A non-integer sliding mode controller to stabilize fractional-order nonlinear systems (original) (raw)
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Sliding Mode Control of Generalized Fractional Chaotic Systems
International Journal of Bifurcation and Chaos, 2006
A sliding mode control technique is introduced for generalized fractional chaotic systems. These systems are governed by a set of fractional differential equations of incommensurate orders. The proposed design method relies on the fact that the stability region of a fractional system contains the stability region of its underlying integer-order model. A sliding mode controller designed for an equivalent integer-order chaotic system is used to stabilize all its corresponding fractional chaotic systems. The design technique is demonstrated using two generalized fractional chaotic models; a chaotic oscillator and the Chen system. The effect of the total fractional order is investigated with respect to the controller effort and the convergence rate of the system response to the origin. Numerical simulations validate the main results of this work.
Communications in Nonlinear Science and Numerical Simulation, 2012
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.
Pramana, 2017
In this paper, a new design of fractional-order sliding mode control scheme is proposed for the synchronization of a class of nonlinear fractional-order systems with chaotic behaviour. The considered design approach provides a set of fractional-order laws that guarantee asymptotic stability of fractional-order chaotic systems in the sense of the Lyapunov stability theorem. Two illustrative simulation examples on the fractional-order Genesio-Tesi chaotic systems and the fractional-order modified Jerk systems are provided. These examples show the effectiveness and robustness of this control solution.
Control of Fractional Order Uncertain Chaotic Unified Systems via Sliding Mode Control
In this paper, a sliding mode control law is designed to control chaos in fractional order unified chaotic systems. 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 Lorenz system by utilizing this method. Simulation results confirm numerical results.
Conformable Fractional Order Sliding Mode Control for a Class of Fractional Order Chaotic Systems
International Journal of Industrial Electronics, Control and Optimization (IECO), 2019
In this paper, a novel conformable fractional order (FO) sliding mode control technique is studied for a class of FO chaotic systems in the presence of uncertainties and disturbances. First, a novel FO nonlinear surface based on conformable FO calculus is proposed to design the FO sliding mode controller. Then, asymptotic stability of the controller is derived by means of the Lyapunov direct method via conformable FO operators. The stability analysis is performed in the sliding and reaching phase. In addition, the realization of reaching phase is guaranteed in finite time and the reaching time is calculated analytically. The proposed control approach has some superiorities. Reduction of the chattering phenomenon, high robustness against the uncertainty and external disturbance, and fast convergence speed are the main advantages of the proposed control scheme. Moreover, it has simple calculations because of using conformable FO operators in the control design. The numerical simulations verify the efficiency of the proposed controller.
Circuits, Systems, and Signal Processing, 2019
In the two last decades, several research works proposed adaptive sliding mode control (SMC) algorithms to deal with fractional-order chaotic systems for control and synchronization. As a contribution to this investigation effort, this paper proposes a new adaptation law for fractional-order SMC addressing the synchronization problem for a class of nonlinear fractional-order systems with chaotic behavior. The main innovation in the proposed control design concerns the choice of a sliding surface with two adjustable parameters, leading easily to an efficient adaptation law for the SMC controller. Stability analysis of the proposed control scheme is performed using the Lyapunov stability theorem. As an illustration of the effectiveness of this synchronization strategy, a simulation example on the fractional-order Arneodo chaotic system is presented and discussed.
International Journal of Industrial Electronics, Control and Optimization (IECO), 2019
This paper deals with the problem of synchronization (anti-synchronization) of fractional nonlinear systems. Here, due to the advantages of fractional calculus and sliding mode control, we provide a new fractional order sliding mode control for synchronization (anti-synchronization) problems. So, in this paper a novel sliding surface is introduced and with and without the existence of uncertainties and external disturbances, finite-time synchronization is achieved by designing a new fractional sliding mode control. This method applied to the class of fractional order nonlinear systems and sufficient conditions for achieving synchronization/anti-synchronization are derived by the use of fractional Lyapunov theory. To show the effectiveness and robustness of the proposal, we applied our method on two identical fractional order financial system to verify the efficacy.
Robust Fuzzy Adaptive Sliding Mode Stabilization for Fractional-Order Chaos
Algorithms, 2018
In this paper, a new adaptive fuzzy sliding mode control (AFSMC) design strategy is proposed for the control of a special class of three-dimensional fractional order chaotic systems with uncertainties and external disturbance. The design methodology is developed in two stages: first, an adaptive sliding mode control law is proposed for the class of fractional order chaotic systems without uncertainties, and then a fuzzy logic system is used to estimate the control compensation effort to be added in the case of uncertainties on the system's model. Based on the Lyapunov theory, the stability analysis of both control laws is provided with elimination of the chattering action in the control signal. The developed control scheme is simple to implement and the overall control scheme guarantees the global asymptotic stability in the Lyapunov sense if all the involved signals are uniformly bounded. In the present work, simulation studies on fractional-order Chen chaotic systems are carried out to show the efficiency of the proposed fractional adaptive controllers.
Control of Fractional Stochastic chaos with Fractional Sliding Mode
In this paper, a Fractional Sliding Mode Control strategy is introduced to realize the control of fractional stochastic chaotic system. Besides modeling stochastic chaos by an Ito stochastic differential form, fractional-order of this chaos system is studied. Based on the Lyapunov stability theory, the stability of the closed-loop system is guaranteed. The Chaos control is also obtained between two fractional-order stochastic and deterministic systems in two cases: 1) A control task; to behave period doubling form, 2) Synchronization job; both in different order. This is the first research to apply the proposed sliding mode controller to control of this type system. The simulation results and the lyapunov exponents demonstrate the feasibility of the proposed control method.
International Journal of Computers Communications & Control, 2011
In this paper, in order to achieve tracking performance of uncertain fractional order chaotic systems an adaptive hybrid fuzzy controller is proposed. During the design procedure, a hybrid learning algorithm combining sliding mode control and Lyapunov stability criterion is adopted to tune the free parameters on line by output feedback control law and adaptive law. A weighting factor, which can be adjusted by the trade-off between plant knowledge and control knowledge, is adopted to sum together the control efforts from indirect adaptive fuzzy controller and direct adaptive fuzzy controller. To confirm effectiveness of the proposed control scheme, the fractional order chaotic response system is fully illustrated to track the trajectory generated from the fractional order chaotic drive system. The numerical results show that tracking error and control effort can be made smaller and the proposed hybrid intelligent control structure is more flexible during the design process.