Class Of Fractional-order Chaotic Systems Research Articles

Correction to: Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology

This note provides a corrigendum to the paper “Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology” [Nonlinear Dynamics, 89 (2017) 1357–1370]. It is pointed out that we have relied on a wrong formula in [2] to compute the upper bounds of the finite settling times of the proposed control methods in [1]. Fortunately, the minor errors appeared in [1] are readily corrected via the Mean Value Theorem for definite integrals. It is proved that the mentioned minor errors do not affect the main results and designs of the control methods in [1].

Multi-switching adaptive synchronization of two fractional-order chaotic systems with different structure and different order

In this work, we combine the active and adaptive control theories, and propose a novel synchronization scheme for a class of fractional-order chaotic systems with different structure and different order. Based on the new version of fractional-order Lyapunov stability theory, we design the adaptive controllers and updating laws of different switching. We use the fractional-order Lorenz chaotic system and the fractional-order Chen chaotic system as examples to analyze the multi-switching synchronization process for fractional-order chaotic systems with different structures and different orders. Finally, numerical simulations are also given to illustrate the effectiveness and validation of the proposed method, and the model uncertainties and external disturbances are added to the considered systems to verify the robustness of the proposed controllers.

Elaboration of a generalized approach to control and to synchronize the fractional-order chaotic systems

In this paper, a new general optimal structure and parameters of the fractional-order feedback control law (OSP-FoF) is developed for controlling and synchronizing a large class of fractional-order chaotic systems (FoCS). The design of the OSP-FoF model is based on bounded-input bounded-output stabilization arguments, small gain theorem (SGT), and the matrix norms. The proposed model is theoretically rigorous and represents a powerful and simple approach which provides a reasonable trade-off between computational overhead, storage space, numerical accuracy and stability analysis for control and synchronization purposes of FoCS. Simulation results for the stabilization and synchronization of fractional-order hyperchaotic systems demonstrate the effectiveness of this newly proposed approach.

A practical synchronization approach for fractional-order chaotic systems

A practical synchronization approach is proposed for a class of fractional-order chaotic systems to realize perfect $$\delta $$ -synchronization, and the nonlinear functions in the fractional-order chaotic systems are all polynomials. The $$\delta $$ -synchronization scheme in this paper means that the origin in synchronization error system is stable. The reliability of $$\delta $$ -synchronization has been confirmed on a class of fractional-order chaotic systems with detailed theoretical proof and discussion. Furthermore, the $$\delta $$ -synchronization scheme for the fractional-order Lorenz chaotic system and the fractional-order Chua circuit is presented to demonstrate the effectiveness of the proposed method.

Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology

This paper is devoted to demonstrate how a class of fractional-order chaotic systems can be controlled in a given finite time using just a single control input. First a novel fractional switching sliding surface is proposed with desired properties such as fast convergence to zero equilibrium and no steady state errors. At the second phase, a smooth reaching control law is derived to guarantee the occurrence of the sliding motion with a finite settling time. Owing to the integration of the control signal discontinuity, chattering oscillations are hindered from the controller. Rigorous stability analysis is performed to validate the design claims. The effects of high frequency external noises as well as modeling errors and dynamic variations are also taken into account, and the robustness of the closed-loop system is ensured. The proposed robust controller is realized for a class of chaotic fractional-order systems with one control input. In accordance, some remarks regarding the inclusion of mismatched uncertainties in the system dynamics are given. The robust functionality and quick convergence property as well as chatter-free attribute of the introduced non-smooth sliding mode technology are demonstrated using oscillation suppression of fractional-order chaotic Lorenz and financial systems.

Optimal synchronization of fractional-order chaotic systems subject to unknown fractional order, input nonlinearities and uncertain dynamic using type-2 fuzzy CMAC

In this paper, a new robust optimal control strategy is presented to synchronize a class of fractional-order chaotic systems with unknown fractional orders, uncertain dynamics and input nonlinearities. The dynamics of the system are estimated by an integer-order model, using proposed non-singleton type-2 fuzzy cerebellar model articulation controller (NST2-CMAC). The free parameters of NST2-CMAC are adjusted based on the adaptation laws which are derived from Lyapunov stability analysis. To show the effectiveness of the proposed method, two simulation examples are provided and the results are compared with other methods.

