Brief PapersObserver-based fuzzy control for nonlinear stochastic systems with multiplicative noise and successive packet dropouts
Introduction
The past few decades have witnessed an ever increasing research interest in the control problem, which is one of the most important robust controller design approaches. The study of control first starts from the deterministic linear systems, and the derivation of the state-space formulation of the standard control [9] makes it possible to develop the control theory for nonlinear systems [16] or stochastic systems [15]. Due to the wide applications of stochastic systems with multiplicative noise in the fields of synthetic biology [1], mathematical finance [32], portfolio selection [33], etc., the linear and nonlinear stochastic problems of such systems have received a great deal of attention in recent years [19], [24], [30], [28], [34]. From the dissipation point of view, the state feedback control problem has been studied for a general class of nonlinear stochastic systems in [34]. In [30], the problem of dynamic output feedback control has been discussed for nonlinear stochastic time-delay systems with missing measurements and logarithmic quantization effects. Generally, the nonlinear stochastic controller design can be achieved by solving second-order coupled Hamilton–Jacobi equations (HJEs) [34] or Hamilton–Jacobi inequalities (HJIs) [30]. However, it is usually difficult to find efficient numerical algorithms to solve such coupled HJEs or HJIs.
For several decades, the Takagi–Sugeno (T–S) fuzzy model has been widely applied for the control design of nonlinear systems [5], [10], [21], [27], [37], since it can approximate any smooth nonlinear system to any specified accuracy by blending a family of local linear models through fuzzy membership functions. Recently, the T–S fuzzy model has been employed to deal with the control and filter problems of nonlinear stochastic systems with multiplicative noise [1], [25], [26]. In [26], based on the T–S fuzzy model, the filter problem of nonlinear stochastic systems has been transformed into the solvability of a set of linear matrix inequalities (LMIs) instead of coupled HJIs, which provides a possible approach to solve the HJIs. Along the same line of [26], a robust optimal reference-tracking design method has been proposed in [1] for synthetic gene networks which described by nonlinear stochastic systems. By using the fuzzy approach, the state feedback control problem has been discussed in [25] for nonlinear stochastic systems with Markov jump parameters. However, up to now, the problem of observer-based fuzzy control for nonlinear stochastic systems with multiplicative noise has not received adequate research attention yet.
On the other hand, the study of networked control systems (NCSs) has gradually become an active area of research owing to their advantages such as low cost, reduced weight, and ease of installation. Because of unreliable measurements or network congestion, some network-induced problems have drawn considerable research interest, such as time delay [14], [36], [38], packet dropouts [6], [20], [22], [31] (also called data missing or missing measurement), quantization effects [8], [13], [29], [30], channel fading [2], [3], [4], [7], which could seriously degrade the system performances. In the literature, there have been several different approaches for modeling the packet dropouts phenomenon in the NCSs. In [20], it has been assumed that the packet dropouts occur according to a time-homogeneous Markov process, and the problem of fault detection has been investigated for NCSs. In [31], the packet dropouts have been modeled by a binary switching random sequence, which means that the measurement signal is either completely available or completely missing. Replacing the lost current measurement by the latest received measurement, the successive packet dropouts model has been proposed in [22], which is more realistic than that in [31]. However, most literature has only been concentrated on the problems of filter and state estimation for NCSs with successive packet dropouts [17], [23], while little attention has been paid on the control problems of such systems due probably to the technical difficulty.
Motivated by the preceding discussion, we aim to investigate the observer-based control problem for a class of T–S fuzzy nonlinear stochastic systems with multiplicative noise and successive packet dropouts. The main contributions of this paper are twofold. (1) The observer-based control problem is considered for a class of T–S fuzzy nonlinear stochastic systems when the system state is not easily available and the measurement output is partly missing. (2) This paper represents the first of few attempts to deal with the control problem for the augmented system induced by successive packet dropouts, and an efficient controller design method is proposed for such systems. The rest of the paper is organized as follows. In Section 2, we give the problem formulation and present several useful lemmas. Section 3 contains the main results of the paper on the observer-based fuzzy control with multiplicative noise and successive packet dropouts. A numerical example is provided to show the effectiveness of the proposed method in Section 4. Section 5 summarizes our conclusions.
