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.     

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.     

Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach

In this paper, an adaptive sliding mode controller for a novel class of fractional-order chaotic systems with uncertainty and external disturbance is proposed to realize chaos control. The bounds of the uncertainty and external disturbance are assumed to be unknown. Appropriate adaptive laws are designed to tackle the uncertainty and external disturbance. In the adaptive sliding mode control (ASMC) strategy, fractional-order derivative is introduced to obtain a novel sliding surface. The adaptive sliding mode controller is shown to guarantee asymptotical stability of the considered fractional-order chaotic systems in the presence of uncertainty and external disturbance. Some numerical simulations demonstrate the effectiveness of the proposed ASMC scheme.     

Optimal Sliding Mode Robust Control for Fractional-Order Systems with Application to Permanent Magnet Synchronous Motor Tracking Control

This paper presents an optimal sliding mode output tracking control scheme for a class of fractional-order uncertain systems. Firstly, an augmented fractional-order system, composed of the original system and the external system, is constructed to transform the optimal output tracking issue into the design problem of linear quadratic regulator. The optimal tracking control problem for the nominal augmented fractional-order system is then studied. Secondly, the fractional-integral sliding mode controller is introduced to robustify the augmented fractional-order system, which satisfy the matching conditions. As a result, the original system output can track the external system output trajectory effectively even the uncertainties exist. Finally, the developed design techniques are applied to the tracking control of fractional-order permanent magnet synchronous motor. The simulation results demonstrate the validity of this approach.     

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.     

超混沌分数阶Bao系统的自适应滑模同步控制

研究超混沌分数阶Bao系统自适应滑模同步,设计出分数阶滑模函数、适应规则和控制器,取得超混沌分数阶Bao系统自适应滑模同步的充分条件,文末用MATLAB数值仿真验证了所得结论.     

A sliding mode control for linear fractional systems with input and state delays

In this paper, a sliding mode control design for fractional order systems with input and state time-delay is proposed. First, we consider a fractional order system without delay for which a sliding surface is proposed based on fractional integration of the state. Then, a stabilizing switching controller is derived. Second, a fractional system with state delay is considered. Third, a strategy including a fractional state predictor input delay compensation is developed. The existence of the sliding mode and the stability of the proposed control design are discussed. Numerical examples are given to illustrate the theoretical developments.     

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.     

一类分数阶金融系统的混沌同步

研究了一类分数阶金融系统的混沌同步问题,基于Lyapunov稳定性理论和分数阶微积分的相关理论,给出了两种实现同步的控制方案,仿真算例表明了方法的有效性.     

Feedback control and hybrid projective synchronization of a fractional-order Newton-Leipnik system

In this work, stability analysis of the fractional-order Newton-Leipnik system is studied by using the fractional Routh-Hurwitz criteria. The fractional Routh-Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh-Hurwitz conditions and using specific choice of linear feedback controllers, it is shown that the Newton-Leipnik system is controlled to its equilibrium points. Moreover, the theoretical basis of hybird projective synchronization of commensurate and incommensurate fractional-order Newton-Leipnik systems is investigated. Based on the stability theorems of fractional-order systems, the controllers for hybrid projective synchroniztion are derived. Numerical results show the effectiveness of the theoretical analysis.     

Van der pol情绪混沌模型的滑模同步

研究了整数阶分数阶van der pol情绪混沌模型的滑模同步问题,利用分数阶微积分给出了情绪模型的主从系统取得同步的充分条件,研究表明,一定条件下,Van der pol情绪模型的主从系统能够达到同步,数值仿真验证了该方法的可行性.     

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.     

Control of non‐integer‐order dynamical systems using sliding mode scheme

This article deals with the problem of control of canonical non‐integer‐order dynamical systems. We design a simple dynamical fractional‐order integral sliding manifold with desired stability and convergence properties. The main feature of the proposed dynamical sliding surface is transferring the sign function in the control input to the first derivative of the control signal. Therefore, the resulted control input is smooth and without any discontinuity. So, the harmful chattering, which is an inherent characteristic of the traditional sliding modes, is avoided. We use the fractional Lyapunov stability theory to derive a sliding control law to force the system trajectories to reach the sliding manifold and remain on it forever. A nonsmooth positive definite function is applied to prove the existence of the sliding motion in a given finite time. Some computer simulations are presented to show the efficient performance of the proposed chattering‐free fractional‐order sliding mode controller. © 2015 Wiley Periodicals, Inc. Complexity 21: 224–233, 2016     

Adaptive type-2 fuzzy backstepping control of uncertain fractional-order nonlinear systems with unknown dead-zone

A new problem of adaptive type-2 fuzzy fractional control with pseudo-state observer for commensurate fractional order dynamic systems with dead-zone input nonlinearity is considered in presence of unmatched disturbances and model uncertainties; the control scheme is constructed by using the backstepping and adaptive technique. To avoid the complexity of backstepping design process, the dynamic surface control is used. Also, Interval type-2 Fuzzy logic systems (IT2FLS) are used to approximate the unknown nonlinear functions. By using the fractional adaptive backstepping, fractional control laws are constructed; this method is applied to a class of uncertain fractional-order nonlinear systems. In order to better control performance in reducing tracking error, the PSO algorithm is utilized for tuning the controller parameters. Stability of the system is proven by the Mittag–Leffler method. It is shown that the proposed controller guarantees the boundedness property for the system and also the tracking error can converge to a small neighborhood of the origin. The efficiency of the proposed method is illustrated with simulation examples.     

Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication

This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractional-order hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.     

分数阶双指数混沌系统的自适应滑模同步

研究了分数阶双指数混沌系统的自适应滑模同步问题.通过设计滑模函数和控制器,构造了平方Lyapunov函数进行稳定性分析.利用Barbalat引理证明了同步误差渐近趋于零,获得了系统取得自适应滑模同步的充分条件.数值仿真结果表明:选取适当的控制器及与滑模函数,分数阶双指数混沌系统取得自适应滑模同步.     

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.     

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.