Robust Control for Uncertain Switched Systems with Interval Nondifferentiable Time-Varying Delays

Abstract

This paper addresses the conditions for robust stabilization of a class of uncertain switched systems with delay. The system to be considered is autonomous and the state delay is time-varying. Using Lyapunov functional approach, restriction on the derivative of time-delay function is not required to design switching rule for the robust stabilization of switched systems with time-varying delays. The delay-dependent stability conditions are presented in terms of the solution of LMIs which can be solved by various available algorithms. A numerical example is given to illustrate the effectiveness of theoretical results.

1. Introduction

The switched system is a type of hybrid systems that consist of a family of a differential or difference equations and a switching rule to indicate which subsystem will be activated at a specific interval of time. For applications, switched systems can be used to describe several physical or chemical processes which are concerned by more than one dynamics: some systems work at some interval time then stop and other systems take over such as the automatic system in airplane, car energy system, traffic system, and machine industrial system, see [1, 2]. Thus, a switching strategy must be designed in the study of stability of switched systems, also see [3–5].

The delay system has been considered in many research, especially the real processes in our world often involve time-delay; that is, the present state depends on the past states which brings more difficulty to investigate the stability of the system, especially time varying delay system, see [6–10]. In general, the following assumption on the derivative of the delay is made, namely ℎ̇(𝑡)<1, see [11, 12] and references cited therein. This assumption may leads to conservativeness; for example, it might not be used when the delay is a fast or a nondifferential time varying function.

Moreover, in study of real world applications, the systems are in general influenced by disturbances which might cause inaccuracy of the data. The system can become unstable or less capable because of disturbance. Consequently, the study of robust stability of switched systems with time varying delay becomes important and has been studied by many researchers, see [7, 12–17].

In this paper, we study the problem of robust stability for a class of switched systems with time-varying delay. Compared with existing results in the literature, the novelty of our results is twofold. Firstly, the state delay is time-varying in which the restriction on the derivative of the time-delay function is not required to design switching rule in term of a dwell time for the robust stability of the system. Secondly, the obtained conditions for the robust stability are delay-dependent and formulated in terms of the solution of standard LMIs which can be solved by various available algorithms [18]. The paper is organized as follows. Section 2 presents notations, definitions, and auxiliary propositions required for the proof of the main results. Switching design for the robust stability of the system with illustrative examples is presented in Sections 3 and 4, respectively. The paper ends with a conclusion followed by cited references.

2. Problem Formulation and Preliminaries

Throughout this paper, the following notations will be used:  ℝ^𝑛---the𝑛dimensional Euclidean space; ℝ^𝑛×𝑛---the  set  of  all𝑛×𝑛real  matrices; ℕ---the  set  of  all  positive  integers; ‖𝑥‖---the  Euclidean  norm  of  vector𝑥∈ℝ^𝑛; diag{⋅}---the  block  diagonal  matrix; 𝐼---the  identity  matrix; 𝐴^𝑇---the  transpose  of  matrix𝐴; 𝐴⁻¹---the  inverse  of  matrix𝐴; ^𝐴𝐵_∗𝐶---∗ the symmetric form of matrix, namely,∗=𝐵^𝑇; ‖𝐴‖=sup{‖𝐴𝑥‖∶𝑥∈ℝ^𝑛,‖𝑥‖=1}forany𝐴∈ℝ^𝑛×𝑛;  𝐶_ℎ2---space  of  continuous  vector-valued  function  defined  on[−ℎ₂,0]; 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃),−ℎ₂≤𝜃≤0,where𝑥_𝑡∈𝐶_ℎ2; ‖𝑥_𝑡‖_𝑐ℎ2=sup_{−ℎ2≤𝜃≤0}‖𝑥(𝑡+𝜃)‖;  𝜆_max(𝐴)=max{Re(𝜆)∶𝜆iseigenvalueof𝐴}; 𝜆_min(𝐴)=min{Re(𝜆)∶𝜆 is eigenvalue of 𝐴}; 𝑀={1,2,…,𝑘}. Consider the following uncertain switched system with time varying delay: 𝑥̇(𝑡)=𝐴_𝜎(𝑡)+Δ𝐴_𝜎(𝑡)(𝑡)𝑥(𝑡)+𝐷_𝜎(𝑡)+Δ𝐷_𝜎(𝑡)(𝑡)𝑥𝑡−ℎ_𝜎(𝑡)(𝑡+𝐵_𝜎(𝑡))+Δ𝐵_𝜎(𝑡)(𝑡)𝑢_𝜎(𝑡)(𝑡)𝑥(𝑡)(𝑡),𝑡>0,=𝜙,𝑡∈−ℎ₂,0, (2.1) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇∈ℝ^𝑛 is the state vector. 𝐴_𝑖,𝐷_𝑖,𝐵_𝑖, 𝑖∈𝑀 are known constant matrices, Δ𝐴_𝑖,Δ𝐷_𝑖,Δ𝐵_𝑖 are uncertainty matrices which are of the form Δ𝐴_𝑖=𝐸_1,𝑖𝐹_1,𝑖(𝑡)𝐻_1,𝑖,Δ𝐷_𝑖=𝐸_2,𝑖𝐹_2,𝑖(𝑡)𝐻_2,𝑖,Δ𝐵_𝑖=𝐸_3,𝑖𝐹_3,𝑖(𝑡)𝐻_3,𝑖,𝐹^𝑇_𝑗,𝑖(𝑡)𝐹_𝑗,𝑖(𝑡)≤𝐼,𝑗=1,2,3, (2.2) where 𝐹_1,𝑖(𝑡),𝐹_2,𝑖(𝑡),𝐹_3,𝑖(𝑡) are unknown matrices, 𝐼 is the identity matrix of appropriate dimension. ℎ_𝑖(𝑡) is the delay function for an 𝑖th subsystem which satisfies the following condition: 0≤ℎ₁≤ℎ_1,𝑖≤ℎ_𝑖(𝑡)≤ℎ_2,𝑖≤ℎ₂. (2.3) Let 𝜙(𝑡) be an initial condition. 𝜎(𝑡)∶𝑅⁺∪{0}→ 𝑀, 𝑡∈[𝑡_𝑘,𝑡_𝑘+1),𝑘∈ℕ, 𝜎(𝑡) is called the switching signal; we have the switching sequence {𝑥_𝑡0;(𝑖₀,𝑡₀),…∣𝑖_𝑘∈𝑀,𝑘=0,1,2,…}, which means that when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), the 𝑖_𝑘th subsystem is activated.

Definition 2.1. 𝑇^∗=inf{𝑡_𝑙−𝑡_𝑙−1} is called the dwell time of switched system.

Definition 2.2. The system (2.1) is said to be stabilizabled if there exists a feedback controller 𝑢(𝑡)∈ℝ^𝑚 such that the closed loop switched systems (without uncertainties) is asymptotically stable.

Definition 2.3. The system (2.1) is said to be robustly stabilizable if there exists a feedback controller 𝑢(𝑡)∈ℝ^𝑚 such that the closed loop uncertain switched systems are robustly stable.

The following lemmas will be used throughout this paper.

