On Stochastic Finite-Time Control of Discrete-Time Fuzzy Systems with Packet Dropout

Abstract

This paper is concerned with the stochastic finite-time stability and stochastic finite-time boundedness problems for one family of fuzzy discrete-time systems over networks with packet dropout, parametric uncertainties, and time-varying norm-bounded disturbance. Firstly, we present the dynamic model description studied, in which the discrete-time fuzzy T-S systems with packet loss can be described by one class of fuzzy Markovian jump systems. Then, the concepts of stochastic finite-time stability and stochastic finite-time boundedness and problem formulation are given. Based on Lyapunov function approach, sufficient conditions on stochastic finite-time stability and stochastic finite-time boundedness are established for the resulting closed-loop fuzzy discrete-time system with Markovian jumps, and state-feedback controllers are designed to ensure stochastic finite-time stability and stochastic finite-time boundedness of the class of fuzzy systems. The stochastic finite-time stability and stochastic finite-time boundedness criteria can be tackled in the form of linear matrix inequalities with a fixed parameter. As an auxiliary result, we also give sufficient conditions on the stochastic stability of the class of fuzzy T-S systems with packet loss. Finally, two illustrative examples are presented to show the validity of the developed methodology.

1. Introduction

Systems where feedback loops are closed via digital communication channel are often regarded as networked control systems (NCSs). Compared with traditional point-to-point controller architectures, the advantages of NCSs include low cost, easy maintenance, and increased system flexibility, and so on. However, the insertion of the communication channels creates discrepancies between the data records to be transmitted and their associated remotely transmitted images and hence raises new interesting problems such as quantization, time delays, and data packet loss, see the references [1–4] and the references cited therein. Among a number of issues arising from such a framework, packet loss of NCSs is an important issue to be addressed and has been receiving great attentions. For instance, Xiong and Lam [5] studied the problem of stability and stabilization of linear systems over networks with bounded packet loss. Bakule and De La Sen [6] tackled the problem of decentralized stabilization of networked complex composite systems with nonlinear perturbations. Wang and Yang [7] investigated the problem of state-feedback control synthesis for networked control systems with packet dropout. For more details of the literature related to networked problems with packet dropout, the reader is referred to [8–16] and the references therein. It is worth noting that Takagi-Sugeno (T-S) fuzzy model has been recognized as a popular and powerful tool in approximating and describing complex nonlinear systems [17] in the sense that it can be able to achieve any degree of accuracy in any convex compact set. The T-S fuzzy model is appealing since it combines the flexible fuzzy logic theory and fruitful linear multivariable system theory into a unified framework and then carries out the control design and analysis via Lyapunov function-based approaches. By choosing appropriate Lyapunov functions including a common Lyapunov function, piecewise Lyapunov function, or fuzzy (nonquadratic) Lyapunov function, the stability and stabilization of the class of fuzzy systems can be determined by solving a set of linear matrix inequalities (LMIs), for instance, see [18–24] and the references cited therein.

It is worth pointing out that classical control theory focuses mainly on the asymptotic behavior of the systems, which, just as was mentioned above, deals with the asymptotic property of system trajectories over an infinite time interval and does not usually specify bounds on the trajectories. In many practical applications, however, many concerned problems are the practical ones which described system state that does not exceed some bound during some time interval, for example, large values of the state are not acceptable in the presence of saturations [25]. In order to deal with these transient performance of control systems, finite-time stability or short-time stability was presented in [26]. Applying linear matrix inequality theory, some appealing results were obtained to ensure finite-time stability, finite-time boundedness, and finite-time stabilization of various systems including linear systems, nonlinear systems, fuzzy systems, and stochastic systems. For instance, Zhang and An [27] addressed the problem of finite-time stability and stabilization of linear stochastic systems. Ambrosino et al. [28] investigated finite-time stability of impulsive dynamic systems. For more details of the literature related to finite-time stability, the reader is referred to [29–35], and the references therein. However, to date and to the best of our knowledge, the problems of stochastic finite-time stability and stabilization of fuzzy systems with packet loss have not been investigated, although results related to fuzzy control systems over networks with packet loss are available in the existing literature, see the references [14–16], which motivates the main purpose of our study.

In this paper, we tackle the stochastic finite-time stability (SFTS) and stochastic finite-time boundedness (SFTB) problems for one family of fuzzy discrete-time systems over networks with packet dropout, parametric uncertainties, and time-varying norm-bounded disturbance. Firstly, we present dynamic model description studied, which, if the data packet loss is assumed to be a time-homogenous Markov process, the discrete-time fuzzy T-S systems with packet loss can be regarded as one class of fuzzy Markovian jump systems. Markov jump systems are referred to as a special family of hybrid systems and stochastic systems, which are very appropriate to model plants whose structure is subject to random abrupt changes, see the papers [36–41]. Thus, the class of fuzzy systems studied can be disposed by the theoretical framework of Markov jumps systems. Then, the concepts of SFTS and SFTB and problem formulation are given. The main contribution of this paper is to design a state-feedback controller which guarantees the resulting closed-loop fuzzy discrete-time system with Markovian jumps SFTS and SFTB. The SFTS and SFTB criteria can be tackled in the form of LMIs with a fixed parameter. As an auxiliary result, sufficient conditions on the stochastic stability of the class of fuzzy T-S systems with packet loss are also presented.

The rest of this paper is organized as follows. Section 2 devotes to the dynamic model description and problem formulation. The results on the SFTS and SFTB are given in Section 3. Section 4 presents numerical examples to demonstrate the validity of the proposed methodology. In Section 5, some conclusions are drawn.

Notations 1. The notation used throughout the paper is fairly standard, ℝ^𝑛, ℝ^𝑛×𝑚, and ℤ⁺ denoting the sets of 𝑛 component real vectors, 𝑛×𝑚 real matrices, and the set of nonnegative integers, respectively. The superscript 𝑇 stands for matrix transposition or vector and 𝔼{⋅} denotes the expectation operator with respect to some probability measure ℙ. In addition, the symbol * denotes the transposed elements in the symmetric positions of a matrix. 𝜆_min(𝑃) and 𝜆_max(𝑃) denote the smallest and the largest eigenvalue of matrix 𝑃, respectively. Notations sup and inf denote the supremum and infimum, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation and Preliminaries

2.1. Physical Plant

The fuzzy model proposed by Takagi and Sugeno [17] is described by fuzzy IF-THEN rules, which represents local near input-output relations of a nonlinear system. The rules of the T-S fuzzy models are of the following forms, here DFS denotes the the discrete-time fuzzy system.

