Decentralized control design for switched fuzzy large-scale systems with H∞ performance

doi:10.1016/j.neucom.2015.03.022

Neurocomputing

Volume 165, 1 October 2015, Pages 330-337

https://doi.org/10.1016/j.neucom.2015.03.022 Get rights and content

Abstract

This paper investigates the $H_{\infty}$ control design problem for a class of switched discrete-time Takagi–Sugeno (T–S) fuzzy large-scale systems. The considered fuzzy large-scale systems consist of several interconnected subsystems with different switching modes and each switching mode is described by T–S fuzzy models. In addition, there exists the asynchronous switching between the system switching modes and the controller switching modes. By using parallel distributed compensation design method, a state feedback decentralized controller with $H_{\infty}$ performance is developed. The sufficient conditions of ensuring the switched control system stability are proposed based on the theory of Lyapunov function and average-dwell time methods, which are formulated in the form of linear matrix inequalities (LMIs). An illustrated numerical example is provided to show the effectiveness of the obtained theoretical results.

Introduction

In the past few decades, the stability analysis and control design problem of switched systems have achieved great progress [1], [2], [3], [4], [5], [6], [7], [8]. Switched control systems are an important class of hybrid systems because they can provide a valid modeling and control approach for many physical systems, such as traffic control systems, power systems, and chemical systems. In general, a switched system consists of a family of continuous-time subsystems or discrete-time dynamical subsystems and a rule which orchestrates the switching among the subsystems. The design of switching rules is crucial in the study of switched systems. Currently, the switching rules can be classified into arbitrary switching and constrained switching. As for the arbitrary switching issue, most of the studies have used a common Lyapunov function method [2], [3] or a multiple Lyapunov function method [4], [5] to check the stability condition of the switched systems. As for the constrained switching signal, an average dwell time (ADT) method is an effective tool to investigate the problems of stability analysis and stabilization of switched systems, for example [6], [7], [8].

It is well known that T–S fuzzy model not only can be applied to stability analysis and controller design [9], [10], [11], [12], [13], [14], but also be useful to precisely describe switched nonlinear systems. Therefore, based on the switched fuzzy systems, many control design approaches have been investigated for switched nonlinear systems, for example [15], [16], [17], [18], [19], [20], [21], and references herein. Among them, [15] proposed a switched fuzzy control design method for a class of nonlinear systems, and the stability conditions are obtained by using the switching Lyapunov function. Wang et al. [16] investigated a switching fuzzy control design for a class of T–S fuzzy systems via a switching fuzzy model and a piecewise Lyapunov function, and relaxed stability conditions are developed [17]. Authors in [18], [19] studied the problem of dynamic output feedback $H_{\infty}$ control for nonlinear systems described by T–S fuzzy systems via a switched dynamic parallel distributed compensation scheme. Recently, [21] has studied control design based on a polynomial fuzzy model, and new and relaxed stability conditions are developed by using a switching polynomial Lyapunov function. It should be mentioned that the aforementioned control design and stability analysis theories are only for the fuzzy T–S systems, instead of switched fuzzy T–S systems. More recently, several control design methods and stability analysis conditions have been explored for switched fuzzy T–S systems [22], [23].

However, the above results are all focused on switched fuzzy T–S systems under synchronous switching. As stated in [24], [25], [26], because the matched controller of each subsystem cannot be operating immediately in practice, there exists a lag between a system and its corresponding controller, which often leads to the phenomena of asynchronous switching between the system and its controller. The asynchronous behaviors which exist among the subsystems usually bring unsatisfactory performance or even make the system out of control. Nevertheless, the above-mentioned results are only limited to simple switched fuzzy systems, thus can be not applied to the switched fuzzy large-scale system. In fact, many physical systems, such as aerospace systems, chemical engineering systems and telecommunication networks are described as some class of large-scale systems, which are composed of interconnections of low-dimensional systems. Hence, the study decentralized control design and stability condition for the switched fuzzy large-scale systems are important in both theory and engineering applications.

Motivated by the aforementioned analysis, this paper presents a decentralized controller for a class of switched discrete-time fuzzy large-scale system. The considered fuzzy large-scale systems consist of several interconnected subsystems with different switching modes and each switching mode is described by T–S fuzzy models. By using parallel distributed compensation design method, an $H_{\infty}$ decentralized state feedback controller is developed. The sufficient conditions ensuring the control system stability are proposed and are formulated in the form of linear matrix inequalities (LMIs). Based on the theory of the Lyapunov function and average-dwell time method, the stability of the switched large-scale control system is proved. Compared with the existing literature, the main contributions of this paper can be summarized as follows:

(1)
This paper first investigated the decentralized control design problem for the switched discrete-time fuzzy large-scale systems. Although in recent years, many decentralized control design methods have been reported for the fuzzy large-scale systems [27], [28], these control methods are not the switching control design problem. It should be mentioned that the switched control design has major difference from that of non-switched control design. The former is much more difficult and challenging than the latter.
(2)
This paper studied the decentralized control design problem and stabilization conditions under asynchronous switching between the controlled systems and the controllers. Note that literature [22], [23] also addressed the same problem. However, the control design and stability analysis are under the conditions of synchronous switching. In addition, the considered switched plants [22], [23] are simple continuous-time fuzzy system, instead of switched discrete-time fuzzy large-scale systems. To the best of our knowledge, to date, there are no results reported on the switched discrete-time fuzzy large-scale systems.

