Design of sliding mode controller for a class of fractional-order chaotic systems

In this paper, a sliding mode control law is designed to control chaos in a class of fractional-order chaotic systems. A class of unknown fractional-order systems is introduced. Based on the sliding mode control method, the states of the fractional-order system have been stabled, even if the system with uncertainty is in the presence of external disturbance. In addition, chaos control is implemented in the fractional-order Chen system, the fractional-order Lorenz system, and the same to the fractional-order financial system by utilizing this method. Effectiveness of the proposed control scheme is illustrated through numerical simulations.     

Hybrid chaos control of continuous unified chaotic systems using discrete rippling sliding mode control

In this paper, a new systematic design procedure to stabilize continuous unified chaotic systems based on discrete sliding mode control (DSMC) is presented. In contrast to the previous works, the concept of rippling control is newly introduced such that the design of DSMC can be simplified and only a single controller is needed to realize chaos suppression. As expected, under the proposed DSMC law, the unified system can be stabilized in a manner of ripple effect, even when the external uncertainty is present. Last, two examples are included to illustrate the effectiveness of the proposed rippling DSMC developed in this paper.     

Finite-time stabilization of a class of chaotic systems via adaptive control method

This paper investigates the stabilization of three dimensional chaotic systems in a finite time by extending our previous method for chaos stabilization. Based on the finite-time stability theory, a control law is proposed to realize finite-time stabilization of three dimensional chaotic systems. In comparison with the previous methods, the controller obtained by our method is simpler than those. Moreover, the method obtained in this paper is suitable for a class of three dimensional chaotic systems. The efficiency of the control scheme is revealed by some illustrative simulations.     

Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique

In this paper, the problem of finite-time chaos synchronization between two different chaotic systems with fully unknown parameters is investigated. First, a new nonsingular terminal sliding surface is introduced and its finite-time convergence to the zero equilibrium is proved. Then, appropriate adaptive laws are derived to tackle the unknown parameters of the systems. Afterwards, based on the adaptive laws and finite-time control idea, an adaptive sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time. It is mathematically proved that the introduced sliding mode technique has finite-time convergence and stability in both reaching and sliding mode phases. Finally, some numerical simulations are presented to demonstrate the applicability and effectiveness of the proposed technique.     

Sliding mode control of uncertain unified chaotic systems

This paper investigates the chaos control of the uncertain unified chaotic systems by means of sliding mode control. A proportional plus integral sliding surface is introduced to obtain a sliding mode control law. To confirm the validity of the proposed method, numerical simulations are presented graphically.     

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.     

Synchronization of fractional‐order chaotic systems with disturbances via novel fractional‐integer integral sliding mode control and application to neuron models

In this paper, a novel fractional‐integer integral type sliding mode technique for control and generalized function projective synchronization of different fractional‐order chaotic systems with different dimensions in the presence of disturbances is presented. When the upper bounds of the disturbances are known, a sliding mode control rule is proposed to insure the existence of the sliding motion in finite time. Furthermore, an adaptive sliding mode control is designed when the upper bounds of the disturbances are unknown. The stability analysis of sliding mode surface is given using the Lyapunov stability theory. Finally, the results performed for synchronization of three‐dimensional fractional‐order chaotic Hindmarsh‐Rose (HR) neuron model and two‐dimensional fractional‐order chaotic FitzHugh‐Nagumo (FHN) neuron model.     

Sliding mode control for uncertain chaotic systems with input nonlinearity

For the sliding mode controller of uncertain chaotic systems subject to input nonlinearity, the upper bound of the norm of uncertainties is commonly used to determine the controller parameter. However, this will cause serious chattering. In order to overcome this drawback, two new sliding mode controllers are proposed to ensure robust synchronization for a classes of chaotic systems with input nonlinearities and external uncertainty. Compared with the existing results, the proposed controllers can effectively reduce the chattering nearby sliding mode and improve the dynamic performance of the systems. Simulation results are provided to verify the proposed methods.     

An adaptive-dynamic sliding mode controller for non-minimum phase systems

This paper presents a new algorithm for designing dynamic sliding-mode controllers. The proposed controller is based on dynamic sliding manifolds to circumvent the difficulties associated with the conventional sliding mode controllers in the face of non-minimum phase systems. Unlike previous works, a proper and easy to implement algorithm is presented for designing the dynamic sliding manifold which facilitates the design of the controller. The output tracking problem in nonlinear non-minimum phase systems with matched and unmatched disturbances and matched nonlinearities is addressed. Then, the performance of the dynamic sliding mode controller is significantly improved by combining the given dynamic sliding manifold with online parameter adaptation. Simulations results are presented to demonstrate the effectiveness of the proposed sliding mode controller in terms of performance, robustness and stability.     

Leader-follower strategy via a sliding mode approach

In this paper, the decision making problem in continuoustime dynamic systems is considered for the situation with two decision makers and a hierarchical decision structure. The leader-follower strategy is studied. To implement the leader's strategy, we propose to use a sliding mode approach, which allows the leader to constrain the state of the system within some manifold of the state space and forces the follower to choose the strategy preferable for the leader. The corresponding sliding manifolds are derived from the classical variational problem formulation for a class of systems whose right-hand side is affine with respect to the two control inputs. Numerical examples are considered with simulations to illustrate the technique.The authors wish to express their thanks to Dr. Vadim Utkin for his helpful discussions.     

