Delay-Dependent Robust Exponential Stability and 𝐻_∞ Analysis for a Class of Uncertain Markovian Jumping System with Multiple Delays

Abstract

This paper deals with the problem of robust exponential stability and 𝐻_∞ performance analysis for a class of uncertain Markovian jumping system with multiple delays. Based on the reciprocally convex approach, some novel delay-dependent stability criteria for the addressed system are derived. At last, numerical examples is given presented to show the effectiveness of the proposed results.

1. Introduction

It is well known that time delay is usually the main reason for instability and poor performance of many practical control systems [1–5]. The stability results for delayed systems can be generally classified into two categories: delay-independent stability criteria and delay-dependent criteria. And the delay-dependent results are often less conservative than the delay-independent ones, especially when the time delays are small. Therefore, much more attention has been focused on study of the delay-dependent stability conditions in recent years. For example, the system transformation method in [6], the descriptor system method in [7], parameter-dependent Lyapunov-Krasovskii functional method in [8], Jensen inequality method in [9], Free-weighting matrix method in [10, 11], integral inequality method in [12], augmented Lyapunov functional method in [13], convex domain method in [14], interval partition method in [15, 16], reciprocally convex method in [17], and so forth. And those approaches have been widely used in the stability analysis for lots of delayed systems in recent years [18–20].

On the other hand, since Markovian jumping systems can model many types of dynamic systems subject to abrupt changes in their structures, such as failure prone manufacturing systems, power systems, and economics systems [21–27], a great deal of results related to stability analysis and synthesis for this class of systems with time delays has been reported in recent years. For example, for the delay-independent results, sufficient conditions for mean squares to stochastic stability were obtained in [28], while exponential stability conditions were proposed in [29]. The robust 𝐻_∞ filtering problem was dealt with in [30]. For the delay-dependent ones, the stability and 𝐻_∞ control results were presented by resorting to some bounding techniques for some cross terms and using model transformation to the original delay system in [31]. The 𝐻_∞ control and Filtering problem were taken into account in [32] using the Free-weighting matrix method. The stability and 𝐻_∞ analysis was proposed in [33] with the idea of delay partition. Filtering problem with a new index was considered in [34] using the reciprocally convex method. It is worth mentioned that inspite of the deep study for the delayed stochastic in recent years as mentioned above, there are few papers that consider the problem of stability analysis for uncertain stochastic systems with multiple delays, which motivates our study.

In this paper, the robust exponential stability and 𝐻_∞ performance analysis for a class of uncertain Markovian system with multiple time-varying delays is investigated. Some new delay-dependent stability conditions are derived. And numerical simulation is given to demonstrate the effectiveness of the result.

Notation. Throughout this paper, for symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite); 𝐼 is the identity matrix with appropriate dimension; 𝑀^𝑇 represents the transpose of the matrix 𝑀; ℰ{⋅} denotes the expectation operator with respect to some probability measure 𝒫; 𝐿₂[0,∞] is the space of square-integrable vector functions over [0,∞]; |⋅| refers to the Euclidean vector norm; ‖⋅‖₂ stands for the usual 𝐿₂[0,∞] norm; (Ω,ℱ,𝒫) is a probability space with Ω the sample space and ℱ is the 𝜎-algebra of subsets of the sample space. Matrices, if not explicitly mentioned, are assumed to have compatible dimensions.

2. System Description and Preliminaries

Consider the following uncertain Markovian jumping system with multiple time-varying delays: 𝑥̇(𝑡)=𝐴₀(𝑟(𝑡))𝑥(𝑡)+,𝑡_𝑚𝑖=1𝐴_𝑖(𝑟(𝑡),𝑡)𝑥𝑡−ℎ_𝑖(𝑡)+𝐷₁(𝑟(𝑡),𝑡)𝑤(𝑡),𝑧(𝑡)=^𝑚_𝑖=0𝐶_𝑖(𝑟(𝑡))𝑥𝑡−ℎ_𝑖(𝑡)+𝐷₂(𝑟(𝑡))𝑤(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡∈[−ℎ,0], (1)(2)(3) where 𝑥(𝑡)∈𝑅^𝑛 is the state; 𝑤(𝑡)∈𝑅^𝑝 is the noise disturbance which is assumed to be an arbitrary signal in 𝐿²([0,∞]); 𝑧(𝑡)∈𝑅^𝑞 is the signal to be estimated; 𝑟(𝑡) is a homogenous stationary Markov chain defined on a complete probability space {Ω,𝐹,𝑃} and taking values in a finite set 𝑆={1,2,…,𝑁} with generator Π=(𝜆_𝑚,𝑛)(𝑚,𝑛∈𝑆) given by 𝑃𝑟()(𝑡)=𝜆𝑡+Δ=𝑗∣𝑟=𝑘_𝑘,𝑗Δ+𝑜(Δ)if𝑘≠𝑗,1+𝜆_𝑘,𝑘Δ+𝑜(Δ)if𝑘=𝑗, (4) where Δ>0 and lim_Δ→0𝑜(Δ)/Δ=0, 𝜆_𝑘,𝑗≥0 is the transition rate from 𝑘 to 𝑗 if 𝑘≠𝑗 and 𝜆_𝑘,𝑘=−∑_𝑘≠𝑗𝜆_𝑘,𝑗. The scalar ℎ_𝑖(𝑡) is the time-varying delay with 0≤ℎ_1𝑖≤ℎ_𝑖(𝑡)≤ℎ_2𝑖, ℎ̇_𝑖(𝑡)≤𝜇, 𝑖=1,2,…,𝑚, for any 𝑡>0, where ℎ_1𝑖,ℎ_2𝑖, and 𝜇 are positive scalar constants; 𝜙(𝑡) is the initial function defined in 𝑡∈[−ℎ,0] with ℎ=max{ℎ₂₁,ℎ₂₂,…,ℎ_2𝑚}; 𝐴_𝑖(𝑟(𝑡)), 𝑖=0,1,…,𝑚, and 𝐷₁(𝑟(𝑡)) are matrix functions with time-varying uncertainties described as 𝐴_𝑖(𝑟(𝑡),𝑡)=𝐴_𝑖(𝑟(𝑡))+Δ𝐴_𝑖(𝑟(𝑡),𝑡), 𝐷₁(𝑟(𝑡),𝑡)=𝐷₁(𝑟(𝑡))+Δ𝐷₁(𝑟(𝑡),𝑡), where 𝐴_𝑖(𝑟(𝑡)),𝐷₁(𝑟(𝑡)) are known constant matrices, while uncertainties Δ𝐴_𝑖(𝑟(𝑡),𝑡), Δ𝐷₁(𝑟(𝑡),𝑡) are assumed to be norm bounded asΔ𝐴_𝑖(𝑟(𝑡),𝑡)Δ𝐷₁(𝑟(𝑡))(𝑟(𝑡𝐹(𝑟(𝑡)𝐻,𝑡=𝐸)),𝑡)_𝑖(𝑟(𝑡))𝐻_𝑑(𝑟(𝑡)),𝑖=0,1,…,𝑚, (5) where 𝐸(𝑟(𝑡)), 𝐻_𝑖(𝑟(𝑡)), 𝐻_𝑑(𝑟(𝑡)), and 𝐶_𝑖(𝑟(𝑡)), 𝐷₂(𝑟(𝑡)) in (2) are known constant matrices with appropriate dimensions. The unknown matrix functions 𝐹(𝑟(𝑡)) are having Lebesgue-measurable elements and satisfying 𝐹(𝑟(𝑡))≤𝐼,∀𝑡>0. (6)

