dha.tex

由matlab开发的hybrid系统的描述语言

\chapter{Discrete Hybrid Automata}
\index{DHA|see{Discrete Hybrid Automata}}%
\index{SAS|see{Switched Affine System}}%
\index{EG|see{Event Generator}}%
\index{FSM|see{Finite State Machine}}%
\index{MS|see{Mode Selector}}%

\label{sect::DHA} \emph{Discrete hybrid automata}\index{Discrete
Hybrid Automata} (DHA) are the interconnection of a finite state
machine and a switched linear dynamic system through a mode
selector  and an event generator (see Figure~\ref{fig:dha}).
\figuraeps{model}{t}{.90\textwidth}{A discrete hybrid automaton
(DHA) is the connection of a finite state machine (FSM) and a
switched affine system (SAS), through a mode selector (MS) and an
event generator (EG). The output signals are omitted for
clarity}{fig:dha}

In the following we will use the fact that any discrete variable
$\alpha \in \{\alpha_1,\ldots,\alpha_j\}$, admits a Boolean
encoding $a \in \bol^{d(j)}$. From now on we will refer to either
 the variable or its encoding with
the same name.

\section{Switched Affine System \emph{(SAS)}} \index{Switched Affine System}
A switched affine
system is a collection of linear affine systems:
\begin{subequations}
\begin{eqnarray}
    x'_r(k) &=& A_{i(k)}x_r(k) + B_{i(k)}u_r(k) + f_{i(k)}\label{eq:sas-x}\\
    y_r(k) &=& C_{i(k)}x_r(k) + D_{i(k)}u_r(k) + g_{i(k)}\label{eq:sas-y}{\rm ,}
\end{eqnarray}%
\label{eq::sas}%
\end{subequations}%
where $k\in \Z^+$ is the time indicator, $'$ denotes the successor
operator ($x'_r(k) = x_r(k+1)$), $x_r \in \XX_r\subseteq\rr^{n_r}$
is the continuous state vector, $u_r \in \UU_r\subseteq\rr^{m_r}$
is the exogenous continuous input vector, $y_r \in
\YY_r\subseteq\rr^{p_r}$ is the continuous output vector,
$\{A_{i},B_{i},f_{i},C_{i},D_{i},g_{i}\}_{i\in\II}$ is a
collection of matrices of suitable dimensions, and the mode $i(k)
\in \II$ is an input signal that chooses the linear state update
dynamics. We denote by $\sharp\II = s$ the number of elements in
$\II$. A SAS of the form~(\ref{eq::sas}) preserves the value of
the state when a switch occurs, however it is possible to
implement reset maps on a SAS, as we will show later in this
section. A SAS can be rewritten as the combination of linear terms
and \emph{if-then-else} rules: the state-update
Equation~(\ref{eq:sas-x}) is equivalent to
\begin{subequations}
\label{eq::z-transform} \beqar
    z_1(k) &=& \left\{
  \begin{array}{ll}
    A_1 x_r(k) + B_1 u_r(k) + f_1, & {\rm if~}  (i(k) = 1), \\
    0, & {\rm otherwise},
  \end{array} \right. \label{eq::z-transform-a}\\
  & \vdots & \nonumber \\
    z_s(k) &=& \left\{
  \begin{array}{ll}
    A_s x_r(k) + B_s u_r(k) + f_s & {\rm if~}  (i(k) = s), \\
    0 & {\rm otherwise}, \\
  \end{array} \right.
  \label{eq::z-transform-b}\\
  x'_r(k) &=& \sum_{i=1}^s z_i(k) \label{eq::z-transform-c}
 \eeqar
\end{subequations}
where $z_i(k)\in \rr ^{n_r},i = 1,\ldots,s$, and~(\ref{eq:sas-y})
admits a similar transformation.

\section{Event Generator \emph{(EG)}} \index{Event Generator} An event generator is a
mathematical object that generates a logic signal according to the
satisfaction of a linear affine constraint:
\begin{equation}
\label{eq::eventgenerator}
    \delta_e(k)= f_{\rm H}(x_r(k),u_r(k),k){\rm ,}
\end{equation}
where $f_{\rm H}: \rr^{n_r} \times \rr^{m_r} \times \Z_{\ge 0}
\rightarrow \DE \subseteq  \bol^{n_e}$ is a vector of descriptive
functions of a linear hyperplane, and $\Z_{\geq
0}\eqdef\{0,1,\ldots\}$ is the set of nonnegative integers. In
particular, {\em time events} are modeled as: $[\delta^i_e(k) = 1]
\leftrightarrow [kT_s \ge t_0]$, where $T_s$ is the sampling time,
while  {\em threshold events} are modeled as: $[\delta^i_e(k) = 1]
\leftrightarrow [a^Tx_r(k) + b^Tu_r(k) \le c]$, where the
superscript $i$ denotes the $i$-th component of a vector.

