\documentclass[reqno]{amsart}
\usepackage{graphicx}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 159, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/159\hfil Exponential stability]
{Exponential stability of switched linear systems with
time-varying delay}
\author[S. Pairote, V. N. Phat\hfil EJDE-2007/159\hfilneg]
{Vu N. Phat, Satiracoo Pairote}
\address{Vu N. Phat \newline
Institute of Mathematics\\
18 Hoang Quoc Viet Road, Hanoi, Vietnam}
\email{vnphat@math.ac.vn}
\address{Satiracoo Pairote \newline
Department of Mathematics, Mahidol Univerity\\
Bangkok 10400, Thailand}
\email{scpsc@mucc.mahidol.ac.th}
\thanks{Submitted September 5, 2007. Published November 21, 2007.}
\subjclass[2000]{34K20, 93C10, 34D10}
\keywords{Switched system; time delay; exponential stability;
\hfill\break\indent Lyapunov equation}
\begin{abstract}
We use a Lyapunov-Krasovskii functional approach to establish the
exponential stability of linear systems with time-varying delay.
Our delay-dependent condition allows to compute simultaneously
the two bounds that characterize the exponential stability rate of
the solution. A simple procedure for constructing switching rule
is also presented.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
As an important class of hybrid systems, switched system is a
family of differential equations together with rules to switch
between them. A switched system can be described by a differential
equation of the form
$$
\dot x = f_\alpha(t,x),
$$
where $\{f_\alpha(.): \alpha \in \Omega\}$, is a family of functions
that is parameterized by some index set $\Omega$, and $\alpha(\cdot)
\in \Omega$ depending on the system state in each time is a
switching rule/signal. The set $\Omega$ is typically a finite set.
Switched systems arise in many practical models in manufacturing,
communication networks, automotive engine control, chemical
processes, and so on; see for example \cite{l1,s1,s4}.
During the previous decade, the
stability problem of switched linear systems has attracted a lot
of attention (see,; e.g. \cite{l2,p2,s3} and the references
therein). The main approach for stability analysis relies on the
use of Lyapunov-Krasovskii functionals and LMI approach for
constructing common Lyapunov function \cite{s2}. Although many important
results have been obtained for switched linear systems, there are
few results concerning the stability of the systems
with time delay. Under commutative assumption on the system
matrices, the authors of \cite{n1} showed that when all subsystems are
asymptotically stable, the switched system is asymptotically
stable under arbitrary switching rule. The paper \cite{z1} studied
the asymptotic stability for switched linear systems with time
delay, but the result was limited to symmetric systems.
In \cite{p1,x1}, delay-dependent asymptotic stability conditions are extended
to switched linear discrete-time linear systems with time delay.
The exponential stability problem was
considered in \cite{z2} for switched linear systems with impulsive effects by using
matrix measure concept and in \cite{z3} for nonholonomic chained systems with strongly
nonlinear input/state driven disturbances and drifts.
In recent paper \cite{k1}, studying a switching system composed
of a finite number of linear delay differential equations, it was
shown that the asymptotic stability of this kind of switching
systems may be achieved by using a common Lyapunov function method
switching rule. There are some other results concerning asymptotic
stability for switched linear systems with time delay, but we do
not find any result on exponential stability even for the switched
systems without delay except \cite{z2,z3}. On the other hand, it is
worth noting that the existing stability conditions for time-delay
systems must be solved upon a grid on the parameter space, which
results in testing a nonlinear Riccati-like equation or a finite
number of LMIs. In this case, the results using finite gridding
points are unreliable and the numerical complexity of the tests
grows rapidly. Therefore, finding simple stability conditions for
switched linear systems with time-delay is of interest.