Lemma 2.4 (Schur complement lemma). Given constant symmetric matrices 𝑄,𝑆and𝑅∈ℝ^𝑛×𝑛, where 𝑅>0,𝑄=𝑄^𝑇, and 𝑅=𝑅^𝑇, one has 𝑆𝑄𝑆_𝑇−𝑅<0⟺𝑄+𝑆𝑅⁻¹𝑆^𝑇<0. (2.4)

Lemma 2.5 (see [13]). Given 𝜀>0 and matrices 𝐷, 𝐸, and 𝐹 with 𝐹^𝑇𝐹≤𝐼, one has 𝐷𝐸𝐹+𝐸^𝑇𝐹^𝑇𝐷^𝑇≤𝜀𝐷𝐷^𝑇+𝜀⁻¹𝐸^𝑇𝐸. (2.5)

Lemma 2.6 (see [13]). Given a positive definite matrix 𝑃∈ℝ^𝑛×𝑛, any symmetric matrix 𝑄∈ℝ^𝑛×𝑛 and 𝑥∈ℝ^𝑛, then 𝜆_min𝑃⁻¹𝑄𝑥^𝑇𝑃𝑥≤𝑥^𝑇𝑄𝑥≤𝜆_max𝑃⁻¹𝑄𝑥^𝑇𝑃𝑥. (2.6)

Lemma 2.7 (see [13]). For any positive semidefinite matrix 𝑀∈ℝ^𝑛×𝑛, a scalar 𝛾>0, and a vector function 𝜔∶[0,𝛾]→ℝ^𝑛×𝑛 such that the integrals concerned are well-defined, one has ^𝛾₀𝜔(𝑠)𝑑𝑠^𝑇𝑀^𝛾₀𝜔(𝑠)𝑑𝑠≤𝛾^𝛾₀𝜔^𝑇(𝑠)𝑀𝜔(𝑠)𝑑𝑠. (2.7)

Lemma 2.8 (Cauchy inequality). For any symmetric positive definite matrix 𝑁∈𝑀^𝑚×𝑚 and 𝑥,𝑦∈ℝ^𝑛, one has ±2𝑥^𝑇𝑦≤𝑥^𝑇𝑁𝑥+𝑦^𝑇𝑁⁻¹𝑦. (2.8)
For simplicity of later presentation, one gives the following notations. Σ_11,𝑖=𝑃_𝑖𝐴^𝑇_𝑖+𝐴_𝑖𝑃_𝑖−𝐵_𝑖𝐵^𝑇_𝑖+2+ℎ²₁+ℎ²₂𝑄_𝑖−𝑅_𝑖;Σ_12,𝑖=𝑃_𝑖𝐴^𝑇_𝑖−0.5𝐵_𝑖𝐵^𝑇_𝑖;Σ_13,𝑖=0.5𝑅_𝑖;Σ_14,𝑖=0.5𝑅_𝑖;Σ_15,𝑖=𝐷_𝑖𝑃_𝑖;Σ_22,𝑖=ℎ²_1,𝑖+ℎ²_2,𝑖𝑅_𝑖+𝛿²_𝑖𝑄_𝑖−2𝑃_𝑖;Σ_25,𝑖=𝐷_𝑖𝑃_𝑖;Σ_33,𝑖=−𝑄_𝑖−0.5𝑅_𝑖−0.5𝑈_𝑖;Σ_35,𝑖=0.5𝑈_𝑖;Σ_44,𝑖=−𝑄_𝑖−0.5𝑅_𝑖−0.5𝑈_𝑖;Σ_45,𝑖=0.5𝑈_𝑖;Σ_55,𝑖=−𝑈_𝑖;𝜆₁=max_𝑖∈𝑀𝜆_max𝑃_𝑖⁻¹;𝜆₂=min_𝑖∈𝑀𝜆_min𝑃_𝑖⁻¹;𝜆₃=max_𝑖∈𝑀𝜆_max𝑃_𝑖;𝜆₄=min_{𝑖∈𝐼𝑘}𝜆_min−Ω_1,𝑖;𝜆₅=min_𝑖∈𝑀𝜆_min−Ω_2,𝑖;𝜆^∗=min𝜆₄,𝜆₅;ℎ_𝑎,𝑖=12ℎ_1,𝑖+ℎ_2,𝑖;𝛿_𝑖=ℎ_2,𝑖−ℎ_1,𝑖;ℎ^∗=min_𝑖∈𝑀ℎ_2,𝑖−ℎ_𝑎,𝑖;𝛿=ℎ₂−ℎ₁. (2.9)

3. Main Results

3.1. Asymptotical Stabilization for Nominal Switched Systems with Interval Time-Varying Delay

The nominal switched systems are given by 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐷_𝜎(𝑡)𝑥𝑡−ℎ_𝜎(𝑡)(𝑡)+𝐵_𝜎(𝑡)𝑢_𝜎(𝑡)(𝑡)𝑥(𝑡)(𝑡),𝑡>0=𝜙,𝑡∈−ℎ₂,0. (3.1) We now state the main result on sufficient condition for stabilization of the switched systems (3.1).

Theorem 3.1. Given 𝛼∈(0,1). If there exists symmetric positive definite matrices 𝑃_𝑖,𝑄_𝑖,𝑅_𝑖,𝑈_𝑖 such that the following conditions hold: Ω_1,𝑖=Ω_𝑖−000−𝐼𝐼^𝑇𝑈_𝑖Ω000−𝐼𝐼<0,_2,𝑖=Ω_𝑖−00𝐼0−𝐼^𝑇𝑈_𝑖Ω00𝐼0−𝐼<0,_𝑖=Σ_11,𝑖Σ_12,𝑖Σ_13,𝑖Σ_14,𝑖Σ_15,𝑖∗Σ_22,𝑖00Σ_25,𝑖∗∗Σ_33,𝑖0Σ_35,𝑖∗∗∗Σ_44,𝑖Σ_45,𝑖∗∗∗∗Σ_55,𝑖, (3.2) and if 𝑇^∗≥1𝜌𝜆ln¹𝛼𝜆₂, (3.3) where 𝜌=min{ℎ^∗𝜆^∗/𝛿𝜆₃,1/2ℎ₂,1/2}, 𝑇^∗ is the dwell time, then for any switching rule satisfying (3.3) the switched system (3.1) is stabilizable under the feedback controller 𝑢_𝑖(𝑡)1=−2𝐵^𝑇_𝑖𝑃_𝑖⁻¹𝑥(𝑡),𝑡≥0. (3.4)