Notations: The notation used in this paper is fairly standard. is the set of all real n-dimensional vectors. is the set of all real matrices. (respectively, ) is a real symmetric positive definite (respectively, positive semi-definite) matrix. is the transpose of matrix A. denotes the smallest eigenvalue of the matrix A. I is the identity matrix. , , denotes the matrix with row full rank such that with is column full rank. is the norm of a vector or matrix. stands for the mathematical expectation of x. . is a block-diagonal matrix. is a filtered probability space where is the family of sub σ-algebras of generated by . Denote as the space of nonanticipatory square-summable stochastic process with respect to . denotes the class of all continuous nondecreasing convex functions μ: such that and for . is the class of functions V(x) that are twice continuously differential with respect to except possibly at the origin. The asterisk ⁎ in a matrix is used to denote the term that is induced by symmetry.
Section snippets
Definitions and preliminaries
In this section, we consider a class of discrete nonlinear stochastic systems with multiplicative noise, and such systems can be described by the following T–S fuzzy model:where , r is the number of IF-THEN rules, are the fuzzy sets, are the premise variables, is the system state, is the control input, is the
Main results
In this section, the following theorem first provides a sufficient condition under which the augmented system (14), (15) with vk=0 is stochastically stable and the controlled output zk satisfies (16) for all nonzero vk under the zero-initial condition. Theorem 1 Given , system (14), (15) is stochastically stable and criterion (16) holds, if there exists a positive definite symmetric matrix satisfying the following inequality:
A numerical example
In this section, a numerical example is given to illustrate the proposed control design method. Consider the following discrete-time T–S fuzzy system of the form (1):where
Conclusions
In this paper, the problem of observer-based control has been investigated for a class of T–S fuzzy nonlinear stochastic systems with multiplicative noise and successive packet dropouts. A sufficient condition that guarantees the stochastic stability and performance constraint of the closed-loop system has been presented. Moreover, it has been shown that the observer-based fuzzy controller design problem for systems with successive packet dropouts is solvable if a set of LMIs is
Acknowledgments
This work is supported by National Natural Science Foundation of China (Nos. 61203053, 61403420, 61573377) and Fundamental Research Fund for the Central Universities (Nos. 15CX02038A, 12CX02010A).
Ming Gao received her M.S. and Ph.D. degrees from Shandong Normal University in 2006 and Jiangnan University in 2009, respectively. She is currently an associate professor with the College of Information and Control Engineering, China University of Petroleum (East China), Qingdao, China. Her current research interests include robust control and stochastic control systems.
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Ming Gao received her M.S. and Ph.D. degrees from Shandong Normal University in 2006 and Jiangnan University in 2009, respectively. She is currently an associate professor with the College of Information and Control Engineering, China University of Petroleum (East China), Qingdao, China. Her current research interests include robust control and stochastic control systems.
Li Sheng received the Ph.D. degree in control theory and control engineering from Jiangnan University in 2010 and received the M.S. degree from Shandong Normal University in 2006. He is currently an associate professor with the College of Information and Control Engineering, China University of Petroleum (East China), Qingdao, China. His current research interests include stochastic control and networked control systems.
Yurong Liu received his B.S. degree in Mathematics from Suzhou University, Suzhou, China, in 1986, the M.S. degree in Applied Mathematics from Nanjing University of Science and Technology, Nanjing, China, in 1989, and the Ph.D. degree in Applied Mathematics from Suzhou University, Suzhou, China, in 2000.
Dr. Liu is currently a Professor in the Department of Mathematics at Yangzhou University, China. He has published more than 50 papers in refereed international journals. His current interests include neural networks, complex networks, nonlinear dynamics, time-delay systems, multiagent systems, and chaotic dynamics.
Zhengmao Zhu received his M.S. degree in engineering from North China Electric Power University (NCEPU) in 2000. He is currently a senior engineer at Science and Technology Research Institute of NCEPU, and working toward the Ph.D. degree in the Control and Computer Engineering College, NCEPU, Beijing. His current research interests include stochastic systems and process control.