Stochastic ℋ_∞ Finite-Time Control of Discrete-Time Systems with Packet Loss

Abstract

This paper investigates the stochastic finite-time stabilization and ℋ_∞ control problem for one family of linear discrete-time systems over networks with packet loss, parametric uncertainties, and time-varying norm-bounded disturbance. Firstly, the dynamic model description studied is given, which, if the packet dropout is assumed to be a discrete-time homogenous Markov process, the class of discrete-time linear systems with packet loss can be regarded as Markovian jump systems. Based on Lyapunov function approach, sufficient conditions are established for the resulting closed-loop discrete-time system with Markovian jumps to be stochastic ℋ_∞ finite-time boundedness and then state feedback controllers are designed to guarantee stochastic ℋ_∞ finite-time stabilization of the class of stochastic systems. The stochastic ℋ_∞ 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 robust stochastic stabilization of the class of linear systems with packet loss. Finally, simulation examples are presented to illustrate the validity of the developed scheme.

1. Introduction

Networked control systems (NCSs) are feedback control systems with control closed loops via digital communication channel. Compared with traditional point-to-point controller architectures, the advantages of NCSs include low cost, high reliability, less wiring, and easy maintenance [1]. In recent years, NCSs have found successful applications in broad range of modern scientific areas such as internet-based control, distributed communication, and industrial automation [2]. However, the insertion of the communication channels creates discrepancies between the data records to be transmitted and their associated remotely transmitted images, which hence makes the traditional control theory confronts new challenges. Among these challenges, random communication delay, data packet dropout, and signal quantization are known to be three main interesting problems for the stability and performance degradation of the controlled networked system. In view of this, many researchers have made to study how to design control systems by packet loss, delay, and quantization, see [3–6] and the references cited therein. Among a number of issues arising from such a framework, packet losses of NCSs are an important issue to be addressed and have received great attention, see [7–15]. Meanwhile, Markovian jumps systems are regarded to be 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 [16–22] and references therein.

It is well known that classical Lyapunov theory focuses mainly on the state convergence property of the systems in infinite time interval, which, just as was mentioned above, does not usually specify bounds on the trajectories in finite interval. However, the main attention in many practical applications is the behavior of the dynamic systems over a specified time interval, for instance, large values of the state are not acceptable in the presence of saturations [23]. To discuss this transient performance of control dynamics, finite-time stability or short-time stability was presented in [24]. Then, some appealing results were found in [25–32]. However, to date and to the best of our knowledge, the problems of stochastic finite-time stability and stabilization of network control systems with packet loss have not fully investigated and still remain challenging, although results related to systems over networks with packet loss are reported in the existing literature, see [6–15, 33–36].

Motivated by the above discussion, in this paper, we address the stochastic ℋ_∞ finite-time boundedness (Sℋ_∞FTB) problems for linear 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 class of linear discrete-time systems with packet loss can be referred as Markovian jump systems. Thus, the class of linear systems investigated could be studied by the theoretical framework of Markov jumps systems. Then, the concepts of stochastic finite-time stability, stochastic finite-time boundedness, and Sℋ_∞FTB and problem formulation are given. The main contribution of this paper is to design a state feedback controller which guarantees the resulting closed-loop discrete-time system with Markovian jumps Sℋ_∞FTB. As an auxiliary result, we also give sufficient conditions on the robust stochastic stabilization of the class of linear systems with packet loss. The Sℋ_∞FTB criteria of the class of Markovian jump systems can be addressed in the form of linear matrix inequalities (LMIs) with a fixed parameter.

The rest of this paper is organized as follows. Section 2 is devoted to the dynamic model description and problem formulation. The results on the Sℋ_∞FTB are presented in Section 3. Section 4 presents numerical examples to demonstrate the validity of the proposed methodology. Finally, in Section 5, the conclusions are given.

