Stochastic Finite-Time 𝐻_∞ Performance Analysis of Continuous-Time Systems with Random Abrupt Changes

Abstract

The problem of 𝐻_∞ control performance analysis of continuous-time systems with random abrupt changes is concerned in this paper. By employing an augmented multiple mode-dependent Lyapunov-Krasovskii functional and using some integral inequalities, new sufficient conditions are obtained relating to finite-time bounded and an 𝐻_∞ performance index. The finite-time 𝐻_∞ control performance problem is solved and desired controller is given to ensure the system trajectory stays within a prescribed bound during a given time interval. At last, two numerical examples are provided to show that our results are less conservative than the existing ones.

1. Introduction

It is well known that Markovian jump systems were introduced when the physical models are always subject to random changes, which can be also regarded as a special class of hybrid systems because of the structures are subject to random abrupt changes [1]. In the recent years, there are a lot of people towards to Markovian jump systems for its widely applications, for example, target tracking, robotics, manufacturing systems, aircraft control, and power systems [2–4]. Markovian jump systems are regarded as a special class of stochastic systems which switches from one to another at different time in the finite operation modes. Many important topics have been studied for Markovian jumping systems such as stability, control synthesis, stabilization, and filter design [5–7].

On the other hand, time delay is very common in practical dynamical systems, for example, networked control systems, chemical processes, communication systems, and so on [8–20]. Therefore, during the past two decades, various research topics have been considered for Markovian jump systems with time-varying delays [8–14]. It worth pointing out that when time delay is small enough in linear Markovian jump systems, the delay-dependent criteria are always less conservative than delay-independent ones. Over the past few years, for Markovian jump systems, many important topics related to delay-dependent have been extensively studied [14, 15].

Generally speaking, finite-time stability is investigated to address these transient performances of control systems in finite-time interval. Up to now, the concept of finite-time stability has been revisited with different systems, and many important results are obtained for finite-time stability and finite-time boundedness [21–26]. However, to the best of authors' knowledge, the stochastic finite-time 𝐻_∞ control for Markovian jump systems has not been fully studied. There is some room for next investigation due to the fact that analysis methods in existing references seem still conservative.

The major contribution of this paper is that we introduce a newly Lyapunov-Krasovskii functional for Markovian jump system. Some sufficient conditions are obtained to ensure the finite-time stability and bounded of the closed-loop Markovian jump systems. Compared with traditional methods of MJSs, it is shown the less conservative results can be obtained and the desired 𝐻_∞ control performance is obtained by employing mode-dependent Lyapunov functional instead of mode-independent Lyapunov functional. The finite-time bounded criterion can be dealt with in the terms of LMIs. Finally, the effectiveness of the developed techniques is also illustrated by two numerical examples.

2. Preliminaries

Given the probability space (Ω,𝐹,𝑃), where Ω, 𝐹, and 𝑃 represent the sample space, the algebra of events, and the probability measure defined on 𝐹, respectively, the following Markovian jump systems over the probability space (Ω,𝐹,𝑃) are considered: 𝑥̇(𝑡)=𝐴_𝑟𝑡𝑥(𝑡)+𝐴_𝜏𝑟𝑡𝑥𝑡−𝜏_𝑟𝑡(𝑡)+𝐵_𝑟𝑡𝑢(𝑡)+𝐷_𝑟𝑡𝜔(𝑡),𝑧(𝑡)=𝐶_𝑟𝑡𝑥(𝑡)+𝐶_𝜏𝑟𝑡𝑥𝑡−𝜏_𝑟𝑡(𝑡)+𝐹_𝑟𝑡𝜔(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡=[−ℎ,0], (1) where 𝑥(𝑡)∈ℛ^𝑛 represents the state vector of Markovian jump system, 𝑢(𝑡)∈ℛ^𝑚 is the control input, 𝑧(𝑡)∈ℛ^𝑞 denotes the controlled output and 𝜑(𝑡), 𝑡=[−ℎ,0], where 𝑟₀∈𝒩 is initial condition. 𝜔(𝑘)∈ℛ^𝑞 denotes the disturbance input which satisfies ^𝑇₀𝜔^⊺(𝑡)𝜔(𝑡)𝑑𝑡≤𝑑. (2)

Firstly, taking value on the finite set 𝒩={1,2,…,𝑁}, let the random form process {𝑟_𝑡,𝑡≥0} be the stochastic process with transition rate matrix Ω={𝜋_𝑖𝑗}, 𝑖,𝑗∈𝒩 and let the transition probabilities also be denoted as follows: Pr𝑟_𝑡+Δ=𝑗∣𝑟_𝑡=𝑖=𝜌_𝑖𝑗+𝜋_𝑖𝑗Δ+𝑜(Δ), (3) where 𝜚_𝑖𝑗=0,if1𝑖≠𝑗if𝑖=𝑗, (4) and Δ>0, 𝜋_𝑖𝑗≥0, for 𝑖≠𝑗, denotes the mode 𝑖 in time 𝑡 to time 𝑡+Δ with mode 𝑗, −𝜋_𝑖𝑖=^𝑁_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗 (5) for each mode 𝑖∈𝒩, lim_Δ→0+(𝑜(Δ)/Δ)=0. 𝜏_𝑖(𝑡) denotes the time-varying delay, which satisfies 0<𝜏_𝑖(𝑡)≤ℎ_𝑖𝜏̇<∞,_𝑖(𝑡)≤𝜇_𝑖, (6) where ℎ=max{ℎ_𝑖,𝑖∈𝒩} is the given upper bound of time-varying delays 𝜏_𝑖(𝑡) and 𝜇=max{𝜇_𝑖,𝑖∈𝒩} is the given upper bound of 𝜏̇_𝑖(𝑡). All the matrices are known matrices with the appropriate dimension.

In this paper, the objective is to design a state feedback controller as follows: 𝑢(𝑡)=𝐾_𝑖𝑥(𝑡), (7) where 𝐾_𝑖 is the controller gains to be designed.

Definition 1. System (1) is said to be finite-time bounded with respect to (𝑐₁,𝑐₂,𝑇,𝑅,𝑑), if condition (2) and the following inequality hold: sup_{−ℎ≤𝜐≤0}𝔼{𝑥^⊺(𝜐)𝑅𝑥(𝜐),𝑥̇^⊺(𝜐)𝑅𝑥̇(𝜐)}≤𝑐₁⟹𝔼{𝑥^⊺(𝑡)𝑅𝑥(𝑡)}<𝑐₂,∀𝑡∈[0,𝑇], (8) where 𝑐₂>𝑐₁≥0 and 𝑅>0.

