Finite-Time Control for Markovian Jump Systems with Polytopic Uncertain Transition Description and Actuator Saturation

Abstract

The problem of finite-time 𝐿₂-𝐿_∞ control for Markovian jump systems (MJS) is investigated. The systems considered time-varying delays, actuator saturation, and polytopic uncertain transition description. The purpose of this paper is to design a state feedback controller such that the system is finite-time bounded (FTB) and a prescribed 𝐿₂-𝐿_∞ disturbance attenuation level during a specified time interval is guaranteed. Based on the Lyapunov method, a linear matrix inequality (LMI) optimization problem is formulated to design the delayed feedback controller which satisfies the given attenuation level. Finally, illustrative examples show that the proposed conditions are effective for the design of robust state feedback controller.

1. Introduction

In the aspect of modeling practical systems with abrupt random changes, such as manufacturing system, telecommunication, and economic systems, MJS have powerful ability. MJS have been extensively studied during the past decades and many systematic results have been obtained [1–3]. The peak-to-peak filtering problem was studied for a class of Markov jump systems with uncertain parameters in [4]. A robust 𝐻₂ state feedback controller for continuous-time Markov jump linear systems subject to polytopic-type parameter uncertainty was designed in [5]. In [6], the authors address the stabilization problem for single-input Markov jump linear systems via mode-dependent quantized state feedback for control.

Actuator saturation which can lead to poor performance of the closed-loop system is another active research area. In practical situations, it may be encountered sometimes. How to preserve the closed-loop system performance in the case of actuator saturation would be more meaningful. In [7], the 𝐻_∞ control problem for discrete-time singular Markov jump systems with actuator saturation was considered. In [8] the stochastic stabilization problem for a class of Markov jump linear systems subject to actuator saturation was considered.

In some practical applications, the behavior of the system over a finite-time interval is mainly considered. Finite-time stable (FTS) and Lyapunov asymptotic stability are independent concepts. The concept of FTS was first introduced in [9]. A system is said to be finite-time stable if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval. FTS of linear time-varying systems was considered in [10]. Sufficient conditions for the solvability of both the state and the output feedback problems are stated. Amato [11] provided a necessary and sufficient condition for the FTS of linear-varying systems with jumps. Recently, robust finite-time 𝐻_∞ control of jump systems was dealt with in [12–14]. In [15], the problems of finite-time stability analysis were investigated for a class of Markovian switching stochastic systems. To the best of authors’ knowledge, however, the problem of finite-time 𝐿₂-𝐿_∞ performance for discrete-time MJS with imprecise transition probabilities and time-varying delays has not been well addressed, which motivates our work.

This paper deals with this problem. More specifically, the actuator is saturation. By using the Lyapunov-Krasovskii functional, a new sufficient condition for stochastic asymptotic stability with finite-time 𝐿₂-𝐿_∞ performance is derived in terms of LMI. Based on this, the existence condition of the desired performance which guarantees finite-time stability and an 𝐿₂-𝐿_∞ performance of the MJS is presented. A numerical example is provided to show the effectiveness of the proposed results.

Throughout the paper, if not explicitly stated, matrices are assumed to have compatible dimensions. The notation 𝑊>(≥,<,≤)0 is used to denote a symmetric positive definite (positive semidefinite, negative, negative semidefinite) matrix. 𝜆_min(⋅) and 𝜆_max(⋅) represent the minimum and maximum eigenvalues of the corresponding matrix, respectively. 𝐼 is the identity matrix with compatible dimensions. ‖⋅‖ refers to the Euclidean norm of vectors and 𝐸[⋅] stands for the mathematical expectation. For a symmetric block matrix, “*” is used as an ellipsis for the terms that are obtained by symmetry.

