Abstract

This paper deals with the problems of exponential stability and guaranteed cost of switched linear systems with mixed time delays. Based on Lyapunov functional method, we present new delay-dependent conditions that guarantee both the exponential stability and an upper bound for some performance index. The criteria are delay-dependent conditions and are given in terms of linear matrix inequalities. Numerical examples are provided to illustrate the effectiveness of the results.

1. Introduction

A switched system is a dynamical system that includes several subsystems and a logical rule that orchestrates switching between these subsystems at each instant of time [1]. The logical rule generates switching signals to determine which subsystem will be activated on a certain time interval. In fact, switching systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models in manufacturing, communication networks automotive engine control, chemical processes, and so on see, for example, [2, 3]. There have been many studies on stability of switching systems [4–6].

The stability analysis of switched time delay has attracted a lot of attention [7–9]. The main approach for stability analysis relies on the use of Lyapunov Krasovskii functionals and LMI approach for constructing common Lyapunov function and switching rules [10–12]. In [13], the asymptotic stability of switched linear time delay symmetric systems has been studied. In [11], a switching system composed by a finite number of linear point time delay differential equations has been studied, and it has been shown that the asymptotic stability may be achieved by using a common Lyapunov function method and minimum switching rule. The results of [11] have been extended in [14] to linear switching system with discrete and distributed delays. The delays considered are time invariant, and the conditions are given in terms of maximal and minimal eigenvalues of certain matrices. The exponential stability problem was considered in [15] for switching linear systems with impulsive effects by using the matrix measure concept. The exponential stability of time delay systems has been considered in [16, 17], and switching linear systems with mixed delays have been studied in [18]. In [19], sufficient stability conditions have been obtained by using a generalisation of Halanay’s inequality.

In this paper, we will focus on exponential stability of switched linear systems with mixed time delays. The subsystems considered are continuous with time-varying delays. The derived conditions are delay dependent and are given in terms of LMIs. The conditions guarantee both the exponential stability and an upper bound of a performance index. The approach is based on Lyapunov Krasovskii functional and allows us to avoid explicitly substitution of the dynamic system equation in the derivative of the Lyapunov function. Furthermore, we use an extended variable to completely avoid bounding treatment of the weighted cross-product of the instantaneous state and the delayed state. As a consequence, the conservatism of the conditions is expected to be reduced with respect to earlier works (e.g. [10, 16, 20]). The conditions allow us to compute easily both the stability factor and the decay rate of the solution. Numerical computations are performed for illustration.

2. Preliminaries

Consider a class of switched linear systems with time-varying delay of the forṁ𝑥(𝑡)=𝐴𝜎𝑥(𝑡)+𝐷𝜎𝑥(𝑡−ℎ(𝑡))+𝐸𝜎𝑡𝑡−𝜏(𝑡)[],𝑥(𝑠)𝑑𝑠,𝑡>0,𝑥(𝑡)=𝜙(𝑡),𝑡∈−𝑟,0(2.1) where 𝑥∈𝑅𝑛 is the state, 𝜎(𝑥)∶𝑅𝑛→𝐼∶={1,2,…,𝑁} is the switching rule which is piece-wise constant function depending on the state in each time, 𝜙∈𝐶([−ℎ,0],𝑅𝑛) is the initial function, with norm ‖𝜙‖=sup𝜃∈[−ℎ,0]‖𝜙(𝜃)‖, 𝐴𝜎,𝐷𝜎,𝐸𝜎∈{[𝐴𝑖,𝐷𝑖,𝐸𝑖],𝑖=1,2,…,𝑁}, 𝐴𝑖, 𝐷𝑖, and 𝐸𝑖 are given matrices. Moreover 𝜎(𝑥)=𝑖 implies that the 𝑖th subsystem is activated, and we have the following subsystem:̇𝑥(𝑡)=𝐴𝑖𝑥(𝑡)+𝐷𝑖𝑥(𝑡−ℎ(𝑡))+𝐸𝑖𝑡𝑡−𝜏(𝑡)𝑥(𝑠)𝑑𝑠,𝑡>0.(2.2)ℎ(𝑡) and 𝜏(𝑡) are unknown time-varying delay terms, but bounded by0≤ℎ(𝑡)≤̇ℎℎ,(𝑡)≤𝜇ℎ,(2.3)0≤𝜏(𝑡)≤𝜏,𝜏(𝑡)≤𝜇𝜏,(2.4) where ℎ, 𝜏, 𝜇ℎ, and 𝜇𝜏 are given nonnegative constants, 𝑟=max(ℎ,𝜏).

