Delay-Dependent Stability Criterion of Arbitrary Switched Linear Systems with Time-Varying Delay

Abstract

This paper deals with the problem of delay-dependent stability criterion of arbitrary switched linear systems with time-varying delay. Based on switched quadratic Lyapunov functional approach and free-weighting matrix approach, some linear matrix inequality criterions are found to guarantee delay-dependent asymptotically stability of these systems. Simultaneously, arbitrary switched linear system can be expressed as a problem of uncertain liner system, so some delay-dependent stability criterions are obtained with the result of uncertain liner system. Two examples illustrate the exactness of the proposed criterions.

1. Introduction

Recently, switched linear systems have got more and more attention in the research community, which consists of a family of liner subsystems described by liner differential or difference equations and a switching law that orchestrates switching between them; see, for example, [1–4]. Simultaneously, systems with delays abound in the world and time-delay systems frequently appear in vast engineering systems [5–7]. Therefore, many papers consider switched linear systems with time constant delay or time-varying delay [8–24]. Naturally, stability is a fundamental property which has been investigated from the very beginning for this class of systems [25]. For stability analysis under arbitrary switching, even when all subsystems of a switched system are asymptotically stable or exponentially stable, it is still possible to construct a divergent trajectory from any initial state for such a switched system [4]. Thus, this paper aims to study the stability of arbitrary switched linear system with time-varying delay.

On one hand, many methods have been developed in the study of arbitrary switched systems such as common quadratic Lyapunov functional approach (CQLF), converse Lyapunov theorem, and switched quadratic Lyapunov functional approach (SQLF) [4, 26–28]. On the other hand, Wu M. and He Y. develop free-weighting matrix approach for stability of liner system and uncertain liner system [29–33]. In this paper, Based on SQLF and free-weighting matrix approach, we consider the linear switched system:𝑥(𝑘+1)=𝐴_𝑟(𝑘)𝑥(𝑘)+𝐴_{𝑑𝑟(𝑘)}𝑥(𝑘−𝑑(𝑘))+𝐵_𝑟(𝑘)𝑢(𝑘),𝑘∈𝐙⁺,𝑟(𝑘)∈Ω, (1.1) where 𝑥(𝑘)∈𝐑^𝑛 is the state, 𝑢(𝑘)∈𝐑^𝑛 is the control input, and 𝑟(𝑘) is a switching rule defined by 𝑟(𝑘)∶𝑁→Ω with Ω={1,2,…,𝑁}. Moreover, 𝑟(𝑘)=𝑖 means the subsystem (𝐴_𝑖,𝐴_𝑑𝑖,𝐵_𝑖) is active. 𝑑(𝑘) is nonnegative differential time-varying functions which denote the time delays and satisfy 0≤𝑑₁≤𝑑(𝑘)≤𝑑₂.

At the same time, the uncertain linear system𝑥(𝑘+1)=(𝐴−Δ𝐴(𝑘))𝑥(𝑘)+𝐴_𝑑−Δ𝐴_𝑑(𝑘)𝑥(𝑘−𝑑(𝑘))+(𝐵−Δ𝐵(𝑘))𝑢(𝑘),𝑘∈𝐙⁺, (1.2) where 𝑥(𝑘)∈𝑅^𝑛 is the state, 𝑢(𝑘)∈𝑅^𝑛 is the control input, 𝐴, 𝐴_𝑑, and 𝐵 are given constant matrices, Δ𝐴(𝑘), Δ𝐴_𝑑(𝑘), and Δ𝐵(𝑘) are the parameter uncertainties matrices which are assumed to be of the formΔ𝐴(𝑘)Δ𝐴_𝑑(𝑘)Δ𝐵(𝑘)=𝐷𝐹(𝑘)𝐸_𝑎𝐸_𝑎𝑑𝐸_𝑏, (1.3) where 𝐸_𝑎, 𝐸_𝑎𝑑, and 𝐸_𝑏 are given constant matrices of appropriate dimensions and 𝐹(𝑘) is the uncertain matrix such that𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼. (1.4) From (1.1) and (1.2), we know that when one subsystem switches to another subsystem, there exist matrixes 𝐴, 𝐴_𝑑, and 𝐵 such that𝐴_𝑟(𝑘)𝐴_{𝑑𝑟(𝑘)}𝐵_𝑟(𝑘)=𝐴−Δ𝐴(𝑘)𝐴_𝑑−Δ𝐴_𝑑(𝑘)𝐵−Δ𝐵(𝑘) (1.5) so system (1.1) be equivalent to system (1.2). The key ideas of this paper are that SQLF is connected with free-weighting matrix approach and arbitrary switched linear system can be expressed as a problem of uncertain liner system.

This paper is organized as follows. In Section 2, we give some basic definitions. We analyze the stability of the system (1.1) with the SQLF and free-weighting matrix approach in Section 3. Based on uncertain liner system, we study the stability of the system (1.1) in Section 4. Some examples are given in Section 5. The last section offers the conclusions of this paper.

