Improved Stability Criteria for Markovian Jump Systems with Time-Varying Delays

Abstract

The delay-dependent stochastic stability problem of Markovian jump systems with time-varying delays is investigated in this paper. Though the Lyapunov-Krasovskii functional is general and simple, less conservative results are derived by using the convex combination method, improved Wirtinger’s integral inequality, and a slack condition on Lyapunov matrix. The obtained results are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are provided to verify the effectiveness and superiority of the presented results.

1. Introduction

Markovian jump systems are a special class of stochastic hybrid systems. Many dynamical systems subject to random abrupt variations, such as mechanical systems, economics, and systems with human operators, can be modeled by Markovian jump systems [1]. Due to their extensive applications in many files, the analysis and synthesis of Markovian jump systems have received much research attention and lots of significant results have been reported; see, for example, [2–8] and the references therein.

Time delay is an inherent characteristic of many dynamic systems such as networked control systems, industrial systems, and process control systems. The systems with or without time delays are convergent when time delays are close to zero; otherwise, they may be divergent. In other words, time delays can degrade the performance of systems designed without considering the delays and can even destabilize the systems. During the past few decades, considerable attention has been paid to the stability analysis of time-delay systems [9–16]. The existing stability criteria for linear systems can be classified into two types: delay-independent ones which are applicable to delays of arbitrary size and delay-dependent ones which include information on the size of delays. In general, delay-dependent stability criteria are less conservative than delay-independent ones especially when the size of the delay is small. Thus, considerable attention has been paid to the delay-dependent stability criteria; see [11–23], for example. As for delay-dependent stability, many methods have been taken for deriving stability criteria, such as free-weighting matrices methods [11], model transformation techniques [12, 13], convex combination methods [14–17], delay decomposition approaches [18], multiple integral approaches [19], and input-output approaches [20]. Recently, the bounding techniques of the cross terms and integral terms in the derivatives of the Lyapunov-Krasovskii functional are widely investigated, such as improved Jensen’s integral inequality [21], reciprocally convex approach [22, 23], and improved Wirtinger’s integral inequality [24]. Some less conservative stability results have been derived by using the above techniques.

In this paper, we develop some new stability criteria by using an improved Wirtinger’s integral inequality and the convex combination method to deal with the cross terms and integral terms in the derivatives of the Lyapunov-Krasovskii functional. In addition, the positive definiteness of some Lyapunov matrix is not required. The obtained results can be applied to both slow and fast time-varying delays. The numerical examples demonstrate the effectiveness and superiority of the presented results.

Notation. Throughout this paper, (Ω,𝐹,𝑃) is a probability space, Ω is the sample space, 𝐹 is the 𝜎-algebra of the sample space, and 𝑃 is the probability measure on 𝐹. 𝔼{⋅} refers to the expectation operator with respect to some probability measure 𝑃. 𝐴>0 (<0) means 𝐴 is a symmetric positive (negative) definite matrix and 𝐴⁻¹ denotes the inverse of matrix 𝐴. 𝐴^𝑇 represents the transpose of 𝐴. Sym(𝑀) stands for 𝑀+𝑀^𝑇. The symbol ∗ in LMIs denotes the symmetric term of the matrix. col{𝑋,𝑌} represents a column vector formed by 𝑋 and 𝑌. Identity matrix, of appropriate dimensions, will be denoted by 𝐼. diag(⋅,⋅) denotes a diagonal matrix. Ω(𝑖,𝑗) means the element in the 𝑖th row and 𝑗th column of the block matrix Ω.

2. Problem Statement

Fix a probability space (Ω,𝐹,𝑃) and consider the following Markovian jump systems: 𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡𝑥((𝑡,𝑥(𝑡)(𝑡)𝑡−𝑑))=𝜙,𝑡∈−𝑑₂,0, (1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector and 𝜙(𝑡) is a compatible vector-valued initial function defined on [−𝑑₂,0]. {𝑟_𝑡,𝑡⩾0} is a continuous-time Markovian process taking values in a finite space 𝑆={1,2,…,𝑁}; 𝐴(𝑟_𝑡) and 𝐴_𝑑(𝑟_𝑡) (𝑟_𝑡=𝑖,𝑖∈𝑆) are real constant matrices with appropriate dimensions which depend on 𝑟_𝑡. The time delay 𝑑(𝑡) satisfies 0<𝑑₁⩽𝑑(𝑡)⩽𝑑₂,𝑑̇(𝑡)⩽𝜇. (2) The evolution of the Markovian process {𝑟_𝑡,𝑡⩾0} is governed by the following transition probability: Pr𝑟_𝑡+Δ=𝑗∣𝑟_𝑡=𝜋=𝑖^𝑖𝑗Δ+𝑜(Δ),𝑗≠𝑖,1+𝜋_𝑖𝑖Δ+𝑜(Δ),𝑗=𝑖, (3) where Δ>0 and lim_Δ→0(𝑜(Δ)/Δ)=0; 𝜋_𝑖𝑗⩾0 for 𝑖≠𝑗 is the transition probability from mode 𝑖 at time 𝑡 to mode 𝑗 at time 𝑡+Δ and 𝜋_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗.

