Delay-Independent Stability of Switched Linear Systems with Unbounded Time-Varying Delays

Abstract

This paper is focused on delay-independent stability analysis for a class of switched linear systems with time-varying delays that can be unbounded. When the switched system is not necessarily positive, we first establish a delay-independent stability criterion under arbitrary switching signal by using a new method that is different from the methods to positive systems in the literature. We also apply this method to a class of time-varying switched linear systems with mixed delays.

1. Introduction

The theory of switched systems has historically assumed a position of great importance in systems theory and has been studied extensively in recent years [1–6]. A switched system is a type of hybrid dynamic system that consists of a family of continuous-time (discrete-time) subsystems and a switching signal, which determines the switching between subsystems. The stability of switched linear systems under arbitrary switching signal is a very important problem, which is usually studied by a common Lyapunov functional approach, especially a common quadratic Lyapunov functional approach [7–10].

Very recently, the stability of positive switched linear system has attracted a lot of attention [11–16]. As usual, a system is said to be positive if its state and outputs are nonnegative whenever the initial condition and inputs are nonnegative. For stability of positive switched linear system under arbitrary switching signal, a common linear copositive Lyapunov function is usually applied [17–20]. A switched linear copositive Lyapunov function has been used in discrete-time positive switched systems in [21]. When the positive switched linear system involves multiple time-varying delays that can be unbounded, it has been proved in [22] that the stability of such systems under any switching signal does not depend on delays if the switched system shares a common linear copositive Lyapunov function, which generalizes the early results in [23, 24].

For the general switched linear systems with unbounded time-varying delays, it is necessary to consider whether the similar delay-independent stability criterion under arbitrary switching signal can also be derived. Note that the system is not necessarily positive; the methods to positive systems in [22] usually do not hold. Consequently, to answer this problem, we need a new approach that is different from those methods to positive systems in the literature.

The main purpose of this paper is to establish a delay-independent stability criterion under arbitrary switching signal for the general switched linear systems with time-varying delays that can be unbounded. Since the switched systems are not necessarily positive, a new method based on some smart techniques of real analysis is proposed. By using this method, we not only present a delay-independent stability criterion for the system, but also extend the main result to a class of time-varying switched system with mixed delays, where one kind of delays is time-varying state delay that can be unbounded and the others are bounded time-varying distributed delay. Another advantage of the new method used in this paper lies in that it imposes less constraint on unbounded state delays than that given in [22] (see the corresponding discussion in Section 2).

Notations. Say 𝐴≽0(≺0) if all elements of matrix 𝐴 are nonnegative (negative). We write 𝐴≽𝐵 if and only if 𝐴−𝐵≽0. Denote by 𝕄 the the set of Metzler matrices whose off-diagonal entries are nonnegative. ℝ^𝑛 is an 𝑛-dimensional real vector space, ℝ^𝑛₊ is the set of positive vectors, and ℝ^𝑛×𝑛 is the set of real 𝑛×𝑛-dimensional matrices. For 𝑥∈ℝ^𝑛, denote ‖𝑥‖=max_{1≤𝑖≤𝑛}{|𝑥_𝑖|}. For positive integers 𝑝, 𝑞, 𝑛, and 𝑚, denote 𝑝={1,2,…,𝑝}, 𝑞={1,2,…,𝑞}, 𝑛={1,2…,𝑛}, and 𝑚={1,2,…,𝑚}.

2. Problem Statements and Preliminaries

Consider the following switched linear system with time-varying delays:𝑥̇(𝑡)=𝐴_0𝜚𝑥(𝑡)+^𝑝_𝑙=1𝐴_𝑙𝜚𝑥𝑡−𝜏_𝑙𝜚(𝑡𝑥(𝑡)(𝑡)[],),𝑡≥0,=𝜙,𝑡∈−𝛼,0 (2.1) where 𝑥∈ℝ^𝑛 is the state; the piecewise continuous function 𝜚∶[0,∞)→𝑚 is the switching signal; 𝐴_𝑙𝑠∈ℝ^𝑛×𝑛 are constant matrices for 𝑙∈𝑝∪{0} and 𝑠∈𝑚; time-varying delays 𝜏_𝑙𝑠(𝑡)≥0, 𝑙∈𝑝, 𝑠∈𝑚, are continuous on [0,∞); 𝜙(𝑡) is the continuous vector-valued initial function on [−𝛼,0] with 𝛼=max_{𝑙∈𝑝,𝑠∈𝑚}sup_𝑡≥0{𝜏_𝑙𝑠(𝑡)−𝑡}.