\section{Finite State Machine (\emph{FSM})} \index{Finite State Machine}
A finite state machine\footnote{In this paper we will only refer
to synchronous finite state machines, where the transitions may
happen only at sampling times. The adjective synchronous will be
omitted for brevity.} (or automaton) is a discrete dynamic process
that evolves according to a logic state update function:
\begin{subequations}
\begin{equation}
    x_b'(k) = f_{\rm B}(x_b(k),u_b(k),\delta_e(k)){\rm ,}
\end{equation}
where $x_b \in \XX_b \subseteq \bol^{n_b}$ is the Boolean state,
$u_b \in \UU_b  \subseteq \bol^{m_b}$ is the exogenous Boolean
input, $\delta_e(k)$ is the endogenous input coming from the EG,
and $f_{\rm B}: \XX_b \times \UU_b \times \DE \rightarrow \XX_b$
is a deterministic logic function. A FSM can be conveniently
represented using an oriented graph. A FSM may also have an
associated Boolean output
\begin{equation}
    y_b(k) = g_{\rm B}(x_b(k),u_b(k),\delta_e(k)){\rm ,}
\end{equation}
\end{subequations}
where $y_b \in \YY_b \subseteq \bol^{p_b}$.

\figuraeps{fsm1}{t}{.4\textwidth}{Example of finite state
machine}{fig:fsm}
\begin{example}
\label{ex:automa}  Figure~\ref{fig:fsm} shows a finite state
machine where $u_b=[u_{b1}~u_{b2}]^T$ is the input vector, and
$\delta = [\delta_1 \ldots \delta_4]^T$ ia a vector of signals
coming from the event generator. The state transition function is:
\begin{equation}
    x_b'(k) = \left\{
    \begin{array}{lcl}
    {\sf Red} & {\rm if } & ((x_b(k) = {\sf Green}) \ANd d_4) \OR ((x_b(k) = {\sf Red}) \ANd \NOT u_{b2} \ANd \NOT \delta_1), \\
    {\sf Green} & {\rm if } & ((x_b(k) = {\sf Red}) \ANd u_{b2}) \OR ((x_b(k) = {\sf Blue}) \ANd \delta_2) \OR \\
    & & ((x_b(k) = {\sf Green}) \ANd \NOT \delta_4 \ANd \NOT (u_{b1}\ANd\delta_3)), \\
    {\sf Blue} & {\rm if } & ((x_b(k) = {\sf Red}) \ANd \delta_1) \OR ((x_b(k) = {\sf Green}) \ANd (u_{b1}\ANd\delta_3)) \OR\\
    && ((x_b(k) = {\sf Blue}) \ANd \NOT \delta_2))\mbox{\rm .} \\
    \end{array}
    \right.
    \label{eq:exauto}
\end{equation}
By associating a Boolean vector $x_b = \smallmat{x_{b1}\\x_{b2}}$
to each state (${\sf Red} = \smallmat{0\\0}$, ${\sf Green} =
\smallmat{0\\1}$, and ${\sf Blue} = \smallmat{1\\0}$), one can
rewrite~(\ref{eq:exauto}) as:
\begin{eqnarray*}
    x_{b1}' &=&
    \NOT ((\NOT x_{b1} \ANd x_{b2}  \ANd d_4) \OR (\NOT x_{b1} \ANd \NOT x_{b2}  \ANd \NOT u_{b2} \ANd \NOT
    \delta_1))\\
    && \OR \NOT ((\NOT x_{b1} \ANd \NOT x_{b2}  \ANd u_{b2}) \OR (x_{b1} \ANd \NOT x_{b2}  \ANd \delta_2) \OR (\NOT x_{b1} \ANd x_{b2}  \ANd \NOT \delta_4 \ANd \NOT
    (u_{b1}\ANd\delta_3)))\\
    && \OR ((\NOT x_{b1} \ANd \NOT x_{b2}  \ANd \delta_1) \OR (\NOT x_{b1} \ANd x_{b2}  \ANd (u_{b1}\ANd\delta_3)) \OR (x_{b1} \ANd \NOT x_{b2}  \ANd \NOT
    \delta_2)), \\
    x_{b2}' &=&
    \NOT ((\NOT x_{b1} \ANd x_{b2}  \ANd d_4) \OR (\NOT x_{b1} \ANd \NOT x_{b2}  \ANd \NOT u_{b2} \ANd \NOT
    \delta_1)) \\
    && \OR ((\NOT x_{b1} \ANd \NOT x_{b2}  \ANd u_{b2}) \OR (x_{b1} \ANd \NOT x_{b2}  \ANd \delta_2) \OR (\NOT x_{b1} \ANd x_{b2}  \ANd \NOT \delta_4 \ANd \NOT
    (u_{b1}\ANd\delta_3)))\\
    && \OR \NOT ((\NOT x_{b1} \ANd \NOT x_{b2}  \ANd \delta_1) \OR (\NOT x_{b1} \ANd x_{b2}  \ANd (u_{b1}\ANd\delta_3)) \OR (x_{b1} \ANd \NOT x_{b2}  \ANd \NOT
    \delta_2)){\rm ,}
\end{eqnarray*}
\end{example}
where the time index, $k$, has been omitted for brevity.