For simplicity, when 𝑟_𝑡=𝑖, 𝑖∈𝑆, the matrices 𝐴(𝑟_𝑡) and 𝐴_𝑑(𝑟_𝑡) are denoted by 𝐴_𝑖 and 𝐴_𝑑𝑖, respectively.

The following definition and lemmas are needed in the proof of our main results.

Definition 1 (see [2]). System (1) is stochastically stable, if, for any initial state (𝑥₀,𝑟₀), the following relation holds for any initial condition (𝑥₀,𝑟₀): lim_𝑡→+∞𝔼‖𝑥(𝑡)‖²∣𝑥₀,𝑟₀=0. (4)

Lemma 2 (see [22]). Let 𝑓₁,𝑓₂,…,𝑓_𝑛∶ℝ^𝑚→ℝ have positive values in an open subset 𝐃 of  ℝ^𝑚. Then, the reciprocally convex combination of 𝑓_𝑖 over 𝐃 satisfies min_{{𝛼𝑖∣𝛼𝑖>0,∑𝑖𝛼𝑖=1}𝑖}1𝛼_𝑖𝑓_𝑖(𝑡)=_𝑖𝑓_𝑖(𝑡)+max_{𝑔𝑖,𝑗(𝑡)𝑖≠𝑗}𝑔_𝑖,𝑗(𝑡) (5) subject to 𝑔_𝑖,𝑗∶ℝ^𝑚⟶ℝ,𝑔_𝑗,𝑖≜𝑔_𝑖,𝑗,𝑓_𝑖(𝑡)𝑔_𝑖,𝑗(𝑡)𝑔_𝑖,𝑗(𝑡)𝑓_𝑗(𝑡)⩾0. (6)

Lemma 3 (see [24]). If 𝜔(𝑦) is a differentiable function in [𝑎,𝑏]→ℝ^𝑛. Then for any given symmetric positive definite matrix 𝑅>0, the following inequality holds: 𝜔̇^𝑏_𝑎^𝑇(𝑡)𝑅𝜔̇(𝑡)1𝑑𝑡⩾𝜈𝑏−𝑎¹𝜈₂^𝑇𝑅𝜈₁𝜈₂, (7) where 𝜈₁=𝜔(𝑏)−𝜔(𝑎), 𝜈₂=𝜔(𝑏)+𝜔(𝑎)−(2/(𝑏−𝑎))∫^𝑏_𝑎𝜔(𝑡)𝑑𝑡, and 𝑅=diag(𝑅,3𝑅).

3. Improved Stability Criterion

In this section, we will present an improved stochastic stability criterion in terms of LMIs by using Lyapunov-Krasovskii functional method and convex combination technique.

Before stating the main results, some notations are given. Let 𝑑₁₂=𝑑₂−𝑑₁,𝜁(𝑡)=col𝑥(𝑡),^𝑡_𝑡−𝑑1𝑥(𝑠)d𝑠,^𝑡−𝑑1_𝑡−𝑑2𝑥(𝑠),𝜉(𝑡)𝑥(𝑡)((𝑡d𝑠=col,𝑥𝑡−𝑑)),𝑥𝑡−𝑑₁,𝑥𝑡−𝑑₂,1𝑑₁^𝑡_𝑡−𝑑1𝑥(𝑠)1d𝑠,𝑑(𝑡)−𝑑₁^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥(𝑠)1d𝑠,𝑑₂−𝑑(𝑡)^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥(𝑠),𝒫d𝑠_𝑖=𝑃_11𝑖𝑃₁₂𝑃₁₃∗𝑃₂₂𝑃₂₃∗∗𝑃₃₃,(),Γ𝑖∈𝑆_𝑇1=𝐼0000000000𝑑₁𝑑(𝑡)𝐼0000000−𝑑₁𝐼𝑑₂(𝑡)𝐼,𝑋−𝑑𝑋=¹¹𝑋₁₂𝑋₂₁𝑃₂₂,Γ₂=𝐴_𝑖𝐴_𝑑𝑖,().00000𝐼0−𝐼000000𝐼−𝐼000𝑖∈𝑆 (8)