Unlike the assumptions on the system matrices [22], we here do not require 𝐴_0𝑠∈𝕄 and 𝐴_𝑙𝑠≽0 for 𝑙∈𝑝 and 𝑠∈𝑚. What is more, we make a less restrictive assumption on time-varying delays 𝜏_𝑙𝑠(𝑡) as follows:(H1)lim_𝑡→+∞𝑡−𝜏_𝑙𝑠(𝑡)=+∞, 𝑙∈𝑝, 𝑠∈𝑚.

We recall to introduce another assumption on 𝜏_𝑙𝑠(𝑡) in [22] as follows:(H1^′) there exist 𝑇>0 and a scaler 0≤𝜃<1 such that 𝜃=sup_{𝑡>𝑇,𝑙∈𝑝,𝑠∈𝑚}𝜏_𝑙𝑠(𝑡)/𝑡.

We show that (H1) is less constrained than (H1^′). In fact, it is not difficult to see that (H1^′) implies (H1). However, (H1) does not yield (H1^′). For example, let 𝜏_𝑙𝑠(𝑡)=𝑡−√𝑡≥0 for 𝑡≥0.

Sincelim_𝑡→+∞𝑡−𝜏_𝑙𝑠(𝑡)=lim_𝑡→+∞√𝑡=+∞,lim_𝑡→+∞𝜏_𝑙𝑠(𝑡)𝑡=lim_𝑡→+∞𝑡−√𝑡𝑡=1, (2.2) we see that (H1) holds while (H1^′) does not hold.

In the sequel, we say system (2.1) is asymptotically stable under arbitrary switching signal, if for any 𝜖>0, there exists 𝛿>0 such that any solution 𝑥(𝑡,𝜙,𝜚) of system (2.1) under arbitrary switching signal satisfies ‖𝑥(𝑡,𝜙,𝜚)‖<𝜖 when ‖𝜙‖<𝛿, and lim_𝑡→+∞‖𝑥(𝑡,𝜙,𝜚)‖=0.

Generally speaking, due to the less constraint on delays 𝜏_𝑙𝑠(𝑡) and system matrices 𝐴_𝑙𝑠≽0 for 𝑙∈𝑝 and 𝑠∈𝑚, the methods for positive systems in the literature usually become invalid. Consequently, a new method should be introduced to analyze the delay-independent stability for system (2.1) under arbitrary switching signal.

3. Main Result

In the sequel, we denote 𝐴_𝑙𝑠=[𝑎^(𝑙𝑠)_𝑖𝑗] and 𝐴_𝑙𝑠=[𝑎^(𝑙𝑠)_𝑖𝑗] for 𝑙∈𝑝∪{0} and 𝑠∈𝑚, where𝑎^(0𝑠)_𝑖𝑗=𝑎^(0𝑠)_𝑖𝑗,𝑖=𝑗,𝑎^(0𝑠)_𝑖𝑗=𝑎^(0𝑠)_𝑖𝑗,𝑖≠𝑗,𝑎^(𝑙𝑠)_𝑖𝑗=𝑎^(𝑙𝑠)_𝑖𝑗,𝑙∈𝑝. (3.1) It is easy to see that 𝐴_0𝑠∈𝕄 and 𝐴_𝑙𝑠≽0 for 𝑠∈𝑚.

We now present the main result of this paper.

Theorem 3.1. Assume that (H1) holds. If there exists a vector 𝜉∈ℝ^𝑛₊ such that 𝒜_𝑠𝜉≺0,𝑠∈𝑚, (3.2) where 𝒜_𝑠=∑^𝑝_𝑙=0𝐴_𝑙𝑠, then system (2.1) is asymptotically stable under arbitrary switching signal.