Notation 1. The notation used throughout the paper is fairly standard, ℝ^𝑛, ℝ^𝑛×𝑚, and ℤ_𝑘≥0 denote 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 respective to some probability measure 𝒫. In addition, the symbol ∗ denotes the transposed elements in the symmetric positions of a matrix, and diag{⋯} stands for a block-diagonal 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

Let us consider a linear discrete-time system (LDS) as follows: 𝑥()=[(𝑘𝑥(𝑘)+[(𝑘𝑢(𝑘)(𝑘),𝑧(𝑘)(𝑘)𝑘+1𝐴+Δ𝐴)]𝐵+Δ𝐵)]+𝐺𝑤=𝐶𝑥+𝐷₁𝑢(𝑘)+𝐷₂𝑤(𝑘), (2.1) where 𝑥(𝑘)∈ℝ^𝑛 is the state, 𝑧(𝑘)∈ℝ^𝑙1 is the measure output, and 𝑢(𝑘)∈ℝ^𝑚 is the control input. The noise signal 𝑤(𝑘)∈ℝ^𝑙2 satisfies ^∞_𝑘=0𝑤^𝑇(𝑘)𝑤(𝑘)<𝑑²,𝑑>0. (2.2) The matrices Δ𝐴(𝑘) and Δ𝐵(𝑘) are uncertain matrices and satisfy [Δ𝐴(𝑘),Δ𝐵(𝑘)]=𝐹Δ(𝑘)𝐸₁,𝐸₂, (2.3) where Δ(𝑘) is an unknown, time-varying matrix function, and satisfies Δ^𝑇(𝑘)Δ(𝑘)≤𝐼,∀𝑘∈ℤ_𝑘≥0. (2.4)

Due 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 (2.5) 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 [37].
Consider the control law for the LDS (2.1) in the form 𝑢(𝑘)=𝛾(𝑘)𝐿𝑥(𝑘), (2.6) where 𝐿 is to be designed the control gain matrix. {𝛾(𝑘),𝑘≥0} is a Markov packet dropout process satisfying (2.5). Then, the resulting closed-loop LDS follows that 𝑥(𝑘+1)=𝐴+𝛾(𝑘)𝑥(𝑘)(𝑘),𝑧(𝑘)=(𝑘)𝐷𝐵𝐿+𝐺𝑤𝐶+𝛾₁𝐿𝑥(𝑘)+𝐷₂𝑤(𝑘), (2.7) where 𝐴=𝐴+Δ𝐴(𝑘) and 𝐵=𝐵+Δ𝐵(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)=𝐴+𝐵𝐿_𝜁𝑥(𝑘)(𝑘),𝑧(𝑘)=+𝐺𝑤𝐶+𝐷₁𝐿_𝜁𝑥(𝑘)+𝐷₂𝑤(𝑘). (2.8) If 𝛾(𝑘)=0, we define the Model 2 at time 𝑘+1 as follows: 𝑥(𝑘+1)=(𝑘)(𝑘),𝑧(𝑘)(𝑘)𝐴𝑥+𝐺𝑤=𝐶𝑥+𝐷₂𝑤(𝑘), (2.9) where the selection of 𝐿_𝜁 in (2.7) 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_𝜁=𝐿₁, otherwise, if 𝑥(𝑘) is at Model 2, which is 𝛾(𝑘−1)=0,𝐿_𝜁=𝐿₀.
Then, (2.7) can be regarded as a closed-loop LDS with Markovian jumps described by 𝑥(𝑘+1)=ℏ(1)𝐴+𝐵𝐿_𝜁𝑥(𝑘)+𝐺𝑤(𝑘)+ℏ(2)(𝑘)(𝑘,𝑧(𝑘)(1𝐴𝑥+𝐺𝑤)=ℏ)𝐶+𝐷₁𝐿_𝜁𝑥(𝑘)+𝐷₂𝑤(𝑘)+ℏ(2)𝐶𝑥(𝑘)+𝐷₂𝑤(𝑘), (2.10) where ℏ(𝑎), 𝑎∈{1,2} denotes the mode indicator function. ℏ(1) corresponds to a mode with feedback, and ℏ(2) corresponds to a mode without feedback. It is noted that it yields ℏ(𝑎)=1 when at time 𝑘+1 be 𝑎∈{1,2} and ℏ(𝑏)=0 for 𝑏≠𝑎. The mode transition probabilities of Markovian jump LDS (2.10) is given by 𝒫𝜂_𝑣(𝑘+1)=𝑣∣𝜂_𝑢(𝑘)=𝑢=𝜋_𝑢𝑣, (2.11) where 𝜋_𝑢𝑣≥0 for all 𝑢,𝑣∈{1,2} and ∑²_𝑣=1𝜋_𝑢𝑣=1. 𝜂_𝑣(𝑘)=1 implies ℏ(1)=1,ℏ(2)=0, which the communication transmission succeeds, and 𝜂_𝑣(𝑘)=2 implies ℏ(1)=0,ℏ(2)=1, which the communication dropout occurs. Thus, compared to (2.5), it follows that 𝜋₁₁=1−𝑝,𝜋₁₂=𝑝,𝜋₂₁=𝑞,𝜋₂₂=1−𝑞.