2. Problem Statement and Preliminaries

Consider a discrete-time MJS with actuator saturation and delay in the state. Let the system dynamics be described by the following: 𝑥(𝑘+1)=𝐴_𝜃1𝑟_𝑘𝑥(𝑘)+𝐴_𝜃2𝑟_𝑘𝑥(𝑘−𝑑)+𝐵_𝜃1𝑟_𝑘𝜎𝑢_𝑘+𝐵_𝜃2𝑟_𝑘𝑤_𝑘,𝑧(𝑘)=𝐶_𝜃1𝑟_𝑘𝑥(𝑘)+𝐶_𝜃2𝑟_𝑘𝑥(𝑘−𝑑)+𝐷_𝜃1𝑟_𝑘𝑤_𝑘, (1) where 𝑥_𝑘∈𝑅^𝑛 is the system state, 𝑧_𝑘∈𝑅^𝑛 is the system output, 𝑢_𝑘∈𝑅^𝑚 is the control input, 𝑤_𝑘∈𝑅^𝑞 is the disturbance input which belongs to 𝐿₂[0,∞) and ∑^∞_𝑘=0𝑤^𝑇_𝑘𝑤_𝑘<𝜅², and𝜅 is a given positive scalar. 𝐴_𝜃1(𝑟_𝑘),𝐴_𝜃2(𝑟_𝑘),𝐵_𝜃1(𝑟_𝑘),𝐵_𝜃2(𝑟_𝑘),𝐶_𝜃1(𝑟_𝑘),𝐷_𝜃1(𝑟_𝑘), and 𝐷_𝜃2(𝑟_𝑘) are appropriately dimensioned real-valued matrices, which belong to the part of convex polyhedron Φ(𝑟_𝑘): Φ𝑟_𝑘=^𝐿_𝑙=1𝜃_𝑙𝐴_𝑙1𝑟_𝑘,𝐴_𝑙2𝑟_𝑘,𝐵_𝑙1𝑟_𝑘,𝐵_𝑙2𝑟_𝑘,𝐶_𝑙1𝑟_𝑘,𝐶_𝑙2𝑟_𝑘,𝐷_𝑙1𝑟_𝑘,𝐷_𝑙2𝑟_𝑘,^𝐿_𝑙=1𝜃_𝑙=1,𝜃_𝑙≥0, (2) where 𝐴_𝑙1(𝑟_𝑘),𝐴_𝑙2(𝑟_𝑘),𝐵_𝑙1(𝑟_𝑘),𝐵_𝑙2(𝑟_𝑘),𝐶_𝑙1(𝑟_𝑘),𝐶_𝑙2(𝑟_𝑘), and 𝐷_𝑙1(𝑟_𝑘) are matrix functions of the random jumping process {𝑟_𝑘} (Figure 1), which is a discrete-time Markov chain taking values in a finite set Ω={1,2,…,𝑆} with transition probabilities: 𝑃𝑟_𝑘+1=𝑗∣𝑟_𝑘=𝑖=𝜋_𝑖𝑗. (3) Here 𝜋_𝑖𝑗≥0 and for any 𝑖,𝑗∈Ω,∑^𝑠_𝑗=1𝜋_𝑖𝑗=1. Assuming that the transition probability 𝜋_𝑖𝑗 is not exactly known, a certain range can only be given 𝜋_𝑣(𝑖,1),𝜋_𝑣(𝑖,2),…,𝜋_𝑣(=𝑖,𝑆)_𝑀𝑚=1𝑣_𝑚𝜋_𝑚(𝑖,1),𝜋_𝑚(𝑖,2),…,𝜋_𝑚(𝑖,𝑆), (4) where 𝑣=[𝑣₁⋯𝑣_𝑀]^𝑇∈𝑅^𝑀 and ∑^𝑀_𝑚=1𝑣_𝑚=1, and the transition probability belongs to the following convex polyhedron: ℵ𝑟_𝑘𝜋=𝑖=Co¹(𝑖,1),𝜋₁(𝑖,2),…,𝜋₁(𝜋𝑖,𝑁)_𝑀(𝑖,1),𝜋_𝑀(𝑖,2),…,𝜋_𝑀(.𝑖,𝑁) (5) When the system operates in the 𝑖th mode (𝑟_𝑘=𝑖), for simplicity, the matrices 𝐴_𝜃1(𝑟_𝑘),𝐴_𝜃2(𝑟_𝑘),𝐵_𝜃1(𝑟_𝑘),𝐵_𝜃2(𝑟_𝑘),𝐶_𝜃1(𝑟_𝑘), and 𝐷_𝜃1(𝑟_𝑘) are denoted as 𝐴_𝜃1𝑖,𝐴_𝜃2𝑖,𝐵_𝜃1𝑖, 𝐵_𝜃2𝑖, 𝐶_𝜃1𝑖, and 𝐷_𝜃1𝑖, respectively. 𝑑 is a positive integer denoting the constant delay of the system state (Figures 2 and 3).

Figure 1 

Jumping mode.

Figure 2 

Response of the system state 𝑥1.

Figure 3 

Response of the system state 𝑥2.

