Adaptive fuzzy sliding-mode control for multi-input multi-output chaotic systems

This paper describes an adaptive fuzzy sliding-mode control algorithm for controlling unknown or uncertain, multi-input multi-output (MIMO), possibly chaotic, dynamical systems. The control approach encompasses a fuzzy system and a robust controller. The fuzzy system is designed to mimic an ideal sliding-mode controller, and the robust controller compensates the difference between the fuzzy controller and the ideal one. The parameters of the fuzzy system, as well as the uncertainty bound of the robust controller, are tuned adaptively. The adaptive laws are derived in the Lyapunov sense to guarantee the asymptotic stability and tracking of the controlled system. The effectiveness of the proposed method is shown by applying it to some well-known chaotic systems.     

Adaptive fuzzy control for a class of chaotic systems with nonaffine inputs

A robust adaptive fuzzy control scheme is presented for a class of chaotic systems with nonaffine inputs, modeling uncertainties and external disturbances by using backstepping approach. Fuzzy logic systems (FLS) are employed to approximate the unknown parts of the virtual control and practical controls. The main characteristics of the scheme are that the number of the online adaptive parameters is no more than two times of the order of chaotic system and the tracking errors are guaranteed to be uniformly asymptotically stable with the aid of additional adaptive compensation terms. Lorenz system, Chen system, Lü system and Liu system are presented to illustrate the feasibility and effectiveness of the proposed control technique.     

Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems

This paper deals with chaos synchronization between two different uncertain fractional order chaotic systems based on adaptive fuzzy sliding mode control (AFSMC). With the definition of fractional derivatives and integrals, 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. Moreover, chattering phenomena in the control efforts can be reduced. The sliding mode design procedure not only guarantees the stability and robustness of the proposed AFSMC, but also 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.     

Adaptive feedback control for a class of chaotic systems

In this paper, the problem of control for a class of chaotic systems is considered. The nonlinear functions of chaotic systems are not necessarily to satisfy the Lipsichtz conditions, but bounded by a polynomial with the gains unknown. Employing adaptive method, the corresponding controller which renders the closed-loop system asymptotically stable is constructed. The designed controller is robust with respect to certain class of disturbances in the chaotic systems. Simulations on unified chaotic systems and Arneodo chaotic system are performed and the results verify the validity of the proposed techniques.     

A simple method of chaos control for a class of chaotic discrete-time systems

In this paper, a simple method is proposed for chaos control for a class of discrete-time chaotic systems. The proposed method is built upon the state feedback control and the characteristic of ergodicity of chaos. The feedback gain matrix of the controller is designed using a simple criterion, so that control parameters can be selected via the pole placement technique of linear control theory. The new controller has a feature that it only uses the state variable for control and does not require the target equilibrium point in the feedback path. Moreover, the proposed control method cannot only overcome the so-called “odd eigenvalues number limitation” of delayed feedback control, but also control the chaotic systems to the specified equilibrium points. The effectiveness of the proposed method is demonstrated by a two-dimensional discrete-time chaotic system.     

Adaptive sliding mode control in a novel class of chaotic systems

This paper proposes a robust adaptive sliding mode control strategy for an introduced class of uncertain chaotic systems. Using the sliding mode control technique and based on Lyapunov stability theory, a time varying sliding surface is determined and an adaptive gain of the robust control law will be tuned to stabilize the new chaotic class. Unlike many well-known methods of the sliding mode control, no knowledge on the bound of uncertainty and disturbance is required. Simulation results are demonstrated for several chaotic examples to illustrate the effectiveness of the proposed adaptive sliding mode control scheme.     

Adaptive fuzzy observer based synchronization design and secure communications of chaotic systems

This paper proposes a synchronization design scheme based on an alternative indirect adaptive fuzzy observer and its application to secure communication of chaotic systems. It is assumed that their states are unmeasurable and their parameters are unknown. Chaotic systems and the structure of the fuzzy observer are represented by the Takagi–Sugeno fuzzy model. Using Lyapunov stability theory, an adaptive law is derived to estimate the unknown parameters and the stability of the proposed system is guaranteed. Through this process, the asymptotic synchronization of chaotic systems is achieved. The proposed observer is applied to secure communications of chaotic systems and some numerical simulation results show the validity of theoretical derivations and the performance of the proposed observer.     

Synchronization of different-order chaotic systems: Adaptive active vs. optimal control

In this paper, an adaptive algorithm is proposed for synchronization of chaotic systems with different orders. A modular adaptive control strategy is applied to make states of the slave system track those of the master, despite the unknown parameters. One of the most advantages of the modularity approach, which is applied for the first time in chaos synchronization, is its flexibility in choosing identification and control modules and designing them completely independently. In this paper, a modified recursive least square algorithm is used to identify the unknown parameters of the slave system, and the control module is designed by means of two different algorithms. First it is designed based on active control method, and then, in order to synchronize with a lower energy, we design an optimal controller. The two methods are applied on a practical case study, and the results are compared. Two different dimensional neuron models, the HR neuron model and the cable model of cylindrical cell, are considered as the master and slave systems, respectively. Simulation results confirm the effectiveness of the proposed method.     

Robust ISS-satisficing variable universe indirect fuzzy control for chaotic systems

In the conventional robust input to state stable (ISS)-satisficing control system, all parameters of the system must be known beforehand, so the application area is limited. In this paper, an attempt is made to create a bridge between two important design techniques, i.e., the robust ISS-satisficing control strategy and the fuzzy control strategy, and the new control method we first proposed has both the inverse optimality of robust ISS-satisficing control and the robust and predictive performance of fuzzy control. By control Lyapunov method, the overall closed-loop system is shown to be stable. In this work, we combine these two control methods, make them learn from the other’s strong points, offset its weakness. The simulation results are given to confirm the control algorithm is feasible and performances well.     