In this paper, we study the exponential stability of a class of
switched linear systems with time-varying delay. The system
studied in this paper is time-varying under a switching rule
dependent on continuous system states. A delay-dependent
condition for the exponential stability are formulated in terms of
a generalized Lyapunov equation for linear systems with
time-varying delay, which allows easily to compute simultaneously
the two bounds that characterize the exponential stability rate of
the solution. A simple procedure for constructing switching rule
is presented. The results obtained in this paper can be partly
considered as extensions of existing results for linear time-delay
systems and for switched linear systems without time delays.
The organization of this paper is as follows. Following the
introduction and the problem motivation, Section 2 presents the
notation, definitions and auxiliary propositions. The main result
is given in Section 3 and followed by an numerical example and
conclusion.
\section{Problem formulation}
The following notations will be used throughout this paper. $\mathbb{R}^+$
denotes the set of all non-negative real numbers; $\mathbb{R}^n$ denotes
the $n-$finite-dimensional Euclidean space, with the Euclidean
norm $\|.\|$ and scalar product $x^Ty$ of two vectors $x, y$.
$\mathbb{R}^{n\times m}$ denotes the set of all $(n\times m)-$matrices;
$\lambda_{\rm max}(A)(\lambda_{\rm min}(A)$, resp.) denotes the maximal
number (the minimum number, resp.) of the real part of eigenvalues
of $A;$ $A^T$ denotes the transpose of the matrix $A;$ $Q\geq 0
(Q >0$, resp.) means $Q$ is semi-positive definite (positive
definite, resp.).
Consider a switched linear system with time-varying delay of the
form
\begin{equation} \label{Sigma-a}
\begin{gathered}
\Sigma_\alpha\hskip2cm \begin{cases}
\dot x(t) = A_{\alpha}x(t) + D_{\alpha}x(t-h(t)), \quad t \in \mathbb{R}^+, \\
x(t) = \phi (t), \quad t \in [-h, 0],
\end{cases}
\end{gathered}
\end{equation}
where $x(t)\in \mathbb{R}^n$ is the continuous trajectory of system,
$A_{\alpha}, D_{\alpha}\in \mathbb{R}^{n\times n}$ are given constant
matrices, $\phi(t)\in C([-h,0],\mathbb{R}^n)$ is the initial function with
the norm $\|\phi\| = \sup_{s\in [-h,0]}\|\phi(s)\|$.
$\alpha(x): \mathbb{R}^n\to \Omega := \{1, 2, \dots, N\}$ is the switching rule,
which is a piece-wise constant function depending on the state in
each time. A switching rule is a rule which determines a switching
sequence for a given switching system. Moreover, $\alpha(x) = i$
implies that the system realization is chosen as $[A_i, B_i]$,
$i =1, 2, \dots, N$. It is seen that the system \eqref{Sigma-a} can be viewed as
an linear autonomous switched system in which the effective
subsystem changes when the state $x(t)$ hits predefined
boundaries; i.e., the switching rule is dependent on the system
trajectory. The time-varying delay function $h(t)$ satisfies the
following assumption
$$
0\leq h(t) \leq h, \quad h >0,\quad \dot h(t)\leq \mu < 1.
$$
This assumption means that the time delay may change from time to
time, but the rate of changing is bounded. Also, due to the upper
bound, the delay can not increase as fast as the time itself. In
fact, the function $h(t)$ can be different for each subsystem;
i.e., it should be denoted by $h_i(t)$, but we assume $h_i(t)$ be
the same value in this paper for convenient formulation.
The exponential stability problem for switched linear system \eqref{Sigma-a}
is to construct a switching rule that makes the system
exponentially stable.
\begin{definition} \label{def2.1} \rm
Switched linear system \eqref{Sigma-a} is
exponentially stable if there exists switching rule $\alpha(.)$
such that every solution $x(t,\phi)$ of the system $\Sigma_\alpha$ satisfies the condition
$$
\exists M >0,\; \delta >0:\;
\|x(t,\phi)\| \leq Me^{-\delta t}\|\phi\|,\quad \forall t\in
\mathbb{R}^+.
$$
The numbers $N >0$ and $\delta >0$ are called the
stability factor and decay rate of the system.