The exponential synchronization of a class of fractional-order chaotic systems with discontinuous input

This paper investigates the exponential synchronization of a class of fractional-order chaotic systems. The response system is controlled by input which may be either discontinuous or continuous variable. Moreover, the input is assumed to be affected by external disturbance. Based on the Mittag-Leffler function, sufficient conditions for achieving exponential synchronization of fractional-order chaotic systems have been derived. Numerical examples are presented by taking the fractional-order chaotic economical system as an example to verify and demonstrate the effectivenessof the proposed schemes.

A New Scheme on Synchronization of Commensurate Fractional-Order Chaotic Systems Based on Lyapunov Equation

This paper proposes a new fractional-order approach for synchronization of a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. A simple but practical method to synchronize many familiar fractional-order chaotic systems has been put forward. A new theorem is proposed for a class of cascade fractional-order systems and it is applied in chaos synchronization. Combined with the fact that the states of the fractional chaotic systems are bounded, many coupled items can be taken as zero items. Then, the whole system can be simplified greatly and a simpler controller can be derived. Finally, the validity of the presented scheme is illustrated by numerical simulations of the fractional-order unified system.

Adaptive sliding mode synchronization for a class of fractional-order chaotic systems with disturbance

This paper studies the fractional-order disturbance observer (FODO)-based adaptive sliding mode synchronization control for a class of fractional-order chaotic systems with unknown bounded disturbances. To handle unknown disturbances, the nonlinear FODO is explored for the fractional-order chaotic system. By choosing the appropriate control gain parameter, the disturbance observer can approximate the disturbance well. On the basis of the sliding mode control technique, a simple sliding mode surface is defined. A synchronization control scheme incorporating the introduced sliding mode surface and the designed disturbance observer is then developed. Under the control of the synchronization scheme, a good synchronization performance is realized between two identical fractional-order chaotic systems with different initial conditions. Finally, the numerical simulation results illustrate the effectiveness of the developed synchronization control scheme for fractional-order chaotic systems in the presence of external disturbances.

Suppressing chaos for a class of fractional-order chaotic systems by adaptive integer-order and fractional-order feedback control

This paper is devoted to studying how to more effectively suppress chaos for a class of fractional-order nonlinear systems. By means of adaptive control theory for integer-order nonlinear system, we propose two simple and novel adaptive feedback methods to control chaos. Rigorous theoretical proof is provided based on some essential properties of fractional calculus and Barbalat's Lyapunov-like stability theorem. It is discovered that both fractional-order feedback controller and integer-order one can guide chaotic trajectories to the unstable equilibrium point. To display the feasibility and validity of presented methods, some typical fractional-order chaotic systems have been chosen as numerical illustration. Furthermore, by comparing two different control techniques, one can find the fractional-order feedback control algorithm is more stable and more flexible.

Projective Synchronization for a Class of Fractional-Order Chaotic Systems with Fractional-Order in the (1, 2) Interval

In this paper, a projective synchronization approach for a class of fractional-order chaotic systems with fractional-order 1 < q < 2 is demonstrated. The projective synchronization approach is established through precise theorization. To illustrate the effectiveness of the proposed scheme, we discuss two examples: (1) the fractional-order Lorenz chaotic system with fractional-order q = 1.1; (2) the fractional-order modified Chua’s chaotic system with fractional-order q = 1.02. The numerical simulations show the validity and feasibility of the proposed scheme.

Robust Stabilization of a Class of Uncertain Fractional-order Chaotic Systems via a Novel Sliding Mode Control Scheme

This paper proposes a novel sliding mode control (SMC) scheme to stabilize a class of fractional-order chaotic systems. Through constructing two sliding mode variables, the control problem of n-dimensional system can be transformed to the equivalent stabilizing problem of a reduced-order system. Subsequently, on the basis of second-order sliding mode (SOSM) technique, a robust control law is designed, which strongly attenuates the chattering phenomenon inherent in traditional sliding mode controller, and guarantees the existence of sliding motion in a finite time. The stability of two sliding mode variables to the origin is proved by conventional and fractional Lyapunov theories, respectively. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach.