Proof. Let 𝑌_𝑖=𝑃_𝑖⁻¹,𝑦(𝑡)=𝑌_𝑖𝑥(𝑡). Using the feedback controller (3.4), we choose a Lyapunov-Krasovskii functional candidates as 𝑉_𝑖𝑥_𝑡=𝑉_1,𝑖𝑥_𝑡+⋯+𝑉_8,𝑖𝑥_𝑡, (3.5) where 𝑉_1,𝑖𝑥_𝑡=𝑥^𝑇(𝑡)𝑌_𝑖𝑥(𝑡),𝑉_2,𝑖𝑥_𝑡=^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑉𝑑𝑠,_3,𝑖𝑥_𝑡=^𝑡_{𝑡−ℎ2,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑉𝑑𝑠,_4,𝑖𝑥_𝑡=ℎ_1,𝑖⁰_−ℎ1,𝑖^𝑡_𝑡+𝑠𝑥̇^𝑇(𝜃)𝑌_𝑖𝑅_𝑖𝑌_𝑖𝑥̇(𝜃)𝑉𝑑𝜃𝑑𝑠,_5,𝑖𝑥_𝑡=ℎ_2,𝑖⁰_−ℎ2,𝑖^𝑡_𝑡+𝑠𝑥̇^𝑇(𝜃)𝑌_𝑖𝑅_𝑖𝑌_𝑖𝑥̇(𝜃)𝑉𝑑𝜃𝑑𝑠,_6,𝑖𝑥_𝑡=𝛿_𝑖^−ℎ1,𝑖_−ℎ2,𝑖^𝑡_𝑡+𝑠𝑥̇^𝑇(𝜃)𝑌_𝑖𝑈_𝑖𝑌_𝑖𝑥̇(𝜃)𝑉𝑑𝜃𝑑𝑠,_7,𝑖𝑥_𝑡=ℎ_1,𝑖⁰_−ℎ1,𝑖^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝜃)𝑉𝑑𝜃𝑑𝑠,_8,𝑖𝑥_𝑡=ℎ_2,𝑖⁰_−ℎ2,𝑖^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝜃)𝑑𝜃𝑑𝑠. (3.6) It is easy to see that 𝑉_𝑖𝑥_𝑡≥𝑐₁‖𝑥(𝑡)‖², (3.7) for some 𝑐₁>0. Taking the derivative of 𝑉_𝑖(𝑥_𝑡) with respect to 𝑡 along any trajectory of solution of (3.1) yields 𝑉̇_1,𝑖𝑥_𝑡=2𝑥^𝑇(𝑡)𝑌_𝑖𝑥̇(𝑡),=𝑦^𝑇(𝑡)𝑃_𝑖𝐴^𝑇_𝑖+𝐴_𝑖𝑃_𝑖𝑦(𝑡)−𝑦^𝑇(𝑡)𝐵^𝑇_𝑖𝐵_𝑖𝑦(𝑡)+2𝑦^𝑇(𝑡)𝐷_𝑖𝑃_𝑖𝑦𝑡−ℎ_𝑖(𝑡),𝑉̇_2,𝑖𝑥_𝑡=𝑦^𝑇(𝑡)𝑄_𝑖𝑦(𝑡)−𝑦^𝑇𝑡−ℎ_1,𝑖𝑄_𝑖𝑦𝑡−ℎ_1,𝑖,𝑉̇_3,𝑖𝑥_𝑡=𝑦^𝑇(𝑡)𝑄_𝑖𝑦(𝑡)−𝑦^𝑇𝑡−ℎ_2,𝑖𝑄_𝑖𝑦𝑡−ℎ_2,𝑖,𝑉̇_4,𝑖𝑥_𝑡=ℎ²_1,𝑖𝑦̇^𝑇(𝑡)𝑅_𝑖𝑦̇(𝑡)−ℎ_1,𝑖2^𝑡_{𝑡−ℎ1,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)ℎ𝑑𝑠−^1,𝑖2^𝑡_{𝑡−ℎ1,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝑑𝑠,𝑉̇_5,𝑖𝑥_𝑡=ℎ²_2,𝑖𝑦̇^𝑇(𝑡)𝑅_𝑖𝑦̇(𝑡)−ℎ_2,𝑖2^𝑡_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)ℎ𝑑𝑠−^2,𝑖2^𝑡_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝑑𝑠,𝑉̇_6,𝑖𝑥_𝑡=𝛿²_𝑖𝑦̇^𝑇(𝑡)𝑈_𝑖𝑦̇(𝑡)−𝛿_𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)𝛿𝑑𝑠−^𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝑑𝑠,𝑉̇_7,𝑖𝑥_𝑡=ℎ²_1,𝑖𝑦^𝑇(𝑡)𝑄_𝑖𝑦(𝑡)−ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠,𝑉̇_8,𝑖𝑥_𝑡=ℎ²_2,𝑖𝑦^𝑇(𝑡)𝑄_𝑖𝑦(𝑡)−ℎ_2,𝑖^𝑡_{𝑡−ℎ2,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠. (3.8)(3.9)(3.10)(3.11)(3.12)(3.13)(3.14)(3.15) Then by applying Lemma 2.7 and Leibniz-Newton formular, we have −ℎ_1,𝑖2^𝑡_{𝑡−ℎ1,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝑑𝑠≤𝑦^𝑇(𝑡)−𝑅_𝑖2𝑦(𝑡)+𝑦^𝑇(𝑡)𝑅_𝑖𝑦𝑡−ℎ_1,𝑖+𝑦^𝑇𝑡−ℎ_1,𝑖−𝑅_𝑖2𝑦𝑡−ℎ_1,𝑖,−ℎ_2,𝑖2^𝑡_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝑑𝑠≤𝑦^𝑇(𝑡)−𝑅_𝑖2𝑦(𝑡)+𝑦^𝑇(𝑡)𝑅_𝑖𝑦𝑡−ℎ_2,𝑖+𝑦^𝑇𝑡−ℎ_2,𝑖−𝑅_𝑖2𝑦𝑡−ℎ_2,𝑖. (3.16) Note that −𝛿_𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)𝛿𝑑𝑠=−^𝑖2^{𝑡−ℎ𝑖(𝑡)}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)𝛿𝑑𝑠−^𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ𝑖(𝑡)}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)ℎ𝑑𝑠=−^2,𝑖−ℎ_𝑖(𝑡)2^{𝑡−ℎ(𝑡)}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)ℎ(𝑡)𝑑𝑠−−ℎ_1,𝑖2×^{𝑡−ℎ𝑖(𝑡)}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)ℎ𝑑𝑠−^𝑖(𝑡)−ℎ_1,𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ𝑖(𝑡)}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)−ℎ𝑑𝑠_2,𝑖−ℎ_𝑖(𝑡)2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ(𝑡)}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)𝑑𝑠. (3.17) Using Lemma 2.7 yields −ℎ_2,𝑖−ℎ_𝑖(𝑡)2^{𝑡−ℎ𝑖(𝑡)}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)≤𝑦𝑑𝑠𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖^𝑇−𝑈_𝑖2𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖,−ℎ_𝑖(𝑡)−ℎ_1,𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ𝑖(𝑡)}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)≤𝑦𝑑𝑠𝑡−ℎ_1,𝑖−𝑦𝑡−ℎ_𝑖(𝑡)^𝑇−𝑈_𝑖2𝑦𝑡−ℎ_1,𝑖−𝑦𝑡−ℎ_𝑖(𝑡). (3.18) Let 