Distributed piecewise filtering design for large-scale networked nonlinear systems

Shao and Chen EURASIP Journal on Advances in Signal Processing (2016) 2016:51 DOI 10.1186/s13634-016-0350-2 EURASIP Journal on Advances in Signal Processing RESEARCH Open Access Distributed piecewise H∞ filtering design for large-scale networked nonlinear systems Zhenhua Shao* and Tianxiang Chen Abstract This paper investigates the problem of distributed piecewise H∞ filtering for discrete-time large-scale nonlinear systems. The considered large-scale system is composed of a number of nonlinear subsystems and exchanges its information through communication network. Each nonlinear subsystem is described by a Takagi-Sugeno (T-S) model, and data-packet dropouts happen intermittently in communication network, and its stochastic variables are assumed to satisfy the Bernoulli random-binary distribution. Our objective is to design a distributed piecewise filter such that the filtering error system is stochastically stable with an H∞ performance. Based on a piecewise Lyapunov function and some convexifying techniques, less conservative results are developed for the distributed piecewise H∞ filtering design of the considered system in the form of linear matrix inequalities (LMIs). The effectiveness of the proposed method is validated by two examples. Keywords: Large-scale fuzzy systems, Distributed H∞ filter, Piecewise Lyapunov function 1 Introduction In practical application, some complex systems, such as transportation systems, power systems, communication networks, and industrial processes, are referred to as large-scale systems [1, 2]. Due to strong interconnection and high dimensionality, large-scale systems lead to severe difficulties for their analysis and control synthesis. To date, three main control approaches, centralized, decentralized, and distributed control, have been proposed for large-scale systems with interconnection. Since the centralized control suffers from the excessive information processing and heavy computational burdens, there has been recently an increasing interest in the use of decentralized control for large-scale systems [3]. The decentralized control is firstly to partition the overall control problem of a large-scale system into several independent or almost independent subproblems. Then, instead of a single controller, a set of independent controllers can be designed to achieve the overall control of large-scale system [4]. However, the decentralized control strategy appears weaker stability margins and performance, especially when the interconnections among subsystems are *Correspondence: High-voltage Key Laboratory of Fujian Province, Xiamen University of Technology, Xiamen 361024, People’s Republic of China strong [5]. In the distributed control, the supplemental feedbacks with the interconnected information are provided for the local controllers to enhance the requirements of stability and performance. As a result, the distributed control avoids those shortages appearing in both centralized and decentralized controls [6, 7]. On the other hand, an important issue is to consider the control problems of nonlinear systems because most control plants are nonlinear. Recently, Takagi-Sugeno (T-S) model has been proved to be a powerful solution to represent any smooth nonlinear functions at any preciseness [8, 9]. The T-S model employs a group of IF-THEN fuzzy rules to describe the global behavior of the nonlinear system in which a number of linear models are connected smoothly by fuzzy membership functions. T-S fuzzy approach combining the merits of both fuzzy logic theory and linear system theory is successfully implemented in embedded microprocessors and is widely applied in a variety of engineering fields [10–13]. During the past few years, a great number of results on function approximation, systematic stability analysis, controller and filtering design for T-S fuzzy systems have been reported in the open literature [14–19]. With the rapid development of digital technology, in the feedback loops, communication networks are often used instead of point-to-point connections due © 2016 Shao and Chen. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Shao and Chen EURASIP Journal on Advances in Signal Processing (2016) 2016:51 to their great advantages, such as simple maintenance and installation, and low cost [20–24]. Unfortunately, the network-induced imperfections, such as quantization errors, packet dropouts, and time delays, can degrade significantly the performance of control systems and may even lead to instability [25–29]. Recently, based on fuzzy/piecewise Lyapunov functions, some results on stability analysis and controller synthesis of fuzzy systems have been presented. It has been demonstrated that the inherently conservatism in common Lyapunov function can be relaxed by using piecewise/fuzzy Lyapunov functions. More recently, T-S fuzzy control has been developed to investigate large-scale nonlinear systems [30–35]. To mention a few, some results on analysis and synthesis methods for decentralized control of large-scale systems have been presented in [30–32]. In [33, 34], the decentralized H∞ filtering problem was studied for the discrete-time large-scale system with time-varying delay. To the best knowledge of the authors, few results on the distributed H∞ filtering design have been given for large-scale networked TS fuzzy systems by using piecewise Lyapunov function, which motivates us for the research presented in this paper. This paper will deal with the distributed H∞ filtering problem for discrete-time large-scale nonlinear systems. The large-scale system is composed of several nonlinear subsystems and exchanges its information through communication network. Each nonlinear subsystem is described by a T-S model, and data-packet dropouts occur intermittently in communication network, and its stochastic variables satisfy the Bernoulli random-binary distribution. Based on a piecewise Lyapunov functional (PLF) and some convexifying techniques, the distributed H∞ filtering design result will be proposed. It will be shown that the filtering error system is stochastically stable with an H∞ performance, and the filtering gains can be given by the form of LMIs. Two simulation examples will be presented to demonstrate the advantage of the proposed methods. Notations. n×m is the n-dimensional Euclidean space and n×m denotes the set of n × m matrices. P > 0 (≥ 0) means that matrix P is positive definite (positive semidefinite). Sym{A} denotes A + AT . In and 0m×n are the n × n identity matrix and m × n zero matrix, respectively. The subscripts n and m × n are omitted when the siz (...truncated)