\end{definition}
\begin{definition}[\cite{s6}] \label{def2.2}\rm
System of symmetric matrices $\{L_i\}$,
$i = 1, 2, \dots, N$, is said to be strictly complete if for every
$0\neq x\in \mathbb{R}^n$ there is $i\in \{1, 2, \dots, N\}$ such that
$x^TL_ix < 0$.
\end{definition}
Let us define the sets
$$
\Omega_i = \{x\in \mathbb{R}^n : x^TL_ix < 0\},\quad i=1, 2, \dots, N.
$$
It is easy to show that the system $\{L_i\}$, $i = 1, 2, \dots, N$, is
strictly complete if and only if $\cup_{i=1}^N\Omega_i =
\mathbb{R}^n\setminus \{0\}$.
\begin{remark} \label{rmk2.1}\rm
As shown in \cite{s5}, a sufficient condition for
the strict completeness of the system
$\{L_i\}$ is that there exist numbers $\tau_i \geq 0$,
$i =1, 2,\dots , N$, such that $\sum_{i=1}^N\tau_i > 0$ and
$$
\sum_{i=1}^N\tau_iL_i < 0,
$$
and in the case if $N =2$, then the above condition is also
necessary for the strict completeness.
\end{remark}
Before presenting the main result, we recall the following well-known
matrix inequality and Lyapunov stability theorem for time delay
systems.
\begin{proposition} \label{prop2.1}
For any $\epsilon >0$, $x,y\in \mathbb{R}^n$, we
have
$$
- 2x^Ty \leq \epsilon^{-1}x^Tx +\epsilon y^Ty.
$$
\end{proposition}
\begin{proposition}[\cite{h1}] \label{prop2.2}
Let $x_t := \{x(t+s), s\in [-h,0]\}$. Consider nonlinear delay
system
\begin{gather*}
\dot x(t) = f(t,x_t),\\
x(t) = \phi(t),\\
f(t,0) = 0, t\in \mathbb{R}^+.
\end{gather*}
If there exists a Lyapunov function $V(t,x_t)$ satisfying
the following conditions:
\begin{itemize}
\item[(i)] There exists $\lambda_1 >0$, $\lambda_2 >0$ such that
$\lambda_1\|x(t)\|^2 \leq V(t,x_t)\leq \lambda_2\|x_t\|^2$,
for all $t\in \mathbb{R}^+,$
\item[(ii)] $\dot{V}_f(t,x_t):= \frac{\partial V}{\partial t}
+\frac{\partial V}{\partial x}f(t,x_t) \leq 0$, for all solutions
$x(t)$ of the system,
\end{itemize}
then the solution $x(t,\phi)$ is bounded: There exists $N >0$
such that $\|x(t,\phi)\| \leq N\|\phi\|$, for all $t\in \mathbb{R}^+$.
\end{proposition}
\section{Main result}
For given positive numbers $\mu, \delta, h, \epsilon$ we set
\begin{gather*}
\beta = ( 1- \mu)^{-1},\\
L_i(P) = A_i^TP + PA_i + \epsilon^{-1}e^{2\delta h}D_i^TPD_i +
(\epsilon\beta + 2\delta)P.\\
S_i^P = \{x\in \mathbb{R}^n:\quad x^TL_i(P)x < 0\},\\
\bar S_1^P = S_1^P,
\quad \bar S_i^P = S_i^P \setminus \bigcup_{j=1}^{i-1}\bar S_j^P, \quad
i = 2, 3, \dots, N,\\
\lambda_{\rm max}(D) = \max_{i=1,2,\dots,N}\lambda_{\rm max}(D_i^TD_i),
\\
M = \sqrt{\frac{\lambda_{\rm max}(P)}{\lambda_{\rm min}(P)}
+\frac{\epsilon^{-1}(e^{2\delta h}-1)\lambda_{\rm max}(P)
\lambda_{\rm max}(D)}{2\delta}}.