Definition 2.2 (stochastic finite-time stability (SFTS)). The closed-loop Markovian jump LDS with 𝑤(𝑘)=0 (2.10) 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_𝑘≥0, if 𝐄𝑥^𝑇(0)𝑅𝑥(0)≤𝛿²_𝑥⟹𝐄𝑥^𝑇(𝑘)𝑅𝑥(𝑘)<𝜖²,∀𝑘∈{1,2,…,𝑁}. (2.12)

Definition 2.3 (stochastic finite-time boundedness (SFTB)). The closed-loop LDS with Markovian jumps (2.10) 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_𝑘≥0, if the relation condition (2.12) holds.

Definition 2.4 (stochastic ℋ_∞ finite-time boundedness (Sℋ_∞FTB)). The closed-loop LDS with Markovian jumps (2.10) is said to be Sℋ_∞FTB with respect to (𝛿_𝑥,𝜖,𝛾,𝑅,𝑁,𝑑), where 0<𝛿_𝑥<𝜖, 𝑅 is a symmetric positive-definite matrix and 𝑁∈ℤ_𝑘≥0, if the closed-loop LDS with Markovian jumps (2.10) is SFTB with respect to (𝛿_𝑥,𝜖,𝑅,𝑁,𝑑) and under the zero-initial condition the output 𝑧(𝑘) satisfies 𝐄^𝑁_𝑗=0𝑧^𝑇𝑗𝑧𝑗<𝛾^2𝑁_𝑗=0𝑤^𝑇𝑗𝑤𝑗 (2.13) for any nonzero 𝑤(𝑘) which satisfies (2.2), where 𝛾 is a prescribed positive scalar.

Lemma 2.5 (see [38]). The linear matrix inequality ^𝑋11∗_𝑋21𝑋22<0 is equivalent to 𝑋₂₂<0 and 𝑋₁₁−𝑋^𝑇₂₁𝑋⁻¹₂₂𝑋₂₁<0, where 𝑋₁₁=𝑋^𝑇₁₁ and 𝑋₂₂=𝑋^𝑇₂₂.

Lemma 2.6 (see [38]). 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 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 𝜚, such that the inequality 𝑋+𝜚𝑌𝑌^𝑇+𝜚⁻¹𝑍^𝑇𝑍<0 holds.

In this paper, the feedback gain matrices 𝐿₁ and 𝐿₀ 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 LDS (2.10) Sℋ_∞FTB.

3. Main Results

In this section, for the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), we will design a state feedback controller that assures Sℋ_∞FTB of the Markovian jump LDS (2.10).

Theorem 3.1. For the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), the closed-loop Markovian jump LDS (2.10) is SFTB with respect to (𝛿_𝑥,𝜖,𝑅,𝑁,𝑑), if there exist scalars 𝜇≥1, 𝛾>0, two symmetric positive-definite matrices 𝑃₁,𝑃₂, and a set of feedback control matrices {𝐿_𝜁,𝜁∈{1,0}}, such that the following inequalities hold:

