dha.tex

由matlab开发的hybrid系统的描述语言

\end{proposition}

\noindent\proof Let the superscript ${\res}$ denote the variables
of $\Sigma^{\res}$. Let
$x_b(k)=[x_b^{\res}(k);x_b^{\res}(k-1);\delta_e^{\res}(k-1);u_b^{\res}(k-1)]$,
where $;$ denotes the concatenation of column vectors,
$\delta_e(k)=\delta_e^{\res}(k)$, $u_b(k)=u_b^{\res}(k)$,
$x_r(k)=x_r^{\res}(k)$, $u_r(k)=u_r^{\res}(k)$,
$y_r(k)=y_r^{\res}(k)$, $y_b(k)=y_b^{\res}(k)$, and define the FSM
\[
    x'_b(k)=f_{\rm B}(x_b(k),u_b(k),\delta_e(k))
=\smallmat{f_{\rm B}^{\res}(x_b^{\res}(k),u_b(k),\delta_e(k))\\x_b^{\res}(k)\\
\delta_e(k)\\u_b(k)}.
\]
The SAS dynamics of $\Sigma$ is defined as in~(\ref{eq::sas}) with
$i(k)\in\{1,2,\ldots,s^{\res},s^{\res}+1,\ldots,s^{\res}+r\}$,
where $s^{\res}$ is the number of modes of $\Sigma^\res$, and
$r\leq s^{\res}(s^{\res}-1)$ is the number of reset maps (we
assume that when the mode switch $h\rightarrow j$ of $\Sigma_0$
does not have an associated reset map
$f^{\res}_{\overrightarrow{hj}}$, then
$f^{\res}_{\overrightarrow{hj}}$ defaults to the $j$-th state
update map). The MS of $\Sigma$ should internally compute
$j=i^\res(k)$, $h=i^\res(k-1)$, compare them, and then choose
either the $j$-th dynamics (if $j=h$, or $j\neq h$ and
$f^{\res}_{\overrightarrow{hj}}$ is not specified) or the reset
dynamics $f^{\res}_{\overrightarrow{hj}}$. Since
$i^\res(k)=f^\res_{\rm
M}(x^\res(k),u_b^\res(k),\delta_e^\res(k))$,
$i^\res(k-1)=f^\res_{\rm
M}(x^\res(k-1),u_b^\res(k-1),\delta_e^\res(k-1))$, it follows that
the mode $i(k)$ is a function of $x_b(k)$, $u_b(k)$,
$\delta_e(k)$. \cvd


In some circumstances, it is desirable to predict the mode switch
and to anticipate the reset by one sampling step, i.e., to reset
the state {\em before} the guardline is actually crossed. Assume
that the event $\delta_e(k)$ triggering the mode switch does not
depend on the continuous input $u_r(k)$, and that the logic input
$u_b(k)$ does not affect the mode selector. In this case,
$i'(k)=f_{\rm M}(x'_b(k),f_{\rm H}(x'_r(k),k))$ only depends on
quantities available at step $k$, and a mode switch $i'(k)\neq
i(k)$ can be predicted already at step $k$. In this case, we can
apply the corresponding reset directly for
$x_r'(k)=A_{\overrightarrow{hj}}x_r(k)+B_{\overrightarrow{hj}}u_r(k)+f_{\overrightarrow{hj}}$
where $h=i(k)$, $j=i'(k)$. This kind of resets will be referred to
as {\em predicted resets}, in order to distinguish them from the
resets described before, that we will call {\em a-posteriori
resets}.

