A Delay-dependent Condition for the Exponential Stability of Switched Linear Systems with Time-varying Delay

where { fα(.) : α ∈ Ω}, is a family of functions that is parameterized by some index set Ω, and α(·) ∈ Ω depending on the system state in each time is a switching rule/signal. The set Ω 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 (D. Liberzon, 2003; A. V. Savkin, 2002; Z. Sun, 2005). During the previous decade, the stability problem of switched linear systems has attracted a lot of attention (see,; e.g. (D. Liberzon, 1999; V. N. Phat, 2006; Z. Sun, 2005) and the references therein). The main approach for stability analysis relies on the use of LyapunovKrasovskii functionals and LMI approach for constructing common Lyapunov function (R. N. Shorten, 1988). 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 (K. S. Narendra, 1994) showed that when all subsystems are asymptotically stable, the switched system is asymptotically stable under arbitrary switching rule. The paper (G. Zhai, 2003) studied the asymptotic stability for switched linear systems with time delay, but the result was limited to symmetric systems. In (V. N. Phat, 2005; G. Xie, 2004), delay-dependent asymptotic stability conditions are extended to switched linear discrete-time linear systems with time delay. The exponential stability problem was considered in (Zairong Xi, 2003) for switched linear systems with impulsive effects by using matrix measure concept and in (G. D. Zong, 2004) for nonholonomic chained systems with strongly nonlinear input/state driven disturbances and drifts. In recent paper (S. Kim, 2006), 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 (Zairong Xi, 2003; G. D. Zong, 2004). 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.