−𝜇𝑃₁∗∗∗0−𝛾²𝜇^−𝑁𝐼∗∗𝐴+𝐵𝐿₁1𝐺−𝑃1−𝑝¹⁻¹∗1𝐴𝐺0−𝑝𝑃₂⁻¹<0, (3.1)−𝜇𝑃₂∗∗∗0−𝛾²𝜇^−𝑁𝐼∗∗𝐴+𝐵𝐿₀1𝐺−𝑞𝑃₁⁻¹∗1𝐴𝐺0−𝑃1−𝑞²⁻¹<0, (3.2)sup_{}𝑎∈1,2𝜆_max𝑃_𝑎𝜇^𝑁𝛿²_𝑥+𝛾²𝑑²<inf_{}𝑎∈1,2𝜆_min𝑃_𝑎𝜖², (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_𝜁=𝐿₁, otherwise if 𝑎=2, then 𝛾(𝑘−1)=0 and 𝐿_𝜁=𝐿₀. Consider the following Lyapunov-Krasovskii functional candidate for the Markov jump LDS (2.10): 𝑉𝑥(𝑘),𝜂_𝑣(𝑘)=𝑎=𝑥^𝑇(𝑘)𝑃_𝑎𝑥(𝑘). (3.4) Then, we have 𝐄{𝑉(𝑘+1)}=𝐄²_𝑣=1𝒫𝜂_𝑣(𝑘+1)=𝑣∣𝜂_𝑣(𝑘)=𝑎×𝑥^𝑇(𝑘+1)𝑃_𝑣𝑥(𝑘+1)=𝜋_𝑎1𝐴+𝐵𝐿_𝜁𝑥(𝑘)+𝐺𝑤(𝑘)^𝑇𝑃₁𝐴+𝐵𝐿_𝜁𝑥(𝑘)+𝐺𝑤(𝑘)+𝜋_𝑎2𝐴𝑥(𝑘)+𝐺𝑤(𝑘)^𝑇𝑃₂𝐴𝑥(𝑘)+𝐺𝑤(𝑘). (3.5) Denote Θ(𝑥(𝑘),𝑤(𝑘),𝑎)≜𝐄{𝑉(𝑘+1)}−𝜇𝑉(𝑘)−𝛾²𝜇^−𝑁𝑤^𝑇(𝑘)𝑤(𝑘). (3.6) Taking into account that if 𝑎=1, then 𝐿_𝜁=𝐿₁, otherwise 𝑎=2, then 𝐿_𝜁=𝐿₀. Noting that 𝜋₁₁=1−𝑝,𝜋₁₂=𝑝,𝜋₂₁=𝑞,𝜋₂₂=1−𝑞. Thus, when 𝑎=1, it follows that Θ(𝑥(𝑘),𝑤(𝑘),1)=1−𝑝𝐴+𝐵𝐿₁𝑥(𝑘)+𝐺𝑤(𝑘)^𝑇𝑃₁𝐴+𝐵𝐿₁𝑥(𝑘)+𝐺𝑤(𝑘)+𝑝𝐴𝑥(𝑘)+𝐺𝑤(𝑘)^𝑇𝑃₂𝐴𝑥(𝑘)+𝐺𝑤(𝑘)−𝜇𝑥^𝑇(𝑘)𝑃₁𝑥(𝑘)−𝛾²𝜇^−𝑁𝑤^𝑇(𝑘)𝑤(𝑘)=𝑥(𝑘)𝑤(𝑘)^𝑇Φ^𝑇₁1−𝑝𝑃₁∗0𝑝𝑃₂Φ₁−𝜇𝑃₁∗0𝛾²𝜇^−𝑁𝐼𝑥(𝑘)𝑤(𝑘), (3.7) where Φ₁=𝐴+𝐵𝐿₁𝐺.𝐴𝐺 (3.8) By Lemma 2.5, it follows from (3.1) and (3.7) that Θ(𝑥(𝑘),𝑤(𝑘),1)<0. (3.9) When 𝑎=2, taking into account condition (3.2), the similar to (3.9), we can derive the following inequality: Θ(𝑥(𝑘),𝑤(𝑘),2)<0. (3.10) Thus, for all 𝑎∈{1,2}, we have Θ(𝑥(𝑘),𝑤(𝑘),𝑎)<0. (3.11) That is to say, for all 𝑎∈{1,2}, it follows that 𝐄{𝑉(𝑘+1)}<𝜇𝑉(𝑘)+𝛾²𝜇^−𝑁𝑤^𝑇(𝑘)𝑤(𝑘). (3.12) By (3.12), it is obvious that 𝐄{𝑉(𝑘+1)}<𝜇𝐄{𝑉(𝑘)}+𝛾²𝜇^−𝑁𝑤^𝑇(𝑘)𝑤(𝑘). (3.13) From (2.2) and (3.13) and noting that 𝜇≥1, we have 𝐄{𝑉(𝑘)}<𝜇^𝑘𝐄{𝑉(0)}+𝛾²𝜇^{−𝑁𝑘−1}_𝑗=0𝜇^{𝑘−𝑗−1}𝑤^𝑇𝑗𝑤𝑗≤𝜇^𝑘𝐄{𝑉(0)}+𝛾²𝜇^−𝑁𝜇^𝑘𝑑². (3.14) 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∈1,2𝜆_max𝑃_𝑎𝐄𝑥^𝑇(0)𝑅𝑥(0)≤sup_{}𝑎∈1,2𝜆_max𝑃_𝑎𝛿²_𝑥. (3.15) 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∈1,2𝜆_min𝑃_𝑎𝐄𝑥^𝑇(𝑘)𝑅𝑥(𝑘). (3.16) Combing with (3.14)–(3.16), we can derive 𝐄𝑥^𝑇(𝑘)𝑅𝑥(𝑘)<sup_𝑎∈{1,2}𝜆_max𝑃_𝑎𝜇^𝑘𝛿²_𝑥+𝛾²𝜇^−𝑁𝜇^𝑘𝑑²inf_𝑎∈{1,2}𝜆_min𝑃_𝑎. (3.17) Noting condition (3.3), it is obvious that 𝐄{𝑥^𝑇(𝑘)𝑅𝑥(𝑘)}<𝜖² for all 𝑘∈{1,2,…,𝑁}. This completes the proof of this theorem.