In this paper, we are interested in establishing conditions guaranteeing the asymptotic stability of switched system (2.1) and finding the least upper bound for the cost function given by𝐽=0+∞𝑥𝑇(𝑡)𝑌𝑥(𝑡)𝑑𝑡,(2.5) where 𝑌>0. Assume that there exists a Hurwitz linear convex combination 𝐴 of 𝐴𝑖, that is,𝐴=𝑁𝑖=1𝜆𝑖𝐴𝑖,0<𝜆i<1,𝑁𝑖=1𝜆𝑖=1.(2.6) Then for a positive definite matrix 𝑍1, there exists a positive definite matrix 𝑅11 satisfying𝐴𝑇𝑅11+𝑅11𝐴=−𝑍1.(2.7) Hence for 𝑥≠0, we can write𝑥𝑇𝐴𝑇𝑅11+𝑅11𝐴𝑥=−𝑥𝑇𝑍1𝑥<0.(2.8) Which implies that𝜆𝑖𝑥𝑇𝐴𝑇𝑖𝑅11+𝑅11𝐴𝑖𝑥<0.(2.9) for at least one 𝑖. So for a positive definite matrix 𝑍, construct 𝑁 region of 𝑅𝑛 as follows:Ω𝑖=𝑥∈𝑅𝑛∶𝑥𝑇𝐴𝑇𝑖𝑅11+𝑅11𝐴𝑖𝑥<−𝑥𝑇𝑍𝑥,𝑖∈𝐼.(2.10) It is clear that ⋃𝑁𝑖=1Ω𝑖=𝑅𝑛⧵{0}.

DenoteΩ1=Ω1,Ω𝑖=Ω𝑖⧵𝑖−1𝑗=1Ω𝑗,𝑖=2,…,𝑁.(2.11) It follows that ⋃𝑁𝑖=1Ω𝑖=𝑅𝑛⧵{0}, Ω𝑖∩Ω𝑗=∅,𝑖≠𝑗. The switching rule is chosen as follows:[𝑆𝑅]:Step 0: let 𝑥(𝑡)=𝜙(𝑡).Step 1: set 𝜎(𝑥)=argmin{𝑥𝑇(𝐴𝑇𝑖𝑅11+𝑅11𝐴𝑖)𝑥}.Step 2: stay in the 𝑖th mode as long as 𝑥(𝑡) in Ω𝑖.Step 3: if 𝑥(𝑡) hits the boundary of Ω𝑖, go to Step 1 to determine the next mode.

Definition 2.1. Given 𝛼>0, the system (2.1) is 𝛼-exponentially stable if there exist a switching rule 𝜎 and a constant 𝛽≥1 such that every solution 𝑥(𝑡,𝜙) of the system satisfies the following inequality: ‖𝑥(𝑡,𝜙)‖≤𝛽𝑒−𝛼𝑡‖𝜙‖,𝑡≥0.(2.12)

The following lemmas will be useful.

Lemma 2.2. For any 𝑋,𝑌∈𝑅𝑛, matrices 𝑀>0,𝐹𝑇𝐹≤𝐼, one has 2𝑋𝑇𝑌≤𝑋𝑇𝑀−1𝑋+𝑌𝑇𝑀𝑌.(2.13)

Lemma 2.3. For any constant matrix 𝑀=𝑀𝑇∈𝑅𝑛×𝑛, 𝑀>0, scalar 𝛾≥𝜂(𝑡)>0, vector function 𝜔∶[0,𝛾]→𝑅𝑛 such that the integrations in the following are well defined, then 𝜂(𝑡)0𝜂(𝑡)𝜔𝑇(𝛽)𝑀𝜔(𝛽)𝑑𝛽≥0𝜂(𝑡)𝜔(𝛽)𝑑𝛽𝑇𝑀0𝜂(𝑡)𝜔(𝛽)𝑑𝛽.(2.14)