\end{gather*}
The main result of this paper is summarized in the following theorem.
\begin{theorem} \label{thm3.1}
Switched linear system \eqref{Sigma-a} is exponentially stable
if there exist positive numbers
$\epsilon, \delta$ and symmetric positive definite matrix
$P\in \mathbb{R}^{n\times n}$ such that one of the following
conditions holds
\begin{itemize}
\item[(i)] The system of matrices $\{L_i(P)\}$, $i =1, 2, \dots, N$, is
strictly complete.
\item[(ii)] There exists $\tau_i \geq 0$, $i =1, 2, \dots, N$,
with $\sum_{i=1}^N\tau_i > 0$ and
\begin{equation} \label{e3.1}
\sum_{i=1}^N\tau_iL_i(P) < 0.
\end{equation}
\end{itemize}
The switching rule is chosen in case (i) as $\alpha (x(t)) = i$
whenever $x(t)\in \bar S_i$, and in case (ii) as
$$
\alpha(x(t)) = \arg \min \{x^T(t)L_i(P)x(t)\}.
$$
Moreover, the solution $x(t,\phi)$ of the system satisfies
$$
\|x(t,\phi)\| \leq Me^{-\delta t}\|\phi\|,\quad t\in \mathbb{R}^+.
$$
In the case $N =2$, the conditions (i) and (ii) are equivalent.
\end{theorem}
\begin{proof}
For $\delta >0$, we utilize the following state transformation
$y(t) = e^{\delta t}x(t)$.
The system \eqref{Sigma-a} is transformed into
\begin{equation}\label{e3.2}
\begin{gathered}
\dot y(t) = \bar A_{\alpha}y(t) + e^{\delta h(t)}
D_{\alpha}y(t-h(t)), t \in \mathbb{R}^+,\\
y(t) = e^{\delta t}\phi (t), \quad t \in [-h, 0],
\end{gathered}
\end{equation}
where $\bar A_{\alpha} := A_{\alpha} +\delta I$. Assume that the
condition (i) of the theorem holds. For every $i =1, 2, \dots, N$,
we consider the Lyapunov-Krasovskii functional
\begin{equation} \label{e3.3}
V(t,y_t) = \langle Py(t),y(t)\rangle + \epsilon^{-1}e^{2\delta
h}\int_{t-h(t)}^t\langle D_i^TPD_iy(s),y(s)\rangle ds.
\end{equation}
It is easy to see that there are positive numbers
$\lambda_1, \lambda _2$ such that
$$
\lambda_1\|y(t)\|^2\leq V(t, y_t)\leq \lambda_2\|y_t\|^2,\quad
\forall t\in \mathbb{R}^+.
$$
The derivative along the trajectory of the system \eqref{e3.2} is
\begin{equation}\label{e3.4}
\begin{aligned}
\dot V(t,y_t) &= 2\langle P\dot y(t),y(t)\rangle +
\epsilon^{-1}e^{2\delta h}\langle D_i^TPD_iy(t),y(t)\rangle \\
&\quad - \epsilon^{-1}e^{\delta h}( 1- \dot h(t))\langle
D_i^TPD_iy(t-h(t)),y(t-h(t))\rangle \\
&\leq \langle (A_i^TP +PA_i + 2\delta P+ \epsilon^{-1}e^{\delta h}
D_i^TPD_i)y(t),y(t)\rangle \\
&\quad + 2e^{2\delta h(t)}\langle PD_iy(t-h(t)),y(t)\rangle \\
&\quad - \epsilon^{-1}e^{\delta h}( 1- \mu)\langle
D_i^TPD_iy(t-h(t)),y(t-h(t))\rangle.