Theorem 3.2. For the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), the closed-loop Markovian jump LDS (2.10) is Sℋ_∞FTB with respect to (𝛿_𝑥,𝜖,𝛾,𝑅,𝑁,𝑑), if there exist scalars 𝜇≥1, 𝛾>0, two symmetric positive-definite matrices 𝑃₁,𝑃₂, and a set of feedback control matrices {𝐿_𝜁,𝜁∈{1,0}}, such that (3.3) and the following inequalities hold: −𝜇𝑃₁∗∗∗∗0−𝛾²𝜇^−𝑁𝐼∗∗∗𝐴+𝐵𝐿₁1𝐺−𝑃1−𝑝¹⁻¹∗∗1𝐴𝐺0−𝑝𝑃₂⁻¹∗𝐶+𝐷₁𝐿₁𝐷₂00−𝐼<0,−𝜇𝑃₂∗∗∗∗0−𝛾²𝜇^−𝑁𝐼∗∗∗𝐴+𝐵𝐿₀1𝐺−𝑞𝑃₁⁻¹∗∗1𝐴𝐺0−𝑃1−𝑞²⁻¹∗𝐶𝐷₂000−𝐼<0, (3.18)(3.19) where 𝑃_𝑎=𝑅^−1/2𝑃_𝑎𝑅^−1/2 for all 𝑎∈{1,2}.