3. Main Results

Let,𝛽1=𝜆min𝑅11𝛽,(3.1)2=1+ℎ2𝜆max(𝑅)+1−𝑒−2𝛼ℎ𝜆2𝛼max𝑒(𝑃)+−2𝛼ℎ+2𝛼ℎ−14𝛼2𝜆max𝑒(𝑄)+−2𝛼𝜏+2𝛼𝜏−14𝛼2𝜆max(𝑆),(3.2)𝛽=𝛽2𝛽1,𝐿𝑖=𝑅11𝐴𝑖+𝐴𝑇𝑖𝑅11+𝑍,⎛⎜⎜⎜⎜⎜⎜⎜⎝𝜇𝑍=ℎ𝑅1200𝜇ℎ𝑅22⎞⎟⎟⎟⎟⎟⎟⎟⎠,𝜇000000𝑍=diagℎ𝑍1,𝜇ℎ𝑍2,𝐺𝑖=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐺𝑖11𝐺𝑖12𝐺𝑖13𝑅11𝐸𝑖+𝐴𝑇𝑖𝐹𝑇3−𝐹0+𝐴𝑇𝑖𝐹𝑇4∗𝐺𝑖22𝐺𝑖23𝐷𝑇𝑖𝐹𝑇3−𝐹1+𝐷𝑇𝑖𝐹𝑇4∗∗𝐺𝑖33𝑅𝑇12𝐸𝑖+𝐸𝑇𝑖𝐹𝑇3−𝐹2+𝐸𝑇𝑖𝐹𝑇4∗∗∗−1−𝜇𝜏𝜏𝑒−2𝛼𝜏−𝐹3∗∗∗∗−𝐹4−𝐹𝑇4⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.3) where “∗” denotes the symmetric part in a symmetric matrix and,𝐺𝑖11=𝑅12+𝑅𝑇12+𝑃+2𝛼𝑅11+ℎ𝑄+𝜏𝑆+𝐹0𝐴𝑖+𝐴𝑇𝑖𝐹𝑇0𝐺−𝑍,𝑖12=−𝑅12+𝑅11𝐷𝑖+𝐹0𝐷𝑖+𝐴𝑇𝑖𝐹𝑇1,𝐺𝑖13=2𝛼𝑅12+𝑅22+𝐴𝑇𝑖𝐹𝑇2+𝐹0𝐸𝑖+𝐴𝑇𝑖𝑅12,𝐺𝑖22=−1−𝜇ℎ𝑒−2𝛼ℎ𝑃+𝐹1𝐷𝑖+𝐷𝑇𝑖𝐹𝑇1+𝜇ℎ𝑍1,𝐺𝑖23=−𝑅22+𝐷𝑇𝑖𝐹𝑇2+𝐷𝑖𝑅12+𝐹1𝐸𝑖,𝐺𝑖331=−ℎ𝑒−2𝛼ℎ𝑄+2𝛼𝑅22+𝐸𝑇𝑖𝐹𝑇2+𝐹2𝐸𝑖+𝜇ℎ𝑍2.(3.4)

Theorem 3.1. For given 𝛼>0, switched linear system (2.1) is 𝛼-exponentially stable if there exist symmetric positive definite matrices 𝑍, 𝑍1, 𝑍2, 𝑆, 𝑃, 𝑄, 𝑅11, 𝑅22, 𝑌, matrices 𝑅12, 𝐹0, 𝐹1, 𝐹2, 𝐹3, and 𝐹4 satisfying. (i)There exist 0<𝜆𝑖<1,𝑖=1,2,…,𝑁 such that ∑𝑁𝑖=1𝜆𝑖=1 and 𝑁𝑖=1𝜆𝑖𝐿𝑖<−𝑌.(3.5)(ii)⎛⎜⎜⎝𝐺𝑖𝑍𝑍𝑇−𝑍⎞⎟⎟⎠<0,𝑖=1,…,𝑁.(3.6)The switching rule is given by [𝑆𝑅], and the solution 𝑥(𝑡,𝜙) of the system satisfies ‖𝑥(𝑡,𝜙)‖≤𝛽𝑒−𝛼𝑡‖𝜙‖,𝑡≥0.(3.7) Furthermore, the cost function in (2.5) satisfies 𝐽≤𝐽∗=⎛⎜⎜⎜⎝𝜙(0)00−ℎ(0)⎞⎟⎟⎟⎠𝜙(𝑠)𝑑𝑠𝑇𝑅⎛⎜⎜⎜⎝𝜙(0)00−ℎ(0)⎞⎟⎟⎟⎠+𝜙(𝑠)𝑑𝑠0−ℎ𝑒2𝛼𝜃𝜙𝑇+(𝜃)𝑃𝜙(𝜃)𝑑𝜃0−ℎ0𝑠𝑒2𝛼𝜃𝜙𝑇(𝜃)𝑄𝜙(𝜃)𝑑𝜃𝑑𝑠+0−𝜏(0)0𝑠𝑒2𝛼𝜃𝜙𝑇(𝜃)𝑆𝜙(𝜃)𝑑𝜃𝑑𝑠.(3.8)