Remark 1. When 𝑚=1, the system with multiple time-varying delays (1)–(3) is actually deduced to the uncertain Markovian jumping system with interval delay, which have been deeply studied in recent years. That is, the obtained results of multiple delayed systems can be directly deduced to the interval delayed systems.

Throughout this paper, we will use the following Definitions and Lemmas.

Definition 2. The uncertain Markovian jumping system with multiple time-varying delays (1)–(3) is said to be robustly exponentially stable in mean square for all admissible uncertainties, if there exist scalars 𝛼₁>0 and 𝛼₂>0 such that for all 𝑡≥0, 𝑥(𝑡,𝑥₀,𝑡₀)²≤𝛼₁𝑒^−𝛼2𝑡sup_{−2ℎ≤𝑠≤0}𝜙(𝑠)², (7) where 𝑥(𝑡,𝑥₀,𝑡₀) is the trivial solution of systems (1)–(3) with 𝑤(𝑡)=0.

Definition 3. Given a scalar 𝛾>0, uncertain Markovian jumping system with multiple time-varying delays (1)–(3) is said to be robustly exponentially stable with a prescribed 𝐻_∞ performance level 𝛾 if it is robustly exponentially stable, and under the zero initial condition, satisfies ‖𝑧‖_𝐸2<𝛾‖𝑤‖₂, (8) for all admissible uncertainties and nonzero 𝑤(𝑡)∈𝐿₂[0,∞), where ‖𝑧‖_𝐸2=^∞₀𝑧(𝑡)²𝑑𝑡^1/2. (9)

Lemma 4 (see [9]). For any constant matrix 𝑀∈𝑅^𝑚×𝑚,  𝑀=𝑀^𝑇>0, scalar 𝛾>0, vector function 𝜔∶[0,𝛾]→𝑅^𝑚 such that the integrations in the following are well defined; then 𝛾^𝛾₀𝜔𝛽^𝑇𝑀𝜔𝛽𝑑𝛽≥^𝛾₀𝜔𝛽𝑑𝛽^𝑇𝑀^𝛾₀𝜔𝛽𝑑𝛽. (10)

Lemma 5 (see [35]). Let 𝐴, 𝐷, 𝐸 be real constant matrices with appropriate dimensions; matrix 𝐹(𝑡) satisfies 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼. For any 𝜀>0, such that 𝑃⁻¹−𝜀𝐷𝐷^𝑇>0, 𝐷𝐹(𝑡)𝐸+𝐸^𝑇𝐹^𝑇(𝑡)𝐷^𝑇≤𝜀⁻¹𝐷𝐷^𝑇+𝜀𝐸^𝑇𝐸. (11)

Lemma 6 (see [36]). Consider system (1) with 0≤ℎ_1𝑖≤ℎ_𝑖(𝑡)≤ℎ_2𝑖, 𝑖=1,2,…,𝑚, for any matrices 𝑍_𝑖∈𝑅^𝑛×𝑛 and 𝑈_𝑖∈𝑅^𝑛×𝑛 satisfying ^{𝑍𝑖𝑈𝑖}_∗𝑍𝑖≥0; the following inequality holds −𝑑_𝑖^{𝑡−ℎ1𝑖}_{𝑡−ℎ2𝑖}𝑥̇(𝑠)^𝑇𝑍_𝑖𝑥̇(𝑠)𝑑𝑠≤𝜉_𝑖(𝑡)^𝑇Ω_𝑖𝜉_𝑖(𝑡), (12) where 𝑑_𝑖=ℎ_2𝑖−ℎ_1𝑖, and 𝜉_𝑖(𝑡)=𝑥𝑡−ℎ_1𝑖^𝑇𝑥𝑡−ℎ_𝑖(𝑡)^𝑇𝑥𝑡−ℎ_2𝑖^𝑇𝑇,Ω_𝑖=−𝑍_𝑖𝑍_𝑖−𝑈_𝑖𝑈_𝑖∗−2𝑍_𝑖+𝑈_𝑖+𝑈^𝑇_𝑖𝑍_𝑖−𝑈_𝑖∗∗−𝑍_𝑖. (13)

3. Main Results

For simplicity, we define𝜒(𝑡)=𝑥(𝑡)^𝑇𝑥𝑡−ℎ₁₁^𝑇𝑥𝑡−ℎ₁(𝑡)^𝑇𝑥𝑡−ℎ₂₁^𝑇⋯𝑥𝑡−ℎ_1𝑚^𝑇𝑥𝑡−ℎ_𝑚(𝑡)^𝑇𝑥(𝑡−ℎ_2𝑚)^𝑇𝑥̇^𝑇𝑇,𝑒₁=10⋯0^𝑇_1×3𝑚+2,𝑒₂=01⋯0^𝑇_1×3𝑚+2,⋮𝑒_3𝑚+2=00⋯1^𝑇_1×3𝑚+2,𝛽₁=10⋯0^𝑇_1×3𝑚+3,𝛽₂=01⋯0^𝑇_1×3𝑚+3,⋮𝛽_3𝑚+3=00⋯1^𝑇_1×3𝑚+3,𝜒₁(𝑡)=𝜒(𝑡)^𝑇𝑤(𝑡)^𝑇𝑇. (𝑡)(14)

3.1. Robust Exponential Stability Analysis

The criteria of the robust exponential stability for the systems (1)–(3) are proposed in the following Theorem.