2. Preliminaries

In this section, with the switched quadratic Lyapunov functional approach, we investigate the stability of the origin of an autonomous switched system given by𝑥(𝑘+1)=𝐴_𝑟(𝑘)𝑥(𝑘),𝑘∈𝐙⁺,𝑟(𝑘)∈Ω. (2.1) Define the indicatorfunction𝜉(𝑘)=𝜉₁(𝑘),…,𝜉_𝑖(𝑘),…,𝜉_𝑁(𝑘)^𝑇, (2.2)with𝜉_𝑖(𝑘)=1if𝑟(𝑘)=𝑖,0,otherwise. (2.3)

Then, the switched system (2.1) can also be written as𝑥(𝑘+1)=^𝑁_𝑖=1𝜉_𝑖(𝑘)𝐴_𝑖𝑥(𝑘). (2.4) This corresponds to the switched Lyapunov function defined as𝑉(𝑘,𝑥(𝑘))=𝑥^𝑇(𝑘)𝑃_𝑟(𝑘)𝑥(𝑘)=𝑥^𝑇(𝑘)^𝑁_𝑖=1𝜉_𝑖(𝑘)𝑃_𝑖𝑥(𝑘) (2.5) with 𝑃_𝑖 is symmetric positive definite matrices. If such a positive-definite Lyapunov function exists andΔ𝑉(𝑘,𝑥(𝑘))=𝑉(𝑘+1,𝑥(𝑘+1))−𝑉(𝑘,𝑥(𝑘)) (2.6) is negative definite along the solutions of (2.1), then the origin of the switched system (2.1) is asymptotically stable. In order to represent, we give the following notation.

Throughout this paper, the superscript 𝑇 stands for the inverse and transpose of a matrix; 𝑅^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices; 𝑃>0 means that the matrix 𝑃 is positive definite; and the symmetric terms in a symmetric matrix are denoted by ⋆, for example,=𝑂𝑀𝑂⋆𝑁𝑀𝑂_𝑇𝑁. (2.7)

Lemma 2.1 (see [4]). If there exist positive definite symmetric matrices 𝑃_𝑖∈𝑅^𝑛×𝑛(𝑃_𝑖=𝑃^𝑇_𝑖), satisfying 𝑃_𝑖𝐴^𝑇_𝑖𝑃_𝑗⋆𝑃_𝑗<0 (2.8) for all 𝑖,𝑗∈Ω, then the switched linear system (2.1) is asymptotically stable.

Lemma 2.2 (see [4]). If there exist positive definite symmetric matrices 𝑃_𝑖∈𝑅^𝑛×𝑛(𝑃_𝑖=𝑃^𝑇_𝑖) and matrices 𝐹_𝑖,𝐺_𝑖∈𝑅^𝑛×𝑛(𝑖∈Ω), satisfying 𝐴_𝑖𝐹^𝑇_𝑖+𝐹_𝑖𝐴^𝑇_𝑖−𝑃_𝑖𝐴_𝑖𝐺_𝑖−𝐹_𝑖⋆𝑃_𝑗−𝐺_𝑖−𝐺^𝑇_𝑖<0 (2.9) for all 𝑖,𝑗∈Ω, then the switched linear system (2.1) is asymptotically stable.

Lemma 2.3 (see [33]). Let 𝑑₁ and 𝑑₂ be positive integers such that 0≤𝑑₁≤𝑑₂. When 𝑢(𝑘)=0, the systems (1.2) is asymptotically stability if there exist symmetric matrices 𝑃=𝑃^𝑇>0, 𝑄=𝑄^𝑇>0, 𝑍=𝑍^𝑇>0, 𝑋=[^𝑋11𝑋12_⋆𝑋22]≥0 and any appropriate dimensional matrices 𝑁₁, 𝑁₂ and 𝜆>0 such that the following LMIs hold, Ψ₁₁+𝜆𝐸^𝑇_𝑎𝐸_𝑎Ψ₁₂+𝜆𝐸^𝑇_𝑎𝐸_ad(𝐴−𝐼)^𝑇𝐻𝑃𝐷⋆Ψ₂₂+𝜆𝐸^𝑇_ad𝐸_ad𝐴^𝑇_𝑑𝑋𝐻0⋆⋆−𝐻𝐻𝐷⋆⋆⋆−𝜆𝐼<0,₁₁𝑋₁₂𝑁₁⋆𝑋₂₂𝑁₂⋆⋆𝑍≥0, (2.10) where Ψ₁₁=(𝑑₂−𝑑₁+1)𝑄+𝑃(𝐴−𝐼)+(𝐴−𝐼)^𝑇𝑃+𝑁₁+𝑁^𝑇₁+𝑑₂𝑋₁₁, Ψ₁₂=𝑃𝐴_𝑑+𝑁^𝑇_𝑍−𝑁₁+𝑑₂𝑋₁₂,Ψ₂₂=−𝑄−𝑁^𝑇₂−𝑁₂+𝑑₂𝑋₂₂, and 𝐻=𝑃+𝑑₂𝑍.

3. Stability Analysis of System (1.1) with SQLF

In this section, firstly, when we do not consider the control input, the linear switched system (1.1) is rewritten as𝑥(𝑘+1)=𝐴_𝑟(𝑘)𝑥(𝑘)+𝐴_{𝑑𝑟(𝑘)}𝑥(𝑘−𝑑(𝑘)),𝑘∈𝑍⁺,𝑟(𝑘)∈Ω. (3.1)

With SQLF and free-weighting matrix approach, we have the following theorem.