Fuzzy Rule 𝑖
IF 𝜃₁(𝑘) is 𝑀_𝑖1 and 𝜃₂(𝑘) is 𝑀_𝑖2⋯𝜃_𝑐(𝑘) is 𝑀_𝑖𝑐, THEN 𝑥(𝑘+1)=𝐴_𝑖+Δ𝐴_𝑖(𝑘)𝑥(𝑘)+𝐵_𝑖+Δ𝐵_𝑖(𝑘)𝑢(𝑘)+𝐺_𝑖+Δ𝐺_𝑖(𝑘)𝑤(𝑘), (2.1) where 𝑖∈𝕄∶={1,2,…,𝑟}, 𝑥(𝑘)∈ℝ^𝑛 is the state, 𝑢(𝑘)∈ℝ^𝑚 is the control input, 𝑟 is the number of IF-THEN rule, 𝑀_𝑖1,𝑀_𝑖2,…,𝑀_𝑖𝑐 are fuzzy variable, and 𝜃(𝑘)=[𝜃₁(𝑘),𝜃₂(𝑘),…,𝜃_𝑐(𝑘)] is the premise variables vector. It is assumed the premise variable does not depend on the input variable 𝑢(𝑘), which is employed to avoid a complicated defuzzification process of fuzzy controllers [22]. The noise signal 𝑤(𝑘)∈ℝ^𝑙 satisfies

^∞_𝑘=0𝑤^𝑇(𝑘)𝑤(𝑘)≤𝑑²,𝑑≥0. (2.2) The matrices Δ𝐴_𝑖(𝑘),Δ𝐵_𝑖(𝑘), and Δ𝐺_𝑖(𝑘) are uncertain matrices and satisfyΔ𝐴_𝑖(𝑘),Δ𝐵_𝑖(𝑘),Δ𝐺_𝑖(𝑘)=𝐹_𝑖Δ_𝑖(𝑘)𝐸_1𝑖,𝐸_2𝑖,𝐸_3𝑖, (2.3) where Δ_𝑖(𝑘) is an unknown, time-varying matrix function and satisfiesΔ^𝑇_𝑖(𝑘)Δ_𝑖(𝑘)≤𝐼,∀𝑘∈ℤ⁺. (2.4) Given a pair of (𝑥(𝑘),𝑢(𝑘)), the final outputs of the fuzzy systems are inferred as follows:𝑥(𝑘+1)=^r_𝑖=1ℎ_𝑖(𝜃(𝑘))𝐴_𝑖+Δ𝐴_𝑖(𝑘)𝑥(𝑘)+𝐵_𝑖+Δ𝐵_𝑖(𝑘)𝑢(𝑘)+𝐺_𝑖+Δ𝐺_𝑖(𝑘)𝑤(𝑘), (2.5) where ℎ_𝑖(𝜃(𝑘)) are fuzzy basis function given byℎ_𝑖(𝜃(𝑘=𝑤))^𝑖(𝜃(𝑘))∑^𝑟_𝑖=1𝑤_𝑖(𝜃(𝑘)),𝑤_𝑖(𝜃(𝑘))=^𝑐_𝑗=1𝑀_𝑖𝑗𝜃_𝑗(𝑘), (2.6) where 𝑀_𝑖𝑗(𝜃_𝑗(𝑘)) represents the grade of membership of 𝜃_𝑗(𝑘) in 𝑀_𝑖𝑗. Then, it can be seen that𝑤_𝑖(𝜃(𝑘))≥0,𝑖∈𝕄,^𝑟_𝑖=1𝑤_𝑖(𝜃(𝑘))>0,∀𝑘≥0. (2.7) Therefore, we can obtainℎ_𝑖(𝜃(𝑘))≥0,𝑖∈𝕄,^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑘))=1,∀𝑘≥0. (2.8)

2.2. Communication Links and Controller

Owing to the existence of the packet dropout of the communication during the transmission, the packet dropout process of the network can be regarded as a time-homogenous Markov process {𝛾(𝑘),𝑘≥0}. Let 𝛾(𝑘)=1 mean that the packet has been successfully delivered to the decoder while 𝛾(𝑘)=0 corresponds to the dropout of the packet. The Markov chain has a transition probability matrix defined byℙ𝛾(𝑘+1)=𝑗∣𝛾(𝑘)=𝑖=1−𝑞𝑞𝑝1−𝑝, (2.9) where 𝑖,𝑗∈𝕎∶={0,1} are the state of the Markov chain. Without loss of generality, let 𝛾(0)=1 and the failure rate 𝑝 and the recovery rate 𝑞 of the channel satisfy 𝑝,𝑞∈(0,1). It is worth noting that a smaller value of 𝑝 and a larger value of 𝑞 indicate a more reliable channel.

Remark 2.1. When the above transition probability matrix is 𝑝1−𝑝𝑝1−𝑝 with 0≤𝑝≤1, the above two-state Markov process is reduced to a Bernoulli process [42].

In this paper, we consider the following fuzzy control law for the DFS (2.5).

Controller Rule 𝑖
IF 𝜃₁(𝑘) is 𝑀_𝑖1 and 𝜃₂(𝑘) is 𝑀_𝑖2⋯𝜃_𝑐(𝑘) is 𝑀_𝑖𝑐, THEN 𝑢(𝑘)=𝛾(𝑘)𝐿_𝑖𝑥(𝑘), (2.10) where 𝐿_𝑖 are to be designed the control gain matrices. {𝛾(𝑘),𝑘≥0} is a Markov packet dropout process satisfying (2.9). Then, we have the controller of DFS as 𝑢(𝑘)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑘))𝛾(𝑘)𝐿_𝑖𝑥(𝑘). (2.11)

2.3. Closed-Loop System

The following closed-loop DFS can be obtained from (2.5) and (2.11):𝑥(𝑘+1)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐴_𝑖+𝛾(𝑘)𝐵_𝑖𝐿_𝑗𝑥(𝑘)+𝐺_𝑖𝑤(𝑘), (2.12) where 𝐴_𝑖=𝐴_𝑖+Δ𝐴_𝑖(𝑘),𝐵_𝑖=𝐵_𝑖+Δ𝐵_𝑖(𝑘), and 𝐺_𝑖=𝐺_𝑖+Δ𝐺_𝑖(𝑘).