Theorem 7. Systems (1)–(3) with 𝑤(𝑡)=0 is robustly exponentially stable if there exist positive matrices 𝑃_𝑘=𝑃^𝑇_𝑘>0, 𝑄_𝑖=𝑄^𝑇_𝑖>0, 𝑅_1𝑖=𝑅^𝑇_1𝑖>0, 𝑅_2𝑖=𝑅^𝑇_2𝑖>0, 𝑆_𝑖=𝑆^𝑇_𝑖>0, 𝑍_𝑖=𝑍^𝑇_𝑖>0, any matrices 𝑈_𝑖𝑀_𝑗 with appropriate dimensions satisfying ^{𝑍𝑖𝑈𝑖}_∗𝑍𝑖≥0,  𝑖=1,2,…,𝑚;  𝑗=1,2,  𝑘=1,2,…,𝑁, and positive scalars 𝜀>0, such that the following LMI holds Θ_11𝑘Θ_12𝑘∗−𝜀𝐼<0,𝑘=1,2,…,𝑁, (15) where Θ_11𝑘=𝑒₁𝑃_𝑘𝑒^𝑇_3𝑚+2+𝑒_3𝑚+2𝑃_𝑘𝑒^𝑇₁+^𝑁_𝑗=1𝑒₁𝜆_𝑘,𝑗𝑃_𝑗𝑒^𝑇₁+𝑒₁^𝑚_𝑖=1𝑄_𝑖𝑒^𝑇₁−^𝑚_𝑖=11−𝜇𝑒_3𝑖𝑄_𝑖𝑒^𝑇_3𝑖+𝑒₁^𝑚_𝑖=1𝑅_1𝑖+𝑅_2𝑖𝑒^𝑇₁−^𝑚_𝑖=1𝑒_3𝑖−1𝑅_1𝑖𝑒^𝑇_3𝑖−1−^𝑚_𝑖=1𝑒_3𝑖+1𝑅_2𝑖𝑒^𝑇_3𝑖+1+^𝑚_𝑖=1ℎ²_1𝑖𝑒_3𝑚+2𝑆_𝑖𝑒^𝑇_3𝑚+2+^𝑚_𝑖=1𝑑²_𝑖𝑒_3𝑚+2𝑍_𝑖𝑒^𝑇_3𝑚+2−^𝑚_𝑖=1𝑒₁−𝑒_3𝑖−1𝑆_𝑖𝑒₁−𝑒_3𝑖−1^𝑇+^𝑚_𝑖=1diag0,Ω_𝑖+𝑒,0₁𝑀₁+𝑒_3𝑚+2𝑀₂⋅𝐴₀(𝑘)𝑒^𝑇₁+𝐴₁(𝑘)𝑒^𝑇₃+⋯+𝐴_𝑚(𝑘)𝑒^𝑇_3𝑚−𝑒^𝑇_3𝑚+2+𝑒₁𝐴^𝑇₀(𝑘)+𝑒₃𝐴^𝑇₁(𝑘)+⋯+𝑒_3𝑚𝐴^𝑇_𝑚(𝑘)−𝑒_3𝑚+2×𝑀^𝑇₁𝑒^𝑇₁+𝑀^𝑇₂𝑒^𝑇_3𝑚+2+𝜀𝐻₀(𝑘)𝑒^𝑇₁+𝐻₁(𝑘)𝑒^𝑇₃+⋯+𝐻_𝑚(𝑘)𝑒^𝑇_3𝑚^𝑇×𝐻₀(𝑘)𝑒^𝑇₁+𝐻₁(𝑘)𝑒^𝑇₃+⋯+𝐻_𝑚(𝑘)𝑒^𝑇_3𝑚,Θ_12𝑘=𝑒₁𝑀₁+𝑒_3𝑚+2𝑀₂𝐸(𝑘). (16) And Ω_𝑖𝑑_𝑖 are defined in (3).

