix+b_iu+f_i$, $i=1,2$, $\delta \in \bol$, $z \in \rr$, $x \in
\rr^n$, $u \in \rr^m$. Note that
(\ref{eq:if-then-else})--(\ref{eq:if-ineq}) are a generalization
of the real product $z=\delta \cdot (ax+bu+f)$ described
in~\cite{Will93}.

\subsection{Continuous Dynamics}
\label{sect.cont} As already mentioned, we will deal
with dynamics described by linear affine %differential or
difference equations
\begin{equation}
  x'(k) = \sum_{i=1}^s z_i(k).
\label{eq:linear}
\end{equation}

\subsection{Mixed Logical Dynamical (MLD) Systems}
\index{Mixed Logical Dynamical Systems} \label{sect.mld}
In~\cite{BM99}, the authors proposed discrete-time hybrid systems
denoted as mixed logical dynamical (MLD) systems. An MLD system is
described by the following relations:
\begin{subequations}
\begin{align}
    x'(k)&=\Aa x(k)+\BB_{1}u(k)+\BB_{2}\delta(k)+\BB_{3}z(k)+\BB_5,\label{MLDS-a}\\
    y(k)&=\Cc x(k)+\DD_{1}u(k)+\DD_{2}\delta(k)+\DD_{3}z(k)+\DD_5,\label{MLDS-b}\\
    \EE_{2}\delta(k)+\EE_{3}z(k)&\leq
    \EE_{1}u(k)+\EE_{4}x(k)+\EE_{5},\label{MLDS-c}\\
    \tilde \EE_{2}\delta(k)+\tilde \EE_{3}z(k)&<\tilde \EE_{1}u(k)+\tilde \EE_{4}x(k)+\tilde \EE_{5} {\rm .}\label{MLDS-d}
\end{align}
\label{MLDS-all}
\end{subequations}
\noindent where $x\in\rr^{n_r}\times\{0,1\}^{n_b}$ is a vector of
continuous and binary states, $u\in \rr^{m_r}\times\{0,1\}^{m_b}$
are the inputs, $y\in \rr^{p_r}\times\{0,1\}^{p_b}$ the outputs,
$\delta\in\{0,1\}^{r_b}$, $z\in\rr^{r_r}$ represent auxiliary
binary and continuous variables, respectively, and $\Aa$, $\BB_1$,
$\BB_2$, $\BB_3$, $\Cc$, $\DD_1$, $\DD_2$, $\DD_3$,
$\EE_1$,\ldots,$\EE_5$ and $\tilde \EE_1$,\ldots,$\tilde \EE_5$
are matrices of suitable dimensions. Given the current state
$x(k)$ and input $u(k)$, the time-evolution of~(\ref{MLDS-all}) is
determined by solving $\delta(k)$ and $z(k)$
from~(\ref{MLDS-c})--(\ref{MLDS-d}), and then updating $x'(k)$ and
$y(k)$ from~(\ref{MLDS-a})--(\ref{MLDS-b}). The equations and
inequalities obtained with methods presented in
 Sections~\ref{sec.purelylogic}, \ref{sect.cli}, \ref{sect.cont} can
 be represented using the MLD framework.
When the problems of synthesis and analysis of MLD models are
tackled by optimization techniques, it is convenient to replace
the strict inequalities as in (\ref{eq:ti3}). We will therefore
consider an MLD model where the matrices $\tilde
\EE_1$,\ldots,$\tilde \EE_5$ are embedded in (\ref{MLDS-c}) as
nonstrict inequalities. For MLD systems, well-posedness is defined
similarly to Definition~\ref{def:well-posed}.

\begin{lemma}
Let $\Sigma_{\rm DHA}$ be a well-posed DHA model defined on a set
of states $\XX\subseteq\rr^n$, a set of inputs
$\UU\subseteq\rr^m$, and a set of outputs $\YY\subseteq\rr^p$.
Then there exists a well-posed MLD model $\Sigma_{\rm MLD}$ such
that $\Sigma_{\rm MLD}\stackrel{\XX,\UU,\YY}
{\rightsquigarrow}\Sigma_{\rm DHA}$ under the identity state
transformation $T(x)=x$. \label{lemma:MLDDHA}
\end{lemma}
\noindent\proof Directly follows from
Sections~\ref{sec.purelylogic}, \ref{sect.cli}, \ref{sect.cont}.
\cvd

\section{Other Computational Models}
 In the previous section we showed the equivalence
relations between DHA and PWA and MLD systems. In this section, we
review other existing models of linear hybrid systems and show
further relationships with DHA.

