Stabilization of a Class of Fractional-Order Chaotic Systems via Direct Adaptive Fuzzy Optimal Sliding Mode Control

Stabilization of a Class of Fractional-Order Chaotic Systems via Direct Adaptive Fuzzy Optimal Sliding Mode Control

Bachir Bourouba (Setif-1 University, Algeria)
DOI: 10.4018/978-1-5225-5418-9.ch010

Abstract

In this chapter a new direct adaptive fuzzy optimal sliding mode control approach is proposed for the stabilization of fractional chaotic systems with different initial conditions of the state under the presence of uncertainties and external disturbances. Using Lyapunov analysis, the direct adaptive fuzzy optimal sliding mode control approach illustrates asymptotic convergence of error to zero as well as good robustness against external disturbances and uncertainties. The authors present a method for optimum tuning of sliding mode control system parameter using particle swarm optimization (PSO) algorithm. PSO is a robust stochastic optimization technique based on the movement and intelligence of swarm, applying the concept of social interaction to problem solving. Simulation examples for the control of nonlinear fractional-order systems are given to illustrate the effectiveness of the proposed fractional adaptive fuzzy control strategy.
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Introduction

Chaos is a complex nonlinear phenomenon which is frequently observed in physical, biological, electrical, mechanical, economical, and chemical systems. A chaotic system is a nonlinear deterministic dynamical system that exhibits special attributes including extraordinary sensitivity to system initial conditions, broad band Fourier spectrum, strange attractor, and fractal properties of the motion in phase space. Due to the existence of chaos in real world systems and many valuable applications in engineering and science, synchronization and stabilization of chaotic systems have attracted significant interests among the researchers in the last decades (Rabah et al., 2017). The control of chaotic systems has been focused on more attentions in nonlinear science due to its potential applications in science and engineering such as circuit, mathematics, power systems, medicine, biology, chemical reactors, and so on. Many researchers have made great contributions. Chaotic behaviors can be a troublemaker when it causes undesired irregularity in dynamical systems. Due to troubles that may arise from unusual behaviors of a chaotic system, chaos control has gained increasing attention in the past few decades. In chaos control, an important objective is to suppress the chaotic oscillations completely or reduce them to the regular oscillations. Nowadays, many control techniques have been implemented in the control of chaotic systems (Tavazoei et al., 2009; Hamamci, 2012; Rabah et al., 2017b).

Chaotic dynamic of fractional order systems starts to attract increasing attention due to its potential applications in secure communication and control processing. An important challenge in chaos theory is the control, including stabilization of fractional order chaotic systems to steady states or regular behavior. Some approaches have been proposed to achieve chaos stabilization in fractional-order chaotic systems.

On the other hand, Fractional calculus is an area of mathematics that handles with differentiation and integration of arbitrary (non-integer) orders. It is a generalization of the ordinary differentiation and integration to non-integer (arbitrary) order. The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented this kind of calculation. a fractional derivative was an ongoing topic in the last 300 years (Bouzeriba et al., 2015), but its application to physics and engineering has been reported only in the recent years (Hamamci, 2012). It has been found that many systems in interdisciplinary fields can be described by fractional differential equations, such as viscoelastic systems, dielectric polarization, electrode–electrolyte polarization, some finance systems, and electromagnetic wave systems, Moreover, applications of fractional calculus have been reported in many areas such as signal processing, image processing, automatic control, and robotics. These examples and many other similar samples perfectly clarify the importance of consideration and analysis of dynamical systems with fractional order models (Zamani et al., 2011; Bourouba et al., 2016).

The sliding mode control methodology is one such robust control technique which has its roots in the relay control. One of the most intriguing aspects of sliding mode is the discontinuous nature of the control action whose primary function is to switch between two distinctively different structures about some predefined manifold such that a new type of system motion called sliding mode exists in a manifold. This peculiar system characteristic is claimed to result in a superb system performance which includes insensitivity to parameter variations and complete rejection of certain class of disturbances. Furthermore, the system possesses new properties which are not present in original system. Sliding mode contains two phases (a) reaching phase in which the system states are driven from any initial state to reach the switching manifolds (the anticipated sliding modes) in finite time and (b) sliding phase in which the system is induced into the sliding motion on the switching manifolds, i.e., the switching manifolds become attractors. The robustness and order reduction property of sliding mode control comes into picture only after the occurrence of sliding mode.

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