Sliding mode adaptive synchronization for a class of fractional-order chaotic systems with uncertainties

Sliding mode adaptive synchronization for a class of fractional-order chaotic systems with uncertainties

Takagi-Sugeno fuzzy control for a wide class of fractional-order chaotic systems with uncertain parameters via linear matrix inequality

In this paper, the authors focus on the application of linear matrix inequality (LMI) in the stabilization of fractional-order chaotic systems and the strict mathematical description and proof of LMI. Based on the theory of the fractional-order interval system, the generalized Takagi-Sugeno fuzzy model is applied to a wide class of fractional-order chaotic systems with uncertain parameters. The sufficient stability condition for the fractional-order systems is presented as a set of LMI for the first time and the strict mathematical norms of LMI are given. Furthermore, a controller is developed to stabilize fractional-order chaotic systems, and may be applied to fractional-order systems with uncertainty. Finally, two representative examples of the three-dimensional fractional-order permanent magnet synchronous motor system and the four-dimensional fractional-order hyperchaotic system based on Chen’s system are provided to demonstrate the effectiveness of theoretical analysis.

Synchronization of the fractional-order chaotic system via adaptive observer

The means to design the observer for a class of fractional-order chaotic systems is investigated. A novel Lyapunov function is proposed and a robust adaptive observer is designed to synchronize a given fractional-order chaotic system. The constructed observer could guarantee the error of state converges to zero asymptotically. Simulation results demonstrate the effectiveness and robustness of the proposed scheme.

Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

Robust Control of a Class of Uncertain Fractional-Order Chaotic Systems with Input Nonlinearity via an Adaptive Sliding Mode Technique

In this paper, the problem of stabilizing a class of fractional-order chaotic systems with sector and dead-zone nonlinear inputs is investigated. The effects of model uncertainties and external disturbances are fully taken into account. Moreover, the bounds of both model uncertainties and external disturbances are assumed to be unknown in advance. To deal with the system’s nonlinear items and unknown bounded uncertainties, an adaptive fractional-order sliding mode (AFSM) controller is designed. Then, Lyapunov’s stability theory is used to prove the stability of the designed control scheme. Finally, two simulation examples are given to verify the effectiveness and robustness of the proposed control approach.

Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems

In this paper, a novel adaptive fractional-order feedback controller is first developed by extending an adaptive integer-order feedback controller. Then a simple but practical method to synchronize almost all familiar fractional-order chaotic systems has been put forward. Through rigorous theoretical proof by means of the Lyapunov stability theorem and Barbalat lemma, sufficient conditions are derived to guarantee chaos synchronization. A wide range of fractional-order chaotic systems, including the commensurate system and incommensurate case, autonomous system, and nonautonomous case, is just the novelty of this technique. The feasibility and validity of presented scheme have been illustrated by numerical simulations of the fractional-order Chen system, fractional-order hyperchaotic Lu system, and fractional-order Duffing system.

Robust stabilization of fractional-order chaotic systems with linear controllers: LMI-based sufficient conditions

This paper is concerned with the problem of robust state feedback controller design to suppress fractional-order chaotic systems. A general class of fractional-order chaotic systems is considered and it is assumed that the systems’ equations depend on bounded uncertain parameters. We transform the chaotic system equations into linear interval systems, and a sufficient stabilizability condition is derived in terms of linear matrix inequality (LMI). Then, an appropriate feedback gain is introduced to bring the chaotic states to the origin. Such design will result in a simple but effective controller. Several numerical simulations have been carried out to verify the effectiveness of the theoretic results.

LMI-based stabilization of a class of fractional-order chaotic systems

Based on the theory of stabilization of fractional-order LTI interval systems, a simple controller for stabilization of a class of fractional-order chaotic systems is proposed in this paper. We consider the structure of the chaotic systems as fractional-order LTI interval systems due to the limited amplitude of chaotic trajectories. We introduce a simple feedback controller for the interval system and then, based on a recently established theorem for stabilization of interval systems, we reach to a linear matrix inequality (LMI) problem. Solving the LMI yields an appropriate decoupling feedback control law which suffices to bring the chaotic trajectories to the origin. Several illustrative examples are given which show the effectiveness of the method.