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\title{\begin{flushleft}
\noindent {\small {\it Proceedings of EPNACS 2011 within ECCS'11
\\Emergent Properties in Natural and Artificial Complex Systems
\\Vienna, Austria - September 15, 2011\\[5mm]
% Copyright $\copyright$ 2009 Watam Press \hfill http://www.watam.org/JNSA/\\[5.0mm]
 }}
% \noindent {\small {\it Journal of Nonlinear Systems and Applications
% (2009)
% 151--154\\
%  Copyright $\copyright$ 2009 Watam Press \hfill http://www.watam.org/JNSA/\\[5.0mm]}}
\end{flushleft}\Large\bf \uppercase{Robust exponential stability of uncertain switched stochastic systems with
time-delay} }
%\author{\Large\bf Author Index}
\author{Author1, Author2 and Author3
 \thanks{Author1 and Author2 are with  Department of
Applied Mathematics, University of Waterloo, Canada. E-mails:
jnsa@uwaterloo.ca, john@uwaterloo.ca}
\thanks{ Author3 is with  Department of Civil and Environmental Engineering,
University of Waterloo, Canada.  E-mail: david@uwaterloo.ca}
\thanks{Manuscript received April 19, 2009; revised January 11, 2010.}}
 \maketitle

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\section{Introduction}
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The following is taken from a paper. ``...''


 Stochastic dynamic modeling plays an
essential role in numerous physics and engineering problems. It can
be applied wherever random properties of a dynamical system have to
be considered. Most of the emphasis is placed on the stability
analysis of the stochastic dynamical systems (see
\cite{Arnold,Hasminskii,Kushner}). Moreover, in many applications,
the physical or chemical processes are governed by more than one
dynamics: the dynamics change among a family of choices with respect
to time $t$ or state $x$. Such processes are often described by
switched systems and have been studied extensively in recent years
(see \cite{Cheng, daaf, Hes, Hu, Lizg, Liberzon,Liu,Man, Matveev,
Mill, Orlov, Orlov2, Schaft}). Time-delay and uncertainties are two
main causes for instability of dynamical systems (see
\cite{Hale,Kolmanovskii}). Numerous studies have been carried out on
stability analysis and stabilization of time-delay systems and
uncertain systems (see
\cite{Cao,Chen,Fridman,Fridman2,Fridman3,Gu,Han,He,Kwon,Li0,Li,Li1,
Mao,Mao2,Moon,Park1,Park2,Verriest,Xu0,Xu,Yue}), some of which have
been done in the scope of stochastic systems or switched systems. To
the best knowledge of the authors, few work has been done for
switched stochastic systems with both uncertainties and time-delay.
In addition, the average dwell time scheme associated with the
multiple Lyapunov functions has been proved to be an important
method for switched systems in \cite{Hespanha} (see also
\cite{Liberzon}). However, these pioneering works have only been
conducted for switched systems without delays and stochastic
perturbations, although switched systems with delays or stochastic
perturbations are in fact very common in practice. Moreover, as long
as the stability of a time-delay system is considered, there exist
both delay-independent criteria and delay-dependent criteria, of
which the latter one is proved to be less conservative and therefore
more important in practical applications, although sometimes more
challenging to obtain (see \cite{Moon}). The objective of this paper
is to achieve delay-dependent stability analysis of delay switched
stochastic systems with uncertainties. The delay-dependent stability
criteria are given in terms of linear matrix inequalities (LMIs) and
average dwell time.