Proof. Noting that Υ₁≜𝐶+𝐷₁𝐿₁^𝑇𝐶+𝐷₁𝐿₁∗𝐷^𝑇₂𝐶+𝐷₁𝐿₁𝐷^𝑇₂𝐷₂=𝐶+𝐷₁𝐿₁^𝑇𝐷^𝑇₂𝐶+𝐷₁𝐿₁𝐷₂≥0,Υ₂≜𝐶^𝑇𝐷𝐶∗_𝑇2𝐶𝐷^𝑇₂𝐷₂=𝐶^𝑇𝐷^𝑇₂𝐶𝐷₂≥0. (3.20)(3.21) Applying Lemma 2.5, it follows from (3.18) and (3.19) that conditions (3.1) and (3.2) hold. Therefore, the Morkovian jump LDS (2.10) is stochastic finite-time boundedness according to Theorem 3.1.
Then, we only need to prove (2.13) satisfied under zero-value initial condition. Let us 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_𝜁=𝐿₁, otherwise if 𝑎=2, then 𝛾(𝑘−1)=0 and 𝐿_𝜁=𝐿₀. Let us choose 𝑉(𝑥(𝑘),𝜂_𝑣(𝑘)=𝑎)=𝑥^𝑇(𝑘)𝑃_𝑎𝑥(𝑘) for the Markovian jump LDS (2.10). We denote Λ(𝑥(𝑘),𝑤(𝑘),𝑎)≜𝐄{𝑉(𝑘+1)}−𝜇𝑉(𝑘)+𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝜇^−𝑁𝑤^𝑇(𝑘)𝑤(𝑘). (3.22) Thus, when 𝑎=1, we have Λ(𝑥(𝑘)(𝑘))=,𝑤,11−𝑝𝐴+𝐵𝐿₁𝑥(𝑘)+𝐺𝑤(𝑘)^𝑇𝑃₁𝐴+𝐵𝐿₁𝑥(𝑘)+𝐺𝑤(𝑘)+𝑝𝐴𝑥(𝑘)+𝐺𝑤(𝑘)^𝑇𝑃₂(𝑘)(𝑘+𝐴𝑥+𝐺𝑤)𝐶+𝐷₁𝐿₁𝑥(𝑘)+𝐷₂𝑤(𝑘)^𝑇𝐶+𝐷₁𝐿₁𝑥(𝑘)+𝐷₂𝑤(𝑘)−𝜇𝑥^𝑇(𝑘)𝑃₁𝑥(𝑘)−𝛾²𝜇^−𝑁𝑤^𝑇(𝑘)𝑤(𝑘)=𝑥(𝑘)𝑤(𝑘)^𝑇Φ^𝑇₁1−𝑝𝑃₁∗0𝑝𝑃₂Φ₁−𝜇𝑃₁∗0𝛾²𝜇^−𝑁𝐼+Υ₁𝑥(𝑘)𝑤(𝑘), (3.23) where Φ₁,Υ₁ are the same as the above. Thus, according to Lemma 2.5, we can obtain from (3.18) and (3.23) Λ(𝑥(𝑘),𝑤(𝑘),1)<0. (3.24) When 𝑎=2, taking into account condition (3.19) and (3.21), the similar to the above deduction, we can derive that the following inequality holds: Λ(𝑥(𝑘),𝑤(𝑘),2)<0. (3.25) Thus, for all 𝑎∈{1,2}, we can obtain Λ(𝑥(𝑘),𝑤(𝑘),𝑎)=𝐄{𝑉(𝑘+1)}−𝜇𝑉(𝑘)+𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝜇^−𝑁𝑤^𝑇(𝑘)𝑤(𝑘)<0. (3.26) According to (3.26), it is obvious that 𝐄{𝑉(𝑘+1)}<𝜇𝐄{𝑉(𝑘)}−𝐄𝑧^𝑇(𝑘)𝑧(𝑘)+𝛾²𝜇^−𝑁𝑤^𝑇(𝑘)𝑤(𝑘). (3.27) From (3.27), we have 𝐄{𝑉(𝑘)}<𝜇^𝑘𝐄{𝑉(0)}−^𝑘−1_𝑗=0𝜇^{𝑘−𝑗−1}𝐄𝑧^𝑇𝑗𝑧𝑗+𝛾²𝜇^{−𝑁𝑘−1}_𝑗=0𝜇^{𝑘−𝑗−1}𝑤^𝑇𝑗𝑤𝑗. (3.28) Under the zero-value initial condition and noting that 𝑉(𝑘)≥0 for all 𝑘∈ℤ_𝑘≥0, we have ^𝑘−1_𝑗=0𝜇^{𝑘−𝑗−1}𝐄𝑧^𝑇𝑗𝑧𝑗<𝛾²𝜇^{−𝑁𝑘−1}_𝑗=0𝜇^{𝑘−𝑗−1}𝑤^𝑇𝑗𝑤𝑗. (3.29) From (3.29) and noting that 𝜇≥1, we have 𝐄^𝑁_𝑗=0𝑧^𝑇𝑗𝑧𝑗=^𝑁_𝑗=0𝐄𝑧^𝑇𝑗𝑧𝑗≤^𝑁_𝑗=0𝐄𝜇^𝑁−𝑗𝑧^𝑇𝑗𝑧𝑗<𝛾²𝜇^𝑁−𝑁_𝑗=0𝜇^𝑁−𝑗𝑤^𝑇𝑗𝑤𝑗≤𝛾^2𝑁_𝑗=0𝑤^𝑇𝑗𝑤𝑗. (3.30) Thus, this completes the proof of the theorem.