Theorem 3.1. Let 𝑑₁ and 𝑑₂ be positive integers such that 0≤𝑑₁≤𝑑₂; the systems (3.1) is asymptotically stability,if there exist symmetric matrices 𝑃_𝑖=𝑃^𝑇_𝑖>0, 𝑃_𝑗=𝑃^𝑇_𝑗>0, 𝑄=𝑄^𝑇≥0, 𝑍=𝑍^𝑇≥0, 𝑋^𝑖𝑗=[^{𝑋𝑖𝑗11𝑋𝑖𝑗12}_{⋆𝑋𝑖𝑗22}]≥0 and any appropriate dimensional matrices 𝑁₁^𝑖𝑗 and 𝑁₂^𝑖𝑗 such that the following LMIs hold, Φ^𝑖𝑗=Φ^𝑖𝑗₁₁Φ^𝑖𝑗₁₂𝑑₂𝐴_𝑖−𝐼^𝑇𝑍⋆Φ^𝑖𝑗₂₂𝑑₂𝐴^𝑇_di𝑍⋆⋆−𝑑₂𝑍Θ<0,_𝑖𝑗=𝑋^𝑖𝑗₁₁𝑋^𝑖𝑗₁₂𝑁₁^𝑖𝑗⋆𝑋^𝑖𝑗₂₂𝑁₂^𝑖𝑗⋆⋆𝑍≥0,𝑖,𝑗∈Ω, (3.2)(3.3) where Φ^𝑖𝑗₁₁=(𝑑₂−𝑑₁+1)𝑄+𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑖−𝑃_𝑖+𝑁₁^𝑖𝑗+(𝑁₁^𝑖𝑗)^𝑇+𝑑₂𝑋^𝑖𝑗₁₁, Φ^𝑖𝑗₁₂=𝐴^𝑇_𝑖𝑃_𝑗𝐴_di+(𝑁₂^𝑖𝑗)^𝑇−𝑁₁^𝑖𝑗+𝑑₂𝑋^𝑖𝑗₁₂ and Φ^𝑖𝑗₂₂=𝐴^𝑇_di𝑃_𝑗𝐴_di−𝑄−(𝑁₂^𝑖𝑗)^𝑇−𝑁₂^𝑖𝑗+𝑑₂𝑋^𝑖𝑗₂₂.