Definition 2 (see [8]). Considering system (1) with the stochastic Lyapunov function 𝑉(𝑥_𝑡,𝑟_𝑡), we get the weak infinitesimal operator as follows: £𝑉𝑥_𝑡,𝑟_𝑡,𝑡=lim_Δ𝑡→01𝔼𝑉𝑥Δ𝑡^𝑡+Δ𝑡,𝑟_𝑡+Δ𝑡𝑥,𝑡+Δ𝑡−𝑉_𝑡=𝜕,𝑖,𝑡𝑉𝑥𝜕𝑡^𝑡+𝜕,𝑖,𝑡𝑉𝑥𝜕𝑥^𝑡𝑥̇()+,𝑖,𝑡𝑡,𝑖_𝑁𝑗=1𝜋_𝑖𝑗𝑉𝑥_𝑡,𝑗,𝑡. (9)

Definition 3. Given a constant scalar 𝑇>0 and for all admissible 𝜔(𝑡) given in condition (2), if the Markovian jump system (1) is finite-time stochastic bounded and controller outputs satisfy condition (7) with attenuation 𝛾>0, 𝔼^𝑇₀𝑧^⊺(𝑡)𝑧(𝑡)𝑑𝑡≤𝛾²𝑒^𝜂𝑇𝔼^𝑇₀𝜔^⊺(𝑡)𝜔(𝑡)𝑑𝑡. (10) The Markovian jump system (1) is called the finite-time stochastic bounded with a disturbance attenuation 𝛾.

Lemma 4 (see [27]). Let 𝑓_𝑖: ℛ^𝑚→ℛ(𝑖=1,2,…,𝑁) have positive values in an open subset 𝒟 of ℛ^𝑚. Then, the reciprocally convex combination of 𝑓_𝑖 over 𝒟 satisfies min_{𝛽𝑖∣𝛽𝑖>0,∑𝑖𝛽𝑖=1𝑖}1𝛽_𝑖𝑓_𝑖(𝑡)=_𝑖𝑓_𝑖(𝑡)+max_{𝑔𝑖,𝑗(𝑡)𝑖≠𝑗}𝑔_𝑖,𝑗(𝑡)𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜𝑔_𝑖,𝑗∶ℛ^𝑚⟶ℛ,𝑔_𝑗,𝑖(𝑡)=𝑔_𝑖,𝑗(𝑡),𝑓_𝑖(𝑡)𝑔_𝑖,𝑗(𝑡)𝑔_𝑖,𝑗(𝑡)𝑓_𝑗(𝑡)≥0. (11)

Lemma 5. For any constant matrix 𝑀∈ℛ^𝑚×𝑚 with 𝑀>0, scalars 𝑎<𝑏≤0, vector function 𝑥∶[𝑎,𝑏]→ℛ^𝑚, such that the integrals in the following are well-defined; then, −𝑎²−𝑏²2^𝑏_𝑎^𝑡_𝑡+𝑠𝑥^⊺(𝑠)𝑀𝑥(𝑠)𝑑𝑠𝑑𝜃≤−^𝑏_𝑎^𝑡_𝑡+𝑠𝑥(𝑠)𝑑𝑠𝑑𝜃^⊺𝑀^𝑏_𝑎^𝑡_𝑡+𝑠𝑥(𝑠)𝑑𝑠𝑑𝜃. (12)

3. Finite-Time 𝐻_∞ Performance Analysis

The issue of stability analysis of Markovian jump system (1) subject to 𝑢(𝑡)=0 is given firstly. Therefore, the finite-time stability is obtained in this section.

Theorem 6. System (1) is called the finite-time bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇), if there exist matrices 𝑃_𝑖>0,𝑄_𝑙𝑖()𝒳>0𝑙=1,2,𝑄>0,_𝑖=𝑋_1𝑖𝑋_2𝑖𝑋_3𝑖𝑋_4𝑖𝑋>0,𝒳=¹𝑋₂𝑋₃𝑋₄𝑌>0,_𝑠>0(𝑠=1,2),𝑍_𝑖>0,𝑍>0,𝐻>0, (13) scalars 𝑐₁<𝑐₂, 𝑇>0, 𝜆_𝑠>0, (𝑠=1,2,…,12), 𝜂>0, and Λ>0, such that ∀𝑖,𝑗∈𝒩 and the inequalities hold as follows: 𝑒^𝑁𝛿ℎ_𝑗=1𝜋_𝑖𝑗𝑄_1𝑗+𝑒^𝑁𝛿ℎ_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗𝑄_2𝑗<𝑄,_𝑁𝑗=1𝜋_𝑖𝑗𝒳_𝑗−𝒳<0,_𝑁𝑗=1𝜋_𝑖𝑗𝑍_𝑗−𝑍<0,𝒳_𝑖ℎ𝒮_𝑖∗𝒳_𝑖ℎ>0,𝑌₁𝑊₁∗𝑌₂>0,𝑌₁𝑊₂∗𝑌₂>0, (14)(15)(16)(17)(18)(19)Ξ_𝑖=Ξ_11𝑖Ξ_12𝑖𝑆_4𝑖Ξ_14𝑖Ξ_15𝑖−𝑆_3𝑖+𝑍_𝑖+𝑍^⊺_𝑖ℎ𝑃_𝑖𝐷_𝑖∗Ξ_22𝑖−𝑆_4𝑖+𝑋_4𝑖ℎ0𝑋_3𝑖+𝑋2ℎ^⊺2𝑖𝑆2ℎ^3𝑖−𝑆^⊺_2𝑖−𝑋_3𝑖−𝑋2ℎ^⊺2𝑖02ℎ∗∗−𝑄_1𝑖−𝑋_4𝑖ℎ−𝑊₂00𝑆^⊺_2𝑖+𝑋_3𝑖+𝑋2ℎ^⊺2𝑖02ℎ∗∗∗Ξ_44𝑖000∗∗∗∗Ξ_55𝑖−𝑆_1𝑖−𝑍_𝑖ℎ²−𝑍^⊺_𝑖ℎ²0𝑋∗∗∗∗∗−^1𝑖ℎ−𝑍_𝑖ℎ²−𝑍^⊺_𝑖ℎ²0∗∗∗∗∗∗−𝛿𝐻<0, (20)𝑐₁Λ+𝑑𝛿𝜆₁₂1𝜂1−𝑒^−𝜂𝑇<𝜆₁𝑒^−𝜂𝑇𝑐₂, (21) where Ξ_11𝑖=^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝛿𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖+𝑒^𝛿ℎ𝑄_1𝑖+𝑄_2𝑖𝑒+ℎ𝑄+^𝛿ℎ−1𝛿𝑋_1𝑖+𝑒^𝛿ℎ−𝛿ℎ𝑒^𝛿ℎ−1𝛿²𝑋₁−𝑋_4𝑖ℎ+ℎ𝑌₁+𝑊₁−𝑍_𝑖−𝑍^⊺_𝑖,Ξ_12𝑖=𝑃_𝑖𝐴_𝜏𝑖+𝑋_4𝑖ℎ−𝑆_4𝑖,Ξ_14𝑖=𝑒^𝛿ℎ−1𝑋2𝛿^2𝑖+𝑋^⊺_3𝑖+𝑒^𝛿ℎ−𝛿ℎ𝑒^𝛿ℎ−12𝛿²𝑋₂+𝑋^⊺₃,Ξ_15𝑖𝑋=−^3𝑖−𝑋2ℎ^⊺2𝑖+𝑍2ℎℎ^𝑖2+𝑍^⊺_𝑖ℎ²,Ξ_22𝑖=−1−𝜇_𝑖𝑄_2𝑖−2𝑋_4𝑖ℎ+𝑆_4𝑖+𝑆^⊺_4𝑖−𝑊₁+𝑊₂,Ξ_44𝑖=𝑒^𝛿ℎ−1𝛿𝑋_4𝑖+𝑒^𝛿ℎ−𝛿ℎ𝑒^𝛿ℎ−1𝛿²𝑋_4𝑖+ℎ𝑌₂+𝑒^𝛿ℎ−𝛿ℎ−1𝛿²𝑍_𝑖+ℎ𝛿²𝑒^𝛿ℎ+𝑒^𝛿ℎ+𝛿ℎ+1𝛿³Ξ𝑍,_55𝑖𝑋=−^4𝑖ℎ−𝑍_𝑖ℎ²−𝑍^⊺_𝑖ℎ²,Λ=𝜆₂+ℎ𝑒^𝛿ℎ𝜆₃+𝜆₄+ℎ²𝑒^𝛿ℎ𝜆₅+ℎ²𝑒^𝛿ℎ𝜆₆+12ℎ³𝑒^𝛿ℎ𝜆₇+ℎ²𝑒^𝛿ℎ𝜆₈+𝜆₉+12ℎ³𝑒^𝛿ℎ𝜆₁₀+16ℎ⁴𝑒^𝛿ℎ𝜆₁₁,𝜆₁=max_𝑖∈𝒩𝜆_max𝑃_𝑖,𝜆₂=max_𝑖∈𝒩𝜆_max𝑃_𝑖,𝜆₃=max_𝑖∈𝒩𝜆_max𝑄_1𝑖,𝜆₄=max_𝑖∈𝒩𝜆_max𝑄_2𝑖,𝜆₅=𝜆_max𝑄,𝜆₆=max_𝑖∈𝒩𝜆_max𝒳_𝑖,𝜆₇=max_𝑖∈𝒩𝜆_max𝒳,𝜆₈=𝜆_max𝑌₁,𝜆₉=𝜆_max𝑌₂,𝜆₁₀=max_𝑖∈𝒩𝜆_max𝑍_𝑖,𝜆₁₁=𝜆_max𝑍,𝜆₁₂=𝜆_max(𝐻),𝑃_𝑖=𝑅^−(1/2)𝑃_𝑖𝑅^−(1/2),𝑄_𝑙𝑖=𝑅^−(1/2)𝑄_𝑙𝑖𝑅^−(1/2)(𝑙=1,2),𝑄=𝑅^−(1/2)𝑄𝑅^−(1/2),𝒳_𝑖=𝑅^−(1/2)𝒳_𝑖𝑅^−(1/2)(𝑙=1,2),𝑋=𝑅^−(1/2)𝑋𝑅^−(1/2),𝑌_𝑠=𝑅^−(1/2)𝑌_𝑠𝑅^−(1/2)(),𝑍𝑠=1,2_𝑖=𝑅^−(1/2)𝑍_𝑖𝑅^−(1/2),𝑍=𝑅^−(1/2)𝑍𝑅^−(1/2). (22)