Theorem 4. Given scalars 0<𝑑₁<𝑑₂ and 𝜇, the time-varying delay system (1) is stochastically stable, if there exist matrix 𝒫_𝑖∈ℝ^3𝑛×3𝑛; symmetric positive definite matrices 𝑅₁∈ℝ^𝑛×𝑛, 𝑅₂∈ℝ^𝑛×𝑛, 𝑄₁∈ℝ^𝑛×𝑛, 𝑄₂∈ℝ^𝑛×𝑛, 𝑄₃∈ℝ^𝑛×𝑛, and 𝑀∈ℝ^𝑛×𝑛; and any matrices 𝑋∈ℝ^𝑛×𝑛, 𝑁_𝑙∈ℝ^𝑛×𝑛, and 𝐺_𝑙∈ℝ^𝑛×𝑛(𝑙=1,2) such that, for all 𝑖∈𝑆,Ξ₁=𝑅₂−𝑀0𝑋₁₁𝑋₁₂∗3𝑅₂−𝑀𝑋₂₁𝑋₂₂∗∗𝑅₂𝑅−𝑀0∗∗03₂−𝑀>0,Ξ_2𝑖=𝑃_11𝑖+2𝑑₁𝑅₁+2𝑑²₁₂𝑑₁+𝑑₂𝑅₂𝑃₁₂−2𝑅₁𝑃₁₃−2𝑑₁₂𝑑₁+𝑑₂𝑅₂∗𝑃₂₂+1𝑑₁𝑄₁+2𝑅₁𝑃₂₃∗∗𝑃₃₃+1𝑑₁₂𝑄₂+2𝑑₁+𝑑₂𝑅₂>0,Ξ_3𝑖=Ω_{15𝑛×5𝑛}Ω_25𝑛×𝑛Ω_35𝑛×𝑛Ω_45𝑛×𝑛Ω_55𝑛×𝑛Ω_65𝑛×𝑛∗−12𝑅₂−𝑀−4𝑋₂₂𝑅000∗∗−12₂−𝑀000∗∗∗−𝑅₁00∗∗∗∗−𝑅₂0∗∗∗∗∗−𝑀<0,Ξ_4𝑖=Ω_{15𝑛×5𝑛}Ω_25𝑛×𝑛Ω_35𝑛×𝑛Ω_45𝑛×𝑛Ω_55𝑛×𝑛Ω_65𝑛×𝑛∗−12𝑅₂−𝑀−4𝑋₂₂𝑅000∗∗−12₂−𝑀000∗∗∗−𝑅₁00∗∗∗∗−𝑅₂0∗∗∗∗∗−𝑀<0, (9)(10)(11)(12)where Ω₁(1,1)=Sym𝑃_11𝑖𝐴_𝑖+𝑃₁₂+^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_11𝑗+𝑄₁−4𝑅₁,Ω₁(1,2)=𝑃_11𝑖𝐴_𝑑𝑖,Ω₁(1,3)=−2𝑅₁−𝑃₁₂+𝑃₁₃,Ω₁(1,4)=−𝑃₁₃,Ω₁(1,5)=6𝑅₁+𝑑₁𝐴^𝑇_𝑖𝑃₁₂+𝑃^𝑇₂₂,Ω₁(2,2)=−1−𝜇𝑄₃−8𝑅₂𝑋+8𝑀+Sym₁₁+𝑋₁₂−𝑋₂₁−𝑋₂₂−𝑁₁+𝐺₁,Ω₁(2,3)=−𝑋^𝑇₁₁−𝑋^𝑇₁₂−𝑋^𝑇₂₁−𝑋^𝑇₂₂−2𝑅₂+𝑁₁−𝑁^𝑇₂+𝐺^𝑇₂Ω+2𝑀,₁(2,4)=−2𝑅₂−𝑋₁₁+𝑋₁₂+𝑋₂₁−𝑋₂₂−𝑁^𝑇₃−𝐺^𝑇₁+𝐺^𝑇₃Ω+2𝑀,₁(2,5)=𝑑₁𝐴^𝑇_𝑑𝑖𝑃₁₂Ω₁(3,3)=−4𝑅₁−𝑄₁+𝑄₂+𝑄₃−4𝑅₂+4𝑀+Sym𝑁₂,Ω₁(3,4)=𝑋₁₁−𝑋₁₂+𝑋₂₁−𝑋₂₂+𝑁₃−𝐺₂,Ω₁(3,5)=6𝑅₁+𝑑₁𝑃^𝑇₂₃−𝑃^𝑇₂₂,Ω₁(4,4)=−𝑄₂−4𝑅₂+4𝑀−Sym𝐺₃,Ω₁(4,5)=−𝑑₁𝑃^𝑇₂₃,Ω₁(5,5)=−12𝑅₁,Ω₂(1,1)=𝑑₁₂𝐴^𝑇_𝑖𝑃₁₃+𝑃₂₃,Ω₂(2,1)=𝑑₁₂𝐴^𝑇_𝑑𝑖𝑃₁₃+2𝑋^𝑇₂₁+2𝑋^𝑇₂₂+6𝑅₆Ω−6𝑀,₂(3,1)=−𝑑₁₂𝑃₂₃−𝑃^𝑇₃₃+6𝑅₆Ω−6𝑀,₂(4,1)=−𝑑₁₂𝑃^𝑇₃₃−2𝑋^𝑇₂₁+2𝑋^𝑇₂₂,Ω₂()Ω5,1=0,₃(1,1)=0,Ω₃(2,1)=6𝑅₂−6𝑀−2𝑋₁₂+2𝑋₂₂,Ω₃(3,1)=2𝑋₁₂+2𝑋₂₂,Ω₃(4,1)=6𝑅₂−6𝑀,Ω₃()Ω5,1=0,₄(1,1)=𝑑₁𝐴^𝑇_𝑖𝑅₁,Ω₄(2,1)=𝑑₁𝐴^𝑇_𝑑𝑖𝑅₁,Ω₄(3,1)=Ω₄(4,1)=Ω₄()Ω5,1=0,₅(1,1)=𝑑₁₂𝐴^𝑇_𝑖𝑅₂,Ω₅(2,1)=𝑑₁₂𝐴^𝑇_𝑑𝑖𝑅₂,Ω₅(3,1)=Ω₅(4,1)=Ω₅()Ω5,1=0,₆(1,1)=0,Ω₆(2,1)=𝑁₁,Ω₆(3,1)=𝑁₂,Ω₆(4,1)=𝑁₃,Ω₆()Ω5,1=0,₂(1,1)=0,Ω₂(2,1)=2𝑋^𝑇₂₁+2𝑋^𝑇₂₂+6𝑅₆Ω−6𝑀,₂(3,1)=6𝑅₆Ω−6𝑀,₂(4,1)=−2𝑋^𝑇₂₁+2𝑋^𝑇₂₂,Ω₂()Ω5,1=0,₃(1,1)=𝑑₁₂𝐴^𝑇_𝑖𝑃₁₃+𝑃₂₃,Ω₃(2,1)=𝑑₁₂𝐴^𝑇_𝑑𝑖𝑃₁₃+6𝑅₂−6𝑀−2𝑋₁₂+2𝑋₂₂,Ω₃(3,1)=−𝑑₁₂𝑃₂₃−𝑃^𝑇₃₃+2𝑋₁₂+2𝑋₂₂,Ω₃(4,1)=−𝑑₁₂𝑃^𝑇₃₃+6𝑅₂−6𝑀,Ω₃()Ω5,1=0,₆(1,1)=0,Ω₆(2,1)=𝐺₁,Ω₆(3,1)=𝐺₂,Ω₆(4,1)=𝐺₃,Ω₆(5,1)=0. (13)