Now, we define two models according to the value of 𝛾(𝑘). If 𝛾(𝑘)=1, we define the Model 1 at time 𝑘+1 as follows:𝑥(𝑘+1)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐴_𝑖+𝐵_𝑖𝐿_𝑗𝜁𝑥(𝑘)+𝐺_𝑖𝑤(𝑘). (2.13) If 𝛾(𝑘)=0, we define the Model 2 at time 𝑘+1 as follows:𝑥(𝑘+1)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐴_𝑖𝑥(𝑘)+𝐺_𝑖𝑤(𝑘), (2.14) where the selection of 𝐿_𝑗𝜁 in (2.13) is according to the model of 𝑥(𝑘) for all 𝜁∈{1,0}, that is to say, if 𝑥(𝑘) is at Model 1, which is 𝛾(𝑘−1)=1,𝐿_𝑗𝜁=𝐿_𝑗1, otherwise, if 𝑥(𝑘) is at Model 2, which is 𝛾(𝑘−1)=0,𝐿_𝑗𝜁=𝐿_𝑗0.

Then, (2.12) can be regarded as a DFS with Markovian jumps described by𝑥(𝑘+1)=𝟏(1)^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐴_𝑖+𝐵_𝑖𝐿_𝑗𝜁𝑥(𝑘)+𝐺_𝑖𝑤(𝑘(2))+𝟏^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐴_𝑖𝑥(𝑘)+𝐺_𝑖𝑤(𝑘), (2.15) where 𝟏(𝑎),𝑎∈{1,2} denotes the mode indicator function. 𝟏(1) corresponds to a mode with feedback and 𝟏(2) corresponds to a mode without feedback. It is noted that it yields 𝟏(𝑎)=1 when at time 𝑘+1 be 𝑎∈{1,2} and 𝟏(𝑏)=0 for 𝑏≠𝑎. The mode transition probabilities of Markovian jump DFS (2.15) is given byℙ𝜂_𝑣(𝑘+1)=𝑣∣𝜂_𝑢(𝑘)=𝑢=𝜋_𝑢𝑣, (2.16) where 𝜋_𝑢𝑣≥0 for all 𝑢,𝑣∈{1,2} and ∑²_𝑣=1𝜋_𝑢𝑣=1. 𝜂_𝑣(𝑘)=1 implies 𝟏(1)=1,𝟏(2)=0, which the communication transmission succeeds, and 𝜂_𝑣(𝑘)=2 implies 𝟏(1)=0,𝟏(2)=1, in which the communication dropout occurs. Thus, compared with (2.9), it follows that 𝜋₁₁=1−𝑝,𝜋₁₂=𝑝,𝜋₂₁=𝑞,𝜋₂₂=1−𝑞.

2.4. Definitions and Lemmas

Definition 2.2 (stochastic finite-time stability (SFTS)). The closed-loop DFS with Markov jumps (2.15) with 𝑤(𝑘)=0 is said to be SFTS with respect to (𝛿_𝑥,𝜖,𝑅,𝑁), where 0<𝛿_𝑥<𝜖, 𝑅 is a symmetric positive-definite matrix, and 𝑁∈ℤ⁺, if 𝔼𝑥^𝑇(0)𝑅𝑥(0)≤𝛿²_𝑥⟹𝔼𝑥^𝑇(𝑘)𝑅𝑥(𝑘)<𝜖²,∀𝑘∈{1,2,…,𝑁}. (2.17)

Definition 2.3 (stochastic finite-time bounded (SFTB)). The closed-loop DFS with Markov jumps (2.15) is said to be SFTB with respect to (𝛿_𝑥,𝜖,𝑅,𝑁,𝑑), where 0<𝛿_𝑥<𝜖, 𝑅 is a symmetric positive-definite matrix, and 𝑁∈ℤ⁺, if 𝔼𝑥^𝑇(0)𝑅𝑥(0)≤𝛿²_𝑥⟹𝔼𝑥^𝑇(𝑘)𝑅𝑥(𝑘)<𝜖²,∀𝑘∈{1,2,…,𝑁}. (2.18)

Lemma 2.4 (see [23]). For any matrices 𝑀_𝑖𝑗,𝑁_𝑖𝑗 for 1≤𝑖,𝑗≤𝑟, and 𝑆>0 with appropriate dimensions, one has 2^𝑟𝑟_𝑖=1^𝑟_𝑗=1^𝑟_𝑠=1𝑡=1ℎ_𝑖ℎ_𝑗ℎ_𝑠ℎ_𝑡𝑀_𝑖𝑗𝑆𝑁_𝑠𝑡≤^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖ℎ_𝑗𝑀^𝑇_𝑖𝑗𝑆𝑀_𝑖𝑗+𝑁^𝑇_𝑖𝑗𝑆𝑁_𝑖𝑗. (2.19) In particular, when 𝑀_𝑖𝑗=𝑁_𝑖𝑗 for all 1≤𝑖,𝑗≤𝑟, one has ^𝑟𝑟_𝑖=1^𝑟_𝑗=1^𝑟_𝑠=1𝑡=1ℎ_𝑖ℎ_𝑗ℎ_𝑠ℎ_𝑡𝑀_𝑖𝑗𝑆𝑀_𝑠𝑡≤^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖ℎ_𝑗𝑀^𝑇_𝑖𝑗𝑆𝑀_𝑖𝑗. (2.20)

Lemma 2.5 (Schur complement lemma, see [43]). The linear matrix inequality ^𝑋11𝑋12_∗𝑋22<0 is equivalent to 𝑋₂₂<0 and 𝑋₁₁−𝑋₁₂𝑋⁻¹₂₂𝑋^𝑇₁₂<0, where 𝑋₁₁=𝑋^𝑇₁₁ and 𝑋₂₂=𝑋^𝑇₂₂.