\section{Problem statement and preliminaries}\label{sec2}
Consider the following stochastic uncertain switched system
\begin{align}
dx(t)&=[(A_i+\Delta A_i)x(t)+(\tilde{A_i}+\Delta\tilde{
A_i})x(t-h)]dt\nonumber\\
&\quad+[(B_i+\Delta B_i)x(t)+(\tilde{B_i}+\Delta\tilde{
B_i})x(t-h)]dw(t),\notag\\
&\qquad\qquad\text{for}\quad
t\geq0,\,\alpha(t)=i,\label{sw_sy}
\\[1\jot]
x(t)&=\phi(t),\,t\in [-h,0],\nonumber
\end{align}
where $x\in\Real^n$ is the state and $h$ is the constant time-delay.
$w(t)$ is a scalar Brownian motion. $\phi$ is the initial data.
$A_i, \tilde{A_i}\in\Real^{n\times n}$, $B_i$ and
$\tilde{B_i}\in\Real^{n\times m}$ are constant real matrices. Let
the switching signal $\alpha:\Real^+\rightarrow
I_K\overset{\triangle}{=}\{1,2,...,K\}$ be a piecewise constant and
continuous from the right deterministic function, with finitely many
discontinuities on every bounded subinterval of $\Real^+$. Without
loss of generality, we can suppose that $\alpha(t)$ is right
continuous at its points of discontinuity. $\Delta A_i$,
$\Delta\tilde{A_i}$, $\Delta B_i$ and $\Delta\tilde{B_i}$ are
time-dependent uncertainties of the form
\begin{align*}
\Delta A_i^T\Delta A_i\leq \widehat{A_i},\quad
\Delta\tilde{A_i}^T\Delta\tilde{A_i}\leq\widehat{\tilde{A_i}},
\\
\Delta B_i^T\Delta B_i\leq\widehat{B_i},\quad
\Delta\tilde{B_i}^T\Delta\tilde{B_i}\leq\widehat{\tilde{B_i}},
\end{align*}
where $\widehat{A_i}$, $\widehat{\tilde{A_i}}$, $\widehat{B_i}$, and
$\widehat{\tilde{B_i}}$ are known constant real matrices of
appropriate dimensions, which describe the uncertainty bounds. We
shall call this kind of uncertainties \emph{admissible
uncertainties}. Note that the assumption on the switching signal
$\alpha$ guarantees that system (\ref{sw_sy}) switches only finite
times on any bounded time interval. It is well-known that, given any
initial data $\phi\in L_{\mathcal{F}_0}^{2,h}$, any single system in
(\ref{sw_sy}) has a unique solution (see \cite{Maobook}). Based on
the assumption on the switching signal, it follows step by step that
the switched system (\ref{sw_sy}) has a unique solution as well,
denoted by $x(t;\phi)$.

Two types of stability are considered in this paper, one is almost
sure exponential stability and the other is exponential stability in
$p$th moment (see \cite{Maobook} for definitions). The main problem
now can be formulated as follows:
\begin{prob}
For all admissible uncertainties, under what conditions will the
uncertain switched system $(\ref{sw_sy})$ be almost surely
exponentially stable? Under what conditions will it be exponentially
stable in mean square?
\end{prob}

\section{Main results}\label{sec3}
In this section, the stability analysis of system (\ref{sw_sy}) is
studied.
\begin{thm}\label{main_thm_1}\
Given $h>0$, the uncertain switched stochastic system
$(\ref{sw_sy})$ is robustly exponentially stable in mean square if
the following two conditions hold:
\begin{enumerate}
\item  For each $i\in I_K$, there exist positive numbers
$\varepsilon_{1,i}$, $\varepsilon_{2,i}$, \ldots,
$\varepsilon_{9,i}$ and $\rho_i$, positive definite matrices $Q_i$,
$H_i$, $P_i$, and arbitrary matrices $G_i$ such that the following
matrix inequalities are satisfied:
\begin{align}
\Psi_i=\begin{bmatrix}
\Psi_{11}&\Psi_{12}&\Psi_{13}\\
\star&\Psi_{22}&\Psi_{23}\\
\star&\star&\Psi_{33}
\end{bmatrix}&<0,\label{ineq11}\\
Q_i&\leq\rho_iI\label{ineq12},
\end{align}

\item Let $T_a$ be the average dwell time of system
(\ref{sw_sy}),
\begin{equation}\label{switch_length_1}
T_a>\frac{\ln\beta}{\eps},
\end{equation}
where

and $\eps$ is the unique root of the equation
\begin{equation}\label{eqn}
\lambda-\eps c_2-c_3\eps h e^{\eps h}=0.
\end{equation}
\end{enumerate}
\end{thm}