Section snippets

Problem formulation and preliminaries

Consider the following discrete-time fuzzy large-scale system S that consists of J interconnected fuzzy subsystems S_i, $i = 1, 2, \dots, J$ . The ith switched T–S fuzzy subsystem S_i is described as the following form: $Switching mode rule : R_{mode}^{m} IF switching mode m is activated, THEN Local plant rule : R_{local}^{r} IF x_{i 1} (k) is G_{imr}^{1} and \dots and x_{{ig}_{i}} (k) is G_{imr}^{g_{i}}, THEN x_{i} (k + 1) = A_{imr} x_{i} (k) + B_{imr} u_{in} (k) + E_{imr} w_{i} (k) + \sum_{p = 1, p \neq i}^{J} R_{pimr} x_{p} (k) z_{i} (k) = C_{imr} x_{i} (k) + D_{imr} u_{in} (k)$ where $x_{i} (t) = {[x_{i 1} (t), x_{i 2} (t), \dots, x_{{ig}_{i}} (t)]}^{T}$ is the state vector of the ith fuzzy subsystem, g_i

Decentralized controller design and stability analysis

In this section, we will give the design of the decentralized fuzzy controllers and the sufficient conditions of guaranteeing the $H_{\infty}$ performance index.

According to [28], [30], by using the parallel distributed compensation (PDC), the switching fuzzy controllers are designed as follows: $u_{im} = \sum_{m = 1}^{M_{i}} \sum_{r = 1}^{r_{m}^{i}} ξ_{im} μ_{imr} K_{imr} x_{i} (k)$ where K_imr is the feedback gain matrix to be determined.

Substituting (7) into (2), the switched fuzzy control system is as follows: $x_{i} (k + 1) = \sum_{m = 1}^{M_{i}} \sum_{n = 1}^{M_{i}} \sum_{r = 1}^{r_{m}^{i}} \sum_{s = 1}^{r_{m}^{i}} ξ_{im} ξ_{in} μ_{imr} μ_{ins} (($

Numerical simulation

In this section, a simulation example is given to demonstrate the validity of the decentralized controller design method.

Consider the fuzzy large-scale system which consists of two interconnected fuzzy subsystems (2). Each subsystem is described by the following four fuzzy rules. $Switching mode rule : R_{mode}^{im} (i, m \in {1, 2}) IF switching mode m is activated, THEN Local plant rule : R_{local}^{ir} (r \in {1, 2}) IF x_{ir} (k) is G_{imr}^{r}, THEN x_{i} (k + 1) = A_{imr} x_{i} (k) + B_{imr} u_{in} (k) + E_{imr} w_{i} (k) + \sum_{p = 1, p \neq i}^{2} R_{pimr} x_{p} (k) z_{i} (k) = C_{imr} x_{i} (k) + D_{imr} u_{in} (k)$ where $A_{111} = [\begin{matrix} 0.9 \end{matrix}$

Conclusion

This paper has considered the $H_{\infty}$ decentralized control design problem for a class of switched discrete-time fuzzy large-scale systems with asynchronous switching modes. Based on PDC design principle, a decentralized controller has been developed, and the sufficient conditions of ensuring the stability of the switched fuzzy control systems have been given. Based on the Lyapunov stability theory and average dwell time method. It has been proved that the proposed decentralized control method can

Acknowledgment

This work is supported by National Natural Science Foundation of China under Grant (No. 61374113).

Tiechao Wang received B.E. degree and M.E. degree in control theory and engineering from Liaoning Institute of Technology, Liaoning, PR China, in 1996 and in 2005, respectively, and Ph.D. degree in control theory and engineering from Institute of Automation, Chinese Academy of Sciences, Beijing, PR China, in 2012.

Currently, he is an Associate Professor of College of Electrical Engineering, Liaoning Institute of Technology, Liaoning, PR China. His research interests include fuzzy control theory

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Currently, he is an Associate Professor of College of Electrical Engineering, Liaoning Institute of Technology, Liaoning, PR China. His research interests include fuzzy control theory and intelligent computing.

Shaocheng Tong received the B.A. degree in mathematics from Jinzhou Normal College, Jinzhou, China, the M.A. degree in fuzzy mathematics from Dalian Marine University, PR China, and the Ph.D degree in fuzzy control from Northeastern University, PR China, in 1982, 1988, and 1997, respectively.

Currently, he is a Professor in the Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, PR China. His research interests include fuzzy control theory, nonlinear adaptive control, and intelligent control.

View full text

Decentralized control design for switched fuzzy large-scale systems with H∞ performance

Abstract

Introduction

Section snippets

Problem formulation and preliminaries

Decentralized controller design and stability analysis

Numerical simulation

Conclusion

Acknowledgment

Automatica

Automatica

Automatica

Neurocomputing

Automatica

Basic problems in stability and design of switched systems

IEEE Control Syst. Mag.

Switching in Systems and Control

Stability analysis and control synthesis for switched systemsa switched Lyapunov function approach

IEEE Trans. Autom. Control

Exponential stability analysis for discrete-time switched linear systems with time-delay

Int. J. Innov. Comput. Inf. Control

l2−l∞ model reduction for switched LPV systems with average dwell time

IEEE Trans. Autom. Control

Average dwell time approach to L2−L∞ control of switched delay systems via dynamic output feedback

IET Control Theory Appl.

Fuzzy sampled-data control for uncertain vehicle suspension systems

IEEE Trans. Cybern.

Output-feedback based H∞ control for active suspension systems with control delay

IEEE Trans. Ind. Electron.

A new approach to stability analysis and stabilization of discrete-time T–S fuzzy time-varying delay systems

IEEE Trans. Syst. Man Cybern. Part B

Decentralized control design for switched fuzzy large-scale systems with $H_{\infty}$ performance

$l_{2} - l_{\infty}$ model reduction for switched LPV systems with average dwell time

Average dwell time approach to $L_{2} - L_{\infty}$ control of switched delay systems via dynamic output feedback

Output-feedback based $H_{\infty}$ control for active suspension systems with control delay