Lemma 2.6 (see [43]). For matrices 𝑋,𝑌 and 𝑍 of appropriate dimensions, where 𝑋 is a symmetric matrix, then 𝑋+𝑌𝐹(𝑡)𝑍+[𝑌𝐹(𝑡)𝑍]^𝑇<0 (2.21) holds for all matrix 𝐹(𝑡) satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼 for all 𝑡∈ℝ, if and only if there exists a positive constant 𝜅, such that the following inequality holds: 𝑋+𝜅𝑌𝑌^𝑇+𝜅⁻¹𝑍^𝑇𝑍<0. (2.22)

In this paper, the feedback gain matrices 𝐿_𝑗1 and 𝐿_𝑗0 with Markov packet dropout of failure rate 𝑝 and recovery rate 𝑞 will be designed to guarantee the states of the closed-loop Markovian jump DFS (2.15) SFTB and SFTS.

3. Main Results

In this section, for the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), we start by developing results that can be used to design a state-feedback controller that assures that the nominal Markovian jump DFS (2.15) is SFTB and SFTS. Then, these results will be extended to the case of uncertain systems. LMI conditions are established to design a state-feedback controller that guarantees that either a nominal Markovian jump DFS or an uncertain Markovian jump DFS of the class we are considering is SFTB and SFTS.

3.1. SFTB and SFTS of the Nominal Markov Jump System

Theorem 3.1. For the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), the closed-loop nominal Markovian jump DFS (2.15) is SFTB with respect to (𝛿_𝑥,𝜖,𝑅,𝑁,𝑑), if there exist a scalar 𝜇≥1, four symmetric positive-definite matrices 𝑃₁,𝑃₂,𝑄₁,𝑄₂, and a set of feedback control matrices {𝐿_𝑗𝜁,𝑗∈𝕄,𝜁∈{1,0}}, such that the following inequalities hold for all 𝑖,𝑗∈𝕊: −𝜇𝑃₁0𝐴_𝑖+𝐵_𝑖𝐿_𝑗1^𝑇𝐴^𝑇_𝑖∗−𝑄₁𝐺^𝑇_𝑖𝐺^𝑇_𝑖1∗∗−𝑃1−𝑝¹⁻¹01∗∗∗−𝑝𝑃₂⁻¹<0,−𝜇𝑃₂0𝐴_𝑖+𝐵_𝑖𝐿_𝑗0^𝑇𝐴^𝑇_𝑖∗−𝑄₂𝐺^𝑇_𝑖𝐺^𝑇_𝑖1∗∗−𝑞𝑃₁⁻¹01∗∗∗−𝑃1−𝑞²⁻¹<0,sup_𝑎∈{1,2}𝜆_max𝑃_𝑎𝛿²_𝑥+sup_𝑎∈{1,2}𝜆_max𝑄_𝑎𝑑²<inf_𝑎∈{1,2}𝜆_min𝑃_𝑎𝜇^−𝑁𝜖², (3.1)(3.2)(3.3) where 𝑃_𝑎=𝑅^−1/2𝑃_𝑎𝑅^−1/2 for all 𝑎∈{1,2}.

