Robust adaptive fuzzy control of unknown chaotic systems

doi:10.1016/j.asoc.2008.11.014

Applied Soft Computing

Volume 9, Issue 3, June 2009, Pages 970-976

https://doi.org/10.1016/j.asoc.2008.11.014 Get rights and content

Abstract

This paper presents a robust adaptive fuzzy control algorithm for controlling unknown chaotic systems. The control approach encompasses a fuzzy system and a robust controller. The fuzzy system is designed to mimic an ideal controller, based on sliding-mode control. The robust controller is designed to compensate for the difference between the fuzzy controller and the ideal controller. The parameters of the fuzzy system, as well as uncertainty bound of the robust controller, are tuned adaptively. The adaptive laws are derived in the Lyapunov sense to guarantee the stability of the controlled system. Numerical simulations show the effectiveness of the proposed approach.

Introduction

Chaotic behavior is commonly detected in a wide variety of dynamical systems. However, in some practical situations, chaotic behavior is undesirable and must be controlled. In the last few years, the research on controlling chaos (identification, synchronization, suppression, and stabilization) has attracted increasing attention from engineering, physics, mathematics, and biology.

Chaos control is to convert chaotic oscillations into desired regular ones with a periodic behavior. Different control algorithms can be broadly classified into two categories: feedback and nonfeedback methods [1]. Feedback control methods essentially make use of the intrinsic properties of chaotic systems, including their sensitivity to initial conditions, to stabilize orbits already existing in the system. In contrast to feedback control techniques, nonfeedback methods make use of a small perturbing external force such as a small driving force, a small noise term, a small constant bias, or a weak modulation to some system parameters.

The development of the field triggered in 1990 when Ott et al. [2] first proposed a new method for controlling a chaotic dynamical system. This method is called the OGY method and is based on stabilizing one of the many unstable periodic orbits embedded in a chaotic attractor, through only small time dependent perturbations in some accessible system parameters. From then onwards, numerous control techniques have been proposed for controlling chaos in different chaotic systems [3], [4], [5], [6]. Several nonlinear control techniques, such as feedback linearization [7], sliding-mode control [8], [9], [10], backstepping [11], [12], and adaptive control algorithms [13], [14] have been also applied to control of chaotic systems.

A key issue that arises in chaos control, particularly in practical applications, is that the system parameters are not known precisely (or unknown), and are perturbed during application. Unlike most conventional control systems whose equilibriums are assumed known and fixed regardless of values of the system parameters, the equilibriums of chaotic systems are a function of their system constant parameters. This suggests that, when the system parameters are not precisely known, because the equilibriums are then unknown, the conventional control theory cannot be directly applied. In addition, external disturbances and measurement noise which may affect the system performance are inevitable. Therefore, development of alternative control approaches to efficiently treat the robust tracking control of chaotic systems involving uncertainties and variations is highly desirable.

Recently, fuzzy logic control (FLC) has attracted increasing attention in chaos control problems [15], [16], [17], [18], [19], [20], [21]. FLC systems provide an effective approach to handle nonlinear systems, especially in the presence of incomplete knowledge of the plant or the situation where precise control action is unavailable. Based on the universal approximation capability of fuzzy systems, many FLC schemes have been developed for unknown nonlinear systems. However, the main drawback of FLC systems comes from the lack of a systematic control design methodology. Particularly, stability analysis of an FLC system is not easy, and parameter tuning is generally a time-consuming procedure, due to the nonlinear and multiparametric nature of fuzzy systems [22].

In this paper, the application of a robust adaptive fuzzy control scheme to the case of unknown or uncertain chaotic systems is proposed. The controller comprises a fuzzy system and a robust controller. The fuzzy system, with online tuned parameters is designed based on the, ideal, sliding-mode control (SMC). The robust controller is designed to compensate for the difference between the fuzzy controller and the ideal controller. The uncertainty bound needed in the robust controller is also adaptively tuned online to avoid unnecessary high gain resulted from using fixed and most often conservative bounds. The adaptive laws are derived in the Lyapunov sense; thus, the asymptotic stability of the controlled system is guaranteed. Finally, two interesting examples are conducted to evaluate the effectiveness of the proposed approach.

Section snippets

Sliding-mode control

Sliding-mode control, based on the theory of variable structure systems, has been widely applied to robust control of nonlinear systems. SMCs offer good stability, robustness, and consistent performance of a controlled system with uncertainties and external disturbances [23], [24]. In general, the design of SMCs consists of two steps. The first is finding a feedback controller which causes the state trajectory to reach the sliding surface s in finite time and thereafter remain on s. The second

Robust adaptive fuzzy control

In order to exploit the advantages, and tackle the disadvantages, of both SMC and FLC systems, a hybrid control scheme is proposed in this study, referred to as adaptive fuzzy sliding-mode control (AFSMC) system. This approach provides a systematic way to design FLC systems while retaining the robustness and asymptotic stability properties of SMC.

Assume that the parameters of the system (1) are well known. Then an ideal controller can be obtained as $u^{*} = g^{- 1} (x) [- f (x) + x_{d}^{(n)} + \sum_{i = 1}^{n - 1} C_{i}^{n - 1} D^{n - i} λ^{i} \tilde{x}] .$ By

Illustrated examples

In this section, two typical chaotic systems are chosen to assess the effectiveness of the proposed control method. The first example is the Duffing oscillator and the second one is atomic force microscope.

Conclusion

Utilizing the universal approximation capability of FLC systems and robustness and stability of SMC, the AFSMC is proposed for controlling unknown chaotic dynamical systems.

The parameters of the proposed AFSMC system, as well as the approximation error bound, are adaptively tuned online. The adaptive laws are obtained in the Lyapunov sense to ensure the stability of the control system.

Unlike the conventional SMCs, the design of the proposed AFSMC is independent of the mathematical model of the

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Robust adaptive fuzzy control of unknown chaotic systems

Abstract

Introduction

Section snippets

Sliding-mode control

Robust adaptive fuzzy control

Illustrated examples

Conclusion

Phys. Lett. A

Chaos Solitons Fractals

Phys. Lett. A

Chaos Solitons Fractals

Phys. Lett. A

Phys. Lett. A

Inform. Sci.

Inform. Sci.

Fuzzy Sets Syst.

Appl. Soft Comp.

Phys. Lett. A

Automatica

Chaos in Nonlinear Oscillators: Controlling and Synchronization

Controlling Chaos

Phys. Rev. Lett.