Consider Figure~\ref{fig:reset-b}. At time $k=5$ the state
$x_r(5)$ and the input $u_r(5)$ are such that $A_1x_r(5)+B_1x_r(5)
+ f_1 \geq 0$ which would generate an event $\delta_e$ at the next
time step. As a consequence of the predicted mode switch, the
state is reset according to the reset map
$f_{\overrightarrow{12}}(x,u)$, i.e., $x(6) =
A_{\overrightarrow{12}} x(5)+B_{\overrightarrow{12}} x(5) +
f_{\overrightarrow{12}}$.


\begin{proposition}
Assume that the event $\delta_e(k)$ does not depend on the
continuous input $u_r(k)$, and that the mode $i(k)$ does not
depend on the logic input $u_b(k)$. Then a DHA $\Sigma^{\res}$
with predicted resets can be rewritten as a DHA $\Sigma$ without
resets. \label{prop:resets-pred}
\end{proposition}

\noindent\proof Let again the superscript ${\res}$ denote the
variables of $\Sigma^{\res}$. By hypothesis, predicted resets
imply that $\delta^{\res}_e(k+1)$ only depends on $k$ and
$x_r^{\res}(k+1)$, which is a function of $x_r^{\res}(k)$,
$u_r^{\res}(k)$, $i^{\res}(k)$. Define the EG for system $\Sigma$
as $\delta_e(k)=[\delta^{\res}_e(k);\delta^{\res}_e(k+1)]$, and
let $x_b(k)=x_b^{\res}(k)$, $u_b(k)=u_b^{\res}(k)$,
$x_r(k)=x_r^{\res}(k)$, $u_r(k)=u_r^{\res}(k)$,
$y_r(k)=y_r^{\res}(k)$, $y_b(k)=y_b^{\res}(k)$. Let the FSM for
system $\Sigma$ be equal to the FSM of $\Sigma^\res$, and define
the SAS dynamics of $\Sigma$ as in the proof of
Proposition~\ref{prop:resets}. The MS of $\Sigma$ should
internally compute $j=i^\res(k+1)$, $h=i^\res(k)$, compare them,
and then choose either the $j$-th dynamics (if $j=h$, or $j\neq h$
and $f^{\res}_{\overrightarrow{hj}}$ is not specified) or the
reset dynamics $f^{\res}_{\overrightarrow{hj}}$. Since
$i^\res(k)=f^\res_{\rm M}(x_b^\res(k),\delta_e^\res(k))$,
$i^\res(k+1)=f^\res_{\rm M}(f_{\rm
B}^\res(x_b^\res(k),u_b^\res(k),\delta_e^\res(k)),\delta_e^\res(k+1))$,
it follows that the mode $i(k)$ of the SAS dynamics of system
$\Sigma$ is a function of $x_b(k)$, $u_b(k)$, $\delta_e(k)$. \cvd

\subfigeps{t}{Reset maps}{fig:reset}{reset-post}{.45\hsize}{A
posteriori resets}{reset}{.45\hsize}{Predictive resets}

\begin{example}
Let us consider a DHA with two modes:
\[
  \begin{array}{rl}
  \mbox{\rm SAS:} &
  x_r'(k) = \left\{
  \begin{array}{lcl}
    x_r(k) + u_r(k) - 1, & {\rm if } & i(k)=1, \\
    2x_r(k), & {\rm if } & i(k)=2,
  \end{array}
  \right. \\
  \mbox{\rm EG:} &
    \delta_e(k) = [x_r(k) \ge 0],
  \\
  \mbox{\rm MS:} &
  i(k) = \left\{
  \begin{array}{lcl}
    1, & {\rm if } & \delta_e(k) = 0, \\
    2, & {\rm if } & \delta_e(k) = 1 {\rm .}
  \end{array}
  \right.
  \end{array}
\]
In order to add the predictive reset map
$f_{\overrightarrow{12}}(x_r,u_r) = 2$ to the model, we first
consider the set $\pp = \{x_r,u_r: a_1x_r+b_1u_r+f_1 \geq 0\}$ of
all the state/input pairs that will trigger the event $\delta_e$
in one step and add an event $\delta_f = a_1x_r+b_1u_r+f_1 \geq 0$
in the EG. If the current mode is $i=1$ and the pair $(x_r,u_r)$
triggers the event $\delta_f$ then the state should be updated
according to the reset map. Summing up, we can write the following
DHA:
\begin{equation}
  \begin{array}{rl}
  \mbox{\rm SAS:} &
  x_r'(k) = \left\{
  \begin{array}{lcl}
    x_r(k) + u_r(k) - 1, & {\rm if } & i(k)=1, \\
    2x_r(k), & {\rm if } & i(k)=2, \\
    2, & {\rm if} & i(k)=3,
  \end{array}
  \right. \\
  \mbox{\rm EG:} &
  \left\{
    \begin{array}{l}
    \delta_e(k) = [x_r(k) \ge 0], \\
    \delta_f(k) = [x_r(k) + u_r(k) - 1 \ge 0],
  \end{array}
  \right.
  \\
  \mbox{\rm MS:} &
  i(k) = \left\{
  \begin{array}{lcl}
    1, & {\rm if } & \smallmat{\delta_e(k)\\\delta_f(k)} = \smallmat{0\\0},\\
    2, & {\rm if } & \delta_e(k) = 1, \\
    3, & {\rm if } & \smallmat{\delta_e(k)\\\delta_f(k)} = \smallmat{0\\1}
  \end{array}
  \right.
  \end{array}
  \label{eq:systemreset}
\end{equation}
which admits a PWA~(\ref{PWA}) representation
(Figure~\ref{fig:regions}) that clearly shows that the reset
condition is another dynamical mode.
\end{example}
\figuraeps{regions}{t}{.62\textwidth}{Equivalent PWA
system}{fig:regions}