\end{aligned}
\end{equation}
Since $P$ is a symmetric positive definite matrix, there is $\bar
P = P^{1/2}$ such that $P = \bar P^T\bar P$. Using
Proposition \ref{prop2.1}, for any $\xi >0$, we have
\begin{align*}
&2e^{\delta h(t)}\langle PD_iy(t-h(t)),y(t)\rangle \\
&= 2y^T(t)\bar P^T\bar Pe^{\delta h(t)}D_iy(t-h(t)) \\
&\leq \xi y^T(t)\bar P^T\bar P y(t) + \xi^{-1}e^{2\delta
h(t)}y^T(t-h(t))D_i^T\bar P^T\bar PD_iy(t-h(t)).
\end{align*}
Taking $\xi = \epsilon\beta >0$, and since $h(t)\leq h$, we obtain
\begin{align*}
&2e^{\delta h(t)}\langle PD_iy(t-h(t)),y(t)\rangle \\
&\leq \epsilon\beta y^T(t)Py(t) +
\epsilon^{-1}(1-\mu)e^{2\delta h}y^T(t-h(t))D_i^TP D_iy(t-h(t)).
\end{align*}
Then from \eqref{e3.4} it follows that
\begin{equation} \label{e3.5}
\begin{aligned}
\dot V(t,y_t))
&\leq \langle (A_i^TP + PA_i +
\epsilon^{-1}e^{2\delta h}D_i^TPD_i + (\epsilon\beta+
2\delta)P)y(t),y(t)\rangle \\
&= y^T(t)L_i(P)y(t).
\end{aligned}
\end{equation}
By the assumption, the system of matrices $\{L_i(P)\}$ is strictly
complete. We have
\begin{equation} \label{e3.6}
\bigcup_{i=1}^N{S}_i^P = \mathbb{R}^n\setminus \{0\}.
\end{equation}
Based on the sets $S_i^P$ we constructing the sets
$\bar S_i^P$ as above and we can verify that
\begin{equation} \label{e3.7}
\bar S_i^P\bigcap \bar S_j^P = \{0\}, i\neq j,\quad \bar S_i^P\bigcup
\bar S_j^P = \mathbb{R}^n\setminus \{0\}.
\end{equation}
We then construct the following switching rule: $\alpha (x(t)) = i$,
whenever $x(t) \in \bar S_i^P$ (this switching rule is well-defined
due to the condition \eqref{e3.7}). From the state transformation
$ y(t) = e^{\delta t}x(t)$ and taking \eqref{e3.5} into account we
obtain
\begin{equation} \label{e3.8}
\dot V(t, y_t) = y^T(t)L_\alpha(P)y(t)
= e^{2\delta t}x^T(t)L_\alpha(P)x(t) \leq 0,\quad \forall t\in \mathbb{R}^+,
\end{equation}
which implies that the solution $y(t)$ of system \eqref{e3.2},
by Proposition \ref{prop2.2}, is bounded. Returning to the state
transformation of $y(t) = e^{\delta t}x(t)$ guarantees the
exponentially stability with
the decay rate $\delta$ of the system \eqref{Sigma-a}. To define the
stability factor $M$, integrating both sides of \eqref{e3.8} from $0$ to
$t$ and using the expression of $V(t,y_t)$ from \eqref{e3.3} we have
$$
\langle Py(t),y(t)\rangle + \epsilon^{-1}e^{\delta
h}\int_{t-h(t)}^t\langle D_i^TPD_iy(s),y(s)\rangle ds \leq V(y(0)).
$$
Since $D_i^TPD_i >0$, and
$$
\lambda_{\rm min}\|y(t)\|^2 \leq \langle Py(t),y(t)\rangle ,
$$
we have
$$
\lambda_{\rm min}\|y(t)\|^2 \leq \lambda_{\rm max}(P)\|y(0)\|^2
+ \epsilon^{-1}e^{2\delta
h}\int_{h(0)}^0\langle D_i^TPD_iy(s),y(s)\rangle ds.
