INTRODUCTION
Recently, switched systems have attracted great attention research in academic
and industrial application. It also has been widely and successfully applied
to a variety of industrial processes such as the automotive industry, chemical
procedure control systems, navigation systems, automobile speed change system,
aircraft control system, air traffic control etc, can be appropriately described
by the switched model (Liberzon and Morse, 1999; Morse,
1996; Wu et al., 2010). Switched systems
are a class of hybrid dynamical systems composing of several continuoustime
or discretetime subsystems and a rule that orchestrates the switching sequences
between them. In the study of switched systems, most works have been centralized
on the problem of stability. Therefore, there have been many researches about
the stable analysis and design of switched systems (Chiou,
2006; Chiou and Cheng, 2009a, b;
Chiou et al., 2009, 2010;
Chiou and Wang, 2010; Chiou et
al., 2011) and the references cited therein.
Two important methods are used to construct the switching law for the stability
analysis of the switched systems. One is the statedriven switching strategy
(Chiou and Cheng, 2009a, b;
Chiou et al., 2010, 2011);
the other is the timedriven switching strategy (Chiou,
2006). The timedriven switching method is formed based on the main concept
of a dwell time. If at least one stable subsystem exits, then the switched system
is stable with a proper dwell time switching law. However, in reality, it is
not unusual to encounter cases in which all subsystems are unstable. Therefore,
Li et al. (2005) proposed that all subsystem can
be unstable for timedriven switching method. We would have to turn to the statedriven
switching strategy, from which many choices of switching laws ensuring stability
exist, even if all the subsystems are unstable. Zhai et
al. (2003) proposed quadratic stabilizability via state feedback for
both continuoustime and discretetime switched linear systems that are composed
of polytopic uncertain subsystems. For continuoustime switched linear systems,
if there exists a common positive definite matrix for stability of all convex
combinations of the extreme points which belong to different subsystem matrices,
then the switched system is quadratically stabilizable via state feedback. The
switching rules can be obtained by using the obtained common positive definite
matrix.
Furthermore, the timedelay phenomenon is also unavoidable in practical systems,
for instance, chemical process, long distance transmission line, hybrid procedure,
electron network etc. Timedelays may cause instability and poor performance
for practical systems (Jasem et al., 2010; Li
and Fang, 2009; Liu et al., 2011). In view
of the aforementioned facts, the stability of switched systems with time delay
is very worthy of research. Basically, current efforts to achieve stability
in timedelay systems can be divided into two categories, namely delayindependent
criteria and delaydependent criteria. In this study, the delayindependent
stability of switched timedelay system is considered. For the delayindependent
criteria, LyapunovRazumikhin functional technique is the most suitable for
application (Hale, 1977). In view of the statedriven
switching method, LyapunovRazumikhin functional approach is applied to analyze
the stability problem for switched timedelay systems. By LyapunovRazumikhin
functional technique (Hsiao et al., 2010) to
construct a statedriven switching strategy such that the switched timedelay
system is delayindependent asymptotically stable. And we combine with linear
matrix inequalities techniques to study the stability of switched timedelay
systems.
There are two basic problems in stability and switching law design of the switched timedelay system. These problems are: stability for arbitrary switching sequences and construction of stabilizing switching sequences. We derive stability conditions that guarantee the switched timedelay system is asymptotically stable for two kinds of switching law.
REPRESENTATION MODELING OF SWITCHED TIMEDELAY SYSTEM
Consider the following switched timedelay system:
where, x (t)εR^{n} is state, A_{σ (x (t))} x (t)εR^{nxn}, B_{σ (x (t))} x (t) R^{nxn}, t_{0}≥0 is the initial time, x_{0} is the initial state, σ (x (t)): R^{n}→{1, 2, ..., N} is a piecewise constant function of time, called a switch signal, i.e., the matrix A_{σ (x (t))} switches between matrices A_{1}, A_{2}, ..., A_{N} belonging to the set A = {A_{1}, A_{2}, ..., A_{N}} and A_{i}, iε{1, 2, ..., N}, the matrix B_{σ (x (t))} switches between matrices B_{1}, B_{2}, ..., B_{N} belonging to the set B = {B_{1}, B_{2}, ..., B_{N}} and B_{i}, iε{1, 2, ..., N}. τ>0 is the timedelay duration. ψ (t) is a vectorvalued initial continuous function defined on the interval [τ, 0] and finally ψ (t), defined on τ≤t≤0, is the initial condition of the state.
Therefore, the switched discrete timedelay system (1) can be described as follows:
where, lε{1, 2, ..., N}.