\section{DHA Trajectories}
\index{Trajectories!DHA} For a given initial condition
$\smallmat{x_r(0)\\x_b(0)}\in\XX_r\times\XX_b$, and input
$\smallmat{u_r(k)\\u_b(k)}\in\UU_r\times\UU_b$, $k\in\Z_{\geq 0}$,
the state trajectory $x(k)$, $k\in\ZZ_{\geq 0}$, of the system is
recursively computed as follows:
 \begin{enumerate}
 \item Initialization: $x(0) = \smallmat{x_r(0)\\x_b(0)}$;
 \item Recursion:
\begin{enumerate}
 \item $\delta_e(k) = f_{\rm H}(x_r(k),u_r(k),k)$;
 \item $i(k) = f_{\rm M}(x_b(k),u_b(k),\delta_e(k))$;
 \item $y_r(k) = C_{i(k)}x_r(k) + D_{i(k)}u_r(k) + g_{i(k)}$;
 \item $y_b(k) =  g_{\rm B}(x_b(k),u_b(k),\delta_e(k))$;
 \item $x_r'(k) = A_{i(k)}x_r(k) + B_{i(k)}u_r(k) + f_{i(k)}$;
 \item $x_b'(k) =  f_{\rm B}(x_b(k),u_b(k),\delta_e(k))$.
 \end{enumerate}
 \end{enumerate}
 \label{def:traj}

\begin{definition}
\label{def:well-posed} A DHA is {\em well-posed} on
$\XX_r\times\XX_b$, $\UU_r\times\UU_b$, $\YY_r\times\YY_b$, if for
all initial conditions
$x(0)=\smallmat{x_r(0)\\x_b(0)}\in\XX_r\times\XX_b$, and for all
inputs $u(k)=\smallmat{u_r(k)\\u_b(k)}\in\UU_r\times\UU_b$, for
all $k\in\Z_{\geq 0}$, the state trajectory
$x(k)\in\XX_r\times\XX_b$ and output trajectory
$y(k)=\smallmat{y_r(k)\\y_b(k)}\in\YY_r\times\YY_b$ are uniquely
defined.
\end{definition}
Definition~\ref{def:well-posed} will be used for other types of
hybrid models that we will introduce later. In general a hybrid
model may not be well-posed, either because the trajectories stop
after a finite time (for instance, the state vector leaves the set
$\XX_r\times\XX_b$) or because of bifurcations (the successor
$x_r'(k)$, $x_b'(k)$ may be multiply defined). In general a DHA
can be ``not well-posed'' only if its trajectories are not defined
after finite time. This means that DHA can not bifurcate.