In system (1), 𝜎(⋅)∶𝑅^𝑚→𝑅^𝑚 is the vector-valued standard saturation function defined as follows: 𝜎(𝑢)=𝜎𝑢₁,𝜎𝑢₂,…,𝜎𝑢_𝑚^𝑇, (6) where 𝜎(𝑢_𝜃)=sign(𝑢_𝜃)min{1,|𝑢_𝜃|}. It is assumed that system (1) is completely controllable. A mode-dependent controller is considered here with the following form: 𝜎(𝑢(𝑘))=𝜎𝐾_𝑖𝑥(𝑘), (7) where 𝐾_𝑖∈𝑅^𝑚×𝑛(∀𝑟_𝑘=𝑖∈Ω) is the controller gain to be determined.

Let 𝑀 be the set of 𝑚×𝑚 diagonal matrices whose diagonal elements are either 1 or 0. Suppose each element of 𝑀 is 𝑀_𝑗, 𝑗=1,…,2^𝑚, and denote 𝑀⁻_𝑗=𝐼−𝑀_𝑗. Note that 𝑀⁻_𝑗 is also an element of 𝑀 if 𝑀_𝑗∈𝑀. Let ℎ_𝑖𝑗 be the 𝑗th row of the matrix 𝐻_𝑖, and define the symmetric polyhedron by 𝜑(𝐻_𝑖)={𝑥(𝑡)∈𝑅^𝑛∶|𝑓_𝑖𝑗𝑥(𝑡)|≤1,𝑖=1,2,…,𝑚}.

Lemma 1 (see [8]). Let 𝐾_𝑖,𝐻_𝑖∈𝑅^𝑚×𝑛 be given matrix. For 𝑥(𝑡)∈𝑅^𝑛, if 𝑥(𝑡)∈𝜑(𝐻_𝑖), then 𝜎𝐾_𝑖𝑥(𝑡)=^2𝑚_𝑟=1𝜁_𝑟𝑀_𝑟𝐾_𝑖+𝑀⁻_𝑟𝐻_𝑖𝑥(𝑡), (8) where 0≤𝜁_𝑟≤1, ∑^2𝑚_𝑟=1𝜁_𝑟=1.
By the connection of (6), (7) and (8), the following closed-loop MJS are obtained: 𝑥(𝑘+1)=𝐴_𝜃1𝑟_𝑘+𝐵_𝜃1𝑟_𝑘×^2𝑚_𝑟=1𝜁_𝑟𝑀_𝑟𝐾_𝑖+𝑀⁻_𝑟𝐻_𝑖𝑥(𝑘)+𝐴_𝜃2𝑟_𝑘𝑥(𝑘−𝑑)+𝐵_𝜃2𝑟_𝑘𝑤_𝑘. (9)

To describe the main objective of this note more precisely, let us now introduce the following definition for the underlying system.

Definition 2 (see [13]). Given a time constant 𝑇>0, the MJS 𝑥(𝑘+1)=𝐴_𝜃1𝑟_𝑘𝑥(𝑘)+𝐴_𝜃2𝑟_𝑘𝑥(𝑘−𝑑) (10) are said to be FTS with respect to ℏ₁ℏ₂𝑇𝑅_𝑖, if there exist positive matrix 𝑅_𝑖>0, scalars ℏ₁>0 and ℏ₂>0, and 𝐸𝑥^𝑇𝑘₁𝑅_𝑖𝑥𝑘₁≤ℏ₁⟹𝐸𝑥^𝑇𝑘₂𝑅_𝑖𝑥𝑘₂≤ℏ₂,𝑘₁∈{−ℎ,…,0},𝑘₂∈{1,2,…,𝑇}. (11)

Definition 3 (see [13]). Given a time constant 𝑇>0, the MJS 𝑥(𝑘+1)=𝐴_𝜃1𝑟_𝑘𝑥(𝑘)+𝐴_𝜃2𝑟_𝑘𝑥(𝑘−𝑑)+𝐵_𝜃2𝑟_𝑘𝑤_𝑘 (12) are said to be finite-time bounded (FTB) with respect to ℏ₁ℏ₂𝑇𝑅_𝑖, if there exist positive matrix 𝑅_𝑖>0 and scalars ℏ₁>0 and ℏ₂>0, and satisfied (11).

In general, FTB and FTS are different. If there is external disturbance in systems, the concept of FTB is used. Conversely, FTS is addressed.