Proof. Let 𝑥𝑡∶={𝑥(𝑡+𝑠),𝑠∈[−𝑟,0]} and consider the following Lyapunov Krasovskii functional: 𝑉𝑥𝑡=𝑉1𝑥𝑡+𝑉2𝑥𝑡+𝑉3𝑥𝑡+𝑉4𝑥𝑡.(3.9) With 𝑉1𝑥𝑡=⎛⎜⎜⎜⎝𝑥(𝑡)𝑡𝑡−ℎ(𝑡)⎞⎟⎟⎟⎠𝑥(𝑠)𝑑𝑠𝑇𝑅⎛⎜⎜⎜⎝𝑥(𝑡)𝑡𝑡−ℎ(𝑡)𝑥⎞⎟⎟⎟⎠,𝑉(𝑠)𝑑𝑠2𝑥𝑡=0−ℎ𝑒2𝛼𝑠𝑥𝑇𝑉(𝑡+𝑠)𝑃𝑥(𝑡+𝑠)𝑑𝑠,3𝑥𝑡=0−ℎ0𝑠𝑒2𝛼𝜃𝑥𝑇𝑉(𝑡+𝜃)𝑄𝑥(𝑡+𝜃)𝑑𝜃𝑑𝑠,4𝑥𝑡=0−𝜏(𝑡)0𝑠𝑒2𝛼𝜃𝑥𝑇(𝑡+𝜃)𝑆𝑥(𝑡+𝜃)𝑑𝜃𝑑𝑠.(3.10) It is easy to verify that 0<𝛽1‖𝑥(𝑡)‖2𝑥≤𝑉𝑡≤𝛽2‖‖𝑥𝑡‖‖2,(3.11) where 𝛽1 and 𝛽2 are defined by (3.1) and (3.2). Computing the first time derivative of 𝑉(𝑥𝑡), we obtain ̇𝑉1𝑥𝑡⎛⎜⎜⎜⎝=2𝑥(𝑡)𝑡𝑡−ℎ(𝑡)𝑥⎞⎟⎟⎟⎠(𝑠)𝑑𝑠𝑇𝑅̇⎛⎜⎜⎜⎝̇𝑥(𝑡)𝑥(𝑡)−1−ℎ(𝑡)𝑥(𝑡−ℎ(𝑡))=2𝑥(𝑡)𝑡𝑡−ℎ(𝑡)⎞⎟⎟⎟⎠𝑥(𝑠)𝑑𝑠𝑇𝑅⎛⎜⎜⎝𝐴𝑖𝑥(𝑡)+𝐷𝑖𝑥(𝑡−ℎ(𝑡))+𝐸𝑖𝑡𝑡−𝜏(𝑡)̇⎞⎟⎟⎠.𝑥(𝑠)𝑑𝑠,𝑥(𝑡)−1−ℎ(𝑡)𝑥(𝑡−ℎ(𝑡))(3.12) Letting 𝜉(𝑡)=[𝑥𝑇(𝑡)𝑥𝑇∫(𝑡−ℎ(𝑡))𝑡𝑡−ℎ(𝑡)𝑥𝑇∫(𝑠)𝑑𝑠𝑡𝑡−𝜏(𝑡)𝑥𝑇(𝑠)𝑑𝑠]𝑇, taking account of (2.3), and using the fact that for some positive definite matrices 𝑍1 and 𝑍2, the following inequalities hold 2̇ℎ(𝑡)𝑥𝑇(𝑡)𝑅12𝑥(𝑡−ℎ(𝑡))≤𝜇ℎ𝑥𝑇(𝑡)𝑅12𝑍1−1𝑅𝑇12𝑥(𝑡)+𝜇ℎ𝑥𝑇(𝑡−ℎ(𝑡))𝑍12̇𝑥(𝑡−ℎ(𝑡)),ℎ(𝑡)𝑥𝑇(𝑡−ℎ(𝑡))𝑅22𝑡𝑡−ℎ(𝑡)𝑥(𝑠)𝑑𝑠≤𝜇ℎ𝑥𝑇(𝑡−ℎ(𝑡))𝑅22𝑍2−1𝑅22𝑥(𝑡−ℎ(𝑡))+𝜇ℎ𝑡𝑡−ℎ(𝑡)𝑥(𝑠)𝑑𝑠𝑇𝑍2𝑡𝑡−ℎ(𝑡).𝑥(𝑠)𝑑𝑠(3.13) We can write ̇𝑉1𝑥𝑡≤𝑥𝑇𝑅(𝑡)11𝐴𝑖+𝐴𝑇𝑖𝑅11𝑥(𝑡)+𝜉𝑇(𝑡)𝑇1𝜉(𝑡)−2𝛼𝑉1𝑥𝑡,(3.14) where 𝑇1=⎛⎜⎜⎜⎜⎜⎝𝑅12+𝑅𝑇12+2𝛼𝑅11+𝜇ℎ𝑅12𝑍1−1𝑅𝑇12𝐷∗∗∗𝑇𝑖𝑅11−𝑅𝑇12𝜇ℎ𝑍1+𝜇ℎ𝑅22𝑍2−1𝑅𝑇22𝑅∗∗22+2𝛼𝑅𝑇12+𝑅𝑇12𝐴𝑖𝑅𝑇12𝐷𝑖−𝑅222𝛼𝑅22∗𝐸𝑇𝑖𝑅110𝐸𝑇𝑖𝑅120⎞⎟⎟⎟⎟⎟⎠.(3.15) And “*” denotes the symmetric part in a symmetric matrix ̇𝑉2𝑥𝑡=𝑥𝑇̇𝑒(𝑡)𝑃𝑥(𝑡)−1−ℎ(𝑡)−2𝛼ℎ(𝑡)𝑥𝑇(𝑡−ℎ(𝑡))𝑃𝑥(𝑡−ℎ(𝑡))−2𝛼𝑉2𝑥𝑡≤𝑥𝑇(𝑡)𝑃𝑥(𝑡)−1−𝜇ℎ𝑒−2𝛼ℎ𝑥𝑇(𝑡−ℎ(𝑡))𝑃𝑥(𝑡−ℎ(𝑡))−2𝛼𝑉2𝑥𝑡,̇𝑉3𝑥𝑡=ℎ𝑥𝑇(𝑡)𝑄𝑥(𝑡)−𝑡𝑡−ℎ𝑒2𝛼(𝜃−𝑡)𝑥𝑇(𝜃)𝑄𝑥(𝜃)𝑑𝜃−2𝛼𝑉3𝑥𝑡≤ℎ𝑥𝑇𝑒(𝑡)𝑄𝑥(𝑡)−−2𝛼ℎℎ𝑡𝑡−ℎ(𝑡)𝑥(𝜃)𝑑𝜃𝑇𝑄𝑡𝑡−ℎ(𝑡)𝑥(𝜃)𝑑𝜃−2𝛼𝑉3𝑥𝑡.(3.16) Taking account of (2.4) and applying Lemma 2.3, we obtain ̇𝑉4𝑥𝑡=𝜏(𝑡)𝑥𝑇(𝑡)𝑄𝑥(𝑡)−(1−̇𝜏(𝑡))𝑡𝑡−𝜏(𝑡)𝑒2𝛼(𝜃−𝑡)𝑥𝑇(𝜃)𝑆𝑥(𝜃)𝑑𝜃−2𝛼𝑉4𝑥𝑡≤𝜏𝑥𝑇(𝑡)𝑄𝑥(𝑡)−1−𝜇𝜏𝜏𝑒−2𝛼𝜏𝑡𝑡−𝜏(𝑡)𝑥(𝜃)𝑑𝜃𝑇𝑆𝑡𝑡−𝜏(𝑡)𝑥(𝜃)𝑑𝜃−2𝛼𝑉4𝑥𝑡.