Proof. Suppose that 𝑦(𝑙)=𝑥(𝑙+1)−𝑥(𝑙), then we have 𝑥(𝑘+1)=𝑥(𝑘)+𝑦(𝑘) and 𝑥(𝑘)=𝑥(𝑘−𝑑(𝑘))+∑^𝑘−1_{𝑖=𝑘−𝑑(𝑘)}𝑦(𝑙).
Combined with (2.2), we consider the following SQLF: 𝑉(𝑘,𝑥(𝑘))=𝑉₁(𝑘,𝑥(𝑘))+𝑉₂(𝑘,𝑥(𝑘))+𝑉₃((𝑘,𝑉𝑘,𝑥))₁(𝑘,𝑥(𝑘))=𝑥^𝑇(𝑘)𝑃_𝑟(𝑘)𝑥(𝑘)=𝑥^𝑇(𝑘)^𝑁_𝑖=1𝜉_𝑖(𝑘)𝑃_𝑖𝑥(𝑘),𝑉₂(𝑘,𝑥(𝑘))=⁰_{𝜃=−𝑑2+1}^𝑘−1_{𝑙=𝑘−1+𝜃}𝑦^𝑇(𝑙)𝑍_𝑟(𝑘)𝑦(𝑙)=⁰_{𝜃=−𝑑2+1}^𝑘−1_{𝑙=𝑘−1+𝜃}𝑦^𝑇(𝑙)^𝑁_𝑖=1𝜉_𝑖(𝑘)𝑍_𝑖𝑦(𝑙),𝑉₃(𝑘,𝑥(𝑘))=^−𝑑1+1_{𝜃=−𝑑2+1}^𝑘−1_{𝑙=𝑘−1+𝜃}𝑥^𝑇(𝑙)𝑄_𝑟(𝑘)𝑥(𝑙)=^−𝑑1+1_{𝜃=−𝑑2+1}^𝑘−1_{𝑙=𝑘−1+𝜃}𝑥^𝑇(𝑙)^𝑁_𝑖=1𝜉_𝑖(𝑘)𝑄_𝑖𝑥(𝑙), (3.4) where 𝑃_𝑖=𝑃^𝑇_𝑖>0,𝑍_𝑖=𝑍^𝑇_𝑖≥0,and𝑄_𝑖=𝑄^𝑇_𝑖≥0.
With (2.6), we obtain Δ𝑉₁(𝑘,𝑥(𝑘))=𝑥^𝑇(𝑘+1)𝑃_𝑟(𝑘+1)𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃_𝑟(𝑘)𝑥(𝑘)=𝑥^𝑇(𝑘)𝐴^𝑇_𝑟(𝑘)𝑃_𝑟(𝑘+1)𝐴_𝑟(𝑘)−𝑃_𝑟(𝑘)𝑥(𝑘)+𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇_{(𝑘)𝑑𝑟}𝑃_𝑟(𝑘+1)𝐴_{𝑑𝑟(𝑘)}𝑥(𝑘−𝑑(𝑘))+𝑥^𝑇(𝑘)𝐴^𝑇_𝑟(𝑘)𝑃_𝑟(𝑘+1)𝐴_{𝑑𝑟(𝑘)}𝑥(𝑘−𝑑(𝑘))+𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇_{(𝑘)𝑑𝑟}𝑃_𝑟(𝑘+1)𝐴_𝑟(𝑘)𝑥(𝑘),Δ𝑉₂(𝑘,𝑥(𝑘))=⁰_{𝜃=−𝑑2}^𝑘_{+1𝑙=𝑘+𝜃}𝑦^𝑇(𝑙)𝑍_𝑟(𝑘+1)𝑦(𝑙)−⁰_{𝜃=−𝑑2+1}^𝑘−1_{𝑙=𝑘−1+𝜃}𝑦^𝑇(𝑙)𝑍_𝑟(𝑘)𝑦(𝑙)=𝑑₂𝑦^𝑇(𝑘)𝑍_𝑟(𝑘+1)𝑦(𝑘)−^𝑘−1_{𝑙=𝑘−𝑑2}𝑦^𝑇(𝑙)𝑍_𝑟(𝑘)𝑦(𝑙)+⁰_{𝜃=−𝑑2+1}^𝑘−1_{𝑙=𝑘+𝜃}𝑦^𝑇(𝑙)𝑍_𝑟(𝑘+1)−𝑍_𝑟(𝑘)𝑦(𝑙) (3.5)(3.6) when 𝑍_𝑟(𝑘+1)=𝑍_𝑟(𝑘), Δ𝑉₂(𝑘,𝑥(𝑘))≤𝑑₂𝑦^𝑇(𝑘)𝑍_𝑟(𝑘+1)𝑦(𝑘)−^𝑘−1_{𝑙=𝑘−𝑑2}𝑦^𝑇(𝑙)𝑍_𝑟(𝑘)𝑦(𝑙),Δ𝑉₃(𝑘,𝑥(𝑘))=^−𝑑1+1_{𝜃=−𝑑2}^𝑘_{+1𝑙=𝑘+𝜃}𝑥^𝑇(𝑙)𝑄_𝑟(𝑘+1)𝑥(𝑙)−^−𝑑1+1_{𝜃=−𝑑2+1}^𝑘−1_{𝑙=𝑘−1+𝜃}𝑥^𝑇(𝑙)𝑄_𝑟(𝑘)𝑥(𝑙)=𝑑₂−𝑑₁+1𝑥^𝑇(𝑘)𝑄_𝑟(𝑘+1)𝑥(𝑘)−^𝑘−𝑑1_{𝑙=𝑘−𝑑2}𝑥^𝑇(𝑙)𝑄_𝑟(𝑘)𝑥(𝑙)+^−𝑑1+1_{𝜃=−𝑑2+1}^𝑘−1_{𝑙=𝑘+𝜃}𝑥^𝑇(𝑙)𝑄_𝑟(𝑘+1)−𝑄_𝑟(𝑘)𝑥(𝑙) (3.7)(3.8) when 𝑄_𝑟(𝑘+1)=𝑄_𝑟(𝑘), Δ𝑉₃(𝑘,𝑥(𝑘))≤𝑑₂−𝑑₁+1𝑥^𝑇(𝑘)𝑄_𝑟(𝑘+1)𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑(𝑘))𝑄_𝑟(𝑘)𝑥(𝑘−𝑑(𝑘)). (3.9)
Suppose that 𝑟(𝑘)=𝑖 and 𝑟(𝑘+1)=𝑗 mean that the subsystem 𝑖 switches to the subsystem 𝑗 in the switching system. As this has to be satisfied under arbitrary switching laws, it follows that this has to hold for the special configuration 𝜉_𝑖(𝑘)=1, 𝜉_𝑙≠𝑖(𝑘)=0, 𝜉_𝑗(𝑘+1)=1, and 𝜉_𝑙≠𝑗(𝑘+1)=0. And supposing that 𝑍_𝑗=𝑍_𝑖=𝑍 and 𝑄_𝑗=𝑄_𝑖=𝑄, we obtain Δ𝑉(𝑘,𝑥(𝑘))≤𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑖−𝑃_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇_𝑑𝑖𝑃_𝑗𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝑑₂𝑦^𝑇(𝑙)𝑍𝑦(𝑙)+𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇_𝑑𝑖𝑃_𝑗𝐴_𝑖𝑥(𝑘)−^𝑘−1_{𝑙=𝑘−𝑑2}𝑦^𝑇(𝑙)𝑍𝑦(𝑙)+𝑑₂−𝑑₁+1𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑(𝑘))𝑄𝑥(𝑘−𝑑(𝑘)). (3.10) By using the Leibniz-Newton formula, for any appropriately dimensioned matrices 𝑁₁^𝑖𝑗 and 𝑁₂^𝑖𝑗, the following equation is true: 2𝑥^𝑇(𝑘)𝑁₁^𝑖𝑗+𝑥^𝑇(𝑘−𝑑(𝑘))𝑁₂^𝑖𝑗×𝑥(𝑘)−𝑥(𝑘−𝑑(𝑘))−^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝑦(𝑙)=0. (3.11) In addition, for any semipositive definite matrix 𝑋^𝑖𝑗=[^{𝑋𝑖𝑗11𝑋𝑖𝑗12}_{⋆𝑋𝑖𝑗22}]≥0, the following equation holds: ^𝑘−1_{𝑙=𝑘−d2}𝜙^𝑇₁(𝑘)𝑋^𝑖𝑗𝜙₁(𝑘)−^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝜙^𝑇₁(𝑘)𝑋^𝑖𝑗𝜙₁(𝑘)=𝑑₂𝜙^𝑇₁(𝑘)𝑋^𝑖𝑗𝜙₁(𝑘)−^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝜙^𝑇₁(𝑘)𝑋^𝑖𝑗𝜙₁(𝑘)≥0, (3.12) where 𝜙₁(𝑘)=[𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))]^𝑇.
With (3.1), (3.10), (3.11), and (3.12), we have Δ𝑉(𝑘,𝑥(𝑘))≤𝜙^𝑇₁(𝑘)Γ^𝑖𝑗𝜙₁(𝑘)−^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝜙^𝑇₂(𝑘,𝑙)Θ^𝑖𝑗𝜙₂(𝑘,𝑙)≤0, (3.13) where Γ^𝑖𝑗=Φ^𝑖𝑗₁₁+𝑑₂𝐴_𝑖−𝐼^𝑇𝑍𝐴_𝑖−𝐼Φ^𝑖𝑗₁₂+𝑑₂𝐼𝐴_𝑖−𝐼^𝑇𝑍𝐴_𝑑𝑖⋆Φ^𝑖𝑗₂₂+𝑑₂𝐴_𝑑𝑖𝑍𝐴_𝑑𝑖,𝜙₂(𝑘,𝑙)=𝜙^𝑇₁(𝑘)𝑦^𝑇(𝑙)^𝑇. (3.14)
And 𝜙₁(𝑘) is defined in (3.12); Φ^𝑖𝑗₁₁, Φ^𝑖𝑗₁₂, and Φ^𝑖𝑗₂₂ are defined in (3.2). Therefore, when Γ^𝑖𝑗<0 and Θ^𝑖𝑗≥0, the system (3.1) is asymptotically stability. Applying Schur's complement, Γ^𝑖𝑗<0 is equivalent to Φ^𝑖𝑗<0,𝑖,𝑗∈Ω. This completes the proof of Theorem 3.1.