The objective of this paper is to design a delayed feedback controller which satisfies the given attenuation level of system (1). The design procedure is given in the next section.

Definition 4. The time-delay MJS (1) is said to be finite-time 𝐿₂-𝐿_∞ control with respect to ℏ₁ℏ₂𝑇𝑅_𝑖 and performance 𝛾, where 𝑅_𝑖>0, 𝛾>0,ℏ₁>0, and ℏ₂>0, if the time-delay MJS (1) is stochastically FTB and under the zero-initial condition the output 𝑧(𝑘) satisfies ‖𝑧(𝑘)‖_∞<𝛾‖𝑤(𝑘)‖₂ (13) for all nonzero 𝑤(𝑘)∈𝐿₂[0,∞) subject to the zero-initial condition.

3. Main Results

In this section, firstly stochastic FTB analysis of nominal time-delay MJS (1) is provided. Then, these results will be extended to the MJS (1) with actuator saturation and uncertain transition probability. LMI conditions are established.

Lemma 5. System (1) with 𝜎(𝑢_𝑘)≡0 is stochastic FTB with respect to ℏ₁ℏ₂𝑑𝑅_𝑖𝑁; if for scalars 𝜍≥1, ℏ₁>0, and ℏ₂>0, there exist symmetric matrices 𝑅_𝑖>0(𝑖∈Ω) and 𝑄_𝑖>0(𝑖∈Ω), such that the following matrix inequalities hold: 𝐴Λ=^{𝑇𝑙1𝑖}𝑃_𝑖𝐴_𝑙1𝑖−𝑃_𝑖𝐴+𝑄∗∗_{𝑇𝑙2𝑖}𝑃_𝑖𝐴_𝑙1𝑖−𝑄+𝐴^𝑇_𝑙2𝑖𝑃_𝑖𝐴_𝑙2𝑖∗𝐵^𝑇_𝑙2𝑖𝑃_𝑖𝐴_𝑙1𝑖𝐵^𝑇_𝑙2𝑖𝑃_𝑖𝐴_𝑙2𝑖𝐵^𝑇_𝑙2𝑖𝑃_𝑖𝐵_𝑙2𝑖−𝐼<0𝜍^𝑘𝜆_max𝑃_𝑟(0)+𝜆_max(𝑄)⋅𝑑𝑐₁≤𝑐₂⋅𝜆_min𝑃_𝑖, (14)(15) where 𝑃_𝑖=∑^𝑆_𝑗=1𝜋_𝑣𝑖𝑗𝑃_𝑗.