Proof. Consider the Lyapunov-Krasovskii functional given by 𝑉𝑥_𝑡,𝑖=𝑉₁𝑥_𝑡,𝑖+𝑉₂𝑥_𝑡,𝑖+𝑉₃𝑥_𝑡,𝑖, (14) where 𝑉₁𝑥_𝑡,𝑖=𝜁^𝑇(𝑡)𝒫𝑟_𝑡𝜁(𝑡),𝑉₂𝑥_𝑡,𝑖=^𝑡_𝑡−𝑑1𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+d𝑠_{𝑡−𝑑1𝑡−𝑑2}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)d𝑠+^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)𝑉d𝑠,₃𝑥_𝑡,𝑖=𝑑₁⁰_−𝑑1^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑅₁𝑥̇(𝑠)d𝑠d𝜃+𝑑₁₂^−𝑑1_−𝑑2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)d𝑠d𝜃. (15)
We first show, for some 𝜖>0, the Lyapunov-Krasovskii functional condition 𝑉(𝑥_𝑡,𝑖)⩾𝜖‖𝑥_𝑡‖² for any initial condition (𝑥₀,𝑟₀). Note that 𝑄₁>0, 𝑄₂>0, 𝑅₁>0, and 𝑅₂>0; it follows easily from Jensen’s inequality that 𝑉𝑥_𝑡,𝑖⩾𝜁^𝑇(𝑡)𝒫_𝑖𝜁(𝑡)+1𝑑₁^𝑡_𝑡−𝑑1𝑥^𝑇(𝑠)d𝑠𝑄₁×^𝑡_𝑡−𝑑1𝑥(𝑠)1d𝑠+𝑑₁₂^𝑡−𝑑1_𝑡−𝑑2𝑥^𝑇(𝑠)d𝑠𝑄₂×^𝑡−𝑑1_𝑡−𝑑2𝑥(𝑠)d𝑠+^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)+2d𝑠𝑑₁𝑑₁𝑥^𝑇(𝑡)−^𝑡_𝑡−𝑑1𝑥^𝑇(𝑠)d𝑠×𝑅₁𝑑₁𝑥(𝑡)−^𝑡_𝑡−𝑑1𝑥(𝑠)+d𝑠2𝑑₁₂𝑑²₂−𝑑²₁𝑑₁₂𝑥^𝑇(𝑡)−^𝑡−𝑑1_𝑡−𝑑2𝑥^𝑇(𝑠)d𝑠×𝑅₂𝑑₁₂𝑥(𝑡)−^𝑡−𝑑1_𝑡−𝑑2𝑥(𝑠)d𝑠=𝜁^𝑇(𝑡)Ξ_2𝑖𝜁(𝑡)+^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)d𝑠. (16) Thus, if 𝑄₃>0 and Ξ_2𝑖>0, then there exists a sufficiently small 𝜖>0 such that 𝑉(𝑥_𝑡,𝑖)⩾𝜖‖𝑥_𝑡‖².
We next show that ℒ𝑉(𝑥_𝑡,𝑖)⩽−𝜖‖𝑥(𝑡)‖² for the sufficiently small 𝜖. Let ℒ be the weak infinitesimal generator of the random process {𝑥_𝑡,𝑟_𝑡}. Calculating the difference of 𝑉(𝑥_𝑡,𝑖) along the trajectories of (1), we have ℒ𝑉𝑥_𝑡,𝑖=2𝜁^𝑇(𝑡)𝒫_𝑖𝜁̇(𝑡)+𝜁^𝑇(𝑡)^𝑁_𝑗=1𝜋_𝑖𝑗𝒫_𝑗𝜁(𝑡)+𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)+𝑥^𝑇𝑡−𝑑₁−𝑄₁+𝑄₂+𝑄₃𝑥𝑡−𝑑₁−1−𝑑̇(𝑡)𝑥^𝑇(𝑡−𝑑(𝑡))𝑄₃𝑥(𝑡−𝑑(𝑡))−𝑥^𝑇𝑡−𝑑₂𝑄₂𝑥𝑡−𝑑₂+𝑑²₁𝑥̇^𝑇(𝑡)𝑅₁𝑥̇(𝑡)+𝑑²₁₂𝑥̇^𝑇(𝑡)𝑅₂𝑥̇(𝑡)−𝑑₁^𝑡_𝑡−𝑑1𝑥̇^𝑇(𝑠)𝑅₁𝑥̇(𝑠)d𝑠−𝑑₁₂^𝑡−𝑑1_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)d𝑠, (17) where 𝜁^𝑇(𝑡)=𝜉^𝑇(𝑡)Γ₁ and 𝜁̇(𝑡)=Γ₂𝜉(𝑡).
Using the Newton-Leibniz formula, for any 𝑁_𝑙 and 𝐺_𝑙(𝑙=1,2,3) with appropriate dimensions, the following are true: 0=2𝜉^𝑇(𝑡)𝒩𝑥𝑡−𝑑₁−𝑥(𝑡−𝑑(𝑡))−^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇(𝑠)d𝑠,0=2𝜉^𝑇(𝑡)𝒢𝑥(𝑡−𝑑(𝑡))−𝑥𝑡−𝑑₂−^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇(𝑠)d𝑠, (18) where 𝒩=[0𝑁^𝑇₁𝑁^𝑇₂𝑁^𝑇₃000]^𝑇, 𝒢=[0𝐺^𝑇₁𝐺^𝑇₂𝐺^𝑇₃000]^𝑇.