If we have 𝑦(𝑘), for any appropriately dimensioned matrices 𝑇₁^𝑖𝑗, 𝑇₂^𝑖𝑗, 𝑁₁^𝑖𝑗, 𝑁₂^𝑖𝑗, 𝑁₂^𝑖𝑗, and 𝑋^𝑖𝑗=𝑋^𝑖𝑗₁₁𝑋^𝑖𝑗₁₂𝑋^𝑖𝑗₁₃⋆𝑋^𝑖𝑗₂₂𝑋^𝑖𝑗₂₃⋆⋆𝑋^𝑖𝑗₃₃≥0, (3.15) the following equations are also true:2𝑥^𝑇(𝑘)𝑇₁^𝑖𝑗+𝑦^𝑇(𝑘)𝑇₂^𝑖𝑗×𝑦(𝑘)−𝐴_𝑖−𝐼𝑥(𝑘)−𝐴_𝑑𝑖𝑥((𝑘2𝑥𝑘−𝑑))=0,_𝑇(𝑘)𝑁₁^𝑖𝑗+𝑦^𝑇(𝑘)𝑁₂^𝑖𝑗+𝑥^𝑇(𝑘−𝑑(𝑘))𝑁₃^𝑖𝑗×𝑥(𝑘)−𝑥(𝑘−𝑑(𝑘))−^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝑦(𝑙)=0,_{𝑘−1𝑙=𝑘−𝑑2}𝜂^𝑇₁(𝑘)𝑋^𝑖𝑗𝜂₁(𝑘)−^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝜂^𝑇₁(𝑘)𝑋^𝑖𝑗𝜂₁(𝑘)=𝑑₂𝜂^𝑇₁(𝑘)𝑋^𝑖𝑗𝜂₁(𝑘)−^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝜂^𝑇₁(𝑘)𝑋^𝑖𝑗𝜂₁(𝑘)≥0, (3.16) where 𝜂₁(𝑘)=[𝑥^𝑇(𝑘)𝑦^𝑇(𝑙)𝑥^𝑇(𝑘−𝑑(𝑘))]^𝑇.

Considering (3.16), similar to the proof of Theorem 3.1, we can obtain the following corollary.