Proof. Choose the following Lyapunov functional: 𝑉(𝑘)=𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)+^𝑘−1_{𝑛=𝑘−𝑑}𝑥^𝑇(𝑛)𝑄𝑥(𝑛). (16) The proof of Lemma 5 is divided into two parts. In the first part, the following inequality is obtained: 𝐸{𝑉(𝑘)}<𝜍^𝑘𝐸{𝑉(0)}+𝜍^𝑘𝑤^𝑇(𝑘)𝑤(𝑘). (17) Then, we compute(𝑘){𝑉((𝑘)=Δ𝑉=𝐸𝑘+1)}−𝑉_𝑆𝑗=1𝜋_𝑣𝑖𝑗𝑥^𝑇(𝑘+1)𝑃_𝑗𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)+^𝑆_𝑗=1𝜋_𝑣𝑖𝑗𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑)𝑄𝑥(𝑘−𝑑)=𝜗^𝑇(𝑘)𝐴^𝑇_𝜃1𝑖𝑃_𝑖𝐴_𝜃1𝑖−𝑃_𝑖𝐴+𝑄∗∗_{𝑇𝜃2𝑖}𝑃_𝑖𝐴_𝜃1𝑖−𝑄+𝐴^𝑇_𝜃2𝑖𝑃_𝑖𝐴_𝜃2𝑖∗𝐵^𝑇_𝜃2𝑖𝑃_𝑖𝐴_𝜃1𝑖𝐵^𝑇_𝜃2𝑖𝑃_𝑖𝐴_𝜃2𝑖𝐵^𝑇_𝜃2𝑖𝑃_𝑖𝐵_𝜃2𝑖𝜗(𝑘)=𝜗^𝑇(𝑘)^𝐿_𝑙=1𝜃_𝑙^𝐿_𝑙=1𝜃_𝑙𝐴^𝑇_𝑙1𝑖𝑃_𝑖𝐴_𝑙1𝑖−𝑃_𝑖𝐴+𝑄∗∗_{𝑇𝑙2𝑖}𝑃_𝑖𝐴_𝑙1𝑖−𝑄+𝐴^𝑇_𝑙2𝑖𝑃_𝑖𝐴_𝑙2𝑖∗𝐵^𝑇_𝑙2𝑖𝑃_𝑖𝐴_𝑙1𝑖𝐵^𝑇_𝑙2𝑖𝑃_𝑖𝐴_𝑙2𝑖𝐵^𝑇_𝑙2𝑖𝑃_𝑖𝐵_𝑙2𝑖𝜗(𝑘), (18)where 𝜗(𝑘)=𝑥(𝑘)𝑥(𝑘−𝑑)𝑤(𝑘).
Note condition (14); it follows that 𝐸{𝑉(𝑘+1)}−𝑉(𝑘)<(𝜍−1)𝑉(𝑘)+𝑤^𝑇(𝑘)𝑤(𝑘),𝜍≥1. (19) Therefore, we obtain that 𝐸{𝑉(𝑘+1)}<𝜍𝑉(𝑘)+𝑤^𝑇(𝑘)𝑤(𝑘). (20) That is, 𝐸{𝑉(𝑥(1),𝑟(1))}<𝜍𝑉(𝑥(0),𝑟(0))+𝑤^𝑇(𝑘)𝑤(𝑘)𝐸{𝑉(𝑥()({𝑉(𝑥(𝑘)(𝑘⋮⋮⋮𝑘+1,𝑟𝑘+1))}<𝜍𝐸,𝑟))}+𝑤^𝑇(𝑘)𝑤(𝑘). (21) By recursive, 𝐸{𝑉(𝑘)}<𝜍^𝑘𝐸{𝑉(0)}+𝐸^𝑘−1_𝜏=0𝜍^{𝑘−𝜏−1}𝑤^𝑇(𝜏)𝑤(𝜏). (22) Then the inequality in (17) is obtained.
In the second part, stochastic FTB is established: 𝐸{𝑉(𝑘)}=𝐸𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)+^𝑘−1_{𝑛=𝑘−𝑑}𝑥^𝑇(𝑛)𝑄𝑥(𝑛)≥𝜆_min𝑃_𝑖𝐸𝑥^𝑇(𝑘)𝑅𝑥(𝑘). (23) On the other hand, 𝜍^𝑘𝐸{𝑉(0)}≤𝜍^𝑘𝜆_max𝑃_𝑟(0)𝐸𝑥^𝑇(0)𝑅𝑥(0)+𝜆_max(𝑄)𝐸⁻¹_𝑛=−𝑑𝑥^𝑇(𝑛)𝑅𝑥(𝑛). (24) From Definition 2, we have 𝜍^𝑘𝐸{𝑉(0)}≤𝜍^𝑘𝜆_max𝑃_𝑟(0)+𝜆_max(𝑄)⋅𝑑ℏ₁+𝜍^𝑘. (25) By (23) and (25), we know 𝐸𝑥^𝑇(𝑘)(𝑘≤𝜍𝑅𝑥)_𝑘𝜆_max𝑃_𝑟(0)+𝜆_max(𝑄)⋅𝑑+1ℏ₁𝜆_min𝑃_𝑖≤ℏ₂. (26) This completes the proof.

Theorem 6. System (1) is finite-time 𝐿₂-𝐿_∞ control and satisfies the given lever 𝛾 with respect to ℏ₁ℏ₂𝑑𝑅_𝑖𝑁; if for scalars 𝜍≥1, ℏ₁>0, and ℏ₂>0, there exist symmetric matrices 𝑅_𝑖>0(𝑖∈Ω) and 𝑄>0, such that the following matrix inequalities hold: Θ₁=𝐴^𝑇_𝑙1𝑖𝑃_𝑖𝐴_𝑙1𝑖−𝑃_𝑖𝐴+𝑄∗∗_{𝑇𝑙2𝑖}𝑃_𝑖𝐴_𝑙1𝑖−𝑄+𝐴^𝑇_𝑙2𝑖𝑃_𝑖𝐴_𝑙2𝑖∗𝐵^𝑇_𝑙2𝑖𝑃_𝑖𝐴_𝑙1𝑖𝐵^𝑇_𝑙2𝑖𝑃_𝑖𝐴_𝑙2𝑖𝐵^𝑇_𝑙2𝑖𝑃_𝑖𝐵_𝑙2𝑖−𝐼<0,Θ₂=−𝑃_𝑖𝐶∗∗∗0−𝑄∗∗00−𝐼∗_𝑙1𝑘𝐶_𝑙2𝑘𝐷_𝑙2𝑘−𝛾²𝐼𝜍^𝑘𝜆_max𝑃_𝑟(0)+𝜆_max(𝑄)⋅𝑑𝑐₁≤𝑐₂⋅𝜆_min𝑃_𝑖. (27)(28)