𝛽=(ℎ_2,𝑖−ℎ_𝑖(𝑡))/(ℎ_2,𝑖−ℎ_1,𝑖)≤1. Then −ℎ_2,𝑖−ℎ_𝑖(𝑡)2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ𝑖(𝑡)}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)𝑑𝑠=−𝛽^{𝑡−ℎ1,𝑖}_{𝑡−ℎ𝑖(𝑡)}ℎ_2,𝑖−ℎ_1,𝑖𝑦̇^𝑇(𝑠)𝑈_𝑖2𝑦̇(𝑠)𝑑𝑠≤−𝛽^{𝑡−ℎ1,𝑖}_{𝑡−ℎ𝑖(𝑡)}ℎ_𝑖(𝑡)−ℎ_1,𝑖𝑦̇^𝑇(𝑠)𝑈_𝑖2𝑦̇(𝑠)𝑦𝑑𝑠≤−𝛽𝑡−ℎ_1,𝑖−𝑦𝑡−ℎ_𝑖(𝑡)^𝑇𝑈_𝑖2×𝑦𝑡−ℎ_1,𝑖−𝑦𝑡−ℎ_𝑖(𝑡).−ℎ_𝑖(𝑡)−ℎ_1,𝑖2^{𝑡−ℎ𝑖(𝑡)}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)𝑑𝑠=−1−𝛽^{𝑡−ℎ𝑖(𝑡)}_{𝑡−ℎ2,𝑖}ℎ_2,𝑖−ℎ_1,𝑖𝑦̇^𝑇(𝑠)𝑈_𝑖2𝑦̇(𝑠)𝑑𝑠≤−1−𝛽^{𝑡−ℎ𝑖(𝑡)}_{𝑡−ℎ2,𝑖}ℎ_2,𝑖−ℎ_𝑖(𝑡)𝑦̇^𝑇(𝑠)𝑈_𝑖2𝑦̇(𝑠)𝑦𝑑𝑠≤−1−𝛽𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖^𝑇𝑈_𝑖2×𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖. (3.19) Therefore from (3.18)-(3.19), we have −𝛿_𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)𝑑𝑠≤𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖^𝑇−𝑈_𝑖2𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖+𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖^𝑇−𝑈_𝑖2𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖−𝛽𝑦𝑡−ℎ_1,𝑖−𝑦𝑡−ℎ_𝑖(𝑡)^𝑇𝑈_𝑖2𝑦𝑡−ℎ_1,𝑖−𝑦𝑡−ℎ_𝑖(𝑡)−1−𝛽𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖^𝑇𝑈_𝑖2×𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖. (3.20) Furthermore, from the following zero equation −𝑃_𝑖𝑦̇(𝑡)+𝐴_𝑖𝑃_𝑖𝑦(𝑡)+𝐷_𝑖𝑃_𝑖𝑦𝑡−ℎ_𝑖(𝑡)−0.5𝐵_𝑖𝐵^𝑇_𝑖𝑦(𝑡)=0, (3.21) we obtain −2𝑦̇^𝑇𝑃_𝑖𝑦̇(𝑡)+2𝑦̇^𝑇𝐴_𝑖𝑃_𝑖𝑦(𝑡)+2𝑦̇^𝑇𝐷_𝑖𝑃_𝑖𝑦𝑡−ℎ_𝑖(𝑡)−2𝑦̇^𝑇0.5𝐵_𝑖𝐵^𝑇_𝑖𝑦(𝑡)=0. (3.22) Hence, from (3.5),(3.8)–(3.16),(3.20), and (3.22), we can get 𝑉̇_𝑖𝑥_𝑡≤𝜉^𝑇(𝑡)Ω_𝑖𝜉(𝑡)−𝛽𝑦𝑡−ℎ_1,𝑖−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈_𝑖2𝑦𝑡−ℎ_1,𝑖−𝑦𝑡−ℎ_𝑖(𝑡)−1−𝛽𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖^𝑇𝑈_𝑖2𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖−ℎ_1,𝑖2^𝑡_{𝑡−ℎ1,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)ℎ𝑑𝑠−^2,𝑖2^𝑡_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)−𝛿𝑑𝑠_𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝑑𝑠−ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠−ℎ_2,𝑖^𝑡_{𝑡−ℎ2,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠=𝜉^𝑇(𝑡)1−𝛽Ω_1,𝑖+𝛽Ω_2,𝑖𝜉(𝑡)−ℎ_1,𝑖2^𝑡_{𝑡−ℎ1,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)−ℎ𝑑𝑠_2,𝑖2^𝑡_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝛿𝑑𝑠−^𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝑑𝑠,−ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠−ℎ_2,𝑖^𝑡_{𝑡−ℎ2,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠, where 𝜉^𝑇(𝑡)=[^{𝑦𝑇̇𝑦(𝑡)𝑇(𝑡)𝑦𝑇(𝑡−ℎ}_1,𝑖^{)𝑦𝑇(𝑡−ℎ}_2,𝑖^{)𝑦𝑇(𝑡−ℎ}_𝑖^(𝑡))].
Suppose 𝜏 is the time when the system switches from state 𝑗 to state 𝑖, that is 𝐼(𝜏⁺) = 𝑖 and 𝐼(𝜏⁻) = 𝑗, where 𝜏⁺ and 𝜏⁻ are the right and the left limit of the time 𝜏, respectively. According to Lemma 2.6, we obtain 𝑉_1,𝑖𝑥_𝜏+=𝑥^𝑇(𝜏)𝑌_𝑖𝑥(𝜏)≤𝜆₁𝜆₂𝜆₂𝑥^𝑇(𝜏)𝑥(𝜏)≤𝜆₁𝜆₂𝑥^𝑇(𝜏)𝑃_𝑗𝑥(𝜏)=𝜆₁𝜆₂𝑉_1,𝑗𝑥_𝜏−. (3.24) We can apply this argument to integral terms in the Lypunov-Krasovskii function, so we get 𝑉_𝑖𝑥_𝜏+≤𝜆₁𝜆₂𝑉_𝑗𝑥_𝜏−. (3.25) Now let 𝜈 be the time when the system switches from state 𝑘 to state 𝑗, that is, 𝐼(𝜈⁺)=𝑗 and 𝐼(𝜈⁻)=𝑘, where 𝜈⁺ and 𝜈⁻ are a right and a left limit of the time 𝜈, respectively. In order to show that the switched system is stable, we need to compare 𝑉_𝑖(𝑥_𝜏+) with 𝑉_𝑗(𝑥_𝜈+) and estimate the upper bound for the term 𝜉^𝑇(𝑡)[(1−𝛽)Ω_1,𝑖+𝛽Ω_2,𝑖]𝜉(𝑡) in the inequality (3.23). Hence we consider two following possible cases.