Proof. Denote 𝜉=[𝜉₁,𝜉₂,…,𝜉_𝑛]^𝑇 and 𝒜_𝑠𝜉=−𝜂_1𝑠,𝜂_2𝑠,…,𝜂_𝑛𝑠^𝑇,𝑠∈𝑚, (3.3) where 𝜂_𝑖𝑠>0 for 𝑖∈𝑛  by (3.2). The remaining proof is divided into two parts.
(i) For any constant 𝜖>0, there exists a constant 𝛿>0 such that ‖𝑥(𝑡,𝜙,𝜚)‖<𝜖 when ‖𝜙‖<𝛿. In the sequel, we denote the 𝑖th element of the solution 𝑥(𝑡,𝜙,𝜚) of system (2.1) by 𝑥_𝑖(𝑡) for 𝑖∈𝑛.
In fact, for any given 𝜖>0, let 𝛿=(𝑑₁/𝑑₂)𝜖, where 𝑑₁=min_𝑖∈𝑛𝜉₁,𝜉₂,…,𝜉_𝑛,𝑑₂=max_𝑖∈𝑛𝜉₁,𝜉₂,…,𝜉_𝑛. (3.4) When ‖𝜙‖<𝛿, we prove that 𝑥_𝑖(𝑡<𝜉)^𝑖𝑑₁𝛿,𝑡≥0,𝑖∈𝑛. (3.5) Note that 𝜉_𝑖/𝑑₁≥1 for 𝑖∈𝑛; then 𝑥_𝑖(0≤𝜙𝜉)<𝛿≤^𝑖𝑑₁𝛿,𝑖∈𝑛. (3.6) By the continuity of the solution of system (2.1), we have that there exists 𝑡′>0 such that 𝑥_𝑖(𝑡<𝜉)^𝑖𝑑₁𝛿,𝑡∈0,𝑡^′,𝑖∈𝑛. (3.7)
We further show that (3.5) holds if ‖𝜙‖<𝛿. Otherwise, there exists 𝑡^∗>0 and at least one index 𝑘∈𝑛 such that 𝑥_𝑘𝑡^∗=𝜉_𝑘𝑑₁𝛿,𝑥_𝑖(𝑡<𝜉)^𝑖𝑑₁𝛿,0≤𝑡<𝑡^∗,𝑖∈𝑛, (3.8) which implies 𝐷₋|𝑥_𝑘(𝑡^∗)|≥0, where 𝐷₋ means the left derivative. Set the left limitation 𝜚(𝑡^∗−)=𝑠₁∈𝑚. By (2.1), (3.1), (3.3), and (3.8), we get 𝐷₋𝑥_𝑘𝑡^∗=𝐷₋𝑥_𝑘𝑡^∗sign𝑥_𝑘𝑡^∗≤𝑎^(0𝑠1)_𝑘𝑘𝑥_𝑗𝑡^∗+^𝑛_{𝑗=1,𝑗≠𝑘}𝑎^(0𝑠1)_𝑘𝑗𝑥_𝑗𝑡^∗+^𝑝_𝑙=1^𝑛_𝑗=1𝑎^(𝑙𝑠1)_𝑘𝑗𝑥_𝑗𝑡^∗−𝜏_𝑙𝑠1𝑡^∗≤𝛿𝑑₁^{𝑝𝑙=0𝑛𝑗=1}𝑎^(𝑙𝑠1)_𝑘𝑗𝜉_𝑗𝛿=−𝑑₁𝜂_𝑘𝑠1<0. (3.9) From (3.9), we get a contradiction with the fact 𝐷₋|𝑥_𝑘(𝑡^∗)|≥0. Therefore, for any 𝜖>0, by choosing 𝛿=(𝑑₁/𝑑₂)𝜖 and using (3.5), we have that 𝑥𝑡,𝜙,𝜚<max_𝑖∈𝑛𝜉_𝑖𝑑₁𝛿≤𝑑₂𝑑₁𝛿=𝜖,𝑡≥0, (3.10) if ‖𝜙‖<𝛿. This completes the proof of part (i).
(ii) For any solution 𝑥(𝑡,𝜙,𝜚) of system (2.1), lim_𝑡→+∞‖𝑥(𝑡,𝜙,𝜚)‖=0.
Let 𝑥_𝑖(𝑡)=𝜉_𝑖𝑦_𝑖(𝑡) for 𝑖∈𝑛. Denote the upper limitation of |𝑦_𝑖(𝑡)| by lim_𝑡→+∞|𝑦_𝑖(𝑡)|=𝛽_𝑖 and the lower limitation of |𝑦_𝑖(𝑡)| by lim_𝑡→+∞|𝑦_𝑖(𝑡)|=𝛽_𝑖 for 𝑖∈𝑛. Set 𝛽_𝑘=max_𝑖∈𝑛{𝛽_𝑖} for some 𝑘∈𝑛 and 𝑐_𝑘𝑠=𝑎^(0𝑠)_𝑘𝑘𝜉_𝑘+^𝑛_{𝑗=1,𝑗≠𝑘}𝑎^(0𝑠)_𝑘𝑗𝜉_𝑗+^𝑝_𝑙=1^𝑛_𝑗=1𝑎^(𝑙𝑠)_𝑘𝑗𝜉_𝑗. (3.11)