It can be shown readily that there exists a matrix 𝑀>0 such that 0⩽2𝜉^𝑇(𝑡)𝒩𝑥𝑡−𝑑₁−𝑥(𝑡−𝑑(𝑡))+2𝜉^𝑇(𝑡)𝒢𝑥(𝑡−𝑑(𝑡))−𝑥𝑡−𝑑₂+𝑑(𝑡)−𝑑₁𝑑₁₂𝜉^𝑇(𝑡)𝒩𝑀⁻¹𝒩^𝑇𝜉(𝑡)+𝑑₂−𝑑(𝑡)𝑑₁₂𝜉^𝑇(𝑡)𝒢𝑀⁻¹𝒢^𝑇𝜉(𝑡)+𝑑₁₂^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑀𝑥̇(𝑠)d𝑠+𝑑₁₂^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑀𝑥̇(𝑠)d𝑠. (19) Adding equalities (18) to the right hand of (17) and considering conditions (2) and (19), we have ℒ𝑉𝑥_𝑡,𝑖⩽2𝜉^𝑇(𝑡)Γ₁𝒫_𝑖Γ₂𝜉(𝑡)+𝜉^𝑇(𝑡)Γ₁^𝑁_𝑗=1𝜋_𝑖𝑗𝒫_𝑗Γ^𝑇₁𝜉(𝑡)+𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)+𝑥^𝑇𝑡−𝑑₁−𝑄₁+𝑄₂+𝑄₃𝑥𝑡−𝑑₁−1−𝜇𝑥^𝑇(𝑡−𝑑(𝑡))𝑄₃𝑥(𝑡−𝑑(𝑡))−𝑥^𝑇𝑡−𝑑₂𝑄₂𝑥𝑡−𝑑₂+𝑑²₁𝑥̇^𝑇(𝑡)𝑅₁𝑥̇(𝑡)+𝑑²₁₂𝑥̇^𝑇(𝑡)𝑅₂𝑥̇(𝑡)+2𝜉^𝑇(𝑡)𝒩𝑥𝑡−𝑑₁−𝑥(𝑡−𝑑(𝑡))+2𝜉^𝑇(𝑡)𝒢𝑥(𝑡−𝑑(𝑡))−𝑥𝑡−𝑑₂+𝑑(𝑡)−𝑑₁𝑑₁₂𝜉^𝑇(𝑡)𝒩𝑀⁻¹𝒩^𝑇𝜉(𝑡)+𝑑₂−𝑑(𝑡)𝑑₁₂𝜉^𝑇(𝑡)𝒢𝑀⁻¹𝒢^𝑇𝜉(𝑡)−𝑑₁^𝑡_𝑡−𝑑1𝑥̇^𝑇(𝑠)𝑅₁𝑥̇(𝑠)d𝑠−𝑑₁₂^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑅₂−𝑀𝑥̇(𝑠)d𝑠−𝑑₁₂^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₂−𝑀𝑥̇(𝑠)d𝑠. (20) Now, we deal with the integral terms in inequality (20) by applying Lemma 3. Consider −𝑑₁^𝑡_𝑡−𝑑1𝑥̇^𝑇(𝑠)𝑅₁𝑥̇(𝑠)𝜈d𝑠⩽−¹(𝑡)𝜈₂(𝑡)^𝑇𝑅₁003𝑅₁𝜈₁(𝑡)𝜈₂(𝑡), (21) where 𝜈₁(𝑡)=𝑥(𝑡)−𝑥(𝑡−𝑑₁), 𝜈₂(𝑡)=𝑥(𝑡)+𝑥(𝑡−𝑑₁)−(2/𝑑₁)∫^𝑡_𝑡−𝑑1𝑥(𝑠)d𝑠; −𝑑₁₂^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)d𝑠−𝑑₁₂^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝜍d𝑠⩽−¹(𝑡)𝜍₂(𝑡)^𝑇𝑑₁₂𝑑(𝑡)−𝑑₁∗𝑑Σ0₁₂𝑑₂(𝑡)Σ𝜍−𝑑¹(𝑡)𝜍₂(𝑡), (22) where Σ=diag(𝑅₂−𝑀,3(𝑅₂−𝑀)), 𝜍₁(𝑡)=𝑥𝑡−𝑑₁((𝑡𝑥−𝑥𝑡−𝑑))𝑡−𝑑₁((𝑡−2+𝑥𝑡−𝑑))𝑑(𝑡)−𝑑₁^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥(𝑠),𝜍d𝑠₂(𝑡)=𝑥(𝑡−𝑑(𝑡))−𝑥𝑡−𝑑₂𝑥(𝑡−𝑑(𝑡))+𝑥𝑡−𝑑₂−2𝑑₂−𝑑(𝑡)^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥(𝑠).d𝑠 (23) Applying Lemma 2 to (22), it yields that if there exists a matrix 𝑋=^𝑋11𝑋12_𝑋21𝑋22 with appropriate dimensions such that (9) holds, then −𝑑₁₂^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)d𝑠−𝑑₁₂^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝜍d𝑠⩽−¹(𝑡)𝜍₂(𝑡)^𝑇Σ𝑋𝜍∗Σ¹(𝑡)𝜍₂(𝑡). (24) Combining (20), (21), and (24), we have ℒ𝑉𝑥_𝑡⩽𝑑(𝑡),𝑖−𝑑₁𝑑₁₂𝜉^𝑇(𝑡)×Υ₁+𝑑²₁Υ^𝑇₃𝑅₁Υ₃+𝑑²₁₂Υ^𝑇₃𝑅₂Υ₃+𝒩𝑀⁻¹𝒩^𝑇(𝑡)+𝑑×𝜉²−𝑑(𝑡)𝑑₁₂𝜉^𝑇(𝑡)×Υ₂+𝑑²₁Υ^𝑇₃𝑅₁Υ₃+𝑑²₁₂Υ^𝑇₃𝑅₂Υ₃+𝒢𝑀⁻¹𝒢^𝑇×𝜉(𝑡), (25) where Υ₁=Ω_{15𝑛×5𝑛}Ω_25𝑛×𝑛Ω_35𝑛×𝑛∗−12𝑅₂−𝑀−4𝑋₂₂∗∗−12𝑅₂,Υ−𝑀₂=Ω_{15𝑛×5𝑛}Ω_25𝑛×𝑛Ω_35𝑛×𝑛∗−12𝑅₂−𝑀−4𝑋₂₂∗∗−12𝑅₂,Υ−𝑀₃=𝐴_𝑖,𝐴_𝑑𝑖,0,0,0,0,0. (26) Using the Schur complement formula to (11) and (12), respectively, we have Υ₁+𝑑²₁Υ^𝑇₃𝑅₁Υ₃+𝑑²₁₂Υ^𝑇₃𝑅₂Υ₃+𝒩𝑀⁻¹𝒩^𝑇Υ<0,₂+𝑑²₁Υ^𝑇₃𝑅₁Υ₃+𝑑²₁₂Υ^𝑇₃𝑅₂Υ₃+𝒢𝑀⁻¹𝒢^𝑇<0. (27) By using convex combination approach [15], inequalities (27) imply that ℒ𝑉(𝑥_𝑡,𝑖)<0, which implies there exists a sufficiently small 𝜖>0 such that ℒ𝑉(𝑥_𝑡,𝑖)⩽−𝜖‖𝑥(𝑡)‖² for any initial condition (𝑥₀,𝑟₀). Therefore, for any 𝑡>0, by Dynkin’s formula, we have 𝔼𝑉𝑥_𝑡,𝑖−𝑉𝑥₀,𝑟₀⩽−𝜖𝔼^𝑡₀‖𝑥(𝑠)‖²d𝑠∣𝑥₀,𝑟₀, (28) which yields 𝔼^𝑡₀‖𝑥(𝑠)‖²d𝑠∣𝑥₀,𝑟₀⩽1𝜖𝑉𝑥₀,𝑟₀<∞. (29) The previous inequality means that lim_𝑡→+∞𝔼{‖𝑥(𝑡)‖²∣(𝑥₀,𝑟₀)}=0. Thus, system (1) is stochastically stable by Definition 1.