Case  1 (0≤ℎ₁≤ℎ_1,𝑖≤ℎ(𝑡)≤ℎ_𝑎,𝑖≤ℎ_2,𝑖≤ℎ₂). Since 0≤𝛽≤1,Ω_1,𝑖<0 and Ω_2,𝑖<0, we have 𝜉^𝑇(𝑡)1−𝛽Ω_1,𝑖+𝛽Ω_2,𝑖𝜉(𝑡)≤𝜉^𝑇(𝑡)𝛽Ω_2,𝑖𝜉(𝑡)ℎ≤−^2,𝑖−ℎ_𝑎,𝑖𝛿_𝑖𝜆₅𝑦(𝑡)²+𝑦(̇𝑡)²+𝑦(𝑡−ℎ_1,𝑖)²+𝑦(𝑡−ℎ_2,𝑖)²+𝑦(𝑡−ℎ_𝑖(𝑡))²ℎ≤−^2,𝑖−ℎ_𝑎,𝑖𝛿_𝑖𝜆₅𝑦(𝑡)²ℎ=−^2,𝑖−ℎ_𝑎,𝑖𝛿_𝑖𝜆₅𝜆₃𝜆₃𝑦(𝑡)²ℎ≤−^2,𝑖−ℎ_𝑎,𝑖𝛿𝜆₅𝜆₃𝑥^𝑇(𝑡)𝑃_𝑖⁻¹𝑃_𝑖𝑃_𝑖⁻¹𝑥(𝑡)ℎ≤−^2,𝑖−ℎ_𝑎,𝑖𝛿𝜆₅𝜆₃𝑉_1,𝑖ℎ≤−^2,𝑖−ℎ_𝑎,𝑖𝛿𝜆^∗𝜆₃𝑉_1,𝑖ℎ≤−^∗𝛿𝜆^∗𝜆₃𝑉_1,𝑖. (3.26)
Case  2 (0≤ℎ₁≤ℎ_1,𝑖≤ℎ_𝑎,𝑖≤ℎ(𝑡)≤ℎ_2,𝑖≤ℎ₂). Since 0≤𝛽≤1,Ω_1,𝑖<0 and Ω_2,𝑖<0, we get 𝜉^𝑇(𝑡)1−𝛽Ω_1,𝑖+𝛽Ω_2,𝑖𝜉(𝑡)≤𝜉^𝑇(𝑡)1−𝛽Ω_1,𝑖𝜉(𝑡)ℎ≤−^𝑎,𝑖−ℎ_1,𝑖𝛿_𝑖𝜆₄𝑦(𝑡)²+𝑦(̇𝑡)²+𝑦(𝑡−ℎ_1,𝑖)²+𝑦(𝑡−ℎ_2,𝑖)²+𝑦(𝑡−ℎ_𝑖(𝑡))²ℎ≤−^𝑎,𝑖−ℎ_1,𝑖𝛿_𝑖𝜆₄𝑦(𝑡)²ℎ=−^𝑎,𝑖−ℎ_1,𝑖𝛿_𝑖𝜆₄𝜆₃𝜆₃‖𝑦(𝑡)‖²ℎ≤−^𝑎,𝑖−ℎ_1,𝑖𝛿𝜆₄𝜆₃𝑥^𝑇(𝑡)𝑃_𝑖⁻¹𝑃_𝑖𝑃_𝑖⁻¹𝑥(𝑡)ℎ≤−^𝑎,𝑖−ℎ_1,𝑖𝛿𝜆₄𝜆₃𝑉_1,𝑖. Note that ℎ_2,𝑖−ℎ_𝑎,𝑖=ℎ_𝑎,𝑖−ℎ_1,𝑖, so we obtain 𝜉^𝑇(𝑡)1−𝛽Ω_1,𝑖+𝛽Ω_2,𝑖𝜉(𝑡)ℎ≤−^2,𝑖−ℎ_𝑎,𝑖𝛿𝜆^∗𝜆₃𝑉_1,𝑖ℎ≤−^∗𝛿𝜆^∗𝜆₃𝑉_1,𝑖. (3.28) Moreover, from the definition of 𝑉_2,𝑖, …, 𝑉_8,𝑖, we can get −ℎ_1,𝑖2^𝑡_{𝑡−ℎ1,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)1𝑑𝑠≤−2⁰_−ℎ1,𝑖^𝑡_𝑡+𝑠𝑥̇^𝑇(𝜃)𝑌_𝑖𝑅_𝑖𝑌_𝑖𝑥̇(𝜃)1𝑑𝜃𝑑𝑠≤−2ℎ_2,𝑖𝑉_4,𝑖,−ℎ_2,𝑖2^𝑡_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)1𝑑𝑠≤−2⁰_−ℎ2,𝑖^𝑡_𝑡+𝑠𝑥̇^𝑇(𝜃)𝑌_𝑖𝑅_𝑖𝑌_𝑖𝑥̇(𝜃)1𝑑𝜃𝑑𝑠≤−2ℎ_1,𝑖𝑉_5,𝑖,−𝛿_𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑈_𝑖𝑦̇(𝑠)1𝑑𝑠≤−2^−ℎ1,𝑖_−ℎ2,𝑖^𝑡_𝑡+𝑠𝑥̇^𝑇(𝜃)𝑌_𝑖𝑈_𝑖𝑌_𝑖𝑥̇(𝜃)1𝑑𝜃𝑑𝑠≤−2𝛿_𝑖𝑉_6,𝑖. (3.29) Since −ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠≤−⁰_−ℎ1,𝑖^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝜃)1𝑑𝜃𝑑𝑠≤−ℎ_1,𝑖𝑉_7,𝑖, (3.30) we can get −ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠−ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)1𝑑𝑠≤−ℎ_1,𝑖𝑉_7,𝑖−ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠. (3.31) So we obtain −ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)1𝑑𝑠≤−2ℎ_1,𝑖𝑉_7,𝑖−12𝑉_2,𝑖. (3.32) Similar to (3.32), we have −ℎ_2,𝑖^𝑡_{𝑡−ℎ2,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)1𝑑𝑠≤−2ℎ_2,𝑖𝑉_3,𝑖−12𝑉_8,𝑖. (3.33) According to (3.23), (3.28)–(3.33), we have 𝑉̇_𝑖𝑥_𝑡≤−minℎ^∗𝜆^∗𝛿𝜆₃,12ℎ_1,𝑖,12ℎ_2,𝑖,12𝛿_𝑖,12𝑉_𝑖𝑥_𝑡ℎ≤−min^∗𝜆^∗𝛿𝜆₃,12ℎ₂,12𝑉_𝑖𝑥_𝑡=−𝜌𝑉_𝑖𝑥_𝑡. (3.34) As a result, we get 𝑉̇_𝑗𝑥_𝑡≤−𝜌𝑉_𝑗𝑥_𝑡, (3.35) which yields 1𝑉_𝑗𝑥_𝑡𝑑𝑉_𝑗𝑥_𝑡≤−𝜌𝑑𝑡. (3.36) Integrating (3.36) on (𝜈,𝜏), we obtain 𝑉_𝑗𝑥_𝜏−≤𝑉_𝑗𝑥_𝜈+𝑒^{−𝜌(𝜏−𝜈)}. (3.37) From (3.25) and (3.37), we have 𝑉_𝑖𝑥_𝜏+≤𝜆₁𝜆₂𝑉_𝑗𝑥_𝜈+𝑒^{−𝜌(𝜏−𝜈)}. (3.38) Since 𝜏−𝜈≥𝑇₀≥1𝜌𝜆ln¹𝛼𝜆₂, (3.39) we obtain from (3.38) that 𝑉_𝑖𝑥_𝜏+≤𝛼𝑉_𝑗𝑥_𝜈+. (3.40) Let 𝑁(𝑡) be the number of time the system is switched on [0,𝑡). Since the switched system (3.1) undergoes infinite times of switches on [0,∞), we obtain lim_𝑡→∞𝑁(𝑡)=∞. Assume that 𝐼(𝑡)=𝑖 and 𝐼(0)=𝑖₀. Then, according to (3.7) and (3.40), we have 𝑐₁‖𝑥(𝑡)‖²≤𝑉_𝑖𝑥_𝑡≤𝛼^𝑁(𝑡)𝑉_𝑖0𝑥₀. (3.41) Since 0<𝛼<1 and lim_𝑡→∞𝑁(𝑡)=∞, it follows from (3.41) that ‖𝑥(𝑡)‖→0 as 𝑡→∞. We conclude that the zero solution of (3.1) is stabilizable.

