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ABSTRACT Finite-time fractional-order adaptive intelligent backstepping sliding mode control of uncertain fractional-order chaotic systems

One of the mathematical topics with more than three centuries history is fractional calculus theory which can be traced back to Leibniz, Riemann, Liouville, Grünwald, and Letnikov [1–3]. Although, it did not attract much attention for a long time, but, in the recent years, due to high modeling accuracy of real physical systems by fractional order equations, these systems have been increasingly used for modeling in many areas such as physics [4] and engineering [5]. In interdisciplinary fields, many systems have been found which can be described by fractional differential equations. For instance, electrochemical processes [6], viscoelastic systems, dielectric polarization and electromagnetic waves [2,7], and some biological systems [8] can be encountered in this category. Applying latent possibilities of fractional calculus, in order to improve the performance of classical controllers, is one of the recent applications of this mathematical topic.

One of the phenomena which are frequently observed in the fractional order nonlinear systems is chaos phenomenon. Chaotic systems are a prominent class of nonlinear systems, which have various special properties, such as phenomenal sensitivity to system initial conditions, chaotic attractors, and fractal motions. Study of chaotic systems with fractional orders is one of the hottest research areas. Up to now, researchers have studied the fractional order nonlinear dynamics of many popular chaotic systems, such as fractional order Lorenz system [9–11], fractional order Rössler system [12,13], fractional order Chen system [12,14], fractional order Lü system [15,16] and fractional order Genesio-Tesi system [17] and some other chaotic systems. One of the most attractive topics in the nonlinear science which has been comprehensively studied in the recent decades is Chaos control and synchronization. Therefore, many control procedures such as sliding mode control (SMC) [18–22], linear control [23], adaptive control theory [24–26], backstepping control [27–30], active control [31], fuzzy sliding mode control [32], adaptive sliding mode control [33,34] and some other methods [35–38] have been successfully applied to chaos control and synchronization. Backstepping method is a recursive procedure which was proposed for designing controls for a special type of nonlinear dynamical systems by choosing Lyapunov functions skillfully and a systematic design approach, and it can guarantee global stability, good tracking and transient performance for most of strict-feedback systems. Therefore, it has become one of the important and popular approaches for nonlinear systems [39,40]. Sliding mode control technique has been known as a powerful strategy to design robust control for linear and nonlinear systems in recent years. The significant superiorities of SMC are its robustness to parameters uncertainty and its immutability to external disturbances. It is also a strong robust procedure for controlling the nonlinear dynamic systems [41–43], especially for uncertain systems. It has been studied extensively and received many applications due to its insensitivity to system parameter variations, external disturbances rejection, fast dynamic and good transient response which made it applicable for the control problem of chaotic system [18–22,32–34]. In this control scheme, for conducting an effective switching surface and guaranteeing the stability of an SMC system, a good estimation of the uncertainty bound, including the unknown dynamics, parameter variation, and external disturbance must be available at the outset of the design. Since, in practice, such bounds cannot be estimated easily, some conservative control strategies are also need to be applied for ensuring stability of the closed loop system.

Due to the universal approximation ability of fuzzy systems and neural networks it has been widely used in chaos control problems [44–46]. Recently, the learning abilities of neural networks to realize to design of fuzzy systems have been recognized as a powerful approach [47]. The neuro-fuzzy-network (NFN) possesses the merits of low-level learning and computation power of neural networks, and the high-level human-like thinking and reasoning of fuzzy theory. However, in complicated situations, where plant parameters are subject to perturbations or when the dynamics of the systems are too complex for a mathematical model to describe, adaptive schemes should be used online to gather data and adjust the control parameters, automatically.

Recently, many significant results have been obtained by using hybrid adaptive intelligent control techniques for uncertain chaotic systems [48–53]. For fractional-order systems, this idea has been extended to some extent, by the researchers, in the literature. For example, in [54], uncertain fractional-order Liu system is controlled via fuzzy fractional-order sliding mode method. In [55], adaptive fuzzy sliding mode control has been employed for synchronization of uncertain fractional order chaotic systems, in which, a fuzzy Lyapunov synthesis approach is proposed to tune free parameters of the adaptive fuzzy controller on line by output feedback control law and adaptive law. Synchronization of uncertain fractional order chaotic systems via adaptive interval type-2 fuzzy sliding mode control was considered in [56] to handle high level uncertainties facing the fuzzy logic controller (FLC) in dynamic fractional order chaotic systems such as uncertainties in inputs to the FLC, uncertainties in control outputs, linguistic uncertainties and uncertainties associated with the noisy training data. In [57], an adaptive fuzzy sliding mode control was designed to synchronize two different uncertain fractional-order time-delay chaotic systems, where, the adaptive time-delay fuzzy-logic system is constructed to approximate the unknown fractional-order time-delay-system functions. In [58], the design of adaptive fuzzy wavelet neural sliding mode controller for uncertain nonlinear systems was studied for a class of high-order nonlinear systems while the structure of the system is unknown and no prior knowledge about uncertainty is available. The proposed scheme was employed to construct equivalent control term and an Adaptive Proportional-Integral (A-PI) controller for implementing switching term to provide smooth control input. Next attempt ([59]) deals with the synchronization for Arneodo chaotic system and response system with unknown nonlinear function. Based on backstepping method, a fuzzy adaptive control scheme by combining fuzzy logic system with parameter was presented to achieve synchronization. In [60], a fractionalorder adaptive intelligent controller was designed based active control method, in which, the unknown boundaries of the lumped uncertainties are estimated via an adaptive neuro-fuzzy approximator.

In this paper, in order to merge the capabilities of backstepping control and sliding mode control, an adaptive intelligent backstepping sliding mode controller is proposed for stabilization of fractional order chaotic systems in finite time. The intelligent neuro-fuzzy controller is used to estimate unknown functions. The use of auxiliary controller is in order to improve velocity and performance of the proposed control system and to dispel the uncertainties, external disturbances and error approximation. These three factors are unknown bounded, and so the controller will take advantage of robust adaptive design. In design of the control law, neuro-fuzzy network parameters and the uncertainties bound estimator, external disturbances bound estimator and approximation error, are adjusted as an adaptive scheme. The asymptotic stability of the controller is shown based on Lyapunov theorem and the finite reaching time to the sliding surfaces is also proved. The results are indicative of the Lyapunov stability of the closed loop system, robustness against uncertainties, external disturbances and approximation errors, while the control signal remains bounded.

The rest of this paper is organized as follows: In Section 2, the definitions and some basic properties of fractional calculus are introduced. Section 3, presents dynamic models of fractional order chaotic systems which are used in this paper and in Section 4 description of the neurofuzzy network structure is presented. Problem description is described in Section 5. Numerical simulation results which confirm the validity and feasibility of the designed controllers, are shown in Section 6. Finally, conclusions are given in Section 7.