There are two basic problems in stability and switching law design of the switched system. These problems are: stability for arbitrary switching sequences and construction of stabilizing switching sequences.
Definition 1: Stability for Arbitrary Switching Sequences (SASS):There exist conditions that guarantee that the switched timedelay system is asymptotically Stable for Arbitrary Switching Sequences (SASS).
Definition 2: Construction of Stabilizing Switching Sequences (CSSS): There exist conditions that guarantee that the switched timedelay system is asymptotically stable for Construction of Stabilizing Switching Sequences (CSSS).
STABILITY ANALYSIS
Sufficient conditions for ensuring delayindependent stability of switched timedelay system (1) will be derived using Lyapunov stability approach with two kinds of switching law.
Stability for arbitrary switching sequences
Theorem 1: There exists a switching law (SASS), the switched timedelay
system (1) is delayindependent asymptotically stable if there exist matrices
p>0 and Q>0 such that:
for 1 = 1, 2, ..., N.
Proof: We select the Lyapunov function as:
The derivative of the Lyapunov function V (x (t)) along the trajectories of switched timedelay system is:
By using Razumikhin theorem (Hale, 1977), we assume that
there exists a real υ>1 such that:
Then,
If:
hold for all i and 1, then
From using Razumikhin theorem, notice that if A_{l}^{T} P+PA_{l}+PB_{l} Qb_{l}^{T} P+p<0 holds, i.e., (1)<0, then by continuity, there exists a υ = 1+δ with δ>0 sufficiently small such that (υ)<0 for all i and 1.
Therefore, if A_{l}^{T} P+PA_{l}+PB_{l} Qb_{l}^{T}
P+p<0 hold, then ,
i. e., the switched timedelay system (1) is delayindependent asymptotically
stable under the switching law 1 (SASS).
Switching law 1(SASS): Switched timedelay system with arbitrary N individual systems is switched to or stay at mode l at arbitrary switching sequences.
Using Schur complement we find that the matrix inequalities (3a) and (3b) are equivalent to the following LMIs, respectively:
where, H= P^{1}
Corollary 1: There exists a switching law 1 (SASS), the switched timedelay system (1) is delayindependent asymptotically stable if there exist matrices p>0 and Q>0 satisfying LMIs (8a) and (8b) for i = 1, 2, ..., r and 1 = 1, 2, ... , N.
Construction of stabilizing switching sequences: For this objective of stable analysis, one helpful lemma is given below.
Switching law 2 (CSSS): Switched timedelay system with arbitrary N individual systems is switched to or stay at mode l at time t if (9) is satisfied at time t.
Lemma 1: There exists a switching law 2 (CSSS) for the switched timedelay system (1) such that the system is asymptotically stable if there exist positive constants α_{i} (1≤i≤N) satisfying:
such that the convex combination of the whole switched timedelay system:
is an asymptotically stable system.
Proof: Since there exist positive numbers α_{i} (1≤i≤N) such that:
is asymptotically stable, there exists a Lyapunov function V (x) such that:
It follows that for any t, at least there exists an iε{1, 2, ..., N} such that:
From (11), it implies that a convex combination of the corresponding Lyapunov function is negative along the trajectory. Thus, the switched timedelay system (1) is asymptotically stable.
Theorem 2: There exists a switching law 2 (CSSS), the switched timedelay system (1) is delayindependent asymptotically stable if there exist matrices p>0 and Q>0 such that:
for 1=1, 2, ..., N.
Proof: By Lemma 1, the switched timedelay system (1) implies that a convex combination system:
We select the Lyapunov function as:
The derivative of the Lyapunov function V (x (t)) along the trajectories of
(1) is:
By using Razumikhin theorem (Hale, 1977), we assume that
there exists a real υ>1 such that:
Then,
If
(υ):= A_{1}^{T} P+PA_{1}+PB_{1} Qb_{1}^{T}
P+p<0 hold for all i and l, then .
From using Razumikhin theorem, notice that if A_{1}^{T} P+PA_{1}+PB_{1} Qb_{1}^{T} P+p<0 holds, i.e., (1)<0, then by continuity, there exists a υ = 1+δ with δ>0 sufficiently small such that (υ)<0 for all i and 1.
Therefore, if :
i. e.,
then,
thus, TS fuzzy switched timedelay system (1) is delayindependent asymptotically
stable under the switching law 2 (CSSS).
We denote:
Using Schur complement we find that the matrix inequalities are equivalent to the following LMIs, respectively:
where, H = P^{1}
Corollary 2: There exists a switching law (CSSS), the switched timedelay system (1) is delayindependent asymptotically stable if there exist matrices p>0 and Q>0 satisfying LMIs (18a) and (18b) for i = 1, 2, ..., r and l = 1, 2, ..., N.