We first show that 𝛽_𝑘=𝛽_𝑘. Assume to the contrary that 𝛽_𝑘>𝛽_𝑘. Choose a sufficiently small 𝜖 satisfying 0<𝜖<min𝛽_𝑘,min_𝑠∈𝑚𝜂_𝑘𝑠𝛽_𝑘max_𝑠∈𝑚𝑐_𝑘𝑠. (3.12) By the definition of 𝛽_𝑘, we have that |𝑦_𝑖(𝑡)|≤𝛽_𝑘+𝜖, 𝑖∈𝑛, hold for sufficiently large 𝑡. Since lim_𝑡→+∞𝑡−𝜏_𝑙𝑠(𝑡)=+∞ for 𝑙∈𝑝 and 𝑠∈𝑚, we have that there exists sufficiently large 𝑡₁>0 such that 𝑦_𝑖(𝑡)≤𝛽_𝑘+𝜖,𝑡≥𝑡₁,𝑦_𝑖𝑡−𝜏_𝑙𝑠(𝑡)≤𝛽_𝑘+𝜖,𝑡≥𝑡₁, (3.13) where 𝑖∈𝑛, 𝑙∈𝑝 and 𝑠∈𝑚.
On the other hand, by the assumption that 𝛽_𝑘>𝛽_𝑘 and the choice of 𝜖, there exists a sufficiently large 𝑡₂>𝑡₁>0 such that 𝑦_𝑘𝑡₂≥𝛽_𝑘𝐷−𝜖>0,₊𝑦_𝑘𝑡₂≥0, (3.14) where 𝐷₊ means the right derivative. Otherwise, we have |𝑦_𝑘(𝑡)|<𝛽_𝑘−𝜖 or 𝐷₊|𝑦_𝑘(𝑡)|<0 eventually, which contradicts with the assumption lim_𝑡→+∞𝑦_𝑘(𝑡)=𝛽_𝑘>𝛽_𝑘=lim_𝑡→+∞𝑦_𝑘(𝑡). By (3.3), it is easy to see that 𝑎^0𝑠_𝑘𝑘<0. Denote the right limitation 𝜚(𝑡₂+)=𝑠₂∈𝑚. By (3.3), (3.13), and (3.14), we get 𝐷₊𝑦_𝑘𝑡₂=𝜉_𝑘⁻¹𝐷₊𝑥_𝑘𝑡₂sign𝑥_𝑘𝑡₂≤𝜉_𝑘^−1𝑛_𝑗=1𝑎^(0𝑠2)_𝑘𝑗𝜉_𝑗𝑦_𝑗𝑡₂+^𝑝_𝑙=1^𝑛_𝑗=1𝑎^(𝑙𝑠2)_𝑘𝑗𝜉_𝑗𝑦_𝑗𝑡₂−𝜏_𝑙𝑠2𝑡₂≤𝜉_𝑘⁻¹𝛽_𝑘^𝑝_𝑙=0^𝑛_𝑗=1𝑎^(𝑙𝑠2)_𝑘𝑗𝜉_𝑗+𝜖𝑐_𝑘𝑠2=𝜉_𝑘⁻¹−𝛽_𝑘𝜂_𝑘𝑠2+𝜖𝑐_𝑘𝑠2𝜔≤−^𝑘𝜉_𝑘<0, (3.15) where 𝜔_𝑘=𝛽_𝑘min_𝑠∈𝑚{𝜂_𝑘𝑠}−𝜖max_𝑠∈𝑚{𝑐_𝑘𝑠}>0 by (3.12). This is a contradiction with the fact that 𝐷₊|𝑦_𝑘(𝑡₂)|≥0. Therefore, 𝛽_𝑘=𝛽_𝑘.
Next, we show that 𝛽_𝑘=𝛽_𝑘=0. Otherwise, 𝛽_𝑘=𝛽_𝑘>0. Then, for sufficiently small 𝜖 satisfying (3.12), there exists 𝑡₃>𝑡₁ such that (3.13) holds, and 𝑦_𝑘(𝑡)≥𝛽_𝑘−𝜖>0,𝑡≥𝑡₃. (3.16) Here, (3.16) is concluded from the property of the lower limitation lim_𝑡→+∞|𝑦_𝑘(𝑡)|=𝛽_𝑘=𝛽_𝑘. Similar to the above analysis, we have 𝐷₊𝑦_𝑘(𝑡𝜔)≤−^𝑘𝜉_𝑘,𝑡≥𝑡₃. (3.17) Integrating (3.17) from 𝑡₃ to 𝑡 on both sides, we get the following contradiction: 𝑦_𝑘(𝑡)≤𝑦_𝑘𝑡₃−𝜔_𝑘𝜉_𝑘𝑡−𝑡₃⟶−∞(𝑡⟶+∞). (3.18) Thus, lim_𝑡→+∞|𝑦_𝑘(𝑡)|=0. By the choice of 𝑘 and the definition of 𝑦_𝑖(𝑡), we have that lim_𝑡→+∞|𝑥_𝑖(𝑡)|=0 for 𝑖∈𝑛, which implies that lim_𝑡→+∞‖𝑥(𝑡,𝜙,𝜚)‖=0.
By (i) and (ii), system (2.1) is asymptotically stable under arbitrary switching signal. This completes the proof of Theorem 3.1.