Proof. Firstly, a novel process is defined in this paper as follows: 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠),𝑠∈[−ℎ,0]. (23) Then, the following Lyapunov-Krasovskii functional is considered: 𝑉𝑥_𝑡,𝑟_𝑡,𝑡=⁵_𝑙=1𝑉_𝑙𝑥_𝑡,𝑟_𝑡,𝑡, (24) where 𝑉₁𝑥_𝑡,𝑟_𝑡,𝑡=𝑥(𝑡)^⊺𝑒^𝛿𝑡𝑃_𝑟𝑡𝑥(𝑡),𝑉₂𝑥_𝑡,𝑟_𝑡,𝑡=^𝑡_𝑡−ℎ𝑒^{𝛿(𝑠+ℎ)}𝑥^⊺(𝑠)𝑄_1𝑟𝑡𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏𝑟𝑡(𝑡)}𝑒^{𝛿(𝑠+ℎ)}𝑥^⊺(𝑠)𝑄_2𝑟𝑡𝑥(𝑠)+𝑑𝑠_{0−ℎ𝑡𝑡+𝜃}𝑒^{𝛿(𝑠+ℎ)}𝑥^⊺(𝑠)(𝑠)𝑉𝑄𝑥𝑑𝑠𝑑𝜃,₃𝑥_𝑡,𝑟_𝑡,𝑡=⁰_−ℎ^𝑡_𝑡+𝜃𝑒^{𝛿(𝑠−𝜃)}𝜂^⊺(𝑠)𝒳_𝑟𝑡𝜂(𝑠)+𝑑𝑠𝑑𝜃_{0−ℎ0𝜃𝑡𝑡+𝜐}𝑒^{𝛿(𝑠−𝜃)}𝜂^⊺(𝑠)(𝑠)𝑉𝒳𝜂𝑑𝑠𝑑𝜐𝑑𝜃,₄𝑥_𝑡,𝑟_𝑡,𝑡=⁰_−ℎ^𝑡_𝑡+𝜃𝑒^{𝛿(𝑠−𝜃)}𝑥^⊺(𝑠)𝑌₁𝑥(𝑠)𝑑𝑠𝑑𝜃+⁰_−ℎ^𝑡_𝑡+𝜃𝑒^{𝛿(𝑠−𝜃)}𝑥̇^⊺(𝑠)𝑌₂𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₅𝑥_𝑡,𝑟_𝑡,𝑡=⁰_−ℎ⁰_𝜃^𝑡_𝑡+𝜐𝑒^{𝛿(𝑠−𝜃)}𝑥̇^⊺(𝑠)𝑍_𝑟𝑡𝑥̇(𝑠)+𝑑𝑠𝑑𝜐𝑑𝜃_{0−ℎ0𝜍0𝜃𝑡𝑡+𝜐}𝑒^{𝛿(𝑠−𝜃)}𝑥̇^⊺(𝑠)×𝑍𝑥̇(𝑠)𝑑𝑠𝑑𝜐𝑑𝜃𝑑𝜍, (25) where 𝜂(𝑡)=[𝑥^⊺(𝑡),𝑥̇^⊺(𝑡)]^⊺.
Letting 𝑖 represent the time 𝑡, that is, 𝑟_𝑡=𝑖∈𝒩, one has £𝑉₁𝑥_𝑡,𝑖,𝑡=𝛿𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑃_𝑖𝑥(𝑡)+2𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑃_𝑖𝑥̇(𝑡)+𝑒^𝛿𝑡𝑥^⊺(𝑡)^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑥(𝑡)=𝑒^𝛿𝑡𝑥^⊺(𝑡)^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝛿𝑃_𝑖𝑥(𝑡)+2𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑃_𝑖×𝐴_𝑖𝑥(𝑡)+𝐴_𝜏𝑖𝑥𝑡−𝜏_𝑖(𝑡)+𝐷_𝑖𝜔(𝑡). (26) Noting 𝜋_𝑖𝑗≥0 for 𝑗≠𝑖 and 𝜋_𝑖𝑖≤0, one has £𝑉₂𝑥_𝑡,𝑖,𝑡=𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑒^𝛿ℎ𝑄_1𝑖+𝑒^𝛿ℎ𝑄_2𝑖+ℎ𝑄𝑥(𝑡)−𝑒^𝛿𝑡𝑥^⊺(𝑡−ℎ)𝑄_1𝑖𝑥()−𝜏̇𝑡−ℎ1−_𝑖(𝑡)𝑒^{𝛿(𝑡+ℎ−𝜏𝑖(𝑡))}𝑥^⊺×𝑡−𝜏_𝑖(𝑡)𝑄_2𝑖𝑥𝑡−𝜏_𝑖(𝑡+_{𝑡𝑡−ℎ})𝑒^𝛿𝑠𝑥^⊺(𝑠)𝑒^𝑁𝛿ℎ_𝑗=1𝜋_𝑖𝑗𝑄_1𝑗𝑥(𝑠)+−𝑄𝑑𝑠_𝑁𝑗=1𝜋_𝑖𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝑒^{𝛿(𝑠+ℎ)}𝑥^⊺(𝑠)𝑄_2𝑗𝑥(𝑠)𝑑𝑠≤𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑒^𝛿ℎ𝑄_1𝑖+𝑒^𝛿ℎ𝑄_2𝑖+ℎ𝑄𝑥(𝑡)−𝑒^𝛿𝑡𝑥^⊺(𝑡−ℎ)𝑄_1𝑖𝑥()−𝜏̇𝑡−ℎ1−_𝑖(𝑡)𝑒^𝛿𝑡𝑥^⊺𝑡−𝜏_𝑖(𝑡)𝑄_2𝑖𝑥𝑡−𝜏_𝑖(𝑡)+^𝑡_𝑡−ℎ𝑒^𝛿𝑠𝑥^⊺(𝑠)𝑒^𝑁𝛿ℎ_𝑗=1𝜋_𝑖𝑗𝑄_1𝑗+𝑒^𝛿ℎ×^𝑁_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗𝑄_2𝑗(𝑠)−𝑄×𝑥𝑑𝑠. (27) It follows from (15) and (28) that £𝑉₂𝑥_𝑡,𝑖,𝑡=𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑒^𝛿ℎ𝑄_1𝑖+𝑒^𝛿ℎ𝑄_2𝑖+ℎ𝑄𝑥(𝑡)−𝑒^𝛿𝑡𝑥^⊺(𝑡−ℎ)𝑄_1𝑖𝑥()−𝜏̇𝑡−ℎ1−_𝑖(𝑡)𝑒^𝛿𝑡𝑥^⊺𝑡−𝜏_𝑖(𝑡)𝑄_2𝑖𝑥𝑡−𝜏_𝑖(𝑡),£𝑉₃𝑥_𝑡,𝑖,𝑡=⁰_−ℎ^𝑡_𝑡+𝜃𝑒^{𝛿(𝑠−𝜃)}𝜂^⊺(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝒳_𝑖𝑗(𝑠)−𝒳×𝜂𝑑𝑠𝑑𝜃+𝑒^𝛿𝑡𝜂^⊺𝒳_𝑖𝜂(𝑡)⁰_−ℎ𝑒^−𝛿𝜐𝑑𝜐−𝑒^𝛿𝑡𝑡_𝑡−ℎ𝜂^⊺(𝑠)𝒳_𝑖𝜂(𝑠)𝑑𝑠+𝑒^𝛿𝑡𝜂^⊺(𝑡)𝒳𝜂(𝑡)⁰_−ℎ⁰_𝜐𝑒^−𝛿𝜐𝑑𝜃𝑑𝜐. (28)(29) By employing Lemma 4, we can obtain that −^𝑡_𝑡−ℎ𝜂^⊺(𝑠)𝒳_𝑖𝜂(𝑠)𝑑𝑠=−^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜂^⊺(𝑠)𝒳_𝑖𝜂(𝑠)𝑑𝑠−^{𝑡−𝜏𝑖(𝑡)}_𝑡−ℎ𝜂^⊺(𝑠)𝒳_𝑖𝜂(𝑠)ℎ𝑑𝑠≤−𝜏_𝑖(𝑡)^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜂(𝑠)𝑑𝑠^⊺𝒳_𝑖ℎ^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜂(𝑠)−ℎ𝑑𝑠ℎ−𝜏_𝑖(𝑡)^{𝑡−𝜏𝑖(𝑡)}_𝑡−ℎ𝜂(𝑠)𝑑𝑠^⊺𝒳_𝑖ℎ^{𝑡−𝜏𝑖(𝑡)}_𝑡−ℎ𝜂(𝑠)𝑑𝑠≤−^{𝑡𝑡−𝜏𝑖(𝑡)}𝜂(𝑠)𝑑𝑠_{𝑡−𝜏𝑖(𝑡)𝑡−ℎ}𝜂(𝑠)𝑑𝑠^⊺𝒳_𝑖ℎ𝒮_𝑖∗𝒳_𝑖ℎ^{𝑡𝑡−𝜏𝑖(𝑡)}𝜂(𝑠)𝑑𝑠_{𝑡−𝜏𝑖(𝑡)𝑡−ℎ}𝜂(𝑠).𝑑𝑠 (30) It follows from (30) and (31) that 𝑉₃𝑥_𝑡,𝑖,𝑡≤𝑒^𝛿𝑡𝜂^⊺(𝑡)×𝑒^𝛿ℎ−1𝛿𝒳_𝑖+𝑒^𝛿ℎ−𝛿ℎ𝑒^𝛿ℎ−1𝛿²𝒳𝜂(𝑡)−𝑒^{𝛿𝑡𝑡𝑡−𝜏𝑖(𝑡)}𝜂(𝑠)𝑑𝑠_{𝑡−𝜏𝑖(𝑡)𝑡−ℎ}𝜂(𝑠)𝑑𝑠^⊺𝒳_𝑖ℎ𝒮_𝑖∗𝒳_𝑖ℎ×^{𝑡𝑡−𝜏𝑖(𝑡)}𝜂(𝑠)𝑑𝑠_{𝑡−𝜏𝑖(𝑡)𝑡−ℎ}𝜂(𝑠).