(3.17) Now let 𝐵𝑖=[𝐴𝑖𝐷𝑖𝐸𝑖0−𝐼] and 𝐹=[𝐹𝑇0𝐹𝑇1𝐹𝑇2𝐹𝑇3𝐹𝑇4]𝑇. We can easily verify that 𝐵𝑖𝜒=0, where 𝜒(𝑡)=[𝜉𝑇(𝑡)̇𝑥𝑇(𝑡)]𝑇, and 𝜒𝑇(𝑡)𝐹𝐵𝑖+𝐵𝑇𝑖𝐹𝑇𝜒(𝑡)=0,𝑖=1,2…,𝑁.(3.18) Taking account of (3.18), and adding and subtracting the term 𝑥𝑇(𝑡)𝑍𝑥(𝑡) with 𝑍 a positive definite matrix, we get ̇𝑉𝑥𝑡𝑥+2𝛼𝑉𝑡≤𝑥𝑇𝑅(𝑡)11𝐴𝑖+𝐴𝑇𝑖𝑅11+𝑍𝑥(𝑡)+𝜒𝑇𝐺(𝑡)𝑖+𝑍𝑍−1𝑍𝑇𝜒(𝑡).(3.19) Since the condition (3.6) holds we have 𝜒𝑇(𝑡)(𝐺𝑖+𝑍𝑍−1𝑍𝑇)𝜒(𝑡)<0, it follows that ̇𝑉𝑥𝑡𝑥+2𝛼𝑉𝑡≤𝑥𝑇(𝑡)𝐿𝑖𝑥(𝑡).(3.20) From condition (i), we have ∑𝑁𝑖=1𝜆𝑖𝐿𝑖<−𝑌, where 1>𝜆𝑖>0, 𝑖=1,2,…,𝑁 and ∑𝑁𝑖=1𝜆𝑖=1. So 𝑁𝑖=1𝜆𝑖min𝑖=1,…,𝑁𝑥𝑇(𝑡)𝐿𝑖𝑥(𝑡)≤𝑁𝑖=1𝜆𝑖𝑥𝑇(𝑡)𝐿𝑖𝑥(𝑡)<−𝑥𝑇(𝑡)𝑌𝑥(𝑡)<0.(3.21) By choosing the switching rule as 𝜎(𝑥)=argmin𝑖=1,...,𝑁𝑥𝑇(𝑡)𝐿𝑖𝑥(𝑡).(3.22) We have ̇𝑉𝑥𝑡𝑥+2𝛼𝑉𝑡≤𝑥𝑇(𝑡)𝐿𝑖𝑥(𝑡)≤𝑁𝑖=1𝜆𝑖𝑥𝑇(𝑡)𝐿𝑖𝑥(𝑡)<−𝑥𝑇(𝑡)𝑌𝑥(𝑡)<0.(3.23) This implies that 𝑉(𝑥𝑡)≤𝑉(𝜙)𝑒−2𝛼𝑡,𝑡≥0. Taking account of (3.11), we obtain 𝛽1‖𝑥(𝑡)‖2𝑥≤𝑉𝑡≤𝑉(𝜙)𝑒−2𝛼𝑡≤𝛽2𝑒−2𝛼𝑡‖𝜙‖2.(3.24) And then, ‖𝑥(𝑡)‖≤𝛽𝑒−𝛼𝑡‖𝜙‖,𝑡≥0. Furthermore, since (3.11) holds, we have ̇𝑉𝑥𝑡≤−𝑥𝑇(𝑡)𝑌𝑥(𝑡).(3.25) Integrating both sides of (3.25) from 0 to 𝑇 and using the initial conditions, we obtain −𝑇0𝑥𝑇≥⎛⎜⎜⎜⎝(𝑡)𝑌𝑥(𝑡)𝑑𝑡𝑥(𝑇)𝑇𝑇−ℎ(𝑡)⎞⎟⎟⎟⎠𝑥(𝑠)𝑑𝑠𝑇𝑅⎛⎜⎜⎜⎝𝑥(𝑇)𝑇𝑇−ℎ(𝑇)⎞⎟⎟⎟⎠−⎛⎜⎜⎜⎝𝑥(𝑠)𝑑𝑠𝑥(0)00−ℎ(0)⎞⎟⎟⎟⎠𝑥(𝑠)𝑑𝑠𝑇𝑅⎛⎜⎜⎜⎝𝑥(0)00−ℎ(0)⎞⎟⎟⎟⎠+𝑥(𝑠)𝑑𝑠𝑇𝑇−ℎ𝑒−2𝛼𝑇𝑒2𝛼𝜃𝑥𝑇(𝜃)𝑃𝑥(𝜃)𝑑𝜃−0−ℎ𝑒2𝛼𝜃𝑥𝑇+(𝜃)𝑃𝑥(𝜃)𝑑𝜃0−ℎ𝑇𝑇+𝑠𝑒−2𝛼𝑇𝑒2𝛼𝜃𝑥𝑇(𝜃)𝑄𝑥(𝜃)𝑑𝜃𝑑𝑠−0−ℎ0𝑠𝑒2𝛼𝜃𝑥𝑇+(𝜃)𝑄𝑥(𝜃)𝑑𝜃𝑑𝑠0−𝜏(𝑇)𝑇𝑇+𝑠𝑒−2𝛼𝑇𝑒2𝛼𝜃𝑥𝑇(𝜃)𝑆𝑥(𝜃)𝑑𝜃𝑑𝑠−0−𝜏(0)0𝑠𝑒2𝛼𝜃𝑥𝑇(𝜃)𝑆𝑥(𝜃)𝑑𝜃𝑑𝑠.(3.26) As the system is asymptotically stable, when 𝑇→∞ we have ⎛⎜⎜⎜⎝𝑥(𝑇)𝑇𝑇−ℎ(𝑡)⎞⎟⎟⎟⎠𝑥(𝑠)𝑑𝑠𝑇𝑅⎛⎜⎜⎜⎝𝑥(𝑇)𝑇𝑇−ℎ(𝑇)⎞⎟⎟⎟⎠𝑥(𝑠)𝑑𝑠⟶0,0−ℎ𝑇𝑇+𝑠𝑒−2𝛼𝑇𝑒2𝛼𝜃𝑥𝑇(𝜃)𝑄𝑥(𝜃)𝑑𝜃𝑑𝑠⟶0,𝑇𝑇−ℎ𝑒−2𝛼𝑇𝑒2𝛼𝜃𝑥𝑇(𝜃)𝑃𝑥(𝜃)𝑑𝜃⟶0,0−𝜏(𝑇)𝑇𝑇+𝑠𝑒−2𝛼𝑇𝑒2𝛼𝜃𝑥𝑇(𝜃)𝑆𝑥(𝜃)𝑑𝜃𝑑𝑠⟶0.(3.27) Hence, we get 𝑇0𝑥𝑇⎛⎜⎜⎜⎝(𝑡)𝑌𝑥(𝑡)𝑑𝑡≤𝜙(0)00−ℎ(0)⎞⎟⎟⎟⎠𝜙(𝑠)𝑑𝑠𝑇𝑅⎛⎜⎜⎜⎝𝜙(0)00−ℎ(0)⎞⎟⎟⎟⎠+𝜙(𝑠)𝑑𝑠0−ℎ𝑒2𝛼𝜃𝜙𝑇+(𝜃)𝑃𝜙(𝜃)𝑑𝜃0−ℎ0𝑠𝑒2𝛼𝜃𝜙𝑇(𝜃)𝑄𝜙(𝜃)𝑑𝜃𝑑𝑠+0−𝜏(0)0𝑠𝑒2𝛼𝜃𝜙𝑇(𝜃)𝑆𝜙(𝜃)𝑑𝜃𝑑𝑠(3.28) which concludes the proof.

