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تعداد صفحات این فایل: ۴۰ صفحه

بخشی از مقاله انگلیسیعنوان انگلیسی:Chaos Synchronization of Uncertain Fractional-Order Chaotic Systems With Time Delay Based on Adaptive Fuzzy Sliding Mode Control~~en~~

Abstract

This paper proposes an adaptive fuzzy sliding mode control (AFSMC) to synchronize two different uncertain fractional-order time-delay chaotic systems, which are infinite dimensional in nature, and time delay is a source of instability. Because modeling the behavior of dynamical systems by fractionalorder differential equations has more advantages than integerorder modeling, the adaptive time-delay fuzzy-logic system is constructed to approximate the unknown fractional-order timedelay-system functions. By using Lyapunov stability criterion, the free parameters of the adaptive fuzzy controller can be tuned online by output-feedback-control law and adaptive law. The sliding mode design procedure not only guarantees the stability and robustness of the proposed AFSMC, but it also guarantees that the external disturbance on the synchronization error can be attenuated. The simulation example is included to confirm validity and synchronization performance of the advocated design methodology.

۱ Introduction

TIME delays are often present in many control systems, such as aircraft and chemical or process control systems either in the state, the control input, or the measurements. The existence of pure time delay, regardless of its presence in a control and/or state, is often the cause of poor performance, undesirable system transient response, and instability. The stabilization problem of time-delay systems is a true challenge and has received considerable attention [1]–[۳]. Over the past decade, various methods have been developed in the analysis and synthesis of uncertain systems with time delay. Based on the Lyapunov theory of stability, the sliding-mode control (SMC) has been extensively used and various results have been obtained, because it offers fast response, good transient response, and it is also insensitive to uncertainty in the system [4]. Some works deal with the control problem of time-delay systems via a predictor-based sliding mode [1], [5]–[۸].

In recent years, fractional calculus deals with derivatives and integrations of arbitrary order [9]–[۱۱] and has found many applications in the fields of physics, applied mathematics, and engineering. It is observed that the description of some systems is more accurate when the fractional derivative is used. For instance, electrochemical processes and flexible structures are modeled by fractional-order models [12], the behavior of some biological systems is explored using fractional calculus [13], and the dielectric polarization, electromagnetic waves, and viscoelastic systems are described by fractional-order differential equations [14], [15]. Nowadays, many fractional-order differential systems behave chaotically, such as the fractionalorder Chua’s system [16], [17], the fractional-order Duffing system [18], [19], the fractional-order Lu system [20], the fractional-order Chen’s system [21], [22], the fractional-order cellular neural network [23], [24], and the fractional-order neural network [25].

Recently, due to its potential applications in secure communication and control processing [26], study of chaos synchronization in fractional-order dynamical systems and related phenomena is receiving increasing attention [27], [28], [45]–[۴۸]. The synchronization problem of fractional-order chaotic systems is first investigated by Deng and Li [20], who carried out synchronization in case of the fractional Lu system. Afterward, ¨ they studied chaos synchronization of the Chen system with a fractional order in a different manner [29]–[۳۱].

SMC is a well-known robust nonlinear control technique [4], which guarantees the stability and robustness of the resulting system. This control strategy makes use of the desired sliding surface in the state space and produces the switched control settings based on the observed plant input–output behavior and on considerations concerning the boundary of modeling uncertainties and unknown disturbances [4], [32]. However, there exists chattering phenomena while implementing an SMC, which may excite high-frequency dynamics. In order to eliminate chattering, Palm [33] noted the similarity between fuzzy controller and sliding-mode controller with a boundary layer and provided a fuzzy-sliding-mode-design approach. This design can lead to a stable closed-loop system that avoids the chattering problem in the SMC.

Unfortunately, not many contributions are available for the problem of the SMC of fractional-order systems with time delays. In [49], some results are obtained without using a fractional sliding manifold. In this paper, we develop new results on SMC of fractional-order systems with time delays. In this paper, the core of innovation is based on the fact that fractional-order expression of chaotic systems is very compact in comparison with conventional mathematics. This makes the fractional calculus easier to find an appropriate function for stability analysis. We incorporate adaptive fuzzy-control scheme with SMC approach to synchronize two nonlinear fractional-order Duffing–Holmes chaotic systems with time delay. In our design procedure, both the drive- and response-system dynamics are represented by the Takagi–Sugeno (T–S) fuzzy-neural-network (FNN) model, which expresses the local dynamics of each fuzzy rule by linear combination of all system states.

This paper is organized as follows. In Section II, an introduction to fractional derivative and its relation to the approximation solution are addressed. A brief description of the T–S FNN is presented in Section III. Section IV proposes in a general way the employment of the adaptive fuzzy SMC to synchronize the fractional-order chaotic system with time delay in the presence of uncertainty and its stability analysis. In Section V, application of the proposed method to fractional-order expression of Duffing–Holmes chaotic system with time delay is investigated. Finally, the simulation results and conclusion are presented in Section VI.

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