Proof. Assume the mode at time 𝑘 be 𝑎∈{1,2}. Taking into account that if 𝑎=1, then we have 𝛾(𝑘−1)=1 and 𝐿_𝑗𝜁=𝐿_𝑗1, otherwise, if 𝑎=2, then 𝛾(𝑘−1)=0 and 𝐿_𝑗𝜁=𝐿_𝑗0. Let us consider the following quadratic Lyapunov-Krasovskii functional candidate for the Markov jump DFS (2.15): 𝑉𝑥(𝑘),𝜂_𝑣(𝑘)=𝑎=𝑥^𝑇(𝑘)𝑃_𝑎𝑥(𝑘). (3.4) We compute 𝔼{𝑉(𝑘+1)}=𝔼²_𝑣=1ℙ𝜂_𝑣(𝑘+1)=𝑣∣𝜂_𝑣(𝑘)=𝑎×𝑥^𝑇(𝑘+1)𝑃_𝑣𝑥(=𝑘+1)_{𝑟𝑟𝑖=1𝑟𝑗=1𝑟𝑠=1𝑡=1}ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))ℎ_𝑠(𝜃(𝑘))ℎ_𝑡(𝜃(𝑘×𝜋))_𝑎1𝐴_𝑖+𝐵_𝑖𝐿_𝑗𝜁𝑥(𝑘)+𝐺_𝑖𝑤(𝑘)^𝑇𝑃₁𝐴_𝑠+𝐵_𝑠𝐿_𝑡𝜁𝑥(𝑘)+𝐺_𝑠𝑤(𝑘)+𝜋_𝑎2𝐴_𝑖𝑥(𝑘)+𝐺_𝑖𝑤(𝑘)^𝑇𝑃₂𝐴_𝑠𝑥(𝑘)+𝐺_𝑠𝑤(𝑘). (3.5) According to Lemma 2.4, we obtain 𝔼{𝑉(𝑘+1)}−𝜇𝑉(𝑘)−𝑤^𝑇(𝑘)𝑄_𝑎𝑤(𝑘)≤Ξ_𝑖𝑗(𝑥(𝑘),𝑤(𝑘),𝑎), (3.6) where Ξ_𝑖𝑗(𝑥(𝑘)(𝑘))=,𝑤,𝑎_{𝑟𝑟𝑖=1𝑗=1}ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘×𝜋))_𝑎1𝐴_𝑖+𝐵_𝑖𝐿_𝑗𝜁𝑥(𝑘)+𝐺_𝑖𝑤(𝑘)^𝑇𝑃₁𝐴_𝑖+𝐵_𝑖𝐿_𝑗𝜁𝑥(𝑘)+𝐺_𝑖𝑤(𝑘)+𝜋_𝑎2𝐴_𝑖𝑥(𝑘)+𝐺_𝑖𝑤(𝑘)^𝑇𝑃₂𝐴_𝑖𝑥(𝑘)+𝐺_𝑖𝑤(𝑘)−𝜇𝑥^𝑇(𝑘)𝑃_𝑎𝑥(𝑘)−𝑤^𝑇(𝑘)𝑄_𝑎𝑤(𝑘). (3.7) Taking into account that if 𝑎=1, then 𝐿_𝑗𝜁=𝐿_𝑗1, otherwise, 𝑎=2, then 𝐿_𝑗𝜁=𝐿_𝑗0. Noting that 𝜋₁₁=1−𝑝,𝜋₁₂=𝑝,𝜋₂₁=𝑞,𝜋₂₂=1−𝑞. Thus, when 𝑎=1, we have Ξ_𝑖𝑗(𝑥(𝑘)(𝑘))=,𝑤,1_{𝑟𝑟𝑖=1𝑗=1}ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘×𝐴))1−𝑝_𝑖+𝐵_𝑖𝐿_𝑗1𝑥(𝑘)+𝐺_𝑖𝑤(𝑘)^𝑇𝑃₁𝐴_𝑖+𝐵_𝑖𝐿_𝑗1𝑥(𝑘)+𝐺_𝑖𝑤(𝑘𝐴)+𝑝_𝑖𝑥(𝑘)+𝐺_𝑖𝑤(𝑘)^𝑇𝑃₂𝐴_𝑖𝑥(𝑘)+𝐺_𝑖𝑤(𝑘)−𝜇𝑥^𝑇(𝑘)𝑃₁𝑥(𝑘)−𝑤^𝑇(𝑘)𝑄₁𝑤(𝑘=_{𝑟𝑟𝑖=1𝑗=1})ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘𝑥(𝑘)𝑤(𝑘)))^𝑇Γ^𝑇_𝑖𝑗11−𝑝𝑃₁0∗𝑝𝑃₂Γ_𝑖𝑗1−𝜇𝑃₁0∗𝑄₁𝑥(𝑘)𝑤(𝑘), (3.8) where Γ_𝑖𝑗1=𝐴_𝑖+𝐵_𝑖𝐿_𝑗1𝐺_𝑖𝐴_𝑖𝐺_𝑖. (3.9) By Lemma 2.5, we can obtain from (3.1) that Ξ_𝑖𝑗(𝑥(𝑘),𝑤(𝑘),1)<0. (3.10) Thus, when 𝑎=1, from (3.6)–(3.10), we can obtain 𝔼{𝑉(𝑘+1)}−𝜇𝑉(𝑘)<𝑤^𝑇(𝑘)𝑄₁𝑤(𝑘). (3.11) When 𝑎=2, taking into account condition (3.2), the similar as the above discussion, we can obtain the following inequality: 𝔼{𝑉(𝑘+1)}−𝜇𝑉(𝑘)<𝑤^𝑇(𝑘)𝑄₂𝑤(𝑘). (3.12) Thus, for all 𝑎∈{1,2}, it follows from (3.11) and (3.12) that 𝔼{𝑉(𝑘+1)}−𝜇𝑉(𝑘)<𝑤^𝑇(𝑘)𝑄_𝑎𝑤(𝑘)≤sup_𝑎∈{1,2}𝜆_max𝑄_𝑎𝑤^𝑇(𝑘)𝑤(𝑘). (3.13) By (3.13), it is obvious that 𝔼{𝑉(𝑘+1)}<𝜇𝔼{𝑉(𝑘)}+sup_𝑎∈{1,2}𝜆_max𝑄_𝑎𝑤^𝑇(𝑘)𝑤(𝑘). (3.14) From (2.2) and (3.14) and noting that 𝜇≥1, we have 𝔼{𝑉(𝑘)}<𝜇^𝑘𝔼{𝑉(0)}+sup_{}𝑎∈1,2𝜆_max𝑄_𝑎^𝑘−1_𝑗=0𝜇^{𝑘−𝑗−1}𝑤^𝑇𝑗𝑤𝑗≤𝜇^𝑘𝔼{𝑉(0)}+sup_{}𝑎∈1,2𝜆_max𝑄_𝑎^𝑘−1_𝑗=0𝜇^𝑘𝑤^𝑇𝑗𝑤𝑗≤𝜇^𝑘𝔼{𝑉(0)}+sup_{}𝑎∈1,2𝜆_max𝑄_𝑎𝜇^𝑘𝑑². (3.15) Let 𝑃_𝑎=𝑅^−1/2𝑃_𝑎𝑅^−1/2 and noting that 𝔼{𝑥^𝑇(0)𝑅𝑥(0)}≤𝛿²_𝑥, we have 𝔼{𝑉(0)}=𝔼𝑥^𝑇(0)𝑃_𝑎𝑥(0𝑥)=𝔼^𝑇(0)𝑅^1/2𝑃_𝑎𝑅^1/2𝑥(0)≤sup_𝑎∈{1,2}𝜆_max𝑃_𝑎𝔼𝑥^𝑇(0)𝑅𝑥(0)≤sup_𝑎∈{1,2}𝜆_max𝑃_𝑎𝛿²_𝑥. (3.16) On the other hand, for all 𝑎∈{1,2}, we have 𝔼{𝑉(𝑘)}=𝔼𝑥^𝑇(𝑘)𝑃_𝑎𝑥(𝑘𝑥)=𝔼^𝑇(𝑘)𝑅^1/2𝑃_𝑎𝑅^1/2𝑥(𝑘)≥inf_𝑎∈{1,2}𝜆_min𝑃_𝑎𝔼𝑥^𝑇(𝑘)𝑅𝑥(𝑘). (3.17) Combine (3.15)–(3.17) together, we can obtain 𝔼𝑥^𝑇(𝑘)𝑅𝑥(𝑘)<sup_𝑎∈{1,2}𝜆_max𝑃_𝑎𝜇^𝑘𝛿²_𝑥+sup_𝑎∈{1,2}𝜆_max𝑄_𝑎𝜇^𝑘𝑑²inf_𝑎∈{1,2}𝜆_min𝑃_𝑎. (3.18) Noting condition (3.3), it follows that 𝔼{𝑥^𝑇(𝑘)𝑅𝑥(𝑘)}<𝜖² for all 𝑘∈{1,2,…,𝑁}. This completes the proof of this Theorem.

The similar proof is as the above Theorem 3.1, if 𝑤(𝑘)=0, we can obtain the results on finite-time stability as follows.