Remark 3.2. In order to improve the results, we can use instead of (3.18) the relation 𝜒𝑇𝐹(𝑡)𝑖𝐵𝑖+𝐵𝑇𝑖𝐹𝑇𝑖𝜒(𝑡)=0,𝑖=1,2,…,𝑁,(3.29) where 𝐹𝑖 is given by 𝐹𝑖=[𝐹𝑇0𝑖𝐹𝑇1𝑖𝐹𝑇2𝑖𝐹𝑇3𝑖𝐹𝑇4𝑖]𝑇. Then we can state the following result.

Theorem 3.3. For given 𝛼>0, switched linear system (2.1) is 𝛼-exponentially stable if there exist symmetric positive definite matrices 𝑍, 𝑍1, 𝑍2, 𝑆, 𝑃, 𝑄, 𝑅11, 𝑅22, 𝑌, matrices 𝑅12, 𝐹0𝑖, 𝐹1𝑖, 𝐹2𝑖, 𝐹3𝑖 and 𝐹4𝑖 satisfying the conditions (3.5) and (3.6), where the matrices 𝐹𝑘,𝑘=0,1,…,4 are replaced with 𝐹𝑘𝑖,𝑘=0,1,…,4, 𝑖=1,…,𝑁. The switching rule is given by [𝑆𝑅], the solution 𝑥(𝑡,𝜙) of the system satisfies (3.7), and the cost function in (2.5) satisfies (3.8).