Proof. On one hand, using Lemma 5 and Schur complement lemma to (15), we have Π=𝑒₁𝑃_𝑘𝑒^𝑇_3𝑚+2+𝑒_3𝑚+2𝑃_𝑘𝑒^𝑇₁+^𝑁_𝑗=1𝑒₁𝜆_𝑘,𝑗𝑃_𝑗𝑒^𝑇₁+𝑒₁^𝑚_𝑖=1𝑄_𝑖𝑒^𝑇₁−^𝑚_𝑖=11−𝜇𝑒_3𝑖𝑄_𝑖𝑒^𝑇_3𝑖+𝑒₁^𝑚_𝑖=1𝑅_1𝑖+𝑅_2𝑖𝑒^𝑇₁−^𝑚_𝑖=1𝑒_3𝑖−1𝑅_1𝑖𝑒^𝑇_3𝑖−1−^𝑚_𝑖=1𝑒_3𝑖+1𝑅_2𝑖𝑒^𝑇_3𝑖+1+^𝑚_𝑖=1ℎ²_1𝑖𝑒_3𝑚+2𝑆_𝑖𝑒^𝑇_3𝑚+2+^𝑚_𝑖=1𝑑²_𝑖𝑒_3𝑚+2𝑍_𝑖𝑒^𝑇_3𝑚+2−^𝑚_𝑖=1𝑒₁−𝑒_3𝑖−1𝑆_𝑖𝑒₁−𝑒_3𝑖−1^𝑇+^𝑚_𝑖=1diag0,Ω_𝑖,0+𝑒₁𝑀₁+𝑒_3𝑚+2𝑀₂×𝐴₀(𝑘,𝑡)𝑒^𝑇₁+𝐴₁(𝑘,𝑡)𝑒^𝑇₃+⋯+𝐴_𝑚(𝑘,𝑡)𝑒^𝑇_3𝑚−𝑒^𝑇_3𝑚+2+𝐴₀(𝑘)𝑒^𝑇₁+𝐴₁(𝑘)𝑒^𝑇₃+⋯+𝐴_𝑚(𝑘)𝑒^𝑇_3𝑚−𝑒^𝑇_3𝑚+2^𝑇×𝑒₁𝑀₁+𝑒_3𝑚+2𝑀₂^𝑇≤𝑒₁𝑃𝑒^𝑇_3𝑚+2+𝑒_3𝑚+2𝑃𝑒^𝑇₁+^𝑁_𝑗=1𝑒₁𝑃_𝑘𝑒^𝑇₁+𝑒₁^𝑚_𝑖=1𝑄_𝑖𝑒^𝑇₁−^𝑚_𝑖=11−𝜇𝑒_3𝑖𝑄_𝑖𝑒^𝑇_3𝑖+𝑒₁^𝑚_𝑖=1𝑅_1𝑖+𝑅_2𝑖𝑒^𝑇₁−^𝑚_𝑖=1𝑒_3𝑖−1𝑅_1𝑖𝑒^𝑇_3𝑖−1−^𝑚_𝑖=1𝑒_3𝑖+1𝑅_2𝑖𝑒^𝑇_3𝑖+1+^𝑚_𝑖=1ℎ²_1𝑖𝑒_3𝑚+2𝑆_𝑖𝑒^𝑇_3𝑚+2+^𝑚_𝑖=1𝑑²_𝑖𝑒_3𝑚+2𝑍_𝑖𝑒^𝑇_3𝑚+2−^𝑚_𝑖=1𝑒₁−𝑒_3𝑖−1𝑆_𝑖𝑒₁−𝑒_3𝑖−1^𝑇+^𝑚_𝑖=1diag0,Ω_𝑖,0+𝑒₁𝑀₁+𝑒_3𝑚+2𝑀₂⋅𝐴₀(𝑘)𝑒^𝑇₁+𝐴₁(𝑘)𝑒^𝑇₃+⋯+𝐴_𝑚(𝑘)𝑒^𝑇_3𝑚−𝑒^𝑇_3𝑚+2+𝑒₁𝐴^𝑇₀(𝑘)+𝑒₃𝐴^𝑇₁(𝑘)+⋯+𝑒_3𝑚𝐴^𝑇_𝑚(𝑘)−𝑒_3𝑚+2×𝑀^𝑇₁𝑒^𝑇₁+𝑀^𝑇₂𝑒^𝑇_3𝑚+2+𝜀𝐻₀(𝑘)𝑒^𝑇₁+𝐻₁(𝑘)𝑒^𝑇₃+⋯+𝐻_𝑚(𝑘)𝑒^𝑇_3𝑚^𝑇×𝐻₀(𝑘)𝑒^𝑇₁+𝐻₁(𝑘)𝑒^𝑇₃+⋯+𝐻_𝑚(𝑘)𝑒^𝑇_3𝑚+𝜀⁻¹𝑒₁𝑀₁+𝑒_3𝑚+2𝑀₂𝐸(𝑘)𝐸(𝑘)^𝑇×𝑒₁𝑀₁+𝑒_3𝑚+2𝑀₂^𝑇=Θ₁₁+𝜀⁻¹Θ^𝑇₁₂Θ₁₂<0. (17)
On the other hand, define a new process 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠), 𝑠∈[−2ℎ,0]. Choose a Lyapunov-Krasovskii functional 𝑉𝑥_𝑡,𝑡,𝑟(𝑡)=⁵_𝑖=1𝑉_𝑖𝑥_𝑡,𝑡,𝑟(𝑡), (18) where 𝑉₁𝑥_𝑡,𝑡,𝑟(𝑡)=𝑥(𝑡)^𝑇𝑃_𝑟(𝑡)𝑥(𝑡),𝑉₂𝑥_𝑡,𝑡,𝑟(𝑡)=^𝑚_𝑖=1^𝑡_{𝑡−ℎ𝑖(𝑡)}𝑥(𝑠)^𝑇𝑄_𝑖𝑥(𝑠)𝑉𝑑𝑠,₃𝑥_𝑡,𝑡,𝑟(𝑡)=^𝑚_𝑖=1^𝑡_{𝑡−ℎ1𝑖}𝑥(𝑠)^𝑇𝑅_1𝑖𝑥(𝑠)+𝑑𝑠_{𝑚𝑖=1𝑡𝑡−ℎ2𝑖}𝑥(𝑠)^𝑇𝑅_2𝑖𝑥(𝑠)𝑉𝑑𝑠,₄𝑥_𝑡,𝑡,𝑟(𝑡)=^𝑚_𝑖=1⁰_−ℎ1𝑖^𝑡_𝑡+𝑠ℎ_1𝑖𝑥̇(𝛼)^𝑇𝑆_𝑖𝑥̇(𝛼)𝑉𝑑𝛼𝑑𝑠,₅𝑥_𝑡,𝑡,𝑟(𝑡)=^𝑚_𝑖=1^−ℎ1𝑖_−ℎ2𝑖^𝑡_𝑡+𝑠𝑑_𝑖𝑥̇(𝛼)^𝑇𝑍_𝑖𝑥̇(𝛼)𝑑𝛼𝑑𝑠. (19)