Proof. System (1) with 𝜎(𝑢_𝑘)≡0 is FTB according to Lemma 5 and inequality (27).
Subsequently, to establish the energy-to-peak performance for the system (1), assume that the initial values for the plant are zeros and consider the following function: ℵ∶=𝐸{𝑉(𝑘)}−^𝑘−1_𝑖=0𝑤^𝑇_𝑖𝑤_𝑖. (29) For any nonzero 𝑤_𝑘∈𝑙₂[0,∞) and 𝑘>0, it follows from (18) that ℵ∶=𝐸^𝑘−1_𝑖=0Δ𝑉(𝑖)−^𝑘−1_𝑖=0𝑤^𝑇_𝑖𝑤_𝑖=𝜗^𝑇(𝑘)Θ₁𝜗(𝑘). (30) It follows from (27) that 𝐸{𝑉(𝑘)}<∑^𝑘−1_𝑖=0𝑤^𝑇_𝑖𝑤_𝑖.
For all the time instants 𝑘>0, the expectation of the output can be evaluated as 𝐸𝑧^𝑇_𝑘𝑧_𝑘=𝐸𝜗^𝑇(𝑘)𝐶_𝜃1𝑘𝐶_𝜃2𝑘𝐷_𝜃1𝑘^𝑇×𝐶_𝜃1𝑘𝐶_𝜃2𝑘𝐷_𝜃1𝑘𝜗(𝐾𝛾)<𝐸²𝜗^𝑇(𝑘)𝑃_𝑖∗00𝑄_𝑖0𝜗(𝑘)∗∗𝐼<𝛾²𝐸^𝑘_𝑖=0𝑤^𝑇_𝑖𝑤_𝑖<𝛾²𝐸^∞_𝑖=0𝑤^𝑇_𝑖𝑤_𝑖=𝛾²‖𝑤‖²₂. (31) Applying Definition 4, the statement of Theorem 6 is true.

Theorem 7. Consider the uncertain time-delay system (1); there exists a state feedback controller 𝜎(𝐾_𝑖𝑥(𝑡)) such that the uncertain time-delay system (1) is finite-time 𝐿₂-𝐿_∞ control with respect to ℏ₁ℏ₂𝑑𝑅_𝑖𝑁, if the following LMIs hold: Λ₁=−𝑋_𝑖00𝜀₁₄⋯𝜀₁₆𝑋_𝑖∗−𝑅0𝜀₂₄⋯𝜀₂₆0∗∗−𝐼𝜀₃₄⋯𝜀₃₆0∗∗∗𝜀₄₄⋯00∗∗∗∗⋱00∗∗∗∗∗𝜀₆₆0∗∗∗∗∗∗−𝑅<0,Λ₂=−𝑃_𝑖𝐶∗∗∗0−𝑄∗∗00−𝐼∗_𝑙1𝑘𝐶_𝑙2𝑘𝐷_𝑙2𝑘−𝛾²𝐼𝜍<0,_𝑘𝜆_max𝑃_𝑟(0)+𝜆_max(𝑄)⋅𝑑𝑐₁≤𝑐₂⋅𝜆_min𝑃_𝑖, (32)(33) where 𝜀₁₄=√𝜋_𝑖1(𝐴_𝑙1𝑖+𝐵_𝑙1𝑖(𝑀_𝑟𝑌_𝑖+𝑀⁻_𝑟𝑍_𝑖), 𝜀₂₄=√𝜋_𝑖1𝑅𝐴_𝑙2𝑖, 𝜀₃₄=√𝜋_𝑖1𝑅𝐵_𝑙2𝑖, 𝜀₄₄=−𝑋₁, 𝜀₁₆=√𝜋_𝑖𝑆(𝐴_𝑙1𝑖+𝐵_𝑙1𝑖(𝑀_𝑟𝑌_𝑖+𝑀⁻_𝑟𝑍_𝑖)_𝑖), 𝜀₂₆=√𝜋_𝑖𝑆𝑅𝐴_𝑙2𝑖, 𝜀₃₆=√𝜋_𝑖𝑆𝑅𝐵_𝑙2𝑖, and 𝜀₆₆=−𝑋_𝑆.

