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

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 314812, 7 pages http://dx.doi.org/10.1155/2014/314812 Research Article Finite-Time Control for Markovian Jump Systems with Polytopic Uncertain Transition Description and Actuator Saturation Zhongyi Tang1,2 1 2 Institute of Automation, Jiangnan University, Wuxi 214122, China Faculty of Electronic and Engineering, Huaiyin Institute of Technology, Huaian 223003, China Correspondence should be addressed to Zhongyi Tang; Received 24 January 2014; Revised 25 April 2014; Accepted 30 April 2014; Published 20 May 2014 Academic Editor: Shuping He Copyright © 2014 Zhongyi Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The problem of finite-time 𝐿 2 -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 𝐿 2 -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 continuoustime 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 timevarying 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 𝐿 2 -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 𝐿 2 -𝐿 ∞ performance is derived in terms of LMI. Based on this, the existence condition of 2 Abstract and Applied Analysis 2.5 2 Jumping mode the desired performance which guarantees finite-time stability and an 𝐿 2 -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. 1.5 1 0.5 0 5 10 15 20 25 30 t (s) 2. Problem Statement and Preliminaries Figure 1: Jumping mode. 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 𝜃1 (𝑟𝑘 ) 𝑥 (𝑘) + 𝐴 𝜃2 (𝑟𝑘 ) 𝑥 (𝑘 − 𝑑) [𝜋V (𝑖, 1) , 𝜋V (𝑖, 2) , . . . , 𝜋V (𝑖, 𝑆)] + 𝐵𝜃1 (𝑟𝑘 ) 𝜎 (𝑢𝑘 ) + 𝐵𝜃2 (𝑟𝑘 ) 𝑤𝑘 , 𝑀 𝑧 (𝑘) = 𝐶𝜃1 (𝑟𝑘 ) 𝑥 (𝑘) + 𝐶𝜃2 (𝑟𝑘 ) 𝑥 (𝑘 − 𝑑) + 𝐷𝜃1 (𝑟𝑘 ) 𝑤𝑘 , (1) where 𝑥𝑘 ∈ 𝑅𝑛 is the system state, 𝑧𝑘 ∈ 𝑅𝑛 is the system output, 𝑢𝑘 ∈ 𝑅𝑚 is the control input, 𝑤𝑘 ∈ 𝑅𝑞 is the disturbance input which belongs to 𝑇 2 𝐿 2 [0, ∞) and ∑∞ 𝑘=0 𝑤𝑘 𝑤𝑘 < 𝜅 , and 𝜅 is a given positive scalar. 𝐴 𝜃1 (𝑟𝑘 ), 𝐴 𝜃2 (𝑟𝑘 ), 𝐵𝜃1 (𝑟𝑘 ), 𝐵𝜃2 (𝑟𝑘 ), 𝐶𝜃1 (𝑟𝑘 ), 𝐷𝜃1 (𝑟𝑘 ), and 𝐷𝜃2 (𝑟𝑘 ) are appropriately dimensioned real-valued matrices, which belong to the part of convex polyhedron Φ(𝑟𝑘 ): Φ (𝑟𝑘 ) 𝐿 = {∑ 𝜃𝑙 [𝐴 𝑙1 (𝑟𝑘 ) , 𝐴 𝑙2 (𝑟𝑘 ) , 𝐵𝑙1 (𝑟𝑘 ) , 𝑙=1 𝐵𝑙2 (𝑟𝑘 ) , 𝐶𝑙1 (𝑟𝑘 ) , 𝐶𝑙2 (𝑟𝑘 ) , = ∑ V𝑚 [𝜋𝑚 (𝑖, 1) , 𝜋𝑚 (𝑖, 2) , . . . , 𝜋𝑚 (𝑖, 𝑆)] , 𝐿 where V = [V1 ⋅ ⋅ ⋅ V𝑀]𝑇 ∈ 𝑅𝑀 and ∑𝑀 𝑚=1 V𝑚 = 1, and the transition probability belongs to the following convex polyhedron: ℵ (𝑟𝑘 = 𝑖) = Co { [𝜋1 (𝑖, 1) , 𝜋1 (𝑖, 2) , . . . , 𝜋1 (𝑖, 𝑁)] } . (5) [𝜋𝑀 (𝑖, 1) , 𝜋𝑀 (𝑖, 2) , . . . , 𝜋𝑀 (𝑖, 𝑁)] When the system operates in the 𝑖th mode (𝑟𝑘 = 𝑖), for simplicity, the matrices 𝐴 𝜃1 (𝑟𝑘 ), 𝐴 𝜃2 (𝑟𝑘 ), 𝐵𝜃1 (𝑟𝑘 ), 𝐵𝜃2 (𝑟𝑘 ), 𝐶𝜃1 (𝑟𝑘 ), and 𝐷𝜃1 (𝑟𝑘 ) are denoted as 𝐴 𝜃1𝑖 , 𝐴 𝜃2𝑖 , 𝐵𝜃1𝑖 , 𝐵𝜃2𝑖 , 𝐶𝜃1𝑖 , and 𝐷𝜃1𝑖 , respectively. 𝑑 is a positive integer denoting the constant delay of the system state (Figures 2 and 3). In system (1), 𝜎(⋅) : 𝑅𝑚 → 𝑅𝑚 is the vector-valued standard saturation function defined as follows: 𝜎 (𝑢) = [𝜎 (𝑢1 ) , 𝜎 (𝑢2 ) , . . . , 𝜎 (𝑢𝑚 )] , where 𝐴 𝑙1 (𝑟𝑘 ), 𝐴 𝑙2 (𝑟𝑘 ), 𝐵𝑙1 (𝑟𝑘 ), 𝐵𝑙2 (𝑟𝑘 ), 𝐶𝑙1 (𝑟𝑘 ), 𝐶𝑙2 (𝑟𝑘 ), and 𝐷𝑙1 (𝑟𝑘 ) are matrix functions of the (...truncated)