Corollary 3.2. For the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), the closed-loop nominal Markovian jump DFS (2.15) with 𝑤(𝑘)=0 is SFTS with respect to (𝛿_𝑥,𝜖,𝑅,𝑁), if there exist a scalar 𝜇≥1, two symmetric positive-definite matrices 𝑃₁,𝑃₂, and a set of feedback control matrices {𝐿_𝑗𝜁,𝑗∈𝕄,𝜁∈{1,0}}, such that the following inequalities hold for all 𝑖,𝑗∈𝕊: −𝜇𝑃₁𝐴_𝑖+𝐵_𝑖𝐿_𝑗1^𝑇𝐴^𝑇_𝑖1∗−𝑃1−𝑝¹⁻¹01∗∗−𝑝𝑃₂⁻¹<0,−𝜇𝑃₂𝐴_𝑖+𝐵_𝑖𝐿_𝑗0^𝑇𝐴^𝑇_𝑖1∗−𝑞𝑃₁⁻¹01∗∗−𝑃1−𝑞²⁻¹<0,sup_𝑎∈{1,2}𝜆_max𝑃_𝑎𝛿²_𝑥<inf_{}𝑎∈1,2𝜆_min𝑃_𝑎𝜇^−𝑁𝜖², (3.19) where 𝑃_𝑎=𝑅^−1/2𝑃_𝑎𝑅^−1/2 for all 𝑎∈{1,2}.

Remark 3.3. If 𝜇=1 is a solution of feasibility problem (3.19), then the closed-loop Markovian jumps DFS (2.15) with 𝑤(𝑡)=0 is SFTS with respect to (𝛿_𝑥,𝜖,𝑅,𝑁) and is also stochastically stable.

Denoting 𝑋₁=𝑃₁⁻¹,𝑋₂=𝑃₂⁻¹,𝐿_𝑗1=𝑌_𝑗1𝑋₁⁻¹,𝐿_𝑗0=𝑌_𝑗0𝑋₂⁻¹, one can obtain the following results on the stochastic finite-time stabilization.

Theorem 3.4. For the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), there exists a state-feedback controller 𝑢(𝑡)=𝐿_𝑗𝜁𝑥(𝑡),𝜁∈{1,0} with 𝐿_𝑗1=𝑌_𝑗1𝑋₁⁻¹,𝐿_𝑗0=𝑌_𝑗0𝑋₂⁻¹ such that the closed-loop nominal DFS with Markovian jump (2.15) is SFTB with respect to (𝛿_𝑥,𝜖,𝑅,𝑁,𝑑), if there exist a scalar 𝜇≥1, four symmetric positive-definite matrices 𝑋₁,𝑋₂,𝑄₁,𝑄₂, and a set of feedback control matrices {𝑌_𝑗𝜁,𝑗∈𝕄,𝜁∈{1,0}}, such that the following inequalities hold for all 𝑖,𝑗∈𝕊: −𝜇𝑋₁0𝐴_𝑖𝑋₁+𝐵_𝑖𝑌_𝑗1^𝑇𝐴_𝑖𝑋₁^𝑇∗−𝑄₁𝐺^𝑇_𝑖𝐺^𝑇_𝑖1∗∗−𝑋1−𝑝¹01∗∗∗−𝑝𝑋₂<0, (3.20)−𝜇𝑋₂0𝐴_𝑖𝑋₂+𝐵_𝑖𝑌_𝑗0^𝑇𝐴_𝑖𝑋₂^𝑇∗−𝑄₂𝐺^𝑇_𝑖𝐺^𝑇_𝑖1∗∗−𝑞𝑋₁01∗∗∗−𝑋1−𝑞²<0, (3.21)sup_𝑎∈{1,2}𝜆_max𝑋_𝑎𝛿²_𝑥+sup_{}𝑎∈1,2𝜆_max𝑄_𝑎𝑑²<inf_{}𝑎∈1,2𝜆_min𝑋_𝑎𝜇^−𝑁𝜖², (3.22) where 𝑋_𝑎=𝑅^−1/2𝑋_𝑎⁻¹𝑅^−1/2 for all 𝑎∈{1,2}.

Corollary 3.5. For the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), there exists a state-feedback controller 𝑢(𝑡)=𝐿_𝑗𝜁𝑥(𝑡),𝜁∈{1,0} with 𝐿_𝑗1=𝑌_𝑗1𝑋₁⁻¹,𝐿_𝑗0=𝑌_𝑗0𝑋₂⁻¹ such that the closed-loop nominal Markovian jump DFS (2.15) with 𝑤(𝑘)=0 is SFTS with respect to (𝛿_𝑥,𝜖,𝑅,𝑁), if there exist a scalar 𝜇≥1, two symmetric positive-definite matrices 𝑋₁,𝑋₂, and a set of feedback control matrices {𝑌_𝑗𝜁,𝑗∈𝕄,𝜁∈{0,1}}, such that the following inequalities hold for all 𝑖,𝑗∈𝕊: −𝜇𝑋₁𝐴_𝑖𝑋₁+𝐵_𝑖𝑌_𝑗1^𝑇𝐴_𝑖𝑋₁^𝑇1∗−𝑋1−𝑝¹01∗∗−𝑝𝑋₂<0,−𝜇𝑋₂𝐴_𝑖𝑋₂+𝐵_𝑖𝑌_𝑗0^T𝐴_𝑖𝑋₂^𝑇1∗−𝑞𝑋₁01∗∗−𝑋1−𝑞²<0,sup_𝑎∈{1,2}𝜆_max𝑋_𝑎𝛿²_𝑥<inf_{}𝑎∈1,2𝜆_min𝑋_𝑎𝜇^−𝑁𝜖², (3.23)(3.24)(3.25) where 𝑋_𝑎=𝑅^−1/2𝑋_𝑎⁻¹𝑅^−1/2 for all 𝑎∈{1,2}.

3.2. Extension to SFTB and SFTS of the Uncertain Markov Jump DFS

By Theorems 3.1 and 3.4 and applying Lemmas 2.5 and 2.6, one can obtain the following results stated.