Let ℒ be the weak infinitesimal generator of the random process {𝑥_𝑡,𝑡≥0}. Then for each 𝑟(𝑡)=𝑘,  𝑘∈𝑆, we have ℒ𝑉₁𝑥_𝑡(𝑡)𝑃,𝑡,𝑘=2𝑥_𝑘𝑥̇(𝑡)+^𝑁_𝑗=1𝑥(𝑡)^𝑇𝜆_𝑘,𝑗𝑃_𝑗𝑥(𝑡)=2𝜒(𝑡)^𝑇𝑒₁𝑃_𝑘𝑒^𝑇_3𝑚+2𝜒(𝑡)+𝜒(𝑡)^𝑇𝑁_𝑗=1𝑒₁𝜆_𝑘,𝑗𝑃_𝑗𝑒^𝑇₁𝜒(𝑡),ℒ𝑉₂𝑥_𝑡(𝑡),𝑡,𝑘≤𝑥^𝑇𝑚_𝑖=1𝑄_𝑖𝑥(𝑡)−1−𝜇^𝑚_𝑖=1𝑥𝑡−ℎ_𝑖(𝑡)^𝑇𝑄_𝑖𝑥𝑡−ℎ_𝑖(𝑡(𝑡))=𝜒^𝑇𝑒₁^𝑚_𝑖=1𝑄_𝑖𝑒^𝑇₁𝜒(𝑡)−𝜒(𝑡)^𝑇𝑚_𝑖=11−𝜇𝑒_3𝑖𝑄_𝑖𝑒^𝑇_3𝑖𝜒(𝑡),ℒ𝑉₃𝑥_𝑡(𝑡),𝑡,𝑘≤𝑥^𝑇𝑚_𝑖=1𝑅_1𝑖+𝑅_2𝑖𝑥(𝑡)−^𝑚_𝑖=1𝑥𝑡−ℎ_1𝑖^𝑇𝑅_1𝑖𝑥𝑡−ℎ_1𝑖−𝑥𝑡−ℎ_2𝑖^𝑇𝑅_2𝑖𝑥𝑡−ℎ_2𝑖=𝜒(𝑡)^𝑇𝑒₁^𝑚_𝑖=1𝑅_1𝑖+𝑅_2𝑖𝑒^𝑇₁𝜒(𝑡)−𝜒(𝑡)^𝑇𝑚_𝑖=1𝑒_3𝑖−1𝑅_1𝑖𝑒^𝑇_3𝑖−1𝜒(𝑡)−𝜒(𝑡)^𝑇𝑚_𝑖=1𝑒_3𝑖+1𝑅_2𝑖𝑒^𝑇_3𝑖+1𝜒(𝑡),ℒ𝑉₄𝑥_𝑡=,𝑡,𝑘_𝑚𝑖=1ℎ²_1𝑖𝑥̇(𝑡)^𝑇𝑆_𝑖𝑥̇(𝑡)−^𝑚_𝑖=1^𝑡_{𝑡−ℎ1𝑖}ℎ_1𝑖𝑥̇(𝑠)^𝑇𝑆_𝑖𝑥̇(𝑠)(𝑡)𝑑𝑠=𝜒^𝑇𝑚_𝑖=1ℎ²_1𝑖𝑒_3𝑚+2𝑆_𝑖𝑒^𝑇_3𝑚+2𝜒(𝑡)−^𝑚_𝑖=1^𝑡_{𝑡−ℎ1𝑖}ℎ_1𝑖𝑥̇(𝑠)^𝑇𝑆_𝑖𝑥̇(𝑠)𝑑𝑠,ℒ𝑉₅𝑥_𝑡=,𝑡,𝑘_𝑚𝑖=1𝑑²_𝑖𝑥̇(𝑡)^𝑇𝑍_𝑖𝑥̇(𝑡)−^𝑚_𝑖=1^{𝑡−ℎ1𝑖}_{𝑡−ℎ2𝑖}𝑑_𝑖𝑥̇(𝑠)^𝑇𝑍_𝑖𝑥̇(𝑠)(𝑡)𝑑𝑠=𝜒^𝑇𝑚_𝑖=1𝑑²_𝑖𝑒_3𝑚+2𝑍_𝑖𝑒^𝑇_3𝑚+2𝜒(𝑡)−^𝑚_𝑖=1^{𝑡−ℎ1𝑖}_{𝑡−ℎ2𝑖}𝑑_𝑖𝑥̇(𝑠)^𝑇𝑍_𝑖𝑥̇(𝑠)𝑑𝑠. (20)
Applying Lemma 4 to ℒ𝑉₄(𝑥_𝑡) results in −^𝑚_𝑖=1^𝑡_{𝑡−ℎ1𝑖}ℎ_1𝑖𝑥̇(𝑠)^𝑇𝑆_𝑖𝑥̇(𝑠)𝑑𝑠≤−^𝑚_𝑖=1^𝑡_{𝑡−ℎ1𝑖}𝑥̇(𝑠)𝑑𝑠^𝑇𝑆_𝑖^𝑡_{𝑡−ℎ1𝑖}𝑥̇(𝑠)𝑑𝑠=−^𝑚_𝑖=1𝑥(𝑡)−𝑥𝑡−ℎ_1𝑖^𝑇𝑆_𝑖𝑥(𝑡)−𝑥𝑡−ℎ_1𝑖≤−𝜒(𝑡)^𝑇𝑚_𝑖=1𝑒₁−𝑒_3𝑖−1𝑆_𝑖𝑒₁−𝑒_3𝑖−1^𝑇𝜒(𝑡), (21) and applying Lemma 6 to ℒ𝑉₅(𝑥_𝑡,𝑡,𝑘), we have that there exists 𝑈_𝑖 with ^{𝑍𝑖𝑈𝑖}_∗𝑍𝑖≥0, 𝑖=1,2,…,𝑚, such that −^𝑚_𝑖=1^{𝑡−ℎ1𝑖}_{𝑡−ℎ2𝑖}𝑑_𝑖𝑓(𝑠)^𝑇𝑍_𝑖𝑓(𝑠)≤𝑑𝑠_𝑚𝑖=1𝜉_𝑖(𝑡)^𝑇Ω_𝑖𝜉_𝑖(𝑡)=𝜒(𝑡)^𝑇𝑚_𝑖=1diag0,Ω_𝑖,0𝜒(𝑡), (22) where 𝜉_𝑖(𝑡) and Ω_𝑖 are defined in (13). Meanwhile, we note that 2𝑥(𝑡)^𝑇𝑀₁+𝑥̇(𝑡)^𝑇𝑀₂×𝐴₀(𝑘,𝑡)𝑥(𝑡)+^𝑚_𝑖=1𝐴_𝑖(𝑘,𝑡)𝑥𝑡−ℎ_𝑖(𝑡)−𝑥̇(𝑡)=0, (23) that is, 2𝜒(𝑡)^𝑇𝑒₁𝑀₁+𝑒_3𝑚+2𝑀₂×𝐴₀(𝑘,𝑡)𝑒^𝑇₁+𝐴₁(𝑘,𝑡)𝑒^𝑇₃+⋯+𝐴_𝑚(𝑘,𝑡)𝑒^𝑇_3𝑚−𝑒^𝑇_3𝑚+2×𝜒(𝑡)=0. (24)
Then, we can deduce from (19)–(24) that ℒ𝑉𝑥_𝑡,𝑡,𝑘≤𝜒(𝑡)^𝑇Π𝜒(𝑡)<0, (25) where Π is defined in (17). Therefore, by Definition 2 and the results in [37], we have that the system (1) is robustly stable. Now, we will prove the robust stochastic exponential stability in mean square