Remark 3.2. For the particular case when 𝑚=1, condition (3.2) holds if and only if 𝒜₁ is a Hurwitz matrix [25]. When 𝑚>1, it requires that all 𝒜_𝑠, 𝑠∈𝑟, share a common 𝜉∈ℝ^𝑛₊ such that −𝒜_𝑠𝜉∈ℝ^𝑛₊. This problem has been studied in [20], where necessary and sufficient conditions for the existence of such a vector 𝜉 were established.

4. Extension to Time-Varying Switched Systems with Mixed Delays

We now extend Theorem 3.1 to a class of time-varying switched system with mixed delays:𝑥̇(𝑡)=𝐴_0𝜚(𝑡)𝑥(𝑡)+^𝑝_𝑙=1𝐴_𝑙𝜚(𝑡)𝑥𝑡−𝜏_𝑙𝜚(𝑡)+^𝑞_𝑟=1𝐵_𝑟𝜚(𝑡)^𝑡_{𝑡−𝜎𝑟𝜚(𝑡)}𝑥(𝑢)𝑥(𝑡)(𝑡),𝑑𝑢,𝑡≥0,=𝜙,𝑡∈−𝛾,0 (4.1) where 𝐴_𝑙𝑠(𝑡)=[𝑎^(𝑙𝑠)_𝑖𝑗(𝑡)], 𝐵_𝑟𝑠(𝑡)=[𝑏^(𝑟𝑠)_𝑖𝑗(𝑡)] are continuous matrix function on [0,∞), delays 𝜎_𝑟𝑠(𝑡) are continuous on [0,∞), and 𝛾=max_{𝑙∈𝑝,𝑟∈𝑞,𝑠∈𝑚}{sup_𝑡≥0{𝜏_𝑙𝑠(𝑡)−𝑡},sup_𝑡≥0{𝜎_𝑟𝑠(𝑡)−𝑡}}.