𝑑𝑠 (31) Consider £𝑉₄𝑥_𝑡,𝑖,𝑡=ℎ𝑥^⊺(𝑡)𝑌₁𝑥(𝑡)−^𝑡_𝑡−ℎ𝑥^⊺(𝑠)𝑌₁𝑥(𝑠)𝑑𝑠+ℎ𝑥̇^⊺(𝑡)𝑌₂𝑥̇(𝑡)−^𝑡_𝑡−ℎ𝑥̇^⊺(𝑠)𝑌₂𝑥̇(𝑠)𝑑𝑠. (32) Moreover, the following two zero equalities with any symmetric matrices 𝑊₁ and 𝑊₂ are considered: 0=𝑥^⊺(𝑡)𝑊₁𝑥(𝑡)−𝑥^⊺𝑡−𝜏_𝑖(𝑡)𝑊₁𝑥𝑡−𝜏_𝑖(𝑡)−2^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥^⊺(𝑠)𝑊₁𝑥̇(𝑠)𝑑𝑠,0=𝑥^⊺𝑡−𝜏_𝑖(𝑡)𝑊₂𝑥𝑡−𝜏_𝑖(𝑡)−𝑥^⊺(𝑡−ℎ)𝑊₂𝑥(𝑡−ℎ)−2^{𝑡−𝜏𝑖(𝑡)}_𝑡−ℎ𝑥^⊺(𝑠)𝑊₂𝑥̇(𝑠)𝑑𝑠. (33)(34) With the above two zero equalities (34) and (35), an upper bound of £𝑉₄(𝑥_𝑡,𝑖,𝑡) is £₄𝑥_𝑡,𝑖,𝑡=𝑥^⊺(𝑡)ℎ𝑌₁+𝑊₁𝑥(𝑡)+ℎ𝑥̇^⊺(𝑡)𝑌₂𝑥̇(𝑡)+𝑥^⊺𝑡−𝜏_𝑖(𝑡)𝑊₂−𝑊₁𝑥𝑡−𝜏_𝑖(𝑡)−𝑥^⊺(𝑡−ℎ)𝑊₂𝑥()−𝑡−ℎ_{𝑡𝑡−𝜏𝑖(𝑡)}𝑥(𝑠)𝑥̇(𝑠)^⊺𝑌₁𝑊₁∗𝑌₂𝑥(𝑠)𝑥̇(𝑠)−^{𝑡−𝜏𝑖(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑥̇(𝑠)^⊺𝑌₁𝑊₁∗𝑌₂𝑥(𝑠)𝑥̇(𝑠). (35) From (19), (20), and (36), one can obtain £₄𝑥_𝑡,𝑖,𝑡=𝑥^⊺(𝑡)ℎ𝑌₁+𝑊₁𝑥(𝑡)+ℎ𝑥̇^⊺(𝑡)𝑌₂𝑥̇(𝑡)+𝑥^⊺𝑡−𝜏_𝑖(𝑡)𝑊₂−𝑊₁𝑥𝑡−𝜏_𝑖(𝑡)−𝑥^⊺(𝑡−ℎ)𝑊₂𝑥(𝑡−ℎ). (36) Now, £𝑉₅(𝑥_𝑡,𝑖,𝑡) is obtained as follows: £𝑉₅𝑥_𝑡=,𝑖,𝑡_{0−ℎ0𝜃𝑡𝑡+𝜐}𝑒^{𝛿(𝑠−𝜃)}𝑥̇^⊺(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑍_𝑗×𝑥̇(𝑠)−𝑍𝑑𝑠𝑑𝜐𝑑𝜃+𝑒^𝛿𝑡𝑥̇^⊺(𝑡)𝑍_𝑖𝑥̇(𝑡)⁰_−ℎ⁰_𝜃𝑒^−𝛿𝜐𝑑𝜐𝑑𝜃−𝑒^𝛿𝑡0_−ℎ^𝑡_𝑡+𝜃𝑥̇^⊺(𝑠)𝑍_𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃+𝑒^𝛿𝑡𝑥̇^⊺(𝑡)𝑍𝑥̇(𝑡)⁰_−ℎ⁰_𝜍⁰_𝜃𝑒^−𝛿𝜃𝑑𝜐𝑑𝜃𝑑𝜍. (37) By using Lemma 5, one has −⁰_−ℎ^𝑡_𝑡+𝜃𝑥̇^⊺(𝑠)𝑍_𝑖𝑥̇(𝑠)2𝑑𝑠𝑑𝜃≤−ℎ^{20−ℎ𝑡𝑡+𝜃}𝑥̇^⊺(𝑠)𝑑𝑠𝑑𝜃𝑍_𝑖⁰_−ℎ^𝑡_𝑡+𝜃𝑥̇(𝑠)𝑥(𝑡)−1𝑑𝑠𝑑𝜃=−2ℎ^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥(𝑠)1𝑑𝑠−ℎ^{𝑡−𝜏𝑖(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠^⊺𝑍_𝑖×𝑥(𝑡)−1ℎ^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥(𝑠)1𝑑𝑠−ℎ^{𝑡−𝜏𝑖(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠. (38) Together with (38) and (39), it implies that £𝑉₅𝑥_𝑡,𝑖,𝑡≤𝑒^𝛿𝑡𝑥̇^⊺(𝑡)𝑒^𝛿ℎ−𝛿ℎ−1𝛿²𝑍_𝑖𝑥̇(𝑡)+𝑒^𝛿𝑡𝑥̇^⊺(𝑡)ℎ𝛿²𝑒^𝛿ℎ+𝑒^𝛿ℎ+𝛿ℎ+1𝛿³𝑍𝑥̇(𝑡)−2𝑒^𝛿𝑡𝑥(𝑡)−1ℎ^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥(𝑠)1𝑑𝑠−ℎ^{𝑡−𝜏𝑖(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠^⊺𝑍_𝑖×𝑥(𝑡)−1ℎ^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥(𝑠)1𝑑𝑠−ℎ^{𝑡−𝜏𝑖(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠. (39) From (26)–(40), we can eventually obtain £𝑉𝑥_𝑡,𝑟_𝑡,𝑡−𝛿𝜔^⊺(𝑡)𝐻𝜔(𝑡)≤𝑒^𝛿𝑡𝜉^⊺(𝑡)Ξ_𝑖𝜉(𝑡), (40) where 𝜉^⊺(𝑡)=𝑥^⊺(𝑡),𝑥^⊺𝑡−𝜏_𝑖(𝑡),𝑥^⊺(𝑡−ℎ),^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥^⊺(𝑠)𝑑𝑠,_{𝑡−𝜏𝑖(𝑡)𝑡−ℎ}𝑥^⊺(𝑠)𝑑𝑠,𝜔^⊺(𝑡). (41) It follows from (45) that 𝔼£𝑉𝑥_𝑡,𝑟_𝑡,𝑡≤𝔼𝜂𝑉𝑥_𝑡,𝑟_𝑡,𝑡+𝛿𝜔^⊺(𝑡)𝐻𝜔(𝑡). (42) Multiplying the above inequality by 𝑒^−𝜂𝑡 yields that 𝔼£𝑒^−𝜂𝑡𝑉𝑥_𝑡,𝑟_𝑡,𝑡≤𝑒^−𝜂𝑡𝛿𝜔^⊺(𝑡)𝐻𝜔(𝑡). (43) Integrating the inequality from 0 to 𝑡, we have 𝑒^−𝜂𝑡𝔼𝑉𝑥_𝑡,𝑟_𝑡,𝑡−𝔼𝑉𝑥₀,𝑟₀,0≤𝛿^𝑡₀𝑒^−𝜂𝑠𝜔^⊺(𝑠)𝐻𝜔(𝑠)𝑑𝑠. (44) Denoting 𝑃_𝑖=𝑅^−(1/2)𝑃_𝑖𝑅^−(1/2), 𝑄_𝑖=𝑅^−(1/2)𝑄_𝑖𝑅^−(1/2), 𝑄=𝑅^−(1/2)𝑄𝑅^−(1/2), 𝒳_𝑖=𝑅^−(1/2)𝒳_𝑖𝑅^−(1/2), 𝒳=𝑅^−(1/2)𝒳𝑅^−(1/2), 𝑌_𝑖=𝑅^−(1/2)𝑌_𝑖𝑅^−(1/2), 𝑌=𝑅^−(1/2)𝑌𝑅^−(1/2), 𝑍_𝑖=𝑅^−(1/2)𝑍_𝑖𝑅^−(1/2), and 𝑍=𝑅^−(1/2)𝑍𝑅^−(1/2) yields that 𝔼𝑉𝑥₀,𝑟₀,0≤max_𝑖∈𝒩𝜆_max𝑃_𝑖𝑥^⊺(0)(0)+𝑅𝑥max_𝑖∈𝒩𝜆_max𝑄_1𝑖+max_𝑖∈𝒩𝜆_max𝑄_2𝑖𝑒^𝛿ℎ×⁰_−ℎ𝑒^𝛿𝑠𝑥^⊺(𝑠)𝑅𝑥(𝑠)𝑑𝑠+𝑒^𝛿ℎ𝜆_max(𝑄)⁰_−ℎ⁰_𝜃𝑒^𝛿𝑠𝑥^⊺(𝑠)𝑅𝑥(𝑠)𝑑𝑠+𝑒^𝛿ℎmax_𝑖∈𝒩𝜆_max𝒳_𝑖⁰_−ℎ⁰_𝜃𝑒^−𝛿𝜃𝜂^⊺(𝑠)𝑅𝜂(𝑠)𝑑𝑠𝑑𝜃+𝑒^𝛿ℎ𝜆_max(𝒳)⁰_−ℎ⁰_𝜃⁰_𝜐𝑒^−𝛿𝜃𝜂^⊺(𝑠)(𝑠)+𝜆𝑅𝜂𝑑𝑠𝑑𝜃𝑑𝜐_max𝑌₁+𝜆_max𝑌₂𝑒^𝛿ℎ×⁰_−ℎ⁰_𝜃𝑒^−𝛿𝜃𝑥^⊺(𝑠)𝑅𝑥(𝑠)𝑑𝑠𝑑𝜃+𝑒^𝛿ℎmax_𝑖∈𝒩𝜆_max𝑍_𝑖⁰_−ℎ⁰_𝜃⁰_𝜐𝑒^−𝛿𝜃𝑥^⊺(𝑠)𝑅𝑥(𝑠)𝑑𝑠𝑑𝜃+𝑒^𝛿ℎ𝜆_max(𝑍)⁰_−ℎ⁰_𝜍⁰_𝜃⁰_𝜐𝑒^−𝛿𝜃𝑥^⊺(𝑠)(𝑠)≤𝑅𝑥𝑑𝑠𝑑𝜃𝑑𝜍max_𝑖∈𝒩𝜆_max𝑃_𝑖+ℎ𝑒^𝛿ℎ×max_𝑖∈𝒩𝜆_max𝑄_1𝑖+max_𝑖∈𝒩𝜆_max𝑄_2𝑖+ℎ²𝑒^𝛿ℎ𝜆_max(𝑍)+ℎ²𝑒^𝛿ℎmax_𝑖∈𝒩𝜆_max𝒳_𝑖+12ℎ³𝑒^𝛿ℎ𝜆_max𝒳+ℎ²𝑒^𝛿ℎ𝜆_max𝑌₁+𝜆_max𝑌₂+12ℎ³𝑒^𝛿ℎmax_𝑖∈𝒩𝜆_max𝑍_𝑖+16ℎ⁴𝑒^𝛿ℎ𝜆_max(𝑍)×sup_{−ℎ≤𝑠≤0}{𝑥^⊺(𝑠)𝑅𝑥(𝑠),𝑥̇^⊺(𝑠)𝑅𝑥̇(𝑠)}=𝑐₁Λ. (45) For scalars 𝜂>0 and 𝑇≥𝑡≥0, (46) turns out to be 𝔼𝑉𝑥_𝑡,𝑟_𝑡,𝑡≤𝔼𝑒^𝜂𝑡𝑉𝑥₀,𝑟₀,0+𝑒^𝜂𝑡𝛿^𝑡₀𝑒^−𝜂𝑠𝜔^⊺(𝑠)𝐻𝜔(𝑠)𝑑𝑠≤𝑒^𝜂𝑇𝑐₁Λ+𝑑𝛿𝑒^𝜂𝑇𝜆_max(𝐻)^𝑇₀𝑒^−𝜂𝑠𝑑𝑠≤𝑒^𝜂𝑇𝑐₁Λ+𝑑𝛿𝜆₁₂1𝜂1−𝑒^−𝜂𝑇. (46) To illustrate the bounded, (26) takes the following form: 𝔼𝑉𝑥_𝑡,𝑟_𝑡,𝑡≥𝔼𝑥^⊺(𝑡)𝑒^𝜆𝑡𝑃_𝑖𝑥(𝑡)≥max_𝑖∈𝒩𝜆_min𝑃_𝑖𝔼𝑥^⊺(𝑡)𝑅𝑥(𝑡)=𝜆₁𝔼𝑥^⊺(𝑡)𝑅𝑥(𝑡). (47) From inequalities (46)–(48), one has 𝔼𝑥^⊺(𝑡)(𝑡≤𝑒𝑅𝑥)^𝜂𝑇𝜆₁𝑐₁Λ+𝑑𝛿𝜆₁₂1𝜂1−𝑒^−𝜂𝑇. (48) Finally, inequalities (24) and (49) guarantee that 𝔼𝑥^⊺(𝑡)𝑅𝑥(𝑡)<𝑐₂. (49)
Therefore, the Markovian jump system (1) is finite-time stochastic bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇).