The following proposition, which can be proved directly by Schur's
lemma \cite{Boyd}, gives an equivalent version of inequality
$(\ref{ineq11})$ which can be readily computed with Matlab's LMI
control toolbox \cite{Gahinet}.
\begin{prop}\label{ready_comp_ineq_1}The matrix
inequality $(\ref{ineq11})$ is equivalent to the following
inequality:
\begin{align}
\Lambda_i=
\begin{bmatrix}
\Lambda_{11} & \Lambda_{12} & \Lambda_{13}  & 0           & 0           & \Lambda_{16} \\
\star        & \Lambda_{22} & \Lambda_{23}  & \Lambda_{24}& 0           & 0          \\
\star        & \star        & \Lambda_{33}  & 0           & \Lambda_{35}& 0        \\
\star        & \star        & \star         & \Lambda_{44}& 0           & 0        \\
\star        & \star        & \star         & \star       & \Lambda_{55}& 0        \\
\star        & \star        & \star         & \star       & \star       & \Lambda_{66} \\
\end{bmatrix}&<0,
\end{align}
where
\begin{align*}
\Lambda_{11}&=H_i+hP_i+Q_i(A_i+G_i)+(A_i^T+G_i^T)Q_i^{}\\
&\quad+(\eps_{1,i}+h\eps_{3,i})\widehat{A_i}+B_i^TQ_iB_i\\
&\quad+(\rho_i+\eps_{5,i}+\eps_{7,i}+\eps_{8,i}^{-1}\rho_i^2)\widehat{B_i},\\
\Lambda_{12}&=Q_i(\tilde{A_i}-G_i)+B_i^TQ_i\tilde{B_i},\\
\Lambda_{22}&=-H_i+(\eps_{2,i}+h\eps_{4,i})\widehat{\tilde{A_i}}+\tilde{B_i}^TQ_i\tilde{B_i}\\
&\quad+(\rho_i+\eps_{6,i}+\eps_{8,i}+\eps_{9,i})\widehat{\tilde{B_i}},\\
\Lambda_{13}&=h(A_i^T+G_i^T)Q_iG_i,\\
\Lambda_{23}&=h(\tilde{A}_i^T-G_i^T)Q_iG_i,\\
\Lambda_{33}&=-hP_i,\\
\Lambda_{24}&=\begin{bmatrix}\tilde{B_i}^TQ_i &
\tilde{B_i}^TQ_i\end{bmatrix},\\
\Lambda_{44}&=\diag{-\eps_{7,i}I,-\eps_{9,i}I},\\
\Lambda_{35}&=\begin{bmatrix}hG_i^TQ_i^{} &
hG_i^TQ_i^{}\end{bmatrix},\\
\Lambda_{55}&=\diag{-\eps_{3,i}hI,-\eps_{4,i}hI},\\
\Lambda_{16}&=\begin{bmatrix}Q_i & Q_i & B_i^TQ_i &
B_i^TQ_i\end{bmatrix},\\
\Lambda_{66}&=\diag{-\eps_{1,i}I,-\eps_{2,i}I,-\eps_{5,i}I,-\eps_{6,i}I}.
\end{align*}
\end{prop}

In the previously stated results, we used multiple Lyapunov
functionals to deal with switched systems. If a single $Q$ can be
found to satisfy inequalities $(\ref{ineq11})$ and $(\ref{ineq12})$
, we have the following corollary of Theorem \ref{main_thm_1}.
\begin{cor}\label{single_P_1}
Given $h>0$, if there exists a positive definite matrix $Q$ such
that the matrix inequalities $(\ref{ineq11})$ and $(\ref{ineq12})$
are satisfied with $Q_i=Q$, then, for any given switching law, the
uncertain switched stochastic system $(\ref{sw_sy})$ is robustly
exponentially stable in mean square.
\end{cor}

\begin{rem} For the almost sure exponential stability, according to \cite{Mao} or
\cite{Maobook}, it can be shown that if the conditions of Theorem
\ref{main_thm_1} are satisfied, then system (\ref{sw_sy}) is almost
surely robust exponentially stable as well. Since under the given
conditions almost sure exponential stability can be easily derived
from exponential stability in mean square, we shall simply state a
system is exponential stability in the following section.
\end{rem}

\begin{rem}To conclude this section, we point out that, for briefness of
this paper, we only focus on system with linear uncertainties.
However, stability results for system with nonlinear uncertainties
can be proved via a similar approach.
\end{rem}