Denoting 𝑋₁=𝑃₁⁻¹,𝑋₂=𝑃₂⁻¹,𝐿₁=𝑌₁𝑋₁⁻¹,𝐿₀=𝑌₀𝑋₂⁻¹ and applying Lemmas 2.5 and 2.6, one can obtain from Theorem 3.2 the following results on the stochastic ℋ_∞ finite-time stabilization.

Theorem 3.3. For the given failure rate 𝑝 and recovery rate 𝑞 with 𝑝,𝑞∈(0,1), there exists a state feedback controller 𝑢(𝑡)=𝐿_𝜁𝑥(𝑡),𝜁∈{1,0} with 𝐿₁=𝑌₁𝑋₁⁻¹ and 𝐿₀=𝑌₀𝑋₂⁻¹ such that the closed-loop Markovian jump LDS (2.10) is Sℋ_∞FTB with respect to (𝛿_𝑥,𝜖,𝛾,𝑅,𝑁,𝑑), if there exist scalars 𝜇≥1, 𝛾>0,𝜖₁>0,𝜖₂>0, two symmetric positive-definite matrices 𝑋₁,𝑋₂, and a set of feedback control matrices {𝐿_𝜁,𝜁∈{1,0}}, such that the following inequalities hold: −𝜇𝑋₁∗∗∗∗∗∗0−𝛾²𝜇^−𝑁𝐼∗∗∗∗∗𝐴𝑋₁+𝐵𝑌₁1𝐺−𝑋1−𝑝¹+𝜖₁𝐹𝐹^𝑇∗∗∗∗𝐴𝑋₁1𝐺0−𝑝𝑋₂+𝜖₁𝐹𝐹^𝑇∗∗∗𝐶𝑋₁+𝐷₁𝑌₁𝐷₂𝐸00−𝐼∗∗₁𝑋₁+𝐸₂𝑌₁0000−𝜖₁𝐸𝐼∗₁𝑋₁00000−𝜖₁𝐼<0,−𝜇𝑋₂∗∗∗∗∗∗0−𝛾²𝜇^−𝑁𝐼∗∗∗∗∗𝐴𝑋₂+𝐵𝑌₀1𝐺−𝑞𝑋₁+𝜖₂𝐹𝐹^𝑇∗∗∗∗𝐴𝑋₂1𝐺0−𝑋1−𝑞²+𝜖₂𝐹𝐹^𝑇∗∗∗𝐶𝑋₂𝐷₂𝐸00−𝐼∗∗₁𝑋₂+𝐸₂𝑌₀0000−𝜖₂𝐸𝐼∗₁𝑋₂00000−𝜖₂𝐼<0,sup_{}𝑎∈1,2𝜆_max𝑋_𝑎𝜇^𝑁𝛿²_𝑥+𝛾²𝑑²<inf_{}𝑎∈1,2𝜆_min𝑋_𝑎𝜖², (3.31)(3.32)(3.33) where 𝑋_𝑎=𝑅^−1/2𝑋_𝑎⁻¹𝑅^−1/2 for all 𝑎∈{1,2}.

Remark 3.4. It is easy to check that condition (3.33) is guaranteed by imposing the conditions for all 𝑎∈{1,2}: 𝜆𝑅⁻¹<𝑋_𝑎<𝑅⁻¹,𝜇^−𝑁𝛾²𝑑²−𝜖²∗𝛿_𝑥−𝜆<0. (3.34) It follows that conditions (3.31), (3.32), and (3.34) are not strict LMIs; however, once we fix the parameter 𝜇, the conditions can be turned into LMI-based feasibility problem:

Remark 3.5. From the above discussion, we can obtain that the feasibility of conditions stated in Theorem 3.3 can be turned into the following LMIs based feasibility problem min𝛾²+𝜖²𝑋₁,𝑋₂,𝑌₁,𝑌₀,𝜖₁,𝜖₂,𝜆s.t.LMIs(3.31),(3.32),and(3.34) (3.35) with a fixed parameter 𝜇. Furthermore, we can also find the parameter 𝜇 by an unconstrained nonlinear optimization approach, which a locally convergent solution can be obtained by using the program fminsearch in the optimization toolbox of Matlab.

Remark 3.6. If we can find feasible solution with the parameter 𝜇=1, by the above discussion, we can obtain that the designed controller can ensure both stochastic finite-time boundedness and robust stochastic stabilization of the family of network control systems.