Remark 7. It should be noted that 𝜏_𝑖(𝑡) and 𝜏̇_𝑖(𝑡) may, respectively, get the different upper bound due to the fact that condition (6) holds. However, 𝜏_𝑖(𝑡) and 𝜏̇_𝑖(𝑡) always lead to conservativeness for 𝜏_𝑖(𝑡)≤ℎ=max{ℎ_𝑖,𝑖∈𝒩} and 𝜏̇_𝑖(𝑡)≤𝜇=max{𝜇_𝑖,𝑖∈𝒩} in [14–18], and this case can be improved with employing the different Lyapunov-Krasovskii functional (26).

Remark 8. It should be pointed out that, in Theorem 6, the novelty of the Lyapunov functional (26) lies in the following: (i) triple-integral terms 𝑉₃(𝑥_𝑡,𝑟_𝑡,𝑡) and 𝑉₅(𝑥_𝑡,𝑟_𝑡,𝑡) and four-integral term 𝑉₅(𝑥_𝑡,𝑟_𝑡,𝑡) are introduced and (ii) the distinct Lyapunov matrices (𝑃_𝑖,𝑄_1𝑖,𝑄_2𝑖,𝒳_𝑖,𝑍_𝑖) are chosen for different system modes 𝑖(𝑖=1,2,…,𝑁).
For the condition 𝑟_𝑡=𝑖, the Markovian jump system given in this paper is followed by 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐴_𝜏𝑖𝑥𝑡−𝜏_𝑟𝑡(𝑡)+𝐷_𝑖𝜔(𝑡),𝑧(𝑡)=𝐶_𝑖𝑥(𝑡)+𝐶_𝜏𝑖𝑥𝑡−𝜏_𝑖(𝑡)+𝐹_𝑖𝜔(𝑡), (50) where 𝐴_𝑖=𝐴_𝑖+𝐵_𝑖𝐾_𝑖. (51)