for system (1). Setting 𝜆₀=𝜆_min{−Π}>0, we have ℒ𝑉𝑥_𝑡,𝑡,𝑘≤𝜒(𝑡)^𝑇Π𝜒(𝑡)≤−𝜆₀‖𝑥(𝑡)‖². (26) Choose 𝑉(𝑥_𝑡,𝑡,𝑘)=𝑒^2𝛼𝑡𝑉(𝑥_𝑡,𝑡,𝑘), where 𝛼>0; then ℒ𝑉𝑥_𝑡,𝑡,𝑘=2𝑘𝑒^2𝛼𝑡𝑉𝑥_𝑡,𝑡,𝑘+𝑒^2𝛼𝑡ℒ𝑥_𝑡,𝑡,𝑘≤2𝑘𝑒^2𝛼𝑡𝑉𝑥_𝑡,𝑡,𝑘−𝜆₀𝑒^2𝛼𝑡‖𝑥(𝑡)‖². (27) Integrating the above inequality (27), we get 𝑉𝑥_𝑡𝑥,𝑡,𝑘≤𝑉₀+,0,𝑘_𝑡02𝑘𝑒^2𝛼𝑠𝑉𝑥_𝑠,𝑠,𝑘−𝜆₀𝑒^2𝛼𝑠‖𝑥(𝑠)‖²𝑑𝑠. (28)
From (19), it can be inferred that 𝑉𝑥_𝑠,𝑠,𝑘≤𝜆_max𝑃_𝑘‖𝑥(𝑠)‖²+^𝑚_𝑖=1𝜆_max𝑄_𝑖+𝜆_max𝑅_1𝑖+𝜆_max𝑅_2𝑖×^𝑠_𝑠−ℎ‖𝑥(𝑣)‖²+𝑑𝑣_𝑚𝑖=1ℎ_1𝑖𝜆_max𝑆_𝑖+𝑑_𝑖𝜆_max𝑍_𝑖^𝑠_𝑠−ℎ𝑥̇(𝑣)^𝑇𝑥̇(𝑣)𝑑𝑣. (29) Note that 𝑥̇(𝑣)^𝑇𝑥̇(𝑣)≤𝑚𝜆_max𝐴^𝑇₀(𝑘,𝑣)𝐴₀(𝑘,𝑣)‖𝑥(𝑣)‖²+𝜆_max𝐴^𝑇₁(𝑘,𝑣)𝐴₁(𝑣)𝑥𝑣−ℎ₁(𝑣)²+𝜆_max𝐴^𝑇_𝑚(𝑘,𝑣)𝐴_𝑚(𝑣)𝑥𝑣−ℎ_𝑚(𝑣)². (30) We denote 𝜚=𝑚[𝜆_max(𝐴^𝑇₀(𝑘,𝑣)𝐴₀(𝑘,𝑣))+𝜆_max(𝐴^𝑇₁(𝑘,𝑣)𝐴₁(𝑘,𝑣))+𝜆_max(𝐴^𝑇_𝑚(𝑘,𝑣)𝐴_𝑚(𝑘,𝑣))]; then 𝑥̇^𝑠_𝑠−ℎ(𝑣)^𝑇𝑥̇(𝑣)𝑑𝑣≤𝜚^𝑠_𝑠−2ℎ‖𝑥(𝑣)‖²𝑑𝑣. (31)
From (29) to (31), we obtain 𝑉𝑥_𝑠,𝑠,𝑘≤Ξ₀‖𝑥(𝑠)‖²+Ξ₁^𝑠_𝑠−2ℎ‖𝑥(𝑣)‖²𝑑𝑣, (32) where Ξ₀=𝜆_max𝑃_𝑘,Ξ₁=^𝑚_𝑖=1𝜆_max𝑄_𝑖+𝜆_max𝑅_1𝑖+𝜆_max𝑅_2𝑖+𝜚^𝑚_𝑖=1ℎ_1𝑖𝜆_max𝑆_𝑖+𝑑_𝑖𝜆_max𝑍_𝑖. (33)
By the similar method, we have 𝑉𝑥₀,0,𝑘≤𝜃sup_{−2ℎ≤𝑠≤0}𝜙(𝑠)², (34) where 𝜃=2ℎΞ₁. Therefore, by (28)–(34), we get 𝑉𝑥_𝑡,𝑡,𝑘≤𝜃sup_{−2ℎ≤𝑠≤0}𝜙(𝑠)²+^𝑡₀2𝛼𝑒^2𝛼𝑠Ξ₀‖𝑥(𝑠)‖²+Ξ₁^𝑠_𝑠−2ℎ‖𝑥(𝑣)‖²𝑑𝑣−𝜆₀𝑒^2𝛼𝑠‖𝑥(𝑠)‖²𝑑𝑠≤𝜃sup_{−2ℎ≤𝑠≤0}𝜙(𝑠)²+2𝛼Ξ₀−𝜆₀^𝑡₀𝑒^2𝛼𝑠‖𝑥(𝑠)‖²𝑑𝑠+𝑒^2𝛼𝑠Ξ₁^𝑡_−2ℎ‖𝑥(𝑣)‖²≤𝑑𝑣𝜃+2ℎ𝑒^2𝛼𝑡Ξ₁sup_{−2ℎ≤𝑠≤0}𝜙(𝑠)²+𝑒^2𝛼𝑡Ξ₁+2𝛼Ξ₀−𝜆₀^𝑡₀‖𝑥(𝑣)‖²𝑑𝑣. (35)
Choose 𝛼₀>0 such that 𝑒^2𝛼0𝑡Ξ₁+2𝛼₀Ξ₀−𝜆₀≤0; (36) then 𝑉𝑥_𝑡,𝑡,𝑘≤𝜃+2ℎ𝑒^2𝛼0𝑡Ξ₁sup_{−2ℎ≤𝑠≤0}𝜙(𝑠)². (37)
Since 𝑉(𝑥_𝑡,𝑡,𝑘)≥𝑒^2𝛼0𝑡𝜆_min(𝑃_𝑘)‖𝑥(𝑡)‖², it can be shown from (37) that ‖𝑥(𝑡)‖²≤𝛿𝑒^{−2𝛼0𝑡}sup_{−2ℎ≤𝑠≤0}𝜙(𝑠)², (38) where 𝛿=𝜃+2ℎ𝑒^2𝛼0𝑡Ξ₁𝜆_min𝑃_𝑘, (39) which implies that system (10) is robustly exponentially stable by Definition 2. This completes the proof.