Control of discrete-time chaotic systems via feedback linearization

An approach for controlling discrete-time chaotic systems by feedback linearization is proposed. This method can not only stabilize unstable periodic orbits embedded in a strange attractor, but also can be applied even if the real trajectory is far from the target one. A Hénon map with different operation conditions is implemented to demonstrate the feasibility of the proposed method.     

Adaptive synchronization of T–S fuzzy chaotic systems with unknown parameters

This paper presents a fuzzy model-based adaptive approach for synchronization of chaotic systems which consist of the drive and response systems. Takagi–Sugeno (T–S) fuzzy model is employed to represent the chaotic drive and response systems. Since the parameters of the drive system are assumed unknown, we design the response system that estimates the parameters of the drive system by adaptive strategy. The adaptive law is derived to estimate the unknown parameters and its stability is guaranteed by Lyapunov stability theory. In addition, the controller in the response system contains two parts: one part that can stabilize the synchronization error dynamics and the other part that estimates the unknown parameters. Numerical examples, including Duffing oscillator and Lorenz attractor, are given to demonstrate the validity of the proposed adaptive synchronization approach.     

Fuzzy impulsive control of chaotic systems based on TS fuzzy model

Combining Takagi–Sugeno (TS) fuzzy model and impulsive control, a new approach to control chaotic systems, namely fuzzy impulsive control, is proposed in this paper. The rigorous stability analysis of the proposed method is given. The effectiveness of the approach is tested on Chua’s circuit, Chen’s system and Rössler’s system.     

Adaptive fuzzy control‐based projective synchronization of uncertain nonaffine chaotic systems

This article aims to introduce a projective synchronization approach based on adaptive fuzzy control for a class of perturbed uncertain multivariable nonaffine chaotic systems. The fuzzy‐logic systems are employed to approximate online the uncertain functions. A Lyapunov approach is used to design the parameter adaptation laws and to demonstrate the boundedness of all signals of the closed‐loop system as well as the convergence of the synchronization errors to bounded residual sets. Finally, numerical simulation results are presented to verify the feasibility and effectiveness of the proposed synchronization system based on fuzzy adaptive controller. © 2014 Wiley Periodicals, Inc. Complexity 21: 180–192, 2015     

An alternative approach to Privault's discrete-time chaotic calculus

In this paper, we present an alternative approach to Privault's discrete-time chaotic calculus. Let Z be an appropriate stochastic process indexed by N (the set of nonnegative integers) and l²(Γ) the space of square summable functions defined on Γ (the finite power set of N). First we introduce a stochastic integral operator J with respect to Z, which, unlike discrete multiple Wiener integral operators, acts on l²(Γ). And then we show how to define the gradient and divergence by means of the operator J and the combinatorial properties of l²(Γ). We also prove in our setting the main results of the discrete-time chaotic calculus like the Clark formula, the integration by parts formula, etc. Finally we show an application of the gradient and divergence operators to quantum probability.     

Adaptive fuzzy sliding mode control scheme for uncertain systems

Most physical systems inherently contain nonlinearities which are commonly unknown to the system designer. Therefore, in modeling and analysis of such dynamic systems, one needs to handle unknown nonlinearities and/or uncertain parameters. This paper proposes a new adaptive tracking fuzzy sliding mode controller for a class of nonlinear systems in the presence of uncertainties and external disturbances. The main contribution of the proposed method is that the structure of the controlled system is partially unknown and does not require the bounds of uncertainty and disturbance of the system to be known; meanwhile, the chattering phenomenon that frequently appears in the conventional variable structure systems is also eliminated without deteriorating the system robustness. The performance of the proposed approach is evaluated for two well-known benchmark problems. The simulation results illustrate the effectiveness of our proposed controller.     

Impulsive synchronization of discrete-time chaotic systems under communication constraints

This paper investigates the problem of impulsive synchronization of discrete-time chaotic systems subject to limited communication capacity. Control laws with impulses are derived by using measurement feedback, where the effect of quantization errors is considered. Sufficient conditions for asymptotic stability of synchronization error systems are given in terms of linear matrix inequalities and algebraic inequalities. Some numerical simulations are given to demonstrate the effectiveness of the method.     

Adaptive fuzzy backstepping control design for a class of pure-feedback switched nonlinear systems

In this paper, an adaptive fuzzy output tracking control approach is proposed for a class of single input and single output (SISO) uncertain pure-feedback switched nonlinear systems under arbitrary switchings. Fuzzy logic systems are used to identify the unknown nonlinear system. Under the framework of the backstepping control design and fuzzy adaptive control, a new adaptive fuzzy output tracking control method is developed. It is proved that the proposed control approach can guarantee that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) and the tracking error remains an adjustable neighborhood of the origin. A numerical example is provided to illustrate the effectiveness of the proposed approach.     

Adaptive fuzzy backstepping control for a class of MIMO switched nonlinear systems with unknown control directions

In this article, an adaptive fuzzy output tracking control approach is proposed for a class of multiple‐input and multiple‐output uncertain switched nonlinear systems with unknown control directions and under arbitrary switchings. In the control design, fuzzy logic systems are used to identify the unknown switched nonlinear systems. A Nussbaum gain function is introduced into the control design and the unknown control direction problem is solved. Under the framework of the backstepping control design, fuzzy adaptive control and common Lyapunov function stability theory, a new adaptive fuzzy output tracking control method is developed. It is proved that the proposed control approach can guarantee that all the signals in the closed‐loop system are bounded and the tracking error remains an adjustable neighborhood of the origin. A numerical example is provided to illustrate the effectiveness of the proposed approach. © 2015 Wiley Periodicals, Inc. Complexity 21: 155–166, 2016