Remark 5. In order to guarantee the Lyapunov-Krasovskii functional 𝑉(𝑥_𝑡,𝑟_𝑡)>0, most authors require the Lyapunov matrix 𝑃_𝑖>0 in 𝑉₁ (see, e.g., [4–8]). The Lyapunov-Krasovskii functional employed in this paper is very simple, but a less conservative result is developed by using the LMI (10) instead of inequality 𝒫_𝑖>0, which can be seen in Section 4. However, it should be pointed out that the provided result cannot be used when the lower delay bound is considered to be zero because of the existence of 1/𝑑₁ in LMI (10).

Remark 6. It is well known that the convex combination approach is effective in reducing conservatism in stability analysis. In some literature, the integral terms −2𝜉^𝑇(𝑡)𝒩∫^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇(𝑠)d𝑠 and −2𝜉^𝑇(𝑡)𝒢∫^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇(𝑠)d𝑠 are estimated by ((𝑑(𝑡)−𝑑₁)/𝑑₁₂)𝜉^𝑇(𝑡)𝒩𝑅₂⁻¹𝒩^𝑇𝜉(𝑡)+𝑑₁₂∫^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)d𝑠 and 𝑑₂−𝑑(𝑡)𝑑₁₂𝜉^𝑇(𝑡)𝒢𝑅₂⁻¹𝒢^𝑇𝜉(𝑡)+𝑑₁₂∫^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)d𝑠, respectively, which make the term 𝑑₁₂∫^𝑡−𝑑1_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)d𝑠 in (17) disappear. In order to obtain a less conservative result, we applied Lemma 3 to deal with the integral term. In this case, integral term 𝑑₁₂∫^𝑡−𝑑1_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)d𝑠 must be reserved. Based on the above consideration, another matrix 𝑀>0 is introduced when we estimate the integral terms −2𝜉^𝑇(𝑡)𝒩∫^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇(𝑠)d𝑠 and −2𝜉^𝑇(𝑡)𝒢∫^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇(𝑠)d𝑠.
When 𝑆={1}, the Markovian jump system (1) reduces to the following linear system with interval time-varying delay: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥((𝑡,𝑥(𝑡)(𝑡)𝑡−𝑑))=𝜙,𝑡∈−𝑑₂,0. (30) Based on the above method, we are now ready to give an improved asymptotic stability criterion for system (30).