3.2. Robust Stabilization for Switched Systems with Interval Time-Varying Delay

Theorem 3.2. Given 𝛼∈(0, 1). If there exists symmetric positive definite matrices 𝑃_𝑖,𝑄_𝑖,𝑅_𝑖,𝑈_𝑖 such that the following conditions hold: Ω_1,𝑖=Ω_𝑖−000−𝐼𝐼^𝑇𝑈_𝑖000−𝐼𝐼<0,Ω_2,𝑖=Ω_𝑖−00𝐼0−𝐼^𝑇𝑈_𝑖00𝐼0−𝐼<0,Ω_3,𝑖=Ω_311,𝑖𝑃_𝑖𝐻^𝑇_1,𝑖𝑃_𝑖𝐻^𝑇_1,𝑖𝐵_𝑖𝐻^𝑇_3,𝑖2𝐵_𝑖𝐻^𝑇_3,𝑖2∗−𝜖₁𝐼000∗∗−𝜖₂𝐼00∗∗∗−𝜖₅𝐼20∗∗∗∗−𝜖₆𝐼2<0,Ω_4,𝑖=Ω_411,𝑖𝑃_𝑖𝐻^𝑇_2,𝑖𝑃_𝑖𝐻^𝑇_2,𝑖∗−𝜖₃𝐼0∗∗−𝜖₄𝐼<0,Ω_𝑖=Λ_11,𝑖Λ_12,𝑖Λ_13,𝑖Λ_14,𝑖Λ_15,𝑖∗Λ_22,𝑖00Λ_25,𝑖∗∗Λ_33,𝑖0Λ_35,𝑖∗∗∗Λ_44,𝑖Λ_45,𝑖∗∗∗∗Λ_55,𝑖, (3.42)(3.43)(3.44)(3.45)(3.46) where Λ_11,𝑖=𝑃_𝑖𝐴^𝑇_𝑖+𝐴_𝑖𝑃_𝑖−0.5𝐵_𝑖𝐵^𝑇_𝑖+2+ℎ²_1,𝑖+ℎ²_2,𝑖𝑄_𝑖−0.5𝑅_𝑖+𝜖₁𝐸^𝑇₁𝐸₁+𝜖₃𝐸^𝑇₂𝐸₂+𝜖₅2𝐸^𝑇₃𝐸₃;Λ_12,𝑖=𝑃_𝑖𝐴^𝑇_𝑖−0.5𝐵_𝑖𝐵^𝑇_𝑖;Λ_13,𝑖=0.5𝑅_𝑖;Λ_14,𝑖=0.5𝑅_𝑖;Λ_15,𝑖=𝐷_𝑖𝑃_𝑖;Λ_22,𝑖=ℎ²_1,𝑖+ℎ²_2,𝑖+𝛿²_𝑖𝑄_𝑖−2𝑅_𝑖+𝜖₂𝐸^𝑇₁𝐸₁+𝜖₄𝐸^𝑇₂𝐸₂+𝜖₆2𝐸^𝑇₃𝐸₃;Λ_25,𝑖=𝐷_𝑖𝑃_𝑖;Λ_33,𝑖=−𝑄_𝑖−0.5𝑅_𝑖−0.5𝑈_𝑖;Λ_35,𝑖=0.5𝑈_𝑖;Λ_44,𝑖=−𝑄_𝑖−0.5𝑅_𝑖−0.5𝑈_𝑖;Λ_45,𝑖=0.5𝑈_𝑖;Λ_55,𝑖=−0.5𝑈_𝑖;Ω_311,𝑖=−0.5𝐵_𝑖𝐵^𝑇_𝑖−0.5𝑅_𝑖;Ω_411,𝑖=−0.5𝑈_𝑖. (3.47) Then for any switching rule satisfying (3.3) the switched system (2.1) is robustly stabilizable under the feedback controller given by (3.4).