Corollary 3.2. Let 𝑑₁ and 𝑑₂ be positive integers such that 0≤𝑑₁≤𝑑₂; the systems (3.1) is asymptotically stability if there exist symmetric matrices 𝑃_𝑖=𝑃^𝑇_𝑖>0, 𝑃_𝑗=𝑃^𝑇_𝑗>0, 𝑄=𝑄^𝑇≥0, 𝑍=𝑍^𝑇≥0, and any appropriate dimensional matrices 𝑇₁^𝑖𝑗, 𝑇₂^𝑖𝑗, 𝑁₁^𝑖𝑗, 𝑁₂^𝑖𝑗, 𝑁₂^𝑖𝑗, and 𝑋^𝑖𝑗=𝑋^𝑖𝑗₁₁𝑋^𝑖𝑗₁₂𝑋^𝑖𝑗₁₃⋆𝑋^𝑖𝑗₂₂𝑋^𝑖𝑗₂₃⋆⋆𝑋^𝑖𝑗₃₃≥0 (3.17) such that the following LMIs hold, Υ^𝑖𝑗=Υ^𝑖𝑗₁₁Υ^𝑖𝑗₁₂Υ^𝑖𝑗₁₃⋆Υ^𝑖𝑗₂₂Υ^𝑖𝑗₂₃⋆⋆Υ^𝑖𝑗₃₃𝑋<0,_𝑖𝑗11𝑋^𝑖𝑗₁₂𝑋^𝑖𝑗₁₃𝑁₁^𝑖𝑗⋆𝑋^𝑖𝑗₂₂𝑋^𝑖𝑗₂₃𝑁₂^𝑖𝑗⋆⋆𝑋^𝑖𝑗₃₃𝑁₃^𝑖𝑗⋆⋆⋆𝑍≥0,𝑖,𝑗∈Ω, (3.18) where Υ^𝑖𝑗₁₁=(𝑑₂−𝑑₁+1)𝑄+𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑖−𝑃_𝑖−𝑇₁^𝑖𝑗(𝐴_𝑖−𝐼)−(𝐴_𝑖−𝐼)^𝑇(𝑇₁^𝑖𝑗)^𝑇+(𝑁₁^𝑖𝑗)^𝑇+𝑁₁^𝑖𝑗+𝑑₂𝑋^𝑖𝑗₁₁, Υ^𝑖𝑗₁₂=𝑇₁^𝑖𝑗−(𝐴_𝑖−𝐼)^𝑇(𝑇₂^𝑖𝑗)^𝑇+(𝑁₂^𝑖𝑗)^𝑇+𝑑₂𝑋^𝑖𝑗₁₂, Υ^𝑖𝑗₁₃=𝐴^𝑇_𝑖𝑃_𝑗𝐴_di−𝑇₁^𝑖𝑗𝐴_di+(𝑁₃^𝑖𝑗)^𝑇−𝑁₁^𝑖𝑗+𝑑₂𝑋^𝑖𝑗₁₃, Υ^𝑖𝑗₂₂=𝑑₂𝑍+(𝑇₂^𝑖𝑗)^𝑇+𝑇₂^𝑖𝑗+𝑑₂𝑋^𝑖𝑗₂₂, Υ^𝑖𝑗₂₃=−𝑇₂^𝑖𝑗𝐴_di−𝑁₂^𝑖𝑗+𝑑₂𝑋^𝑖𝑗₂₃, and Υ^𝑖𝑗₃₃=𝐴^𝑇_di𝑃_𝑗𝐴_di−𝑄−𝑁₃^𝑖𝑗+(𝑁₃^𝑖𝑗)^𝑇+𝑑₂𝑋^𝑖𝑗₃₃.

Next, we consider the design of a switched state feedback:𝑢(𝑘)=𝐾_𝑟(𝑘)𝑥(𝑘). (3.19) Ensuring stability of the closed-loop switched system:𝑥(𝑘+1)=𝐴_𝑟(𝑘)+𝐵_𝑟(𝑘)𝐾_𝑟(𝑘)𝑥(𝑘)+𝐴_{𝑑𝑟(𝑘)}𝑥(𝑘−𝑑(𝑘)),𝑘∈𝐙⁺,𝑟(𝑘)∈Ω. (3.20)

Based on Theorem 3.1, we obtain the following theorem.

Theorem 3.3. Let 𝑑₁ and 𝑑₂ be positive integers such that 0≤𝑑₁≤𝑑₂. Under arbitrary switch, the systems (1.1) is asymptotically stability if there exist symmetric matrices 𝑃_𝑖=𝑃^𝑇_𝑖>0, 𝑃_𝑗=𝑃^𝑇_𝑗>0, 𝑄=𝑄^𝑇≥0, 𝑍=𝑍^𝑇≥0, 𝑌^𝑖𝑗=[^{𝑌𝑖𝑗11𝑌𝑖𝑗12}_{⋆𝑌𝑖𝑗22}]≥0, and any appropriate dimensional matrices 𝑀₁^𝑖𝑗 and 𝑀₂^𝑖𝑗 such that the following LMIs hold, Ξ^𝑖𝑗=Ξ^𝑖𝑗₁₁Ξ^𝑖𝑗₁₂𝑑₂𝐴_𝑖𝐿_𝑖+𝐵_𝑖𝑉_𝑖−𝑉_𝑖^𝑇⋆Ξ^𝑖𝑗₂₂𝑑₂𝐿_𝑖𝐴^𝑇_di⋆⋆𝑑₂𝑅𝑌≥0,_𝑖𝑗11𝑌^𝑖𝑗₁₂𝑀₁^𝑖𝑗⋆𝑌^𝑖𝑗₂₂𝑀₂^𝑖𝑗⋆⋆𝐿𝑅⁻¹𝐿≥0,𝑖,𝑗∈Ω, (3.21) where Ξ^𝑖𝑗₁₁=(𝑑₂−𝑑₁+1)𝑊_𝑖+(𝐴_𝑖𝐿_𝑖)^𝑇𝑃_𝑗(𝐴_𝑖𝐿_𝑖)+(𝐴_𝑖𝐿_𝑖)𝑃_𝑗(𝐵_𝑖𝑉_𝑖)+(𝐵_𝑖𝑉_𝑖)^𝑇𝑃_𝑗(𝐴_𝑖𝐿_𝑖)+(𝐵_𝑖𝑉_𝑖)^𝑇𝑃_𝑗(𝐵_𝑖𝑉_𝑖)−𝐿_𝑖+(𝑀₁^𝑖𝑗)^𝑇+𝑀₁^𝑖𝑗+𝑑₂𝑌^𝑖𝑗₁₁, Ξ^𝑖𝑗₁₂=(𝐴_𝑖𝐿_𝑖)^𝑇𝑃_𝑗(𝐴_di𝐿_𝑖)+(𝐵_𝑖𝑉_𝑖)^𝑇𝑃_𝑗(𝐴_di𝐿_𝑖)+(𝑀₂^𝑖𝑗)^𝑇−𝑀₁^𝑖𝑗+𝑑₂𝑌^𝑖𝑗₁₂, and Ξ^𝑖𝑗₂₂=(𝐴_di𝐿_𝑖)^𝑇𝑃_𝑗(𝐴_di𝐿_𝑖)−𝑊−𝑀₂^𝑖𝑗−(𝑀₂^𝑖𝑗)^𝑇+𝑑₂𝑌^𝑖𝑗₂₂.