Theorem 3.6. For the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), there exists a state-feedback controller 𝑢(𝑡)=𝐿_𝑗𝜁𝑥(𝑡),𝜁∈{1,0} with 𝐿_𝑗1=𝑌_𝑗1𝑋₁⁻¹,𝐿_𝑗0=𝑌_𝑗0𝑋₂⁻¹ such that the closed-loop uncertain DFS with Markovian jump (2.15) is SFTB with respect to (𝛿_𝑥,𝜖,𝑅,𝑁,𝑑), if there exist a scalar 𝜇≥1, two sets of positive scalars {𝜖_1𝑖,𝑖∈𝕄}, {𝜖_2𝑖,𝑖∈𝕄}, four symmetric positive-definite matrices 𝑋₁,𝑋₂,𝑄₁,𝑄₂, and a set of feedback control matrices {𝑌_𝑗𝜁,𝑗∈𝕄,𝜁∈{1,0}}, such that (3.22) and the following inequalities hold for all 𝑖,𝑗∈𝕊: −𝜇𝑋₁0𝐴_𝑖𝑋₁+𝐵_𝑖𝑌_j1^𝑇𝐴_𝑖𝑋₁^𝑇𝐸_1𝑖𝑋₁+𝐸_2𝑖𝑌_𝑗1^𝑇𝐸_1𝑖𝑋₁^𝑇∗−𝑄₁𝐺^𝑇_𝑖𝐺^𝑇_𝑖𝐸^𝑇_3𝑖𝐸^𝑇_3𝑖1∗∗−𝑋1−𝑝¹+𝜖_1𝑖𝐹_𝑖𝐹^𝑇_𝑖1000∗∗∗−𝑝𝑋₂+𝜖_1𝑖𝐹_𝑖𝐹^𝑇_𝑖00∗∗∗∗−𝜖_1𝑖𝐼0∗∗∗∗∗−𝜖_1𝑖𝐼<0,−𝜇𝑋₂0𝐴_𝑖𝑋₂+𝐵_𝑖𝑌_𝑗0^𝑇𝐴_𝑖𝑋₂^𝑇𝐸_1𝑖𝑋₂+𝐸_2𝑖𝑌_𝑗0^𝑇𝐸_1𝑖𝑋₂^𝑇∗−𝑄₂𝐺^𝑇_𝑖𝐺^𝑇_𝑖𝐸^𝑇_3𝑖𝐸^𝑇_3𝑖1∗∗−𝑞𝑋₁+𝜖_2𝑖𝐹_𝑖𝐹^𝑇_𝑖1000∗∗∗−𝑋1−𝑞²+𝜖_2𝑖𝐹_𝑖𝐹^𝑇_𝑖00∗∗∗∗−𝜖_2𝑖𝐼0∗∗∗∗∗−𝜖_2𝑖𝐼<0, (3.26)(3.27) where 𝑋_𝑎=𝑅^−1/2𝑋_𝑎⁻¹𝑅^−1/2 for all 𝑎∈{1,2}.

Corollary 3.7. For the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), there exists a state-feedback controller 𝑢(𝑡)=𝐿_𝑗𝜁𝑥(𝑡),𝜁∈{1,0} with 𝐿_𝑗1=𝑌_𝑗1𝑋₁⁻¹,𝐿_𝑗0=𝑌_𝑗0𝑋₂⁻¹ such that the closed-loop uncertain DFS with Markovian jump (2.15) with 𝑤(𝑘)=0 is SFTS with respect to (𝛿_𝑥,𝜖,𝑅,𝑁), if there exist a scalar 𝜇≥1, two sets of positive scalars {𝜖_1𝑖,𝑖∈𝕄}, {𝜖_2𝑖,𝑖∈𝕄}, two symmetric positive-definite matrices 𝑋₁,𝑋₂, and a set of feedback control matrices {𝑌_𝑗𝜁,𝑗∈𝕄,𝜁∈{0,1}}, such that (3.25) and the following inequalities hold for all 𝑖,𝑗∈𝕊: −𝜇𝑋₁𝐴_𝑖𝑋₁+𝐵_𝑖𝑌_𝑗1^𝑇𝐴_𝑖𝑋₁^𝑇𝐸_1𝑖𝑋₁+𝐸_2𝑖𝑌_𝑗1^𝑇𝐸_1𝑖𝑋₁^𝑇1∗−𝑋1−𝑝¹+𝜖_1𝑖𝐹_𝑖𝐹^𝑇_𝑖1000∗∗−𝑝𝑋₂+𝜖_1𝑖𝐹_𝑖𝐹^𝑇_𝑖00∗∗∗−𝜖_1𝑖𝐼0∗∗∗∗−𝜖_1𝑖𝐼<0,−𝜇𝑋₂𝐴_𝑖𝑋₂+𝐵_𝑖𝑌_𝑗0^𝑇𝐴_𝑖𝑋₂^𝑇𝐸_1𝑖𝑋₂+𝐸_2𝑖𝑌_𝑗0^𝑇𝐸_1𝑖𝑋₂^𝑇1∗−𝑞𝑋₁+𝜖_2𝑖𝐹_𝑖𝐹^𝑇_𝑖1000∗∗−𝑋1−𝑞²+𝜖_2𝑖𝐹_𝑖𝐹^𝑇_𝑖00∗∗∗−𝜖_2𝑖𝐼0∗∗∗∗−𝜖_2𝑖𝐼<0, (3.28)(3.29) where 𝑋_𝑎=𝑅^−1/2𝑋_𝑎⁻¹𝑅^−1/2 for all 𝑎∈{1,2}.

Remark 3.8. It is easy to check that conditions (3.22) and (3.25) are guaranteed by imposing the conditions for all 𝑎∈{1,2}, respectively: 𝜆₁𝑅⁻¹<𝑋_𝑎<𝑅⁻¹,0<𝑄_𝑎<𝜆₂𝜆𝐼,²𝑑²−𝜇^−𝑁𝜖²𝛿_𝑥∗−𝜆₁<0,𝜆𝑅⁻¹<𝑋_𝑎<𝑅⁻¹,−𝜇^−𝑁𝜖²𝛿_𝑥∗−𝜆<0. (3.30)(3.31) Conditions (3.26)–(3.31) are not strict LMIs, however, once we fix the parameter 𝜇, the conditions can be turned into LMIs based feasibility problem.

Remark 3.9. From the above discussion, we can obtain that the feasibility of conditions stated in Theorem 3.6 and Corollary 3.7 can be turned into the following LMIs based feasibility problem with a fixed parameter 𝜇, respectively: min𝜖²𝑋₁,𝑋₂,𝑌_𝑗1,𝑌_𝑗0,𝜖_1𝑖,𝜖_2𝑖,𝑄₁,𝑄₂,𝜆₁,𝜆₂s.t.(3.26),(3.27),and(3.30)min𝜖²𝑋₁,𝑋₂,𝑌_𝑗1,𝑌_𝑗0,𝜖_1𝑖,𝜖_2𝑖(),()().,𝜆s.t.3.283.29,and3.31 (3.32) Furthermore, we can also find the parameter 𝜇 by an unconstrained nonlinear optimization approach, in which a locally convergent solution can be obtained by using the program fminsearch in the optimization toolbox of MATLAB.