Assume that(H2) there exist constants 𝜃_𝑟𝑠>0 such that 0≤𝜎_𝑟𝑠(𝑡)≤𝜃_𝑟𝑠 for 𝑟∈𝑞 and 𝑠∈𝑚;(H3) there exist constant matrices 𝐴_0𝑠=[𝑎^(0𝑠)_𝑖𝑗]∈𝕄 and 𝐴_𝑙𝑠=[𝑎^(𝑙𝑠)_𝑖𝑗]≽0,𝐵_𝑟𝑠=[𝑏̂^(𝑟𝑠)_𝑖𝑗]≽0 such that, for 𝑡≥0 and 𝑖,𝑗∈𝑛,𝑎^(0𝑠)_𝑖𝑗(𝑡)≤𝑎^(0𝑠)_𝑖𝑗𝑎,𝑖=𝑗,_{(0𝑠)𝑖𝑗}(𝑡)≤𝑎^(0𝑠)_𝑖𝑗𝑎,𝑖≠𝑗,_{(𝑙𝑠)𝑖𝑗}(𝑡)≤𝑎^(𝑙𝑠)_𝑖𝑗,𝑙∈𝑝,𝑏^(𝑟𝑠)_𝑖𝑗(𝑡)≤𝑏̂^(𝑟𝑠)_𝑖𝑗,𝑟∈𝑞. (4.2) When 𝑥_𝑖(𝑡)≠0, a straightforward computation based on (4.2) yields that𝐷_±𝑥_𝑖(𝑡)=sign𝑥_𝑖(𝑡)^𝑛_𝑗=1𝑎^(0𝜚±)_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑝_𝑙=1^𝑛_𝑗=1𝑎^(𝑙𝜚±)_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑙𝜚±(𝑡)+^𝑞_𝑟=1^𝑛_𝑗=1𝑏^(𝑟𝜚±)_𝑖𝑗(𝑡)^𝑡_{𝑡−𝜎±𝑟𝜚(𝑡)}𝑥_𝑗(𝑢)𝑑𝑢≤^𝑛_𝑗=1𝑎^(0𝜚±)_𝑖𝑗𝑥_𝑗(𝑡)+^𝑝_𝑙=1^𝑛_𝑗=1𝑎^(𝑙𝜚±)_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑙𝜚±(𝑡+_𝑞𝑟=1)𝜃_𝑟𝜚±^𝑛_𝑗=1𝑏̂^(𝑟𝜚±)_𝑖𝑗max_{𝑡−𝜃±𝑟𝜚≤𝑢≤𝑡}𝑥_𝑗(𝑢), (4.3) where 𝜚_±=𝜚(𝑡±). Then, similar to the analysis in Theorem 3.1, it is not difficult to get the following stability criterion for system (4.1).

Theorem 4.1. Assume that (H1)–(H3) hold. If there exists a vector 𝜉∈ℝ^𝑛₊ such that ℬ_𝑠𝜉≺0,𝑠∈𝑚, (4.4) where ℬ_𝑠=∑^𝑝_𝑙=0𝐴_𝑙𝑠+∑^𝑞_𝑟=1𝜃_𝑟𝑠𝐵_𝑟𝑠, then system (4.1) is asymptotically stable under arbitrary switching signal.