$$
We have
\begin{align*}
\int_{-h(0)}^0\langle D_i^TPD_iy(s),y(s)\rangle ds
&\leq \lambda_{\rm max}(P)\lambda_{\rm max}(D)\|\phi\|^2
\int_{-h(0)}^0e^{2\delta s}ds\\
& = \frac{\lambda_{\rm max}(P)\lambda_{\rm max}(D)}{2\delta}(1-e^{-2\delta
h(0)})\|\phi\|^2 \\
& \leq \frac{\lambda_{\rm max}(P)\lambda_{\rm max}(D)}{2\delta}(1-e^{-2\delta
h})\|\phi\|^2,
\end{align*}
we have
$$
\|y(t)\|^2 \leq [\frac{\lambda_{\rm max}(P)}{\lambda_{\rm min}(P)}
+\frac{\epsilon^{-1}e^{2\delta
h}\lambda_{\rm max}(P)\lambda_{\rm max}(D)}{2\delta}(1-e^{-2\delta
h})]\|\phi\|^2.
$$
Therefore,
$$
\|y(t)\| \leq M\|\phi\|,\quad \forall t\in \mathbb{R}^+,
$$
where
$$
M = \sqrt{\frac{\lambda_{\rm max}(P)}{\lambda_{\rm min}(P)}
+\frac{\epsilon^{-1}(e^{2\delta
h}-1)\lambda_{\rm max}(P)\lambda_{\rm max}(D)}{2\delta}}.
$$
We now assume the condition (ii), then we have
$$
\sum_{i=1}^N\tau_iL_i(P) < 0.
$$
where $\tau_i \geq 0$, $i =1, 2, \dots, N$,
$\sum_{i=1}^N\tau_i > 0$. Since the numbers $\tau_i$ are non-negative and
$\sum_{i=1}^N\tau_i >0$, there
is always a number $\epsilon > 0$ such that for any nonzero $y(t)$ we have
$$
\sum_{i=1}^N\tau_iy^T(t)L_i(P)y(t) \leq -\epsilon y^T(t)y(t).
$$
Therefore,
\begin{equation} \label{e3.9}
\sum_{i=1}^N\tau_i\min_{i=1,\dots,N}\{y^T(t)L_i(P)y(t)\} \leq
\sum_{i=1}^N\tau_iy^T(t)L_i(P)y(t)\leq -\epsilon y^T(t)y(t).
\end{equation}
The Lyapunov-Krasovskii functional $V(.)$ is defined by
\eqref{e3.3} and the switching rule is designed as follows
$$
\alpha(x(t)) = \arg \min_{i=1,\dots,N}\{x^T(t)L_i(P)x(t)\}.
$$
Combining \eqref{e3.5} and \eqref{e3.9} gives
$$
\dot V(t, y_t)) \leq - \eta \|y(t)\|^2 \leq 0,\quad \forall
t\in \mathbb{R}^+,
$$
where $\eta = \epsilon(\sum_{i=1}^N\tau_i)^{-1}$. This implies
that all the solution $y(t)$ of system \eqref{e3.2} are bounded. The
proof is then completed by the same way as in the part (i).
\end{proof}
\begin{remark} \label{rmk3.1} \rm
Note that
conditions (i) and (ii) involve linear Lyapunov-type matrix
inequality, which is easy to solve. The following simple
procedure can be applied to construct switching rule and define
the stability factor and decay rate of the system.
\begin{itemize}
\item[Step 1.] Define the matrices $L_i(P)$.
\item[Step 2.] Find the solution $P$ of the generalized
Lyapunov inequality \eqref{e3.1}.
\item[Step 3.] Construct the sets $S_i^P$, and then
$\bar S_i^P,$ and verify the condition \eqref{e3.6}, \eqref{e3.7}.