Theorem 9. System (53) is finite-time stochastic bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇) with a disturbance attenuation, if there exist matrices 𝑃_𝑖>0,𝑄_𝑙𝑖()𝒳>0𝑙=1,2,𝑄>0,_𝑖=𝑋_1𝑖𝑋_2𝑖𝑋_3𝑖𝑋_4𝑖𝑋>0,𝒳=¹𝑋₂𝑋₃𝑋₄𝑌>0,_𝑠>0(𝑠=1,2),𝑍_𝑖>0,𝑍>0, (52) scalars 𝑐₁<𝑐₂, 𝑇>0, 𝜆_𝑠>0, (𝑠=1,2,…,12), 𝜂>0 and Λ>0, such that for all 𝑖,𝑗∈𝒩, (15)–(20) and the following inequalities hold:Σ_𝑖=Σ_11𝑖Ξ_12𝑖𝑆_4𝑖Ξ_14𝑖Ξ_15𝑖−𝑆_3𝑖+𝑍_𝑖+𝑍^⊺_𝑖ℎ𝑃_𝑖𝐷_𝑖𝐶^⊺_𝑖∗Ξ_22𝑖−𝑆_4𝑖+𝑋_4𝑖ℎ0𝑋_3𝑖+𝑋2ℎ^⊺2𝑖𝑆2ℎ^3𝑖−𝑆^⊺_2𝑖−𝑋_3𝑖−𝑋2ℎ2ℎ^⊺2𝑖0𝐶^⊺_𝜏𝑖∗∗−𝑄_1𝑖−𝑋_4𝑖ℎ−𝑊₂00𝑆^⊺_2𝑖+𝑋_3𝑖+𝑋2ℎ2ℎ^⊺2𝑖00∗∗∗Ξ_44𝑖0000∗∗∗∗Ξ_55𝑖−𝑆_1𝑖−𝑍_𝑖ℎ²−𝑍^⊺_𝑖ℎ²𝑋00∗∗∗∗∗−^1𝑖ℎ−𝑍_𝑖ℎ²−𝑍^⊺_𝑖ℎ²00∗∗∗∗∗∗−𝛾²𝐼𝐹^⊺_𝑖∗∗∗∗∗∗∗−𝐼<0, (53)𝑐₁Λ+𝑑𝛾²1𝜂1−𝑒^−𝜂𝑇<𝜆₁𝑒^−𝜂𝑇𝑐₂, (54) where Σ_11𝑖=^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝛿𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖+𝑒^𝛿ℎ𝑄_1𝑖+𝑄_2𝑖𝑒+ℎ𝑄+^𝛿ℎ−1𝛿𝑋_1𝑖+𝑒^𝛿ℎ−𝛿ℎ𝑒^𝛿ℎ−1𝛿²𝑋₁−𝑋_4𝑖ℎ+ℎ𝑌₁+𝑊₁−𝑍_𝑖−𝑍^⊺_𝑖. (55)

