Class Of Fractional-order Chaotic Systems Research Articles

Fixed-time control of a class of fractional-order chaotic systems via backstepping method

This paper investigates the fixed-time control of a class of fractional-order systems via the backstepping method. A new fractional-order fixed-time stability theorem, which is a generalization of the integer order stability theorem, is presented. By using the proposed stability theorem, the fixed-time control problem of a class of fractional-order chaotic systems is investigated. Some fixed-time convergence criteria which have some pretty properties such as no singularity and no chattering are presented via backstepping method. Simulation results are given to show the effectiveness of the presented results.

On chaos control of nonlinear fractional chaotic systems via a neural collocation optimization scheme and some applications

In this paper, we study chaos control of a class of fractional-order chaotic systems where the dynamic control system depends on the Caputo fractional derivatives. We first propose an infinite horizon optimal control problem related to the given fractional chaotic system. With the help of an approximation, we replace the Caputo derivative to integer order derivative. We then convert the obtained infinite horizon optimal control problem into an equivalent finite horizon one. Based on the Pontryagin minimum principle (PMP) for optimal control problems and by constructing an error function, we define an unconstrained minimization problem. In the optimization problem, we use trial solutions for state, costate and control functions where these trial solutions are constructed by using a two-layered perceptron neural network. A learning procedure of the proposed neural network with convergence properties are also given. Some numerical results are introduced to explain our main results. Three applicable examples on chaos control of Malkus waterwheel, finance fractional chaotic models and fractional-order Geomagnetic Field models are finally considered.

Qualitative Analysis of Class of Fractional-Order Chaotic System via Bifurcation and Lyapunov Exponents Notions

This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.

Synchronization in a Class of Fractional-order Chaotic Systems via Feedback Controllers: A Comparative Study

Synchronization in a Class of Fractional-order Chaotic Systems via Feedback Controllers: A Comparative Study

Neural Network Command Filtered Control of Fractional-Order Chaotic Systems.

An adaptive neural network (NN) backstepping control method based on command filtering is proposed for a class of fractional-order chaotic systems (FOCSs) in this paper. In order to solve the problem of the item explosion in the classical backstepping method, a command filter method is adopted and the error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method. Moreover, an adaptive neural network method for unknown FOCSs is proposed. Compared with the existing control methods, the advantage of the proposed control method is that the design of the compensation signals eliminates the filtering errors, which makes the control effect of the actual system improve well. Finally, two examples are given to prove the effectiveness and potential of the proposed method.

Efficient Learning Control of Uncertain Fractional-Order Chaotic Systems With Disturbance.

In this brief, the problem of synchronization control is investigated for a class of fractional-order chaotic systems with unknown dynamics and disturbance. The controller is constructed using neural approximation and disturbance estimation where the system uncertainty is modeled by neural network (NN) and the time-varying disturbance is handled using disturbance observer (DOB). To evaluate the estimation performance quantitatively, the serial-parallel estimation model is constructed based on the compound uncertainty estimation derived from NN and DOB. Then, the prediction error is constructed and employed to design the composite fractional-order updating law. The boundedness of the system signals is analyzed. The simulation results show that the proposed new design scheme can achieve higher synchronization accuracy and better estimation performance.

Novel fractional-order chaotic systems of different order and multiswitching synchronisation

This paper gives multiswitching synchronisation scheme for a class of fractional-order chaotic systems by combining active and adaptive control theories. Adaptive controllers have been designed by using different laws of switching and fractional-order Lyapunov stability theory. We have also constructed a new fractional-order Duffing system. The fractional-order Duffing system and fractional-order Rabinovich–Fabrikant system have been taken as the drive system and the response system respectively. Applications have been demonstrated. Theoretical analysis and numerical simulations are also given to verify the robustness of the proposed controllers.

Fuzzy neural network-based chaos synchronization for a class of fractional-order chaotic systems: an adaptive sliding mode control approach

This paper deals with chaos synchronization problem between two different uncertain fractional-order (FO) chaotic systems with disturbance based on FO Lyapunov stability analysis method. A T–S fuzzy neural network model as a universal approximator is constructed to approximate those uncertain terms and unknown parameters. An adaptive sliding mode control scheme is established, and the adaptive sliding mode control design procedure is proposed, which not only guarantees the stability and robustness of the proposed control method, but also guarantees that the external disturbance on the synchronization error can be attenuated. Finally, simulation results show applicability and feasibility of the proposed control strategy.