The state feedback controller is designed as 𝜎(𝐾_𝑖𝑥(𝑡))=∑^2𝑚_𝑟=1𝜁_𝑟(𝑀_𝑟𝐾_𝑖+𝑀⁻_𝑟𝐻_𝑖)𝑥(𝑡).

Proof. Noting condition (27) and 𝑃_𝑖=Γ_𝑖𝜅Γ^𝑇_𝑖, where 𝜅=diag{𝑃₁,…,𝑃_𝑆}, Γ_𝑖=[𝑝_𝑖1𝐼,…,𝑝_𝑖𝑆𝐼] thus Θ₁ can be rewritten as Θ₁=−𝑃_𝑖+𝐴+𝑄∗∗0−𝑄∗00−𝐼_{𝑇𝑙1𝑖}𝐴^𝑇_𝑙2𝑖𝐵^𝑇_𝑙2𝑖𝜅𝐴_𝑙1𝑖𝐴_𝑙2𝑖𝐵_𝑙2𝑖<0. (34) Using Schur complement, it can be obtained −𝑃+𝑄00𝐴^𝑇_𝑙1𝑖Γ_𝑖∗−𝑄0𝐴^𝑇_𝑙2𝑖Γ_𝑖∗∗−𝐼𝐵^𝑇_𝑙2𝑖Γ_𝑖∗∗∗−𝜅⁻¹<0. (35) Let 𝑋_𝑖=𝑃_𝑖⁻¹,𝑅=𝑄⁻¹, 𝑌_𝑖=𝐾_𝑖𝑋_𝑖, and 𝑍_𝑖=𝐻_𝑖𝑋_𝑖. Pre- and postmultiplying (35) by diag{𝑋_𝑖,𝑅,𝐼,𝐼} and then using Schur complement, then inequality (32) is obtained. Implying Theorem 6, we can conclude that the corresponding closed-loop system is finite-time 𝐿₂-𝐿_∞ control. This completes the proof.

4. Numeral Example

To illustrate the proposed results, a numerical example is considered for finite-time 𝐿₂-𝐿_∞ control. The system is described by (1) and assumed to have two modes; Ω={1,2}. The mode switching is governed by a Markov chain that has the following transition probability matrix: 𝑃=.0.20.80.40.6 (36) The system matrices are as follows: 𝐴₁₁₁=𝐴₂₁₁=,𝐴0.30.102−0.6630.3₁₁₂=𝐴₂₁₂=,𝐴0.80.0539−0.86550.8₁₂₁=𝐴₂₂₁=,𝐴0.50.06−0.8430.5₁₂₂=𝐴₂₂₂=,𝐵0.90.0766−0.76610.9₁₁₁=𝐵₂₁₁=,𝐵0.00050.0539₂₁₂=𝐵₁₁₂=,𝐵0.0050.1078₁₂₁=𝐵₂₂₁=,𝐵0.00450.0539₁₂₂=𝐵₂₂₂=,𝐶0.00450.1078₁₁₁=𝐶₂₁₁=𝐶₁₁₂=𝐶₂₁₂=,𝐶00.2₁₂₁=𝐶₂₂₁=𝐶₁₂₂=𝐶₂₂₂=,𝐷0.30₁₁₁=𝐷₂₁₁=𝐷₁₁₂=𝐷₂₁₂=0.3. (37)

Assume 𝐿₂-𝐿_∞ performance of level 𝛾=0.3; by applying Theorem 7, we can explicitly compute the optimally achievable closed-loop 𝐿₂-𝐿_∞ performance 𝛾 from Theorem 7 as 𝛾=0.2056. Response of the system state is depicted in Figures 2 and 3.

5. Conclusion

The problem of finite-time 𝐿₂-𝐿_∞ control for MJS has been studied. By using the Lyapunov functional approach, a sufficient condition is derived such that the closed-loop MJS are stochastic FTB and satisfy the given level. The controller can be obtained by using the exiting LMI optimization techniques. Finally, numerical and simulation results demonstrate the effectiveness of the results of the paper.

Conflict of Interests

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

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 61273087.