Proof. Construct Lyapunov-Krasovskii functional as in (3.5), we can proof this theorem by using a similar argument as in the proof of Theorem 3.1. By replacing 𝐴_𝑖,𝐷_𝑖,𝐵_𝑖 in (3.8) with 𝐴_𝑖+𝐸_1,𝑖𝐹_1,𝑖(𝑡)𝐻_1,𝑖, 𝐷_𝑖+𝐸_2,𝑖𝐹_2,𝑖(𝑡)𝐻_2,𝑖, 𝐵_𝑖+𝐸_3,𝑖𝐹_3,𝑖(𝑡)𝐻_3,𝑖, respectively, we obtain 𝑉̇_𝑖𝑥_𝑡≤𝑦^𝑇(𝑡)𝑃_𝑖𝐴_𝑖+𝐸_1,𝑖𝐹_1,𝑖(𝑡)𝐻_1,𝑖^𝑇+𝐴_𝑖+𝐸_1,𝑖𝐹_1,𝑖(𝑡)𝐻_1,𝑖𝑃_𝑖𝑦(𝑡)+2𝑦^𝑇(𝑡)𝐷_𝑖+𝐸_2,𝑖𝐹_2,𝑖(𝑡)𝐻_2,𝑖𝑃_𝑖𝑦𝑡−ℎ_𝑖(𝑡)−𝑦^𝑇(𝑡)𝐵^𝑇_𝑖𝐵_𝑖𝑦(𝑡)−𝑦^𝑇(𝑡)𝐸_3,𝑖𝐹_3,𝑖(𝑡)𝐻_3,𝑖𝐵^𝑇_𝑖𝑦(𝑡)+𝑦^𝑇(𝑡)2+ℎ²_1,𝑖+ℎ²_2,𝑖𝑄_𝑖𝑦(𝑡)−𝑦^𝑇𝑡−ℎ₁𝑄_𝑖𝑦𝑡−ℎ₁−𝑦^𝑇𝑡−ℎ₂𝑄_𝑖𝑦𝑡−ℎ₂+𝑦̇^𝑇(𝑡)ℎ²_1,𝑖+ℎ²_1,𝑖𝑅_𝑖+𝛿²_𝑖𝑈_𝑖𝑦̇(𝑡)+𝑦^𝑇(𝑡)−𝑅_𝑖2𝑦(𝑡)+𝑦^𝑇(𝑡)𝑅_𝑖𝑦𝑡−ℎ_1,𝑖+𝑦^𝑇𝑡−ℎ_1,𝑖−𝑅_𝑖2𝑦𝑡−ℎ_1,𝑖+𝑦^𝑇(𝑡)−𝑅_𝑖2𝑦(𝑡)+𝑦^𝑇(𝑡)𝑅_𝑖𝑦𝑡−ℎ_2,𝑖+𝑦^𝑇𝑡−ℎ_2,𝑖−𝑅_𝑖2𝑦𝑡−ℎ_2,𝑖+𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖^𝑇−𝑈_𝑖2𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖+𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖^𝑇−𝑈_𝑖2𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖−𝛽𝑦𝑡−ℎ_1,𝑖−𝑦𝑡−ℎ_𝑖(𝑡)^𝑇−𝑈_𝑖2𝑦𝑡−ℎ_1,𝑖−𝑦𝑡−ℎ_𝑖(𝑡)−1−𝛽𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖^𝑇−𝑈_𝑖2𝑦𝑡−ℎ_𝑖(𝑡)−𝑦𝑡−ℎ_2,𝑖−2𝑦̇^𝑇𝑃_𝑖𝑦̇(𝑡)+2𝑦̇^𝑇𝐴_𝑖+𝐸_1,𝑖𝐹_1,𝑖(𝑡)𝐻_1,𝑖𝑃_𝑖𝑦(𝑡)+2𝑦̇^𝑇𝐷_𝑖+𝐸_2,𝑖𝐹_2,𝑖(𝑡)𝐻_2,𝑖𝑃_𝑖×𝑦𝑡−ℎ_𝑖(𝑡)−2𝑦̇^𝑇0.5𝐵_𝑖𝐵^𝑇_𝑖𝑦(𝑡)−𝑦̇^𝑇𝐸_3,𝑖𝐹_3,𝑖(𝑡)𝐻_3,𝑖𝐵^𝑇_𝑖𝑦(𝑡)−ℎ_1,𝑖2^𝑡_{𝑡−ℎ1,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)ℎ𝑑𝑠−^2,𝑖2^𝑡_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)−𝛿𝑑𝑠_𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝑑𝑠−ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠−ℎ_2,𝑖^𝑡_{𝑡−ℎ2,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠. And using Lemma 2.5, we can get the following upper bounds for the uncertain terms in (3.48): 2𝑦^𝑇(𝑡)𝐸_1,𝑖𝐹_1,𝑖(𝑡)𝐻_1,𝑖𝑃_𝑖𝑦(𝑡)≤𝜖_1,𝑖𝑦^𝑇(𝑡)𝐸^𝑇_1,𝑖𝐸_1,𝑖𝑦(𝑡)+𝜖⁻¹_1,𝑖𝑦^𝑇(𝑡)𝑃_𝑖𝐻^𝑇_1,𝑖𝐻_1,𝑖𝑃_𝑖𝑦(𝑡),2𝑦̇^𝑇(𝑡)𝐸_1,𝑖𝐹_1,𝑖(𝑡)𝐻_1,𝑖𝑃_𝑖𝑦(𝑡)≤𝜖_2,𝑖𝑦̇^𝑇(𝑡)𝐸^𝑇_1,𝑖𝐸_1,𝑖𝑦̇(𝑡)+𝜖⁻¹_2,𝑖𝑦^𝑇(𝑡)𝑃_𝑖𝐻^𝑇_1,𝑖𝐻_1,𝑖𝑃_𝑖𝑦(𝑡),2𝑦^𝑇(𝑡)𝐸_2,𝑖𝐹_2,𝑖(𝑡)𝐻_2,𝑖𝑃_𝑖𝑦𝑡−ℎ_𝑖(𝑡)≤𝜖_3,𝑖𝑦^𝑇(𝑡)𝐸^𝑇_2,𝑖𝐸_2,𝑖𝑦(𝑡)+𝜖⁻¹_3,𝑖𝑦^𝑇𝑡−ℎ_𝑖(𝑡)𝑃_𝑖𝐻^𝑇_2,𝑖𝐻_2,𝑖𝑃_𝑖𝑦𝑡−ℎ_𝑖(𝑡,2𝑦̇_𝑇)(𝑡)𝐸_2,𝑖𝐹_2,𝑖(𝑡)𝐻_2,𝑖𝑃_𝑖𝑦𝑡−ℎ_𝑖(𝑡)≤𝜖_4,𝑖𝑦̇^𝑇(𝑡)𝐸^𝑇_2,𝑖𝐸_2,𝑖𝑦̇(𝑡)+𝜖⁻¹_4,𝑖𝑦^𝑇𝑡−ℎ_𝑖(𝑡)𝑃_𝑖𝐻^𝑇_2,𝑖𝐻_2,𝑖𝑃_𝑖𝑦𝑡−ℎ_𝑖(𝑡),−𝑦^𝑇(𝑡)𝐸_3,𝑖𝐹_3,𝑖(𝑡)𝐻_3,𝑖𝐵^𝑇_𝑖𝑦(𝑡)≤𝜖_5,𝑖2𝑦^𝑇(𝑡)𝐸^𝑇_3,𝑖𝐸_3,𝑖𝑦(𝑡)+𝜖⁻¹_5,𝑖2𝑦^𝑇(𝑡)𝐵_𝑖𝐻^𝑇_3,𝑖𝐻_3,𝑖𝐵_𝑖𝑦(𝑡),−𝑦̇^𝑇(𝑡)𝐸_3,𝑖𝐹_3,𝑖(𝑡)𝐻_3,𝑖𝐵^𝑇_𝑖𝑦(𝑡)≤𝜖_6,𝑖2𝑦̇^𝑇(𝑡)𝐸^𝑇_3,𝑖𝐸_3,𝑖𝑦̇(𝑡)+𝜖⁻¹_6,𝑖2𝑦^𝑇(𝑡)𝐵_𝑖𝐻^𝑇_3,𝑖𝐻_3,𝑖𝐵_𝑖𝑦(𝑡). (3.49) According to (3.48)-(3.49), it follows that 𝑉̇_𝑖𝑥_𝑡≤𝜉^𝑇(𝑡)1−𝛽Ω_1,𝑖+𝛽Ω_2,𝑖𝜉(𝑡)+𝑦^𝑇(𝑡)Ψ_3,𝑖𝑦(𝑡)+𝑦^𝑇𝑡−ℎ_𝑖(𝑡)Ψ_4,𝑖𝑦𝑡−ℎ_𝑖(𝑡)−ℎ_1,𝑖2^𝑡_{𝑡−ℎ1,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)ℎ𝑑𝑠−^2,𝑖2^𝑡_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)−𝛿𝑑𝑠_𝑖2^{𝑡−ℎ1,𝑖}_{𝑡−ℎ2,𝑖}𝑦̇^𝑇(𝑠)𝑅_𝑖𝑦̇(𝑠)𝑑𝑠,−ℎ_1,𝑖^𝑡_{𝑡−ℎ1,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠−ℎ_2,𝑖^𝑡_{𝑡−ℎ2,𝑖}𝑥^𝑇(𝑠)𝑌_𝑖𝑄_𝑖𝑌_𝑖𝑥(𝑠)𝑑𝑠, (3.50) where Ψ_3,𝑖=Ω_311,𝑖+𝜀⁻¹_1,𝑖𝑃_𝑖𝐻^𝑇_1,𝑖𝐻_1,𝑖𝑃_𝑖+𝜖⁻¹_2,𝑖𝑃_𝑖𝐻^𝑇_1,𝑖𝐻_1,𝑖𝑃_𝑖+𝜖⁻¹_5,𝑖2𝐵_𝑖𝐻^𝑇_3,𝑖𝐻_3,𝑖𝐵^𝑇_𝑖+𝜖⁻¹_6,𝑖2𝐵_𝑖𝐻^𝑇_3,𝑖𝐻_3,𝑖𝐵^𝑇_𝑖,Ψ_4,𝑖=Ω_311,𝑖+𝜖⁻¹_3,𝑖𝑃_𝑖𝐻^𝑇_2,𝑖𝐻_2,𝑖𝑃_𝑖+𝜖⁻¹_4,𝑖𝑃_𝑖𝐻^𝑇_2,𝑖𝐻_2,𝑖𝑃_𝑖. (3.51) Applying Lemma 2.4, the LMIs Ψ_3,𝑖<0 and Ψ_4,𝑖<0 are equivalent to Ω_3,𝑖<0 and Ω_4,𝑖<0, respectively. Similarly to the previous theorem we can conclude that the switched system (2.1) is robustly stabilizable.

Remark 3.3. Our proposed method can remove the conservative restrictions on the derivatives of the time-varying delays, meanwhile traditional design methods require the condition ℎ̇(𝑡)≤ℎ_𝐷 (see [14, 19, 20]). So our method can deal with fast time-varying delays. Moreover, the improved method need not to introduce any free-weighting matrix variables which turn out to be less conservative than results in [14, 20, 21].