\item[Step 4.] The switching signal $\alpha(.)$ is chosen as
$\alpha(x) = i$, whenever $x\in \bar S_i^P$ or as
$$
\alpha(x(t)) = \arg \min \{x^T(t)L_i(P)x(t)\}.
$$
\end{itemize}
\end{remark}
\begin{example} \label{exa3.1} \rm
Consider the switched linear system defined by
$$
\dot{x}(t) = A_i x(t) + D_i x(t-h(t)), \quad i = 1,2
$$
where
\begin{gather*}
(A_1,D_1 ) = \left( \begin{bmatrix}
0.39 & 0.09 \\
0.09 & -1.49 \end{bmatrix},
\begin{bmatrix}
0.39 & 0.03 \\
0.03 & -0.29 \end{bmatrix} \right),
\\
(A_2,D_2 ) =
\left( \begin{bmatrix}
-1.99 & -0.13 \\
-0.13 & 0.59 \end{bmatrix},
\begin{bmatrix}
-0.09 & -0.01 \\
-0.01 & 0.19 \end{bmatrix} \right),
\\
h(t)= \sin^2(0.1t).
\end{gather*}
We have $ h = 1$, $\dot h(t) = 0.1 \sin(0.2t)$ and then $ \mu = 0.1$.
By choosing the positive definite matrix $P>0$ as
$$
P = \begin{bmatrix}
1 & 0.1 \\
0.1 & 1 \end{bmatrix}
$$
and $\epsilon = 0.1$, $\delta = 0.001$, one can verify condition
\eqref{e3.1} with $\tau_1 = \tau_2 = 0.5$. The switching regions are
given as
\begin{gather*}
\overline{S}_1 = \{(x_1,x_2): 2.468x_1^2 -0.002x_1 x_2- 2.015x_2^2 <0 \}, \\
\overline{S}_2 = \{(x_1,x_2): 2.468x_1^2 -0.002x_1 x_2- 2.015x_2^2 >0 \}.
\end{gather*}
According to Theorem \ref{thm3.1}, the system with the switching rule
$\alpha(x(t)) = i $ if $x(t) \in \overline{S}_i$ is exponentially stable.
\begin{figure}[ht] \label{fig1}
\begin{center}
\includegraphics[width=0.45\textwidth]{fig1}
\includegraphics[width=0.45\textwidth]{fig2}
\end{center}
\caption{ Region $S_1$ (left), and Region $S_2$ (right)}
\end{figure}
In this case, it can be checked that
$$
(L_1(P),L_2(P)) =
\left( \begin{bmatrix}
2.468 & -0.001 \\
-0.001 & -2.015 \end{bmatrix},
\begin{bmatrix}
-3.809 & -0.416 \\
-0.416 & 1.626 \end{bmatrix}
\right).
$$
Moreover, the sum
$$
\tau_1 L_1(P) + \tau_2 L_2(P) =
\begin{bmatrix}
-0.6705 & -0.2085 \\
-0.2085 & -0.1945 \end{bmatrix}
$$
is negative definite; i.e. the first entry in the first row and
the first column $-0.6705 < 0$ is negative and the determinant of
the matrix is positive. The sets $S_1$ and $S_2$ (without bar) are
given as
\begin{gather*}
S_1 = \{(x_1,x_2): 2.468x_1^2 -0.002x_1 x_2- 2.015x_2^2 <0 \}, \\
S_2 = \{(x_1,x_2): -3.809x_1^2 -0.831x_1 x_2+ 1.626x_2^2 <0 \}.
\end{gather*}
These sets are equivalent to
$$
S_1 = \{(x_1,x_2): (x_2 + 1.107 x_1)(x_2 - 1.106 x_1) >0 \},
$$
and
$$
S_2 = \{(x_1,x_2): (x_2 + 1.296 x_1)(x_2 - 1.807 x_1) <0 \};
$$
see figure \ref{fig1}.
Obviously, the union of these sets is equal to $\mathbb{R}^2$.