Lemma 2: For any matrices A_{1}, A_{2}, ..., A_{N} with the same dimensions, the following inequality folds for any positive constant ε:
Proof: For a positive constant ε, A and B with the same dimension, it is a obvious fact that:
In view of inequality (19), we have:
Lemma 3: Let α_{l} = 1/(1+ε^{1})^{11}
(1+ε), l = 1, 2, ..., N1 and α_{N} = 1/(1+ε^{1})^{N1},
there exists any positive constant ε such that α_{l}ε[0,
1] (1≤l≤N) and:
Proof: Obviously, if there exists a positive constant ε then 0<α_{l}<1,
l = 1, 2, ..., N and
(α_{1}, α_{2}, ..., α_{N1}) is a geometric progression).
In the light of Lemma 2 and 3, we have:
Therefore, we denote:
Corollary 3: There exists a switching law (CSSS), the switched timedelay system (1) is delayindependent asymptotically stable if there exist matrices P>0 and Q>0 satisfying LMIs (23a) and (23b) for l = 1, 2, ..., N.
where, H = P^{1}.
Therefore, the result of Corollary 3 can simplify parameter and reduce α_{1}, α_{2}, ..., α_{N} to ε. Thus, it is easy to analyze the switched timedelay systems.
EXAMPLE
Example 1: Consider the switched timedelay system composed of two subsystems given as:
Subsystem 1:
Subsystem 2:
The switched timedelay system with two subsystems (N = 2), in view of the stability conditions of Corollary 1, inequalities (8) can be written as follows:
By using LMI tool of MATLAB software, we find:
Therefore, for arbitrary constant delay τ, the switched timedelay system
(24) can be stabilized by arbitrary switching sequences (SASS).
With timedelay τ = 2 sec, swithing during [0, 4] sec and initial value
x (0) = [20 10]^{T}, the trajectories of the switched timedelay system
(24) is shown in Fig. 1a. The switching sequence is shown
in Fig. 1b.

Fig. 1a: 
State responses of system (24) 

Fig. 1b: 
Switching sequences of system (24) 
Example 2: Consider the switched timedelay system composed of two subsystems given as:
Subsystem 1:
Subsystem 2:
The switched timedelay system with two subsystems (N = 2), in view of the stability conditions of Corollary 3, inequalities (23) can be written as follows:
We choose ε = 2 and use LMI tool of MATLAB software, then we get:
Therefore, for arbitrary constant delay τ, the switched timedelay system
can be stabilized by Construction of Stabilizing Switching Sequences (CSSS).

Fig. 2a: 
State response of system (25) 

Fig. 2b: 
Switching sequences of system (25) 
Switching law (CSSS): Switched timedelay system with arbitrary N individual systems is switched to or stay at mode l at time t if (26) is satisfied at time t.
With timedelay τ = 2 sec, swithing during [0, 4] sec and initial value
x (0) = [20 10]^{T}, the trajectories of the switched timedelay system
(24) is shown in Fig. 2a. The switching sequence is shown
in Fig. 2b. Therefore, if we select positive constant ε
and satisfy inequalities (23) then the system is delayindependent asymptotically
stable for construction of stabilizing switching sequences (CSSS).
Example 2 is exploited to illustrate the proposed schemes, stability conditions
that guarantee the switched discrete timedelay system is delayindependent
asymptotically stable for construction of stabilizing switching law. In the
light of Corollary 2 and Corollary 3, we simplify parameter and reduce α_{1},
α_{2}, ..., α_{N} to ε. Thus, it is easy to analyze
the switched timedelay system. And we constructively design a switching rule
which can guarantee the stability of the switched systems. Otherwise, the particular
method can be applied to cases whose individual subsystems are unstable.
CONCLUSION
The study adopted LyapunovRazumikhin stability theorem to study the delayindependence stability analysis of a class of switched timedelay system. We derive stability conditions that guarantee the switched timedelay system is delayindependent asymptotically stable for arbitrary switching sequences and construction of stabilizing switching law. The main advantages of our approach are that simplify parameter and reduce α_{1}, α_{2}, ..., α_{N} to ε then it is easy to analyze the switched timedelay system, can be applied to individual subsystems whose includes unstable subsystems, can extend to the case of arbitrary subsystems of switched delay system and develop the simple switching rule to stabilize the switched timedelay system and construct LMIbased design procedures for stability analysis. The subject is interesting and important and it will be attracting increasing attention in future.
ACKNOWLEDGMENTS
This work is supported by the National Science Council, Taiwan, Republic of China, under grand No. NSC 992221E218002.