Corollary 7. Given scalars 0<𝑑₁<𝑑₂ and 𝜇, the time-varying delay system (30) is asymptotically stable, if there exist matrix 𝑃_𝜄𝜅=𝑃^𝑇_𝜅𝜄∈ℝ^𝑛×𝑛, (𝜄,𝜅=1,2,3); symmetric positive definite matrices 𝑅₁∈ℝ^𝑛×𝑛, 𝑅₂∈ℝ^𝑛×𝑛, 𝑄₁∈ℝ^𝑛×𝑛, 𝑄₂∈ℝ^𝑛×𝑛, 𝑄₃∈ℝ^𝑛×𝑛, and 𝑀∈ℝ^𝑛×𝑛; and any matrices 𝑋∈ℝ^𝑛×𝑛, 𝑁_𝑙∈ℝ^𝑛×𝑛, and 𝐺_𝑙∈ℝ^𝑛×𝑛(𝑙=1,2) such thatΨ₁=𝑅₂−𝑀0𝑋₁₁𝑋₁₂∗3𝑅₂−𝑀𝑋₂₁𝑋₂₂∗∗𝑅₂𝑅−𝑀0∗∗03₂Ψ−𝑀>0,₂=𝑃₁₁+2𝑑₁𝑅₁+2𝑑²₁₂𝑑₁+𝑑₂𝑅₂𝑃₁₂−2𝑅₁𝑃₁₃−2𝑑₁₂𝑑₁+𝑑₂𝑅₂∗𝑃₂₂+1𝑑₁𝑄₁+2𝑅₁𝑃₂₃∗∗𝑃₃₃+1𝑑₁₂𝑄₂+2𝑑₁+𝑑₂𝑅₂Ψ>0,₃=Φ_{15𝑛×5𝑛}Φ_25𝑛×𝑛Φ_35𝑛×𝑛Φ_45𝑛×𝑛Φ_55𝑛×𝑛Φ_65𝑛×𝑛∗−12𝑅₂−𝑀−4𝑋₂₂𝑅000∗∗−12₂−𝑀000∗∗∗−𝑅₁00∗∗∗∗−𝑅₂0Ψ∗∗∗∗∗−𝑀<0,₄=Φ_{15𝑛×5𝑛}Φ_25𝑛×𝑛Φ_35𝑛×𝑛Φ_45𝑛×𝑛Φ_55𝑛×𝑛Φ_65𝑛×𝑛∗−12𝑅₂−𝑀−4𝑋₂₂𝑅000∗∗−12₂−𝑀000∗∗∗−𝑅₁00∗∗∗∗−𝑅₂0∗∗∗∗∗−𝑀<0, (31)where Φ₁(1,1)=Sym𝑃₁₁𝐴+𝑃₁₂+𝑄₁−4𝑅₁,Φ₁(1,2)=𝑃₁₁𝐴_𝑑,Φ₁(1,𝑠)=Ω₁(),(),Φ1,𝑠𝑠=3,4₁(1,5)=6𝑅₁+𝑑₁𝐴^𝑇𝑃₁₂+𝑃^𝑇₂₂,Φ₁(2,𝑠)=Ω₁(),()Φ2,𝑠𝑠=1,2,3,4₁(2,5)=𝑑₁𝐴^𝑇_𝑑𝑃₁₂Φ₁(3,𝑠)=Ω₁(),(),Φ3,𝑠𝑠=3,4,5₁(4,𝑠)=Ω₁(),(),Φ4,𝑠𝑠=4,5₁(5,5)=Ω₁(),Φ5,5₂(1,1)=𝑑₁₂𝐴^𝑇𝑃₁₃+𝑃₂₃,Φ₂(2,1)=𝑑₁₂𝐴^𝑇_𝑑𝑃₁₃+2𝑋^𝑇₂₁+2𝑋^𝑇₂₂+6𝑅₆Φ−6𝑀,₂(𝑠,1)=Ω₂(),(),Φ𝑠,1𝑠=3,4,5₃(𝑠,1)=Ω₃(),(),Φ𝑠,1𝑠=1,2,3,4,5₄(1,1)=𝑑₁𝐴^𝑇𝑅₁,Φ₄(2,1)=𝑑₁𝐴^𝑇_𝑑𝑅₁,Φ₄(𝑠,1)=Ω₄(),(),Φ𝑠,1𝑠=3,4,5₅(1,1)=𝑑₁₂𝐴^𝑇𝑅₂,Φ₅(2,1)=𝑑₁₂𝐴^𝑇_𝑑𝑅₂,Φ₅(𝑠,1)=Ω₅(),(),Φ𝑠,1𝑠=3,4,5₂(𝑠,1)=Ω₂(),(),Φ𝑠,1𝑠=1,2,3,4,5₃(1,1)=𝑑₁₂𝐴^𝑇𝑃₁₃+𝑃₂₃,Φ₃(2,1)=𝑑₁₂𝐴^𝑇_𝑑𝑃₁₃+6𝑅₂−6𝑀−2𝑋₁₂+2𝑋₂₂,Φ₃(𝑠,1)=Ω₃(),(),Φ𝑠,1𝑠=3,4,5₆(𝑠,1)=Ω₆(𝑠,1),(𝑠=1,2,3,4,5). (32)