%\subsection{Piecewise Affine (PWA) Systems}

\subsection{Linear Complementarity (LC) Systems}
\index{Linear Complementarity Systems}\index{LC|see{Linear
Complementarity Systems}} Linear complementarity (LC) systems are
given in discrete-time by the equations
 \begin{subequations}\label{eq.lcs}
\begin{eqnarray}
{x}'(k) & = & Ax(k) + B_1 u(k) +B_2 w(k), \label{diff}\\
y(k) & = & Cx(k) + D_1 u(k) + D_2 w(k), \label{output} \\
v(k) & = & E_1 x(k) + E_2 u(k) + E_3 w(k) + E_4, \label{supply}\\
0 \leq v(k) & \bot & w(k)\geq 0 \label{comp}
\end{eqnarray}
\end{subequations}
with $v(k), w(k) \in \rr^q$ and where $\bot$ denotes the
orthogonality of vectors (i.e.\ $v(k)\bot w(k)$ means that
$v^T(k)w(k)=0$). We call $v(k)$ and $w(k)$ the complementarity
variables. $A$, $B_i$, $C$, $D_i$ and $E_i$ are real
matrices~\cite{vdSS98,HSW00,CHS01,Hee:99,vdSS00}.

In \cite{HDB01} the relationships among the model classes
mentioned above and two others, {\em min-max-plus-scaling} (MMPS)
and {\em extended linear complementarity} (ELC) systems, were
discussed. As ELC systems are of similar nature as LC systems, we
will not define them here, but refer to \cite{dSdM99,HDB01}. MMPS
systems are obtained by choosing the state-update function, the
output function, and constraints as (nested) combinations of the
operations maximization, minimization, addition and scalar
multiplication.  More detail on this class can be found in
\cite{dSvdB01,HDB01}.

\begin{fact}
\label{thm:big5} PWA, MLD, LC, ELC, and MMPS models are equivalent
classes of hybrid models (certain equivalences require assumptions
on the boundedness of input, state, and auxiliary variables or  on
well-posedness).
\end{fact}
\noindent\proof See \cite{HDB01} for full detail on assumptions,
relationships, and a constructive proof.\cvd

\begin{psfrag}
    \psfragscanon \psfrag{L}{{\sf Lemma~\ref{lemma:DHAPWA}}}
    \psfragscanon \psfrag{M}{{\sf Lemma~\ref{lemma:MLDDHA}}}
    \psfragscanon \psfrag{F}{{\sf Fact~\ref{thm:big5}}}
\figuraeps{equivalences}{t}{.50\textwidth}{Conceptual scheme for
the proof of Theorem~\ref{thm:bi6}}{fig:equivalences}
\end{psfrag}

\begin{theorem}
\label{thm:bi6} Let $\XX$, $\UU$, $\YY$ be sets of states, inputs,
and outputs respectively, and assume that $\XX$, $\UU$ are
bounded. Then DHA, PWA, MLD, LC, ELC, and MMPS well-posed models
are equivalent to each other on $\XX$, $\UU$, $\YY$.
\end{theorem}
\noindent\proof By referring to Figure~\ref{fig:equivalences},
mutual equivalences among PWA, MLD, LC, ELC, and MMPS on bounded
$\XX$, $\UU$, $\YY$ follows from Fact~\ref{thm:big5}. The
equivalence $\Sigma_{\rm DHA}\stackrel{\XX,\UU,\YY}
{\rightsquigarrow}\Sigma_{\rm PWA}$ follows by
Lemma~\ref{lemma:DHAPWA}, while the equivalence $\Sigma_{\rm
MLD}\stackrel{\XX,\UU,\YY} {\rightsquigarrow}\Sigma_{\rm DHA}$
follows by Lemma~\ref{lemma:MLDDHA}. Therefore, any equivalence
relation can be stated for any ordered pairs of models.

\cvd

Thanks to the equivalences mentioned above, it is clear that
\hysdel{} is a tool that allows generating several different
hybrid models of a given hybrid system. In particular, \hysdel{}
generates MLD models, which can be immediately (and efficiently)
translated into PWA systems~\cite{Bem02a}, or LC/ELC/MMPS systems
using the constructive methods reported in~\cite{HDB01,HDB01b}.
Note that by Propositions~\ref{prop:resets}
and~\ref{prop:resets-pred} also DHA models with resets are
equivalent to any of the other classes of hybrid models.