3.2. Robust 𝐻_∞ Exponential Stability Analysis

The criteria of the robust exponential stability with 𝐻_∞ performance for the systems (1)–(3) are proposed in the following Theorem.

Theorem 8. Given a scalar 𝛾>0, the systems (1)–(3) are robustly exponentially stable with a prescribed 𝐻_∞ performance level 𝛾 if there exist matrices 𝑃_𝑘=𝑃^𝑇_𝑘>0, 𝑄_𝑖=𝑄^𝑇_𝑖>0, 𝑅_1𝑖=𝑅^𝑇_1𝑖>0, 𝑅_2𝑖=𝑅^𝑇_2𝑖>0, 𝑆_𝑖=𝑆^𝑇_𝑖>0, 𝑍_𝑖=𝑍^𝑇_𝑖>0, any matrices 𝑈_𝑖𝑀_𝑗 with appropriate dimensions satisfying ^{𝑍𝑖𝑈𝑖}_∗𝑍𝑖≥0, 𝑖=1,2,…,𝑚; 𝑗=1,2,  𝑘=1,2,…,𝑁, and positive scalars 𝜀>0, such that the following LMI holds Ω_11𝑘Ω_12𝑘Ω_13𝑘∗−𝜀𝐼0∗∗−𝐼<0, (40) where Ω_11𝑘=𝛽₁𝑃_𝑘𝛽^𝑇_3𝑚+2+𝛽_3𝑚+2𝑃_𝑘𝛽^𝑇₁+^𝑁_𝑗=1𝛽₁𝜆_𝑘,𝑗𝑃_𝑗𝛽^𝑇₁+𝛽₁^𝑚_𝑖=1𝑄_𝑖𝛽^𝑇₁−^𝑚_𝑖=11−𝜇𝛽_3𝑖𝑄_𝑖𝛽^𝑇_3𝑖+𝛽₁^𝑚_𝑖=1𝑅_1𝑖+𝑅_2𝑖𝛽^𝑇₁−^𝑚_𝑖=1𝛽_3𝑖−1𝑅_1𝑖𝛽^𝑇_3𝑖−1−^𝑚_𝑖=1𝛽_3𝑖+1𝑅_2𝑖𝛽^𝑇_3𝑖+1+^𝑚_𝑖=1ℎ²_1𝑖𝛽_3𝑚+2𝑆_𝑖𝛽^𝑇_3𝑚+2+^𝑚_𝑖=1𝑑²_𝑖𝛽_3𝑚+2𝑍_𝑖𝛽^𝑇_3𝑚+2−^𝑚_𝑖=1𝛽₁−𝛽_3𝑖−1𝑆_𝑖𝛽₁−𝛽_3𝑖−1^𝑇+^𝑚_𝑖=1diag0,Ω_𝑖,0+𝛽₁𝑀₁+𝛽_3𝑚+2𝑀₂⋅𝐴₀(𝑘)𝛽^𝑇₁+𝐴₁(𝑘)𝛽^𝑇₃+⋯+𝐴_𝑚(𝑘)𝛽^𝑇_3𝑚−𝛽^𝑇_3𝑚+2+𝐷₁(𝑘)𝛽^𝑇_3𝑚+3+𝛽₁𝐴^𝑇₀(𝑘)+𝛽₃𝐴^𝑇₁(𝑘)+⋯+𝛽_3𝑚𝐴^𝑇_𝑚(𝑘)−𝛽_3𝑚+2+𝛽_3𝑚+3𝐷₁(𝑘)^𝑇⋅𝑀^𝑇₁𝛽^𝑇₁+𝑀^𝑇₂𝛽^𝑇_3𝑚+2+𝜀𝐻₀(𝑘)𝛽^𝑇₁+𝐻₁(𝑘)𝛽^𝑇₃+⋯+𝐻_𝑚(𝑘)𝛽^𝑇_3𝑚^𝑇×𝐻₀(𝑘)𝛽^𝑇₁+𝐻₁(𝑘)𝛽^𝑇₃+⋯+𝐻_𝑚(𝑘)𝛽^𝑇_3𝑚−𝛾²𝛽_3𝑚+3𝛽^𝑇_3𝑚+3.Ω_12𝑘=𝛽₁𝑀₁+𝛽_3𝑚+2𝑀₂𝐸(𝑘),Ω_13𝑘=𝐶₀(𝑘)𝛽^𝑇₁+𝐶₁(𝑘)𝛽^𝑇₃+⋯+𝐶_𝑚(𝑘)𝛽^𝑇_3𝑚+𝐷₂(𝑘)𝛽^𝑇_3𝑚+3^𝑇. (41)