4. Illustrative Examples

In this section, we present two examples to illustrate the proposed methods.

Example 4.1. Consider a DFS (2.15) with parameters as 𝐴₁=20.4000.70.300−0.6,𝐵₁=110101,𝐺₁=,𝐹10.10100.1₁=0.0100.0200.2000.010,𝐸₁₁=0.0100.0200.02000.10,𝐸₂₁=0.1000.100.2,𝐸₃₁=,𝐴0.20.10000.1₂=0.830−0.200.750.20−0.4,𝐵₂=100101,𝐺₂=,𝐹0.50.10100.1₂=0.100.0100.10000.1,𝐸₁₂=0.100.0200.2000.10,𝐸₂₂=0.2000.100.3,𝐸₃₂=,0.10.100.100.1 (4.1) and 𝑑=2,Δ_𝑖(𝑘)=diag{Δ_1𝑖(𝑘),Δ_2𝑖(𝑘),Δ_3𝑖(𝑘)}, where Δ_𝑗𝑖 satisfies |Δ_𝑗𝑖(𝑘)|≤1 for all 𝑖=1,2, 𝑗=1,2,3 and 𝑘∈ℤ⁺. The ℎ₁ and ℎ₂ are fuzzy sets defined as ℎ₁(𝑥(𝑘=(𝑘)))sin𝑥𝑥(𝑘)(𝑘)(𝑘)ℎ,𝑥≠0;1,𝑥=0,₂(𝑥(𝑘))=1−ℎ₁(𝑥(𝑘)). (4.2) Moreover, we assume the failure rate 𝑝=0.4 and the recovery rate 𝑞=0.8. Then, we choose 𝑅=𝐼₃,𝛿_𝑥=1,𝑁=5,𝜇=2, Theorem 3.6 yields 𝜖=21.8239 and 𝑋₁=0.9837−0.06710.0587−0.06710.4318−0.03510.0587−0.03510.5113,𝑋₂=,𝑌0.99940.00780.01290.00780.4969−0.07150.0129−0.07150.5220₁₁=𝑌₂₁=,𝑌−1.8795−0.7470−0.17540.0142−0.0668−0.2793₁₀=𝑌₂₀=,𝑄−1.6200−0.83610.05920.0464−0.0911−0.1958₁=3.1164003.1164,𝑄₂=,𝜖3.1139003.1139₁₁=0.6146,𝜖₁₂=2.3435,𝜖₂₁=0.6014,𝜖₂₂𝜆=2.1335,₁=0.4147,𝜆₂=3.1177. (4.3) Thus, we can obtain the following state-feedback controller gains: 𝐿₁₁=𝐿₂₁=,𝐿−2.0379−2.0938−0.23250.0404−0.2016−0.5563₁₀=𝐿₂₀=.−1.6030−1.6832−0.07170.0559−0.2491−0.4119 (4.4) Furthermore, let 𝑅=𝐼₃,𝛿_𝑥=1,𝑁=5, by Theorem 3.6, the optimal bound with minimum value of 𝜖² relies on the parameter 𝜇. We can find feasible solution when 1.20≤𝜇≤44.43. Figure 1 shows the optimal value with different value of 𝜇. Noting that when 𝜇=1.4, it yields the optimal value 𝜖=13.7246. Then, by using the program fminsearch in the optimization toolbox of MATLAB starting at 𝜇=1.4, the locally convergent solution can be derived as 𝐿₁₁=𝐿₂₁=,𝐿−1.8847−1.8653−0.61490.0512−0.1196−0.5385₁₀=𝐿₂₀=.−1.5200−1.6018−0.38260.0426−0.1279−0.4005 (4.5) with 𝜇=1.3591 and the optimal value 𝜖=13.6304.

Figure 1 

The local optimal bound of 𝜖.

Example 4.2. Consider a DFS (2.15) with 𝑤(𝑡)=0 and 𝐴₁=1.50.4000.70.300.5−1,𝐴₂=10.40−0.300.750.60.8−0.4. (4.6) In addition, the other matrices parameters, fuzzy basis functions, and the failure rate 𝑝 and the recovery rate 𝑞 are the same as Example 4.1.

Then, let 𝑅=𝐼₃,𝛿_𝑥=1,𝑁=6, by Corollary 3.7, the optimal bound with minimum value of 𝜖² relies on the parameter 𝜇. We can find feasible solution when 1≤𝜇≤23.61. Thus, the above system is stochastically stable and when 𝜇=1, it yields the optimal value 𝜖=4.5131 and the following optimized state-feedback controller gains:𝐿₁₁=𝐿₂₁=,𝐿−1.6149−0.3843−0.8106−0.0447−0.42690.2344₁₀=𝐿₂₀=.−1.3138−0.2433−0.7052−0.0960−0.33080.2755 (4.7)

5. Conclusions

This paper addresses the stochastic finite-time stability and stochastic finite-time boundedness problems for one family of fuzzy systems over networks with packet dropout. Taking into account the data packet drop being a time homogenous Markov process, thus the discrete-time DFS with packet loss can be described by one class of fuzzy Markovian jump systems. Then, state-feedback controllers are designed to guarantee the resulting closed-loop fuzzy discrete-time system with Markovian jumps SFTS and SFTB. The SFTS and SFTB criteria can be tackled in the form of linear matrix inequalities with a fixed parameter. As an auxiliary result, we also give sufficient conditions on the stochastic stability of the class of fuzzy systems with data packet dropout. Finally, simulation results are also given to demonstrate the validity of the proposed methodology.

Acknowledgments

The authors would like to thank the reviewers and editors for their very helpful comments and suggestions which could have improved the presentation of the paper. The paper was supported by the National Natural Science Foundation of China under Grant 60874006, supported by Doctoral Foundation of Henan University of Technology under Grant 2009BS048, supported by Foundation of Henan Educational Committee under Grant 2011A120003, and supported by Foundation of Henan University of Technology under Grant 09XJC011.