Proof. Considering the Lyapunov-Krasovskii functional in Theorem 6 and from Schur's Lemma, it turns out to be £𝑉𝑥_𝑡,𝑖,𝑡+𝑧^⊺(𝑡)𝑧(𝑡)−𝛾²𝜔^⊺(𝑡)𝜔(𝑡)≤𝜉^⊺(𝑡)Θ_𝑖𝜇_𝑝𝑖,ℎ_𝑞𝑖𝜉(𝑡). (56)
Thanks to (54), we have 𝔼£𝑉𝑥_𝑡,𝑖,𝑡≤𝔼𝜂𝑉𝑥_𝑡,𝑖,𝑡+𝛾²𝜔^⊺(𝑡)𝜔(𝑡)−𝔼𝑧^⊺(𝑡)𝑧(𝑡). (57) Multiplying the (58) by 𝑒^−𝜂𝑡, (58) can be written as 𝔼£𝑒^−𝜂𝑡𝑉𝑥_𝑡,𝑖,𝑡≤𝑒^−𝜂𝑡𝛾²𝜔^⊺(𝑡)𝜔(𝑡)−𝑧^⊺(𝑡)𝑧(𝑡). (58) Under the condition of zero initial and 𝔼[𝑉(𝑥_𝑡,𝑖,𝑡)]>0, one has ^𝑇₀𝑒^−𝜂𝑡𝛾²𝜔^⊺(𝑡)𝜔(𝑡)−𝑧^⊺(𝑡)𝑧(𝑡)𝑑𝑡≤𝔼^𝑇₀£𝑒^−𝜂𝑡𝑉𝑥_𝑡,𝑖,𝑡𝑑𝑡≤𝑉𝑥₀,𝑟₀,0=0. (59) Using the Dynkin formula, it results that 𝔼^𝑇₀𝑒^−𝜂𝜐𝑧^⊺(𝜐)𝑧(𝜐)𝑑𝜐≤𝛾²𝔼^𝑇₀𝑒^−𝜂𝜐𝜔^⊺(𝜐)𝜔(𝜐)𝑑𝜐. (60) Finally, it is easy to obtains that 𝔼^𝑇₀𝑧^⊺(𝜐)𝑧(𝜐)𝑑𝜐≤𝛾²𝑒^𝜂𝑇𝔼^𝑇₀𝜔^⊺(𝜐)𝜔(𝜐)𝑑𝜐. (61) Therefore, the Markovian jump system (53) is finite-time stochastic bounded with an performance 𝛾.