Consider the following uncertain switched system:𝑥̇(𝑡)=𝐴_0𝜚+Δ𝐴_0𝜚(𝑡)𝑥(𝑡)+^𝑝_𝑙=1𝐴_𝑙𝜚+Δ𝐴_𝑙𝜚(𝑡)𝑥𝑡−𝜏_𝑙𝜚(𝑡)+^𝑞_𝑟=1𝐵_𝑟𝜚+Δ𝐵_𝑟𝜚(𝑡)^𝑡_{𝑡−𝜎𝑟𝜚(𝑡)}𝑥(𝑢)𝑥(𝑡)(𝑡),𝑑𝑢,𝑡≥0=𝜙,𝑡∈−𝛾,0 (4.5) where Δ𝐴_𝑙𝑠(𝑡)=[Δ𝑎^(𝑙𝑠)_𝑖𝑗(𝑡)] and Δ𝐵_𝑟𝑠(𝑡)=[Δ𝑏^(𝑙𝑠)_𝑖𝑗(𝑡)] are uncertain matrices satisfyingΔ𝑎^(𝑙𝑠)_𝑖𝑗(𝑡)≤Δ𝑎^(𝑙𝑠)_𝑖𝑗,Δ𝑏^(𝑙𝑠)_𝑖𝑗(𝑡)≤Δ𝑏^(𝑙𝑠)_𝑖𝑗,𝑡≥0. (4.6) Set𝑎^(0𝑠)_𝑖𝑗=𝑎^(0𝑠)_𝑖𝑗+Δ𝑎^(0𝑠)_𝑖𝑗,𝑖=𝑗,𝑎^(0𝑠)_𝑖𝑗=𝑎^(0𝑠)_𝑖𝑗+Δ𝑎^(0𝑠)_𝑖𝑗,𝑖≠𝑗,𝑎^(𝑙𝑠)_𝑖𝑗=𝑎^(𝑙𝑠)_𝑖𝑗+Δ𝑎^(𝑙𝑠)_𝑖𝑗,𝑙∈𝑝,𝑏̂^(𝑟𝑠)_𝑖𝑗=𝑏^(𝑟𝑠)_𝑖𝑗+Δ𝑏^(𝑟𝑠)_𝑖𝑗,𝑟∈𝑞. (4.7) Then, based on the same analysis as above, we have the following result for the uncertain system (4.5).

Theorem 4.2. Assume that (H1) and (H2) hold. If there exists a vector 𝜉∈ℝ^𝑛₊ such that (4.4) holds, then system (4.5) is asymptotically stable under arbitrary switching signal.

5. A Numerical Example

To illustrate Theorem 3.1, we present a simple numerical example of system (2.1) with𝐴₀₁=−3−0.50.5−1.5,𝐴₁₁=,𝐴−10.5−0.50.5₀₂=−1.50.25−0.25−1.5,𝐴₁₂=,0.5−0.250.25−0.5 (5.1) and 𝜏₁₁(𝑡)=𝜏₁₂(𝑡)=0.4𝑡+1 for 𝑡≥0. By (3.1), we have that𝒜₁=−211−1,𝒜₂=−10.50.5−1. (5.2) It is not difficult to verify that (H1) holds and there exists a vector 𝜉=[1,1.5]^𝑇 such that 𝒜_𝑠𝜉≺0 for 𝑠=1,2. Therefore, by Theorem 3.1, we know that system (2.1) is asymptotically stable under arbitrary switching signal. Since system (2.1) is not positive, Theorem  2 in [22] is invalid for this case. The state of the system is given in Figure 1.

Figure 1 

The state of the system under stochastic initial condition and switching.

It is not difficult to work out an example of Theorem 4.1. We omit it here due to the similarity with the above example.

6. Conclusion

In this paper, we investigate the delay-independent stability of the nonpositive switched linear systems with time-varying delays. By using a new method that is different from those methods to positive systems, we show that the stability of the system is also independent of delays if the switched system shares a common linear copositive Lyapunov function. We also apply this method to a class of time-varying switched linear systems with mixed delays, which generalizes some existing results in the literature.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants nos. 60704039 and 61174217 and the Natural Science Foundation of Shandong Province under Grants nos. ZR2010AL002 and JQ201119.