\section{Application}\label{sec4}
In this section, we present an example to illustrate our main
results. The following example shows that sometimes only multiple
Lyapunov functionals would work for the stability analysis.
\begin{ex}\label{ex2}Consider the switched system given by
\begin{align}
dx(t)&=\left[(A_1+\Delta A_1)x(t)+(\tilde{A}_1+\Delta\tilde{A}_1)x(t-h)\right]dt\notag\\
&\quad+g_1(t,x(t),x(t-h))dw(t),\quad\,\alpha(t)=1,\label{sw3}\\
\text{and}\nonumber\\
dx(t)&=\left[(A_2+\Delta
A_2)x(t)+(\tilde{A}_2+\Delta\tilde{A}_2)x(t-h)\right]dt\notag\\
&\quad+g_2(t,x(t),x(t-h))dw(t),\quad\,\alpha(t)=2\label{sw4},
\end{align}
where
\begin{align*}
A_1&=\begin{bmatrix}-1.9 & -3.3\\
7.6 & -11.3
\end{bmatrix},\quad\tilde{A_1}=\begin{bmatrix}-4 & 8.9\\
1.8 & -2.5
\end{bmatrix},\\
A_2&=\begin{bmatrix}-5.4 & 3.9\\
0 & -7.2
\end{bmatrix},\quad\tilde{A_2}=\begin{bmatrix} 5 & 2.6\\
-2.2 & -2.1
\end{bmatrix},
\end{align*}
\begin{align*}
\Delta A_1^T\Delta A_1&\leq\widehat{A}_1=0.01I,\quad
\Delta\tilde{A}_1^T\Delta\tilde{A}_1\leq\widehat{\tilde{A}}_1=0.01I,\\
\Delta A_2^T\Delta A_2&\leq\widehat{A}_2=0.01I,\quad
\Delta\tilde{A}_2^T\Delta\tilde{A}_2\leq\widehat{\tilde{A}}_2=0.01I,
\end{align*}
and
\begin{align*}
&\tr{g_1^T(t,x(t),x(t-h))g_1(t,x(t),x(t-h))}\\
&\quad\leq0.1\norm{x(t)}^2+0.1\norm{x(t-h)}^2,\\
&\tr{g_2^T(t,x(t),x(t-h))g_2(t,x(t),x(t-h))}\\
&\quad\leq0.1\norm{x(t)}^2+0.1\norm{x(t-h)}^2.
\end{align*}

\begin{table}
\caption{Stability bounds of time-delay and average dwell time
(Example \ref{ex2})}\label{table2}
\begin{center}
{\tt
\begin{tabular}{|c||c|c|c|c|}
\hline $T_0\,\,(\times 10^3)$& $0$ & $0.0856$ & $0.2303$ & $0.7960$\\
\hline $h_0$& $0$ & $0.5$ & $1.0$ & $1.5$\\
\hline $T_0\,\,(\times 10^3)$ &$1.7645$& $3.0717$ & $5.1803$ & $9.2734$\\
\hline $h_0$& $2.0$&$2.5$ & $3.0$ & $3.5$\\
\hline $T_0\,\,(\times 10^3)$ &$23.4052$& $238.2837$ & $$ & $$\\
\hline $h_0$& $4.0$&$4.4$ & $$ & $$\\
\hline
\end{tabular}
}
\end{center}
\end{table}

It can be shown using Matlab's LMI control toolbox that there exists
no single $Q$ to suit the inequalities (\ref{ineq11}) and
(\ref{ineq12}). However, if we apply Theorem \ref{main_thm_1}
instead of Corollary \ref{single_P_1}, we can see that the switched
system given by systems (\ref{sw3}) and (\ref{sw4}) is robustly
exponentially stable for $0\leq h\leq 4.4$. For different average
dwell time lower bounds $T_0$, the delay upper bounds $h_0$
guaranteeing the exponential stability of the system are listed in
Table \ref{table2}.
\end{ex}


\section*{Acknowledgements}
The research for this work was supported, in part, by the Natural
Sciences and Engineering Research Council of Canada.

\footnotesize

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