4. Finite-Time 𝐻_∞ Control

Theorem 10. System (53) is finite-time stochastic bounded with respects to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇) with an disturbance attenuation, if there exists matrices 𝑃_𝑖>0,𝑃_𝑖,𝑄_𝑙𝑖>0(𝑙=1,2),𝐾_𝑖𝒳,𝑄>0,_𝑖=𝑋_1𝑖𝑋_2𝑖𝑋_3𝑖𝑋_4𝑖𝑋>0,𝒳=¹𝑋₂𝑋₃𝑋₄𝑌>0,_𝑠>0(𝑠=1,2),𝑍_𝑖>0,𝑍>0, (62) scalars 𝑐₁<𝑐₂, 𝑇>0, 𝜆_𝑠>0, (𝑠=1,2,…,12), 𝜂>0 and Λ>0, such that for all 𝑖,𝑗∈𝒩, (15)–(20) and the following inequalities hold:Θ_11𝑖Ξ_12𝑖𝑆_4𝑖Ξ_14𝑖Ξ_15𝑖−𝑆_3𝑖+𝑍_𝑖+𝑍^⊺_𝑖ℎ𝑃_𝑖𝐷_𝑖𝐶^⊺_𝑖∗Ξ_22𝑖−𝑆_4𝑖+𝑋_4𝑖ℎ0𝑋_3𝑖+𝑋2ℎ^⊺2𝑖𝑆2ℎ^3𝑖−𝑆^⊺_2𝑖−𝑋_3𝑖−𝑋2ℎ2ℎ^⊺2𝑖0𝐶^⊺_𝜏𝑖∗∗−𝑄_1𝑖−𝑋_4𝑖ℎ−𝑊₂00𝑆^⊺_2𝑖+𝑋_3𝑖+𝑋2ℎ2ℎ^⊺2𝑖00∗∗∗Ξ_44𝑖0000∗∗∗∗Ξ_55𝑖−𝑆_1𝑖−𝑍_𝑖ℎ²−𝑍^⊺_𝑖ℎ²𝑋00∗∗∗∗∗−^1𝑖ℎ−𝑍_𝑖ℎ²−𝑍^⊺_𝑖ℎ²00∗∗∗∗∗∗−𝛾²𝐼0∗∗∗∗∗∗∗−𝐼<0, (63)𝑃_𝑖𝐵_𝑖=𝐵_𝑖𝑃_𝑖𝑐₁Λ+𝑑𝛾²1𝜂1−𝑒^−𝜂𝑇<𝜆₁𝑒^−𝜂𝑇𝑐₂, (64)(65) where Θ_11𝑖=^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝛿𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖+𝐴^⊺_𝑖𝑃_𝑖+𝐾^⊺_𝑖𝐵^⊺_𝑖+𝑒^𝛿ℎ𝑄_1𝑖+𝑄_2𝑖𝑒+ℎ𝑄+^𝛿ℎ−1𝛿𝑋_1𝑖+𝑒^𝛿ℎ−𝛿ℎ𝑒^𝛿ℎ−1𝛿²𝑋₁−𝑋_4𝑖ℎ+ℎ𝑌₁+𝑊₁−𝑍_𝑖−𝑍^⊺_𝑖, (66) and the state feedback gain matrices considered in this paper could be designed as follows: 𝐾_𝑖=𝑃_𝑖⁻¹𝐾_𝑖,∀𝑖=1,2,…,𝑁. (67)

Proof. This proof can be completed in view of Theorem 9 with 𝑃_𝑖𝐵_𝑖=𝐵_𝑖𝑃_𝑖 and 𝑃_𝑖𝐾_𝑖=𝐾_𝑖.

5. Illustrative Example

Example 1. Considering the following example with parameters 𝐴₁=−0.90.5−0.32−0.8,𝐴_𝜏1=,𝐵−0.5−0.30.3−0.2₁=−1.050.8−0.15−1.3,𝐶₁=,0.6−0.40.35−0.41 (68) the transition probabilities matrix is given as follows: Ω=.−0.20.20.8−0.8 (69)
Given the different upper bounds of ℎ and 𝛿, the results of the maximum upper bound of decay rates 𝛿 and maximum values of ℎ for different time delays are obtained in Tables 1 and 2, respectively. This example indicates fully that the method proposed in the paper plays an important role in reducing conservatism. It can be also seen that our results in this paper show significant improvement over the results obtained in [11, 12]. This clearly shows that our results have less conservatism in the above two cases.

Table 1 

Comparison of the upper bounds of the decay rate for different delays.

Table 2 

Comparison of the allowable values of time delay ℎ for different decay rates.

Example 2. Consider the Markovian jump system (1) where 𝐴₁=−0.81.523,𝐴_𝜏1=,𝐵−0.451−0.52₁=−10.20.5−0.1,𝐷₁=,𝐶0.20.1₁=0.2000.1,𝐶_𝜏1=,𝐸0.0300.010.02₁=0.0200.010.01,𝐷₁=,𝐴0.010.001₂=−21.214,𝐴_𝜏2=,𝐵−11.20−0.5₂=−110.5−2,𝐷₂=,𝐶0.20.3₂=0.10.0200.1,𝐶_𝜏2=,𝐸0.0200.10.02₂=0.0400.10.01,𝐹₂=,0.040.01 (70) and corresponding transition rate matrix is Ω=.−1.21.21−1 (71)
Assuming that 𝑅=𝐼, 𝑇=2, 𝑐₁=1, and 𝑑=0.01, by suing LMI toolbox, Theorem 10 provides the following controller gains: 𝐾₁=,𝐾−11.5351−8.121013.223010.5612₂=.−21.212315.5613−23.231816.4518 (72)

6. Conclusions

We have presented the problems of finite-time stochastic 𝐻_∞ performance analysis of continuous-time systems with random abrupt changes in this paper. By using a different Lyapunov-Krasovskii functional, several sufficient conditions are provided to ensure the Markovian jump system is finite-time stochastic bounded. The controller gains can be dealt with by LMIs toolbox and optimization techniques. At last, two numerical examples are proposed to illustrate the effective and advantage of the developed theories.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Basic Research Program of China (2010CB732501) and the Fund of Sichuan Provincial Key Laboratory of signal and information processing, Xihua University (SZJJ2009-002, SGXZD0101-10-1).