Proof. Implying Lemma 5 and Schur complement lemma to (40), we obtain Π^′=𝛽₁𝑃_𝑘𝛽^𝑇_3𝑚+2+𝛽_3𝑚+2𝑃_𝑘𝛽^𝑇₁+^𝑁_𝑗=1𝛽₁𝜆_𝑘,𝑗𝑃_𝑗𝛽^𝑇₁+𝛽₁^𝑚_𝑖=1𝑄_𝑖𝛽^𝑇₁−^𝑚_𝑖=11−𝜇𝛽_3𝑖𝑄_𝑖𝛽^𝑇_3𝑖+𝛽₁^𝑚_𝑖=1𝑅_1𝑖+𝑅_2𝑖𝛽^𝑇₁−^𝑚_𝑖=1𝛽_3𝑖−1𝑅_1𝑖𝛽^𝑇_3𝑖−1−^𝑚_𝑖=1𝛽_3𝑖+1𝑅_2𝑖𝛽^𝑇_3𝑖+1+^𝑚_𝑖=1ℎ²_1𝑖𝛽_3𝑚+2𝑆_𝑖𝛽^𝑇_3𝑚+2+^𝑚_𝑖=1𝑑²_𝑖𝛽_3𝑚+2𝑍_𝑖𝛽^𝑇_3𝑚+2−^𝑚_𝑖=1𝛽₁−𝛽_3𝑖−1𝑆_𝑖𝛽₁−𝛽_3𝑖−1^𝑇+^𝑚_𝑖=1diag0,Ω_𝑖+𝛽,0₁𝑀₁+𝛽_3𝑚+2𝑀₂×𝐴₀(𝑘,𝑡)𝛽^𝑇₁+𝐴₁(𝑘,𝑡)𝛽^𝑇₃+⋯+𝐴_𝑚(𝑘,𝑡)𝛽^𝑇_3𝑚−𝛽^𝑇_3𝑚+2+𝐷₁(𝑘,𝑡)𝛽^𝑇_3𝑚+3+𝐴₀(𝑘,𝑡)𝛽^𝑇₁+𝐴₁(𝑘,𝑡)𝛽^𝑇₃+⋯+𝐴_𝑚(𝑘,𝑡)𝛽^𝑇_3𝑚−𝛽^𝑇_3𝑚+2+𝐷₁(𝑘,𝑡)𝛽^𝑇_3𝑚+3^𝑇×𝛽₁𝑀₁+𝛽_3𝑚+2𝑀₂^𝑇−𝛾²𝛽_3𝑚+3𝛽^𝑇_3𝑚+3. (42)
Set 𝐽(𝑡)=^𝑡₀𝑧(𝑠)^𝑇𝑧(𝑠)−𝛾²𝑤(𝑠)^𝑇𝑤(𝑠)𝑑𝑠. (43) Then, it is easy to have 𝐽(𝑡)=^𝑡₀𝑧(𝑠)^𝑇𝑧(𝑠)−𝛾²𝑤(𝑠)^𝑇𝑤(𝑠(𝑥(𝑠)−{𝑉(𝑥(𝑡)≤)+ℓ𝑉,𝑠,𝑘)𝑑𝑠,𝑡)}_𝑡0𝑧(𝑠)^𝑇𝑧(𝑠)−𝛾²𝑤(𝑠)^𝑇𝑤(𝑠)+ℓ𝑉(𝑥(𝑠),𝑠,𝑘)𝑑𝑠, (44) where 𝑉(𝑥(𝑡),𝑡,𝑘) is defined in (18). Similar to the proof of Theorem 7, we can obtain 𝑧(𝑡)^𝑇𝑧(𝑡)−𝛾²𝑤(𝑡)^𝑇𝑤(𝑡)+ℓ𝑉(𝑥(𝑡),𝑡)≤𝜒₁(𝑡)^𝑇Π^′𝜒₁(𝑡), (45) where Π^′ is given in (42) and 𝜒₁(𝑡) is defined in (14). Then, it follows from (40) and (45) that 𝐽(𝑡)<0. (46) This implies that for any nonzero 𝑣(𝑡)∈𝐿₂[0,∞], ‖𝑧‖_𝐸2<𝛾‖𝑤‖₂. (47) Therefore, by Definition 3, the system is robustly exponentially stable with a prescribed 𝐻_∞ performance level 𝛾. This completes the proof.

4. Numerical Example

In this section, we provide an example to demonstrate the effectiveness of the proposed method.

Let 𝑚=2 and 𝑁=2; consider the systems (1)–(3) with parameters as follows. Mode 1 𝐴₀(1)=−500.5−6,𝐴₁(1)=,𝐴−201−3₂(1)=0.10.20.10.5,𝐷₁(1)=,𝐶0.10.20.10.1₀(1)=0.10.10.20.1,𝐶₁(1)=,𝐶−0.100.10.2₂(1)=−0.10.100.3,𝐷₂(1)=,𝐸(1)=−0.1−0.30.1−0.20.10.10.20.3,𝐻₀(1)=,𝐻0.10.20.10.1₁(1)=−0.30.40.5−0.1,𝐻₂(1)=,𝐻0.20.20.30.1_𝑑(1)=−0.10.40.3−0.1,𝜆₁₁=−0.5,𝜆₁₂=0.5. (48) Mode 2 𝐴₀(2)=−201−4,𝐴₁(2)=,𝐴−111−4₂(2)=0.2−0.20−0.3,𝐷₁(2)=,𝐶−0.10.2−0.10.3₀(2)=0.2−0.10.1−0.1,𝐶₁(2)=,𝐶−0.20−0.10.1₂(2)=0.10.20−0.1,𝐷₂(2)=,𝐸(2)=0.10.2−0.1−0.30.1−0.10.1−0.3,𝐻₀(2)=,𝐻0.20.10.20.3₁(2)=−0.10.50.3−0.3,𝐻₂(2)=,𝐻0.10.10.60.2_𝑑(2)=0.2−0.30.2−0.1,𝜆₂₁=0.3,𝜆₂₂=−0.3. (49)And 𝛾=2, 𝜇=0.5, ℎ₁₁=0.1, ℎ₂₁=0.4, ℎ₁₂=0.4, ℎ₂₂=0.5. Then, by solving the LMI (15) with the constraints in Theorem 7, we obtain 𝑃₁=,𝑃59.13768.23568.235626.8113₂=,𝑄229.503850.501650.5016124.2305₁=,𝑄37.1817−0.5668−0.566841.5837₂=,𝑅28.15046.93996.93997.1618₁₁=,𝑅18.11950.91230.912314.7239₂₁=,𝑅31.6879−9.1485−9.148554.6448₁₂=,𝑅31.99766.70976.709710.5708₂₂=,𝑆32.57816.89476.894710.6649₁=,𝑆35.1285−0.1159−0.115925.3955₂=,𝑍0.97870.22680.22680.0975₁=,𝑍52.7205−8.8649−8.864970.5952₂=,𝑈1.74310.40570.40570.1638₁=,𝑈1.780718.406418.8018−53.4269₂=,𝑀−10.2878−1.9164−1.8946−5.3788₁=,𝑀42.47135.05868.491225.0911₂=,11.71471.12501.93533.9491𝜀=12.8944. (50)

If we fix the lower bound of ℎ₁(𝑡) and ℎ₂(𝑡), that is, ℎ₁₂=0.3 and ℎ₂₂=0.5, for the different ℎ₁₁, we can get the upper bounds of ℎ₂₁ as in Table 1.

Table 1 

The upper bound of ℎ₂₁ for different ℎ₁₁.

If we fix the lower bound of ℎ₁(𝑡) and ℎ₂(𝑡), that is, ℎ₁₁=0.2 and ℎ₂₁=0.6, for the different ℎ₁₂, we can get the upper bounds of ℎ₂₂ as in Table 2.

Table 2 

The upper bound of ℎ₂₂ for different ℎ₁₂.

5. Conclusion

The robust exponential stability and 𝐻_∞ performance analysis for uncertain Markovian jumping system with multiple time-varying delays has been investigated based on the reciprocally convex approach. Some new delay-dependent stability conditions are obtained in term of LMIs. Numerical example has been proposed to illustrate the effectiveness of result.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported National Natural Science Foundation of China under Grant nos. 61004046, 61104117, and 61304064.