\end{example}
\begin{example} \label{exa3.2} \rm
Consider switched linear uncertain delay system \eqref{Sigma-a}
with any initial function $\phi(t)$,
where $N =2, h(t) = \sin^2 0.5t$, and
\begin{gather*}
A_1 = \begin{pmatrix} a_1 &0\\0& a_2\end{pmatrix},\quad
A_2 = \begin{pmatrix} a_3 &0\\0&a_4\end{pmatrix},
\\
D_1 =\begin{pmatrix} 1 &1\\1&1\end{pmatrix},\quad
D_2 = \begin{pmatrix} e^{-2} &e^2\\ e^2& e^{-2}\end{pmatrix},
\end{gather*}
where
\begin{gather*}
a_1 = - 1.875e^4 - 3, \quad a_2 = - 3.75e^4 - 3,\\
a_3 = - 0.125e^8 -0.3125e^4 - 3.5, \quad a_4 = - 2e^8 - 5e^4 -3.5.
\end{gather*}
We have $h =1$, and since
$\dot h(t) = 0.5\sin t$, $\mu = 0.5, \beta = 2$. Taking
$\epsilon = 1, \delta =2$ and
$$
P = \begin{pmatrix}0.4 e^{-4} &0\\0&0.1e^{-4}\end{pmatrix},
$$
we find
$$
L_1(P) = \begin{pmatrix} -1 &0.5\\0.5&-0.25\end{pmatrix},\quad
L_2(P) = \begin{pmatrix} -0.25 &0.5\\0.5&-1\end{pmatrix}.$$
Therefore,
\begin{gather*}
\boldsymbol{S}_1^P =
\{(x_1,x_2)\in \mathbb{R}^2: x_1\neq 0.5x_2 \},\\
\boldsymbol{S}_2^P = \{(x_1,x_2)\in \mathbb{R}^2:
x_2 \neq 0.5x_1 \}.
\end{gather*}
So
$$
\boldsymbol{S}_1^P\bigcup
\boldsymbol{S}_2^P = \mathbb{R}^2\setminus \{0\}.
$$
Therefore, taking
$$
\boldsymbol{\bar S}_1^P =
\boldsymbol S_1^P \quad \boldsymbol{\bar S}_2^P = \{(x_1,x_2)\in
\mathbb{R}^2: \quad x_1 = 0.5x_2, x_1\neq 0, x_2\neq 0 \},
$$
we have
$$
\boldsymbol{\bar S}_1^P \cup \boldsymbol{\bar S}_2^P =
\mathbb{R}^2\setminus \{0\},\quad \boldsymbol{\bar S}_1^P \cap
\boldsymbol{\bar S}_2^P = \emptyset.
$$
Then, the switching signal is chosen as
$$
\alpha(x) = \begin{cases} 1, &\text{if } x \in \boldsymbol{\bar S}_1^P,\\
2, &\text{if } x \in \boldsymbol{\bar S}_2^P.
\end{cases}
$$
To find the exponent factor $M$, computing
\begin{gather*}
\lambda_{\rm max}(P) = 0.4e^{-4}, \quad
\lambda_{\rm min}(P) = 0.1e^{-4},\\
\lambda_{\rm max}(D_1^TD_1) = 4\quad
\lambda_{\rm max}(D_2^TD_2) = e^4 +e^{-4}+ 2,
\end{gather*}
we have $ M \approx 3.085.$
Therefore , the solution of the system satisfies the inequality
$$
\|x(t,\phi)\| \leq 3.085 \|\phi\|e^{-2t},\quad \forall
t\geq 0.
$$
\end{example}
\subsection*{Conclusions}
We have presented a delay-dependent condition for the exponential
stability of switched linear systems with time-varying delay. The
condition is formulated in terms of a generalized Lyapunov
equation, which allows easily to compute simultaneously the two
bounds that characterize the exponential stability rate of the
solution. A simple procedure for constructing switching rule has
been given.
\subsection*{Acknowledgments}
This paper was supported by the Basic Program in Natural Sciences,
Vietnam and Thailand Research Fund Grant.
The authors would like to thank anonymous referees for his/her valuable
comments and remarks that improved the paper.
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