Proof. To the system (3.1), 𝐴_𝑖 is replaced by 𝐴_𝑖+𝐵_𝑖𝐾_𝑖 in (3.2). Simultaneously, two parts of inequality (3.2) multiply the same matrix diag[𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑍⁻¹] and two parts of inequality (3.3) multiply the same matrix diag[𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹]. Suppose that 𝐿_𝑖=𝑃_𝑖⁻¹, 𝑊_𝑖=𝑃_𝑖⁻¹𝑄𝑃_𝑖⁻¹, 𝑌^𝑖𝑗=diag[𝑃_𝑖⁻¹,𝑃_𝑖⁻¹]𝑋^𝑖𝑗diag[𝑃_𝑖⁻¹,𝑃_𝑖⁻¹], 𝑅=𝑍⁻¹, 𝑀₁^𝑖𝑗=𝑃_𝑖⁻¹𝑁₁^𝑖𝑗𝑃_𝑖⁻¹, 𝑀₂^𝑖𝑗=𝑃_𝑖⁻¹𝑁₂^𝑖𝑗𝑃_𝑖⁻¹, and 𝑉_𝑖=𝐾_𝑖𝑃_𝑖⁻¹; then we obtain (3.21).This completes the proof of Theorem 3.3.

4. Stability Analysis of System (1.1) with Uncertain Liner System

In this section, results of uncertain liner system are extended to arbitrary switched linear system for arbitrary switched linear system can be expressed as a problem of uncertain liner system. When 𝑢(𝑘)=0, (1.5) are rewritten asΔ𝐴(𝑘)Δ𝐴_𝑑(𝑘)=𝐴−𝐴_𝑟(𝑘)𝐴_𝑑−𝐴_{𝑑𝑟(𝑘)}. (4.1) Then the system (3.1) is rewritten as𝑥(𝑘+1)=𝐴−𝐴−𝐴_𝑟(𝑘)𝑥(𝑘)+𝐴_𝑑−𝐴_𝑑−𝐴_{𝑑𝑟(𝑘)}𝑥(𝑘−𝑑(𝑘)),𝑘∈𝑍⁺,𝑟(𝑘)∈Ω. (4.2)

Combined with Lemma 2.3, we easily have the following theorem.

Theorem 4.1. Let 𝑑₁ and 𝑑₂ be positive integers such that 0≤𝑑₁≤𝑑₂. Under arbitrary switch, the system (4.2) is asymptotically stability if there exist matrices 𝑃=𝑃^𝑇>0, 𝑄=𝑄^𝑇>0, 𝑍=𝑍^𝑇>0, 𝑋=[^𝑋11𝑋12_⋆𝑋22]≥0 and any appropriate dimensional matrices 𝐴,𝐴_𝑑,𝐷, 𝐸_𝑎,𝐸_𝑎𝑑,𝐹_𝑖, 𝑁₁, 𝑁₂, and 𝜆>0 such that the LMIs (2.10) and the following LMIs hold, 𝐴−𝐴_𝑖𝐴_𝑑−𝐴_di=𝐷𝐹_𝑖𝐸_𝑎𝐸_ad𝐹^𝑇_𝑖𝐹_𝑖≤𝐼,𝑖∈Ω. (4.3)

Next, we consider the design of a switched state feedback. With (4.1) and (4.2), the system (3.20) is rewritten as𝑥(𝑘+1)=𝐴−𝐴−𝐴_𝑟(𝑘)+𝐵_𝑟(𝑘)𝐾_𝑟(𝑘)𝑥(𝑘)+𝐴_𝑑−𝐴_𝑑−𝐴_{𝑑𝑟(𝑘)}𝑥(𝑘−𝑑(𝑘)),𝑘∈𝐙⁺,𝑟(𝑘)∈Ω. (4.4)

Combined with Theorem 4.1, we easily have the following theorem.

Theorem 4.2. Let 𝑑₁ and 𝑑₂ be positive integers such that 0≤𝑑₁≤𝑑₂. Under arbitrary switch, the systems (4.4) is asymptotically stability if there exist matrices 𝑃=𝑃^𝑇>0, 𝑄=𝑄^𝑇>0, 𝑍=𝑍^𝑇>0, 𝑋=[^𝑋11𝑋12_⋆𝑋22]≥0, and any appropriate dimensional matrices 𝐴,𝐴_𝑑,𝐷, 𝐸_𝑎,𝐸_𝑎𝑑,𝐹_𝑖, 𝑁₁, 𝑁₂ and 𝜆>0 such that the LMIs (2.10) and (4.3), and the following LMIs hold, 𝐴−𝐴_𝑖+𝐵_𝑖𝐾_𝑖𝐴_𝑑−𝐴_di=𝐷𝐹_𝑖𝐸_𝑎𝐸_ad,𝑖∈Ω. (4.5)