Event-triggered sliding mode control for singular systems with disturbance

This paper investigates the event-triggered sliding mode control (SMC) problem for singular systems with disturbance. Firstly, an event-triggered sliding mode control law is designed to guarantee the reachability of sliding surface. Different from the related methods, in order to deal with the difficulty caused by event-triggered SMC strategy, a novel Lemma is proposed in this paper. Secondly, the admissibility of sliding motion is presented, which is used to solve the controller gain. Then, a positive lower bound of the inter execution time can be guaranteed and the Zeno behavior is avoided. Finally, two simulation examples are presented to show the effectiveness of derived theoretical results.     

Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller

This paper proposes a novel fractional-order sliding mode approach for stabilization and synchronization of a class of fractional-order chaotic systems. Based on the fractional calculus a stable integral type fractional-order sliding surface is introduced. Using the fractional Lyapunov stability theorem, a single sliding mode control law is proposed to ensure the existence of the sliding motion in finite time. The proposed control scheme is applied to stabilize/synchronize a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. Some numerical simulations are performed to confirm the theoretical results of the paper. It is worth noticing that the proposed fractional-order sliding mode controller can be applied to control a broad range of fractional-order dynamical systems.     

Projective synchronization of different fractional-order chaotic systems with non-identical orders

This paper investigates the projective synchronization (PS) of different fractional order chaotic systems while the derivative orders of the states in drive and response systems are unequal. Based on some essential properties on fractional calculus and the stability theorems of fractional-order systems, we propose a general method to achieve the PS in such cases. The fractional operators are introduced into the controller to transform the problem into synchronization problem between chaotic systems with identical orders, and the nonlinear feedback controller is proposed based on the concept of active control technique. The method is both theoretically rigorous and practically feasible. We present two examples that illustrate the effectiveness and applications of the method, which include the PS between two 3-D commensurate fractional-order chaotic systems and the PS between two 4-D fractional-order hyperchaotic systems with incommensurate and commensurate orders, respectively. Abundant numerical simulations are given which agree well with the analytical results. Our investigations show that PS can also be achieved between different chaotic systems with non-identical orders. We have further reviewed and compared some relevant methods on this topic reported in several recent papers. A discussion on the physical implementation of the proposed method is also presented in this paper.     

Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty

A novel type of control strategy combining the fractional calculus with terminal sliding mode control called fractional terminal sliding mode control is introduced for a class of dynamical systems subject to uncertainties. A fractional-order switching manifold is proposed and the corresponding control law is formulated based on the Lyapunov stability theory to guarantee the sliding condition. The proposed fractional-order terminal sliding mode controller ensures the finite time stability of the closed-loop system. Finally, numerical simulation results are presented and compared to illustrate the effectiveness of the proposed method.     

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.     

Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control

This paper investigates the global synchronization of a class of third-order non-autonomous chaotic systems via the master–slave linear state error feedback control. A sufficient global synchronization criterion of linear matrix inequality (LMI) and several algebraic synchronization criteria for single-variable coupling are proven. These LMI and algebraic synchronization criteria are then applied to two classes of well-known third-order chaotic systems, the generalized Lorenz systems and the gyrostat systems, proving that the local synchronization criteria for the chaotic generalized Lorenz systems developed in the existing literature can actually be extended to describe global synchronization and obtaining some easily implemented synchronization criteria for the gyrostat systems.     

Finite‐time stabilization of a class of chaotic systems with matched and unmatched uncertainties: An LMI approach

This article investigates the sliding mode control method for a class of chaotic systems with matched and unmatched uncertain parameters. The proposed reaching law is established to guarantee the existence of the sliding mode around the sliding surface in a finite‐time. Based on the Lyapunov stability theory, the conditions on the state error bound are expressed in the form of linear matrix inequalities. Simulation results for the well‐known Genesio's chaotic system are provided to illustrate the effectiveness of the proposed scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 14–19, 2016     

Comments on “Design of sliding mode controller for a class of fractional-order chaotic systems” [Commun Nonlinear Sci Numer Simulat 17 (2012) 356-366]

Some comments on the paper [Yin C, Zhong S-M, Chen W-F. Design of sliding mode controller for a class of fractional-order chaotic systems. Commun Nonlinear Sci Numer Simulat 17 (2012) 356-366] are pointed out in this note. Besides, recently introduced fractional-order Lyapunov stability theorems are used to prove the finite-time occurrence of the sliding motion.     

Chaos in fractional-order Genesio-Tesi system and its synchronization

In this paper we study the chaotic dynamics of fractional-order Genesio-Tesi system. Theoretically, a necessary condition for occurrence of chaos is obtained. Numerical investigations on the dynamics of this system have been carried out and properties of the system have been analyzed by means of Lyapunov exponents. It is shown that in case of commensurate system the lowest order of fractional-order Genesio-Tesi system to yield chaos is 2.79. Further, chaos synchronization of fractional-order Genesio-Tesi system is investigated via two different control strategies. Active control and sliding mode control are proposed and the stability of the controllers are studied. Numerical simulations have been carried out to verify the effectiveness of controllers.