Terminal observer and disturbance observer for the class of fractional-order chaotic systems

In this paper, a terminal fractional-order observer and a terminal disturbance observer is proposed to estimate internal states and external disturbances of the class of fractional-order chaotic systems. The estimation of states within fixed time is achieved by employing a nonlinear feedback in terms of the observer error. The fixed convergence time is not relevant to the initial conditions and can be adjusted to any desired values by tuning the designable parameters. Finally, the numerical simulations are performed on fractional-order chaotic Liu, Chen, and Financial systems to validate the theoretical results. Moreover, some numerical simulations are provided to compare the obtained theoretical results with the other methods in the literature.

Robust fuzzy control for fractional-order systems with estimated fraction-order

In this paper, a fuzzy control method is proposed for a class of fractional-order chaotic systems. The dynamics of the system are unknown and are perturbed by the external disturbances. Also, the value of the fractional-order is assumed to be unknown. The type-2 fuzzy systems (T2FSs) are employed to estimate the unknown functions in the dynamics of the system. The parameters of T2FS and the value of fractional-order are estimated by unscented Kalman filter. The upper bound of the approximation error is online estimated, and a new fractional-order compensator is designed to eliminate the effect of the uncertainties and to guarantee the closed-loop stability. The effectiveness of the proposed method is shown by simulations, and the results are compared with some other techniques. It is shown that the proposed method results in better performance in the presence of unknown fractional-order and unknown perturbed dynamics.

Conformable Fractional Order Sliding Mode Control for a Class of Fractional Order Chaotic Systems

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.

Linear feedback synchronization and anti-synchronization of a class of fractional-order chaotic systems based on triangular structure

This paper is devoted to investigating the synchronization and the anti-synchronization of a class of fractional-order chaotic systems (FOCSs). Based on triangular structure, two more cases are considered to improve the applicability: FOCSs can be transformed into the structure by appropriate non-singular coordinate transformations completely or partly. Different from the previous works, the controllers designed by linear feedback technique combining Mittag-Leffler stability with triangular structure in this paper are single and linear ones, which can guarantee the synchronization and the anti-synchronization of complex FOCSs effectively. Several illustrative examples are presented to verify the efficiency of the obtained results.

Backstepping-Based Adaptive Fuzzy Synchronization Control for a Class of Fractional-Order Chaotic Systems with Input Saturation

In existing backstepping control schemes of fractional-order systems, either using the optional control method or ignoring the fractional-order derivatives of command functions as the analytic calculation of virtual controllers makes the operation becoming prohibitive. This paper proposes an adaptive fuzzy backstepping synchronization control method for a class of factional-order chaotic systems with input saturation and unknown external disturbance, where the fractional-order derivative of the virtual control function is treated as a part of an unknown function. During the backstepping design, the unknown function is approximated by a fuzzy logic system in each step. Then, by using the fractional Lyapunov stability criterion, asymptomatic convergence of the synchronization error can be guaranteed by the proposed control method. Three numerical simulation results confirm the effectiveness of the proposed method.

Intelligent fuzzy controller for chaos synchronization of uncertain fractional-order chaotic systems with input nonlinearities

ABSTRACTIn this research work, a novel fuzzy adaptive control is proposed to achieve a projective synchronization for a class of fractional-order chaotic systems with input nonlinearities (dead-zone together with sector nonlinearities). These master-slave systems under consideration are supposed to be with distinct models, different fractional-orders, unknown models, and dynamic external disturbances. The proposed control law consists of two main terms, namely: a fuzzy adaptive control term for appropriately approximating the uncertainties and a fractional-order variable-structure control term for robustly dealing with these inherent input nonlinearities. A Lyapunov approach is used to derive the updated laws and to prove the stability of the closed-loop control system. At last, a set of computer simulation results is carried out to illustrate and further validate the theoretical findings.