5. Examples

Example 5.1. Consider the following switched delay systems with two subsystems 𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘)),𝑘∈𝐙⁺,𝑖∈Ω, (5.1) where 𝐴₁=0.1000.5,𝐴_𝑑1=−0.20−0.1−0.1,𝐴₂=0.2000.3,𝐴_𝑑2=−0.100.3−0.2, (5.2) and Ω={1,2}.
When 𝑑₁=𝑑₂ that is, 𝑑(𝑘)=𝑑₂, 𝑑₂ is without limit. To time-varying delay 𝑑(𝑘), when 𝑑₁ is given, 𝑑₂ is a maximum value of the solvability of LMIs (3.2) and (3.3), and some results are in Table 1.
In this example, the switching system has two subsystems, so there are there switches that are between subsystem 1 and subsystem 2, between subsystem 1 and subsystem 1, and between subsystem 2 and subsystem 2. According to Theorem 3.1, when 𝑑₁=1 and 𝑑₂=3, solving the LMIs (3.2) and (3.3) leads to 𝑄=29.9101−3.0324−3.032414.4926𝑍=5.34740.03410.03416.8596,𝑃₁=,𝑃143.7206−9.2029−9.2029102.1555₂=144.9590−8.5971−8.597193.0312,𝑁₁¹²=,𝑁−1.25920.5437−0.4002−2.1476₂₁₂=1.9723−0.14010.45883.0624,𝑁₁²¹=,𝑁−1.3023−0.86190.8282−2.1259₂₂₁=2.20160.3203−0.64053.2331,𝑁₁¹¹=,𝑁−1.25530.6225−0.4427−2.2909₂₁₁=1.9636−0.19490.51263.0790,𝑁₁²²=,𝑁−1.3199−0.70260.7078−2.0469₂₂₂=2.20490.2493−0.53193.1800,𝑋¹²=,𝑋10.23480.0282−0.87220.30920.02828.28570.3092−0.5114−0.87220.30926.3470−0.87160.3092−0.5114−0.87164.7227₂₁=,𝑋10.15290.2694−0.4215−0.65880.26948.2610−0.6588−0.6227−0.4215−0.65885.03570.9594−0.6588−0.62270.95943.9515₁₁=,𝑋10.23690.0251−0.87880.36590.02517.79270.3659−0.4471−0.87880.36596.3347−0.89280.3659−0.4471−0.89284.7126₂₂=.10.14670.2491−0.4220−0.54940.24918.4372−0.5494−0.7209−0.4220−0.54945.24770.8376−0.5494−0.72090.83764.0014 (5.3)
It can be seen from Figure 1 that when 𝑑₁=1 and 𝑑₂=3, all the state solutions corresponding to the 10 random initial points are convergent asymptotically to the unique equilibrium 𝑥^∗={0,0}.

Table 1 

Allowable upper bound of 𝑑₂ with given 𝑑₁.

(a)

(b)

Figure 1 

Global convergence of states 𝑥1 and 𝑥2 in Example 5.1, when 𝑑1=1 and 𝑑2=3.

Example 5.2. Consider the following switched delay systems with two subsystems: 𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘)),𝑘∈𝐙⁺,𝑖∈Ω, (5.4) where 𝐴₁=0.38500.00900.01800.5880,𝐴_𝑑1=,𝐴−0.41500.0090−0.0820−0.3120₂=0.39700.01200.01500.5820,𝐴_𝑑2={}.−0.40300.0120−0.0850−0.3180,Ω=1,2 (5.5) When there exit matrixes 𝐴=0.40000.00000.00000.6000,𝐴_𝑑=,−0.40000.0000−0.1000−0.3000𝐷=0.03000.00000.00000.0300,𝐸_𝑎=,𝐸1.00000.00000.00001.0000_𝑎𝑑=1.00000.00000.00001.0000,𝐹₁=,𝐹−0.50000.30000.6000−0.4000₂=,−0.10000.40000.5000−0.6000 (5.6) which satisfied (4.3), and 𝑑₁=1 and 𝑑₂=3, based on Theorem 4.1 solving the LMIs (2.10) leads to ()∗()∗,()∗𝑃=1.0𝑒+0041.0888−0.0129−0.01290.2758,𝑄=1.0𝑒+0032.51930.03790.03790.4439𝑍=1.0𝑒+0031.49270.08090.08090.4826,𝑁₁=,𝑁−472.4815−6.3448−53.9979−181.8198₂=,478.2798−0.008645.6698185.5393𝑋=216.7792−41.8304−134.0982−18.7083−41.8304113.2493−18.7083−62.9775−134.0982−18.7083174.9568−10.6693−18.7083−62.9775−10.669384.3589,𝜆=148.1177. (5.7)
It can be seen from Figure 2 that when 𝑑₁=1 and 𝑑₂=3, all the state solutions corresponding to the 10 random initial points are convergent asymptotically to the unique equilibrium 𝑥^∗={0,0}.

(a)

(b)

Figure 2 

Global convergence of states 𝑥1 and 𝑥2 in Example 5.2, when 𝑑1=1 and 𝑑2=3.

6. Conclusions

This paper was dedicated to the delay-dependent stability of arbitrary switched linear systems with time-varying delay. We obtain two main results. Firstly, using switched quadratic Lyapunov functional approach and free-weighting matrix approach, less conservative LMI conditions have been proposed. Secondly, based on the result of uncertain liner system, some delay-dependent stability criterions are obtained.

Acknowledgments

The authors would like to thank the editor and the anonymous reviewers for their detailed comments which greatly contributed to this paper.