\section{Mode Selector (\emph{MS})} \index{Mode Selector}
The logic state $x_b(k)$,
 the Boolean inputs $u_b(k)$, and the events
$\delta_e(k)$ select the dynamic mode $i(k)$ of the SAS through a
Boolean function $f_{\rm M}: \XX_b \times \UU_b \times \DE
\rightarrow \II$, which is therefore called {\em mode selector}.
The output of this function
\begin{equation}
  i(k)=f_{\rm M}(x_b(k),u_b(k),\delta_e(k))
    \label{eq:MS}
\end{equation}
is called {\em active mode}. We say that a {\em mode switch}
occurs at step $k$ if $i(k)\neq i(k-1)$. Note that contrarily to
continuous time hybrid models, where switches can occur at any
time, in our discrete-time setting a mode switch can only occur at
sampling instants.


\section{Reset Maps}
\index{Reset Maps} In correspondence with a mode switch $i(k)=j$,
$i(k-1)=h\neq j$, $h,j\in\II$, instead of evolving
$x'_r(k)=A_jx_r(k)+B_ju_r(k)+f_j$ it is possible to associate a
reset of the continuous state vector
\begin{equation}
    x_r'(k)=\phi_{\overrightarrow{hj}}(x_r(k),u_r(k))=A_{\overrightarrow{hj}}x_r(k)+B_{\overrightarrow{hj}}u_r(k)+f_{\overrightarrow{hj}}
\end{equation}
the function $\phi_{\overrightarrow{hj}}$ is called \emph{reset
map}. The reset can be considered as a special dynamics that only
acts for one sampling step. Figure~\ref{fig:reset-a} shows how a
reset affects the state evolution: At time $k=5$ the system is in
mode $i(5)=1$, at time $k=6$ the state
$x_r(6)=A_1x_r(5)+B_1u_r(5)+f_1$ enters the region $x_r\ge 0$.
This generates an event $\delta_e(6)$ through the EG, which in
turn causes the MS to change the system dynamics to $i(6)=2$. The
mode switch $1\rightarrow 2$ resets
$x_r(7)=A_{\overrightarrow{12}}x_r(6)+
B_{\overrightarrow{12}}u_r(6)+ f_{\overrightarrow{12}}$. If the
state $x_r(7)$ after reset belongs again to the region where the
mode $2$ is active, $i(7)=2$, the successor state is
$x_r(8)=A_2x_r(7)+B_2u_r(7)+f_2$. It might even happen that
$x_r(7)$ belongs to another region, say a region where mode $3$ is
active, $i(7)=3$: In this case, since $i(6)\neq i(7)$, a further
reset $2\rightarrow 3$ is applied,
$x_r(8)=A_{\overrightarrow{23}}x_r(7)+
B_{\overrightarrow{23}}u_r(7)+f_{\overrightarrow{23}}$.

\newcommand{\res}{{\rm a}}

\begin{proposition}
A DHA $\Sigma^{\res}$ with reset conditions can be rewritten as a
DHA $\Sigma$ without reset conditions. \label{prop:resets}