The Robust Control and Synchronization of a Class of Fractional-Order Chaotic Systems with External Disturbances via a Single Output

This paper investigates the stabilization and synchronization of a class of fractional-order chaotic systems which are affected by external disturbances. The chaotic systems are assumed that only a single output can be used to design the controller. In order to design the proper controller, some observer systems are proposed. By using the observer systems some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived. Numerical examples are presented by taking the fractional-order generalized Lorenz chaotic system as an example to show the feasibility and validity of the proposed method.

Design of New Fractional Sliding Mode Control Due to Complete Synchronization of Commensurate and Incommensurate Fractional Order Chaotic Systems

Synchronization between two nonlinear chaotic systems is an interesting problem in both theoretical as well as practical point of view. In this manuscript, the goal is introducing a fractional sliding surface and then design of a fractional sliding mode controller so as to synchronize two fractional nonlinear chaotic systems. The proposed method is performed to synchronize a class of fractional-order chaotic systems in the presence of uncertainties and external disturbances. Stability of the controlled system investigated and analysed by Lyapunov stability theory. The method applied on different example and numerical simulations are performed to show the applicability of the proposal. It is worth mentioning that the novel fractional sliding mode controller can be applied in order to control a broad range of fractional order dynamic systems.

Generalized fractional-order time-delayed feedback control and synchronization designs for a class of fractional-order chaotic systems

ABSTRACTIn this paper, we develop a fractional-order time-delayed feedback control (Fo-TDFC) approach combined with an optimal knowledge base to stabilize and synchronize a large class of fractional-order chaotic systems (FoCS). The proposed control laws are formulated based on the Lyapunov stability theory and the multiobjective optimization process of the closed-loop controlled system. The design and optimization of the Fo-TDFC laws are theoretically rigorous and represent a powerful tool to control and synchronize FoCS. Finally, numerical simulation results are provided to illustrate the effectiveness of the proposed method by considering the fractional-order energy resources demand-supply chaotic system as an illustrative example.

Chaos Suppression of a Class of Fractional-Order Chaotic Systems with Order Lying in (1, 2)

It is shown that fractional-order (FO) nonlinear systems can also show higher nonlinearity and complex dynamics. FO chaotic systems have wider applications in secure communication, signal processing, financial field due to FO chaos has larger key space and more complex random sequences than integer-order chaos. Thanks to the lack of the effective analytical methods and controller design methods of integer-order chaotic systems can not be applied directly to FO chaos systems, to control chaos of FO chaotic systems is a very interesting and difficult problem, especially for FO chaotic system with order α:1<α<2. Based on the stability theory of FO systems and the linear state feedback control, an LMI criterion for controlling a class of fractional-order chaotic systems with fractional-order α:1<α<2 is addressed in this paper. The proposed method can be easily verified and resolved by using the Matlab LMI toolbox. Moreover, the proposed controller is linear, easy to implement and overcome some defects in the recent literature, which have improved the existing results. The method employed in this letter can effectively avoid control cost and inaccuracy in the literatures, and can be be applied to FO hyperchaos systems and synchronization controller design of FO chaotic system. Theoretical analysis and numerical simulations are presented to demonstrate the validity and feasibility of the proposed methods.

Parameter identification and optimisation for a class of fractional-order chaotic system with time delay

Parameter identification and optimisation for a class of fractional-order chaotic system with time delay

Parameter identification and optimisation for a class of fractional-order chaotic system with time delay

The fractional-order chaotic systems have more complex dynamic characteristics than the integer order chaotic systems, which can more reflect the physical properties of the actual system and more practical values, whereas it is difficult to control the synchronisation for fractional chaotic systems. Chaotic system identification is the basis of chaos control and performance analysis. In order to identify the parameters of the chaotic systems with time delay, a novel particle swarm optimisation with increasing inertia weight is proposed and then the issue is settled by solving an optimisation problem. The identification of parameters mainly includes the system order, the time delay parameter and the coefficient parameters. An estimation-correction algorithm based on linear interpolation method is used to solve the fractional-order delay differential equation. The Mackey-Glass chaotic system is conducted and comparisons with other two widely used particle swarm optimisations and the differential evolution algorithm indicate the effectiveness of the proposed method and an improvement in identification accuracy as well as convergence speed.