If ℎ(𝑡)≠0 and 𝜏(𝑡)=0, the system (2.2) is reduced to the system as follows:̇𝑥(𝑡)=𝐴𝑖𝑥(𝑡)+𝐷𝑖𝑥(𝑡−ℎ(𝑡)),t>0.(3.30)

In this case, we have the following corollary.

Corollary 3.4. For given 𝛼>0, switched linear system (3.30) is 𝛼-exponentially stable if there exist symmetric positive definite matrices 𝑍, 𝑍1, 𝑍2, 𝑆, 𝑃, 𝑄, 𝑅11, 𝑅22, 𝑌, matrices 𝑅12, 𝐹𝑘𝑖, 𝑘=0,…,4, 𝑖=1,…,𝑁 satisfying condition (3.5) and the following LMI: ℑ𝑖℘℘𝑇−𝑍<0,𝑖=1,…,𝑁,(3.31) where, ℑ𝑖=⎛⎜⎜⎜⎜⎜⎜⎝ℑ𝑖11ℑ𝑖122𝛼𝑅12+𝑅22+𝐴𝑇𝑖𝐹𝑇2𝑖+𝐴𝑇𝑖𝑅12−𝐹0𝑖+𝐴𝑇𝑖𝐹𝑇3𝑖∗ℑ𝑖22−𝑅22+𝐷𝑇𝑖𝐹𝑇2𝑖+𝐷𝑖𝑅12−𝐹1𝑖+𝐷𝑇𝑖𝐹𝑇3𝑖1∗∗−ℎ𝑒−2𝛼ℎ𝑄+2𝛼𝑅22+𝜇ℎ𝑍2−𝐹2𝑖∗∗∗−𝐹3𝑖−𝐹𝑇3𝑖⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎝𝜇<0,℘=ℎ𝑅1200𝜇ℎ𝑅22⎞⎟⎟⎟⎟⎟⎠0000.(3.32) And where “∗” denotes the symmetric part and ℑ𝑖11=𝑅12+𝑅𝑇12+𝑃+2𝛼𝑅11+ℎ𝑄+𝐹0𝑖𝐴𝑖+𝐴𝑇𝑖𝐹𝑇0𝑖ℑ−𝑍,𝑖12=−𝑅12+𝑅11𝐷𝑖+𝐹0𝑖𝐷𝑖+𝐴𝑇𝑖𝐹𝑇1𝑖,ℑ𝑖22=−1−𝜇ℎ𝑒−2𝛼ℎ𝑃+𝐹1𝑖𝐷𝑖+𝐷𝑇𝑖𝐹𝑇1𝑖+𝜇ℎ𝑍1.(3.33)

Remark 3.5. In [16], the results are given in terms of a set of generalized Lyapunov equations type, and in [18], the results are expressed in terms of generalized algebraic Riccati equation type, while in this paper, the results are expressed in terms of linear matrix inequalities.

4. Examples

In this section, two examples will be presented for illustration and comparison.