Remark 8. Theorem 4 and Corollary 7 can be applied to both slow and fast time-varying delays. But when 𝜇 is unknown, the above results cannot be used directly to check the stability. From the construction of Lyapunov-Krasovskii functional, it can be seen by setting 𝑄₃=0 in Theorem 4 and Corollary 7 that the corresponding conclusions are valid for the case when 𝜇 is unknown.

4. Numerical Examples

In this section, three numerical examples will be presented to show the validity of the main results derived above.

Example 1. Consider the Markovian jump systems (1) with 𝐴₁=−2.30.81−2.9,𝐴_𝑑1=,𝐴0.81.20.7−3.5₂=−1.90.20.6−0.8,𝐴_𝑑2=.1.3−2.60.5−1.4 (33) Consider 𝜋₁₁=−𝜋₁₂=−0.5, 𝜋₂₂=−𝜋₂₁=−3. According to [20], for 𝑑₁=0.2, one obtains 𝑑₂=0.520, while by Theorem 4 in this paper we can obtain 𝑑₂=0.605 such that the system is stochastically stable. More comparisons are shown in Table 1, which indicate that Theorem 4 is much less conservative than that in [20].

Table 1 

Admissible upper bounds 𝑑₂ with varying 𝑑₁.

Example 2. Consider the Markovian jump systems (1) with 𝐴₁=,𝐴−3.48880.8057−0.6451−3.2684_𝑑1=,𝐴−0.8620−1.2919−0.6841−2.0729₂=,𝐴−2.48980.28951.3396−0.0211_𝑑2=.−2.83060.4978−0.8436−1.0115 (34) This example has been taken from [4]. To compare the stochastic stability condition in Theorem 4 with that in [25, 26], we choose 𝜋₂₂=0.8, 𝜇=0.9. Using Theorem 4 of our paper, the admissible upper bound 𝑑₂ for different 𝑑₁ and 𝜋₁₁ can be found in Table 2. It can be seen from Table 2 that Theorem 4 in our paper is less conservative.

Table 2 

Admissible upper bounds 𝑑₂ with varying 𝜋₁₁ and 𝑑₁.

Example 3. Consider the systems (30) with the following parameters: 𝐴=−2.00.00.0−0.9,𝐴_𝑑=.−1.00.0−1.0−0.1 (35) This system is a well-known delay-dependent stable system where the maximum allowable delay 𝑑_max=6.1725 [24]. For known and unknown 𝜇, the admissible upper bounds 𝑑₂ for different 𝑑₁, which guarantee the asymptotical stability of the system (30), are listed in Tables 3 and 4, respectively. It can be seen from Tables 3 and 4 that the stability results obtained in this paper are less conservative than those in [14, 16, 17, 22, 27, 28].

Table 3 

Admissible upper bounds 𝑑₂ with varying 𝜇 and 𝑑₁.

Table 4 

Admissible upper bounds 𝑑₂ with unknown 𝜇 and 𝑑₁.

5. Conclusion

In this paper, the problem of stochastic stability for a class of Markovian jump systems has been investigated. By using the convex combination technique and the improved integral inequality, some less conservative delay-dependent stability criteria are established in terms of linear matrix inequalities. The numerical examples demonstrate the effectiveness and superiority of the presented results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Doctoral Research Foundation of Southwest University of Science and Technology (Grant no. 13zx7141).