4. Numerical Examples

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

Example 4.1. Consider a Morkovian jump LDS (2.10) with parameters as𝐴=,1.500.20.5,𝐵=1001,𝐺=1201𝐶=110.51,𝐷₁=1111,𝐷₂=,10.503𝐹=0.10.10.20,𝐸₁=0.3000.4,𝐸₂=,0.400.50.2 (4.1) and 𝑑=1,Δ(𝑘)=diag{Δ₁(𝑘),Δ₂(𝑘)}, where Δ_𝑖(𝑘) satisfies |Δ_𝑖(𝑘)|≤1 for all 𝑖∈{1,2} and 𝑘∈ℤ_𝑘≥0. Moreover, we assume the failure rate 𝑝=0.3 and the recovery rate 𝑞=0.6.
Then, we chose 𝑅=𝐼₃,𝛿_𝑥=1,𝑁=5, and 𝜇=1.8, Theorem 3.3 yields to 𝛾=27.1939, 𝜖=28.7912, and 𝑋₁=0.30690.20900.20900.8775,𝑋₂=,𝑌0.2612−0.1408−0.14080.6357₁=−0.5248−0.4726−0.0747−0.6270,𝑌₀=,𝜖−0.42550.08770.0762−0.4695₁=1.6732,𝜖₂=0.6003,𝜆=0.2127. (4.2) Thus, we can obtain the following state feedback controller gains 𝐿₁=−1.6031−0.15670.2903−0.7837,𝐿₀=.−1.7652−0.2529−0.1206−0.7653 (4.3) Furthermore, let 𝑅=𝐼₃,𝛿_𝑥=1, and 𝑁=5, by Theorem 3.3, the optimal bound with minimum value of 𝛾²+𝜖² relies on the parameter 𝜇. We can find feasible solution when 1.12≤𝜇≤34.13. Figures 1 and 2 show the optimal value with different value of 𝜇. Then, by using the program fminsearch in the optimization toolbox of Matlab starting at 𝜇=1.8, the locally convergent solution can be derived as 𝐿₁=−1.5669−0.11660.3827−0.8124,𝐿₀=,−1.6611−0.16680.0456−0.7598 (4.4) with 𝜇=1.5694 and the optimal value 𝛾=26.2353 and 𝜖=27.4932.

Figure 1 

The local optimal bound of 𝛾.

Figure 2 

The local optimal bound of 𝜖.

Example 4.2. Consider a Morkovian jump LDS (2.10) with 𝐴=,0.500.50.8,𝐵=1021 (4.5) and the failure rate 𝑝=0.3 and the recovery rate 𝑞=0.9. In addition, the other matrices parameters are the same as Example 4.1.
Then, let 𝑅=𝐼₃ and 𝛿_𝑥=1, by Theorem 3.3, we can find feasible solution when 𝜇=1. Furthermore, when 𝜇=1, it yields the optimal value 𝛾=4.7057 and 𝜖=5.0791 and the following optimized state feedback controller gains: 𝐿₁=−0.3412−0.0380−0.4900−1.1045,𝐿₀=.−0.5649−0.06850.6100−0.7019 (4.6) Thus, the above LDS with Morkovian jumps is stochastically stable and the calculated minimum ℋ_∞ performance 𝛾 satisfies ‖𝑇_𝑤𝑧‖<4.7057.

5. Conclusions

This paper addresses the Sℋ_∞FTB control problems for one family of linear discrete-time systems over networks with packet dropout. Under assuming packet loss being a time homogenous Markov process, the class of linear discrete-time systems can be regarded as Markovian jump systems. Sufficient conditions are given for the resulting closed-loop linear discrete-time Markovian jump system to be Sℋ_∞FTB, and state feedback controllers are designed to guarantee Sℋ_∞FTB of the class of linear systems with Markov jumps. The Sℋ_∞FTB 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 robust stochastic stabilization of the class of linear discrete-time systems with data packet dropout. Finally, simulation results are also given to show the validity of the proposed approaches.

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 the Doctoral Foundation of Henan University of Technology under Grant 2009BS048, supported by the Foundation of Henan Educational Committee under Grants 2011A120003 and 2011B110009, and supported by the Foundation of Henan University of Technology under Grant 09XJC011.