4. Numerical Examples

In this section, we provide some examples to illustrate the effectiveness of our results in Theorems 3.1 and 3.2.

Example 4.1. Consider the nominal switched systems with interval time-varying delay 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐷_𝑖𝑥𝑡−ℎ_𝑖(𝑡)+𝐵_𝑖𝑢_𝑖(𝑡),𝑖=1,2, (4.1) with 𝐴₁=−2000.5,𝐷₁=−0.20.10.1−0.1,𝐵₁=01,𝐴₂=−3100.1,𝐷₂=−0.10.10.1−0.1,𝐵₂=01. (4.2) And we choose 𝛼=0.9,ℎ₁(𝑡)=0.1+|sin𝑡|,ℎ₂(𝑡)=0.1+|cos𝑡|, so we get ℎ₁=0.1,ℎ₂=1.1. Then, by using the LMI control toolbox in Matlab, solutions of LMIs (3.2) are given by 𝑃₁=0.8273−0.0052−0.00520.8167,𝑄₁=0.24450.00400.00400.0665,𝑅₁=,𝑈0.12380.00300.00300.5581₁=0.2249−0.0084−0.00840.3873,𝑃₂=3.47351.24441.24444.2337,𝑄₂=,𝑅0.71320.15630.15630.1691₂=0.44290.97770.97772.9654,𝑈₂=,0.56840.42590.42591.8023 (4.3) with stabilizing controllers 𝑢₁(𝑡)=−0.0039−0.6122𝑥(𝑡),𝑢₂(𝑡)=0.0473−0.1320𝑥(𝑡). (4.4) By computation, we obtain 𝜆^∗=0.0461, 𝜆₁=1.2276, 𝜆₂=0.1940, 𝜆₃=5.1547, ℎ^∗=0.5. Thus, 𝜌=0.0054 and from (3.3) we obtain 𝑇₀≥361.1694. (4.5) Hence, from Theorem 3.1, the switched systems (4.1) with arbitrary switching law subject to (4.5) are stabilizable under the feedback controllers which are shown in (4.4).
By choosing initial condition as 𝜙(𝑡)=^sin(𝑡)_sin(𝑡),𝑡∈[−1.1,0], the trajectories of solutions of the switched system and the trajectories of solutions of subsystems 1 and 2 for this example are shown in Figures 1, 2, 3, and 4.
In Tables 1 and 2 we give comparison of maximum allowable value of ℎ₂ of asymptotic stability of nominal switched systems obtained in [14, 22] and Theorem 3.1. As we can see that for some of ℎ₁, the maximum allowable bounds for ℎ₂ obtained from Theorem 3.1 are greater than that obtained in [14, 22]. More important, the differentiability of the time delay ℎ(𝑡) is not required in our theorem.

Table 1 

Maximum allowable upper bounds ℎ₂ of the time-varying delay for different values of the lower bounds ℎ₁ in example 1 of  [8].

Table 2 

Maximum allowable upper bounds ℎ₂ of the time-varying delay for different values of the lower bounds ℎ₁ in example 2 of [4].

Figure 1 

The trajectory of solution of system 𝑖=1.

Figure 2 

The trajectory of solution of system 𝑖=2.

Figure 3 

The trajectory of solution of system i = 1 under the feedback controller (4.4).

Figure 4 

The trajectory of solution of system i = 2 under the feedback controller (4.4).

Example 4.2. Consider the following uncertain switched systems with interval time-varying delay 𝑥̇(𝑡)=𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑡)+𝐷_𝑖+Δ𝐷_𝑖𝑥𝑡−ℎ_𝑖(𝑡)+𝐵_𝑖+Δ𝐵_𝑖𝑢_𝑖(𝑡),𝑖=1,2, (4.6) with 𝐴₁=−2100.1,𝐷₁=0.10.10.1−0.1,𝐵₁=0,𝐴0.1₂=−3000.1,𝐷₂=−0.10.10.1−0.1,𝐵₂=0,0.1𝐸_1,𝑖=𝐸_2,𝑖=𝐸_3,𝑖=0.020.01−0.010.01,𝐻_1,𝑖=𝐻_2,𝑖=0.02000.01,𝐻_3,𝑖=00.01,𝑖=1,2. (4.7)(4.8) As an illustration, we choose 𝛼=0.9,𝜖_𝑗,𝑖=1 for 𝑗=1,2,…6,𝑖=1,2, ℎ(𝑡)=0.1+|sin𝑡|,ℎ₂(𝑡)=0.1+|cos𝑡|,𝐹_1,1(𝑡)=𝐹_1,2(𝑡)=𝐹_2,1(𝑡)=𝐹_2,2(𝑡)=.sin𝑡00sin𝑡 (4.9) In this case, we can take ℎ₁=0.1,ℎ_𝑀=1.1. Then, by using the LMI control toolbox in Matlab, solutions of LMIs (3.42) and (3.45) are given by 𝑃₁=0.04380.00460.00460.0081,𝑄₁=0.01530.00050.00050.0001,𝑅₁=,𝑈0.00370.00050.00050.0010₁=0.0122−0.0005−0.00050.0006,𝑃₂=0.0272−0.0010−0.00100.0088,𝑄₂=,𝑅0.00590.00010.00010.0001₂=0.0006−0.0002−0.00020.0022,𝑈₂=,0.0038−0.0010−0.00100.0012 (4.10) with stabilizing controllers 𝑢₁(𝑡)=0.6907−6.5520𝑥(𝑡),𝑢₂(𝑡)=−0.1997−5.6729𝑥(𝑡). (4.11) By computation, we obtain 𝜆^∗=0.000007,𝜆₁=132.7976,𝜆₂=22.5048,𝜆₃=0.0444,ℎ^∗=0.5 Thus, 𝜌=7.8829×10⁻⁵, and from (3.3) we obtain 𝑇₀≥2.3855×10⁴. (4.12) Hence, from Theorem 3.2, the switched systems (4.6) with arbitrary switching law subject to (4.12) are robustly stabilizable under the feedback controllers which are shown in (4.11).
By choosing the same initial condition as in Example 4.1, the trajectories of solutions of the switched system and the trajectories of solutions of subsystems 1 and 2 for this example are shown in Figures 5, 6, 7, and 8.

Figure 5 

The trajectory of solution of system 𝑖=1.

Figure 6 

The trajectory of solution of system 𝑖=2.

Figure 7 

The trajectory of solution of system 𝑖=1 under the feedback controller (4.12).

Figure 8 

The trajectory of solution of system 𝑖=2 under the feedback controller (4.12).

5. Conclusion

In this paper, we study the problem of robust stabilization for a class of switched systems with time-varying delay. Comparing with some existing results in the literature, the novelty of our results is twofold. Firstly, the state delay is time-varying in which the restriction on the derivative of the time-delay function is not required to design switching rule for the robust stability of the system. Secondly, the obtained conditions for the robust stability are delay-dependent and formulated in terms of the solution of standard LMIs which can be solved by various available algorithms. Numerical example is given to illustrate the effectiveness of the theoretical result.

Acknowledgments

The first author is supported by the Graduate School, Chiang Mai University and Thai Government Scholarships in the area of science and technology (Ministry of Science and Technology). The second author is supported by the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand.