Robust H∞ fuzzy control of dithered chaotic systems☆
Introduction
Chaotic phenomena have been observed in numerous physical systems, which can lead to irregular performance and possibly catastrophic failures [1]. Chaos is a well-known nonlinear phenomenon, and it is the seemingly random behavior of a deterministic system that is characterized by sensitive dependence on initial conditions [2]. Besides, chaos is occasionally preferable, but usually intrinsically unpredictable as it can restrict the operating range of many physical devices and reduce performance. Therefore, the ability to control chaos is of much practical importance. Based on these properties, chaos has received a great deal of interest among scientists from various research fields [3], [4], [5], [6]. After the pioneering work of Ott, Grebogi and Yorke (OGY) [7], controlling chaos has become a challenging topic in the field of nonlinear dynamics [8]. Mohammad and Vahid [9] proposed a new method for the stabilization of dynamic output-feedback synchronization of chaotic systems using T–S fuzzy models of the drive and response systems. According to the concept of dynamic parallel distributed compensation (DPDC), the proposed design minimizes the L2 gain of the fuzzy control system with respect to a bounded disturbance that is a function of the drive system states. Moreover, a robust adaptive fuzzy control algorithm to control unknown chaotic systems was presented by Amir and Amir [10]. On the basis of sliding-mode control, the fuzzy system was designed to mimic an ideal controller.
Due to the unique merits in solving complex nonlinear system identification and control problems, neural-network-based modeling has become an active research field in the past few years. Neural networks (NN) consist of simple elements operating in parallel; these elements are inspired by biological nervous systems. As in nature, the connections between elements largely determine the network function. You can train a neural network to perform a particular function by adjusting the values of the connections (weights) between elements. Hence, the nonlinear system can be approximated as close as desired by the NN model via repetitive training. There have been some successful applications of NN in recent years (see [11], [12], [13], [14], [15], [16]). For instance, Limanond and Si [11] applied neural networks to optimal etch time control design for a reactive ion etching process. Enns and Si [13] advanced an NN-based approximate dynamic programming control mechanism to helicopter flight control. Despite several promising empirical results and its nonlinear mapping approximation properties, the rigorous closed-loop stability results for systems using NN-based controllers are still difficult to establish. Therefore, an LDI state-space representation was introduced to deal with the stability analysis of NN models (see [17]).
Fuzzy control has attracted a great deal of attention from both academic and industrial communities over the past decade, and there have been many successful applications. For example, Wang et al. [18] presented a new measurement system that comprises a model-based fuzzy logic controller, an arterial tonometer and a micro syringe device for the noninvasive monitoring of the continuous blood pressure wave form in the radial artery. A good tracking performance control scheme, a hybrid fuzzy neural-network control for nonlinear motor-toggle servomechanisms, was given by Wai [19]; Hwang et al. [20] developed the trajectory tracking of a car-like mobile robot using network-based fuzzy decentralized sliding-mode control; a hybrid fuzzy-PI speed controller for permanent magnet synchronous motor was proposed in Sant and Rajagopal [21]; and Spatti et al. [22] introduced a fuzzy control strategy for voltage regulation in electric power distribution systems: this real-time controller would act on power transformers equipped with under-load tap changers.
In spite of the successes of fuzzy control, many basic problems remain to be solved. Stability is one of the most important issues in analysis and design of control systems [23]. Recently, significant research efforts have been devoted to these issues (see [24], [25], [26] and the references therein). However, all of them have neglected the modeling errors between the fuzzy combination of T–S fuzzy models and the nonlinear systems under control. The existence of modeling errors is a potential source of instability for control designs based on the assumption that the fuzzy model exactly matches the nonlinear plant [27]. In recent years, novel approaches to overcome the influence of modeling errors in the field of model-based fuzzy control for nonlinear systems have been offered by Kiriakidis [27], Chen et al. [28], [29] and Cao and Frank [30], [31].
Generally speaking, the influence of the external disturbance will worsen the performance of control systems. However, the external disturbances are not considered in Hsiao [32]. In fact, some noises or disturbances always exist that may cause instability. Accordingly, the external disturbances must be taken into account and the stability of uniformly ultimately bounded (UUB) instead of asymptotical stability is discussed in this work. How to reduce the effect of external disturbances is an important issue of control design. The H∞ control problem for nonlinear systems has received considerable attention over the last few years [28], [29], [30], [33], [34], [35]. In this study, a fuzzy controller with guaranteed H∞ control performance was synthesized to suppress the chaos and attenuate the influence of the external disturbance.
If the fuzzy controller cannot suppress the chaos, a dither (as an auxiliary of the fuzzy controller) was simultaneously injected to quench the chaotic motion instead of redesigning the fuzzy controller. The application of dither, a high frequency signal, is an effective and convenient alternative among the approaches in chaos control. It has been long known that the injection of dither into a nonlinear system may improve its performance (see [36], [37], [38], [39], [40], [41], [42], [43] and the references therein). Steinberg and Kadushin [36] presented a rigorous analysis of stability in a general nonlinear system with a dither control. According to the relaxed method, the relaxed system can be asymptotically stabilized by adjusting appropriately the dither's parameters. The fact that a dither of sufficiently high frequency may lead to the output of the relaxed system and the one of the dithered system as close as desired was indicated by Mossaheb [39]. Tsai et al. [1] proposed a compound optimization strategy called the island-based random-walk algorithm (IRA) to identify the relaxed models of the dithered chaotic systems as T–S fuzzy modes. The algorithm is composed of a set of communicating random-walk optimization procedures concatenated with the down-hill simplex method. Recently, numerous reports on the success of dither applications in control systems have appeared in the literature. Feeny and Moon [41] applied dithers to meld the behavior of stick-slip oscillator system in quenching the chaos inherent and showed that the high-frequency excitation effectively removes the discontinuity for the low-frequency behavior. Iannelli et al. [42], [43] further pointed out that the use of dithers could narrow the discontinuous nonlinearities of feedback systems.
Almost all the existing research works of controlling chaos made use of fuzzy models to approximate the chaotic systems (see [1], [2], [10], [49] and the references therein). Although using fuzzy models to approximate the chaotic systems is more simple than the Neural networks (NN), the NN models will approach the chaotic systems by iterative training and adjusting the weights. That is to say, the modeling errors of NN models will be much less than those of fuzzy models. However, a literature search indicates that the use of dither to overcome the influence of modeling error via NN-based approach has not been discussed yet. Hence, a combining scheme of fuzzy controllers and dithers via NN-based approach was proposed to control chaos and improve systems' performance in this study. According to this scheme, we can synthesize a fuzzy controller and find an appropriate dither to tame the chaotic system and achieve the H∞ control performance at the same time. If the dither's frequency is high enough, the trajectory of the dithered system and that of its corresponding mathematical model—the relaxed system can be made as close as desired. This makes it possible to obtain a rigorous prediction of the dithered chaotic system's behavior based on the one of the relaxed system.
The remainder of this paper is organized as follows: In Section 2, the problem formulation is provided. In Section 3, a robustness design of fuzzy control is introduced. In Section 4, an NN-based approach to tame the chaotic system via dither and fuzzy controller is presented. The design algorithm is shown in Section 5. In Section 6, the effectiveness of the proposed approach is illustrated by a numerical example. Finally, the conclusions are drawn in Section 7.
Section snippets
Problem formulation
We begin with considering a chaotic system N:where X(t) denotes the state vector, U(t) is a control input vector, ∂(t) denotes the external disturbance with a known upper bound and f(⋅) is a nonlinear vector-valued function which satisfies the assumptions of general continuity and boundedness given in [36].
The stability of uniformly ultimately bounded (UUB) for a system with disturbances is defined as follows.
Definition (2.1) [28]: The solutions of a dynamic
Robustness design of fuzzy control and stability analysis
In this section, the control of chaos is examined under the influence of modeling errors.
Dithered chaotic system and relaxed model
A high frequency signal, commonly referred to as dither d(t), with a finite switching number η, is injected into the chaotic system N. Thus, the dithered chaotic system Nd can be described as
The algorithm for constructing the dither is given as follows [32], [36]. The time interval [0, T] is divided into an arbitrary number η of equal subintervals. The beginning of the first interval, the end of the first interval, the end of the second interval and the end of
Algorithm
The complete design procedure shown in Fig. 4 can be summarized as follows. Problem Given a chaotic system N, how can we synthesize a fuzzy controller and find an appropriate dither to suppress the chaos?
This problem can be solved using the following steps:
Step 1: Construct the neural-network (NN) model of the chaotic system N, and then convert the dynamics of the NN model into an LDI state-space representation.
Step 2: Based on the state-feedback control scheme, a fuzzy controller (2.12) is synthesized
A numerical example
Example: Chua's oscillator circuit is a nonlinear electronic circuit, which contains five linear elements and a nonlinear resistor. The purpose of this section is to synthesize a fuzzy controller to tame the following Chua's oscillator circuit. A dither d(t) with sufficiently high frequency is injected into the Chua's oscillator circuit if the fuzzy controller cannot quell the chaos.
The dynamics of Chua's oscillator circuit with control inputs is governed by the following state equations [49]:
Conclusion
A combining scheme of fuzzy controllers and dithers via NN-based approach is presented in this work. On the basis of this scheme, we can synthesize a fuzzy controller and find an appropriate dither to tame the chaotic system. A stability criterion was derived in terms of Lyapunov's direct method to guarantee that the trajectory of the chaotic system can be steered into a periodic orbit or a steady state. According to the stability condition of this criterion, a fuzzy controller was then
Feng-Hsiag Hsiao was born in Tainan, Taiwan, ROC, in 1960. He received the B.S. degree in electronic engineering from Chung Yuan Christian University, Chung-Li, Taiwan, in 1983, the M.S. degree in electrical engineering from Tatung University, Taipei, Taiwan, in 1985, and the Ph.D. degree in electrical engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 1991. He is currently a Professor with the Department of Electrical Engineering, National University of Tainan, Tainan,
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Feng-Hsiag Hsiao was born in Tainan, Taiwan, ROC, in 1960. He received the B.S. degree in electronic engineering from Chung Yuan Christian University, Chung-Li, Taiwan, in 1983, the M.S. degree in electrical engineering from Tatung University, Taipei, Taiwan, in 1985, and the Ph.D. degree in electrical engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 1991. He is currently a Professor with the Department of Electrical Engineering, National University of Tainan, Tainan, Taiwan. His research interests are in the area of fuzzy control, neural network, large-scale control, and the dither problem.
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The authors wish to express sincere gratitude to Prof. Heskes for his help and the anonymous reviewers for their constructive comments and helpful suggestions, which led to substantial improvements of this paper.