Example 4.1 (see [18]). Consider the uncertain switched linear systems (2.1), where 𝑁=2, ℎ=1, 𝜏=1 and 𝐴1=−201−32,𝐷1=1−11−1,𝐸1=,𝐴111−32=4−11−32,𝐷2=1−13−4,𝐸2=.−11−14(4.1) We can see that each subsystem is unstable. Applying the results of [18], it is found that the system is asymptotically stable with decay rate 𝛼=0.5. Applying the results of [14], by choosing 𝑄=8∗𝐼, and applying Theorem 3.1, the critical delay of asymptotic stability (without decay rate) is found as 0.5722. Therefore, with the results of [14], we can't conclude about stability of the considered system. Note that both [14, 18] study the systems with constant delays. It is well known that the results for time varying delays are more conservative than those with constant delays. Applying our results, we let 𝜇ℎ=0 and 𝜇𝜏=0, by Theorem 3.1, with 𝜆1=𝜆2=0.5, we obtain a decay rate 𝛼=0.984 with the following solutions: 𝑃=3.1461−3.6923−3.69235.1228,𝑅=10−2⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠,112.4135−41.2719−0.1118−3.2593−41.271973.7166−0.20171.6277−1.0907−0.20170.0337−0.0079−3.25931.6277−0.00790.1106𝑍=17.153−8.754−8.75422.110,𝑄=0.250.00240.00240.6492,𝐹0=,𝐹0.15460.0216−0.11530.36481=10−2−0.87093.37592.7474−4.8734,𝐹2=10−2,𝐹0.14750.53182.27160.1949,𝑆=2.1245−0.4151−0.41517.84353=10−21.4000−0.0729−3.5071−0.2284,𝐹4=10−2,𝑍0.78300.2873−0.20200.07081=2.975002.975,𝑍2=2.975002.975,𝑌=10−2.0.4050−0.0579−0.05790.2367(4.2) And the matrices 𝐿1=−25.336−2.413−2.41324.233,𝐿2=25.3202.4152.415−24.243.(4.3) Satisfy 𝜆1𝐿1+𝜆2𝐿2=10−3−811−5<−𝑌.
The sets Ω1 and Ω2 are given by Ω1=𝑥1,𝑥2∈𝑅2∶−25.336𝑥21−4.826𝑥1𝑥2+24.233𝑥22,Ω<02=𝑥1,𝑥2∈𝑅2∶25.32𝑥21+4.83𝑥1𝑥2−24.243𝑥22.<0(4.4) It can be seen that Ω1∪Ω2=𝑅2⧵{0}, therefore, the switching regions are given by: Ω1=Ω1,Ω2=Ω2⧵Ω1, and the switching rule is given by ⎧⎪⎨⎪⎩𝜎=1if𝑥(𝑡)∈Ω1,2if𝑥(𝑡)∈Ω2.(4.5) Moreover the solution of the system satisfies ‖𝑥(𝑡,𝜙)‖≤3.6297𝑒−0.984𝑡‖𝜙‖,𝑡≥0. For the initial conditions 𝑥1(𝑡)=𝑒𝑡+1 and 𝑥2(𝑡)=0 for 𝑡∈[−1,0], we obtain a guaranteed cost 𝐽∗=14.5632. When we apply Theorem 3.3, we obtain the stability with decay rate 𝛼=1.186, stability factor 𝛽=2.3345, and the guaranteed cost 𝐽∗=19.244.

Example 4.2. Consider the following system: 𝐴01=−302−54,𝐴02=2−22−34,𝐴𝑑1=2−2−2−3.9,𝐴𝑑2=6−2515.31.(4.6) Applying the results of [20], when the time delay is constant, that is, ℎ(𝑡)=ℎ, we obtain the stability bound ℎ=0.063 with the parameters 𝛽1=15 and 𝛽2=0.8. Applying our results, for comparison, we set 𝜇ℎ=0 and 𝛼=0, then we apply Corollary 3.4 with 𝜆1=𝜆2=0.5, it is found that the system is asymptotically stable independent of delay. This shows the improvements of our approach. For ℎ=250 s, we have the following results: 𝑌=51.57348.32158.32153.6640,𝐿1=103−1.00450.00400.00400.6358,𝐿2=,𝜆798.1735−37.3230−37.3230−650.48351𝐿1+𝜆2𝐿2=−103.1467−16.6431−16.6431−7.3280<−𝑌.(4.7) The switching regions Ω1 and Ω1 are given by Ω1=𝑥1,𝑥2∈𝑅2∶−1004.5𝑥21+8𝑥1𝑥2+635.8𝑥22,<0Ω2=𝑥1,𝑥2∈𝑅2∶−1004.5𝑥21+8𝑥1𝑥2+635.8𝑥22,𝑥≥01,𝑥2≠(0,0).(4.8) The state trajectories are depicted in Figure 1 for ℎ(𝑡)=0.5 s and in Figure 2 for ℎ(𝑡)=250 s. It is clear that as the delay increases as the states require much time for convergence to zeros. Now, letting ℎ=0.5 s, 𝛼=0.4, 𝜆1=0.3, 𝜆2=0.7, and the initial conditions 𝑥1(𝑡)=𝑒𝑡+1 and 𝑥2(𝑡)=0 for 𝑡∈[−0.5,0], we obtain a guaranteed cost 𝐽∗=80.004.


Figure 1 
State trajectories for ℎ = 0 . 5  s.

Figure 2 
State trajectories for ℎ = 2 5 0  s.

5. Conclusion

By using Lyapunov Krasovskii approach, the problems of exponential stability and guaranteed cost are investigated for switched linear systems with mixed time delays. Via a designed switching rule, the results allow to the bounds that characterise the exponential stability, that is, the stability factor and the decay rate for the solution. The results are delay dependent and are expressed in terms of linear matrix inequalities. Some numerical examples are given to illustrate these results presented in this paper that have significant improvement over existing ones.