Class Of Fractional-order Nonlinear Systems Research Articles

LMI-based criterion for global Mittag-Leffler lag quasi-synchronization of fractional-order memristor-based neural networks via linear feedback pinning control

This paper addresses global Mittag-Leffler lag quasi-synchronization (GMLQS) of fractional-order memristor-based neural networks (FMNNs). Firstly, for a general class of fractional-order nonlinear systems with interval uncertainties, a kind of linear feedback pinning controller, which works by feeding back partial state errors to the partial controlled variables, is designed to achieve global Mittag-Leffler ultimate boundedness (GMUB). Correspondingly, a GMUB criterion in the form of linear matrix inequality (LMI) is derived with the aid of a Lyapunov function and a newly established fractional-order differential inequality. Then, the linear feedback pinning controller is employed for GMUB of the synchronization error system, which is equivalent to GMLQS of FMNNs. Since there exists the propagation delay between drive-response FMNNs and the synchronization error system can not be obtained straightforwardly, an identical equation about Caputo’s fractional derivatives is established to overcome this difficulty. Moreover, the detailed pinning control scheme design procedures for a prescribed GMLQS performance are presented, where the performance involves both ultimate error bound and transient behavior. In the control scheme, an auxiliary LMI and an optimization objective function are introduced, which can significantly reduce control cost. Finally, a numerical example is presented to show the feasibility of the control scheme. The results obtained in this paper have improved the existing criterion on quasi-synchronization of FMNNs, and will also provide a novel insight into pinning control of fractional-order nonlinear systems.

Non-fragile exponential polynomial observer design for a class of nonlinear fractional-order systems with application in chaotic communication and synchronisation

ABSTRACTThis paper addresses the design of a non-fragile exponential polynomial observer for a class of fractional-order nonlinear systems. Existence of the observer is proven and a sufficient condition for the stability of the state estimation error dynamics with a predetermined exponential convergence rate is derived employing the Lyapunov stability theorem. The exponential stability criterion is proposed in the form of linear matrix inequalities (LMIs). Moreover, some numerical examples have been provided to illustrate the effectiveness of the proposed approach. The synchronisation of fractional-order Lorenz systems has been investigated using the proposed method. Then, the proposed method has been applied to the chaotic communication problem of fractional-order chaotic systems with four scroll attractors.

Almost Disturbance Decoupling for a Class of Fractional-Order Nonlinear Systems with Zero Dynamics

The problem of almost disturbance decoupling is addressed for fractional-order nonlinear systems. A new definition for the norm is proposed to describe the effect of disturbances on the output tracking error for fractional-order systems. Based on the Lyapunov stability theory and the backstepping design method, a tracking controller is constructed to make the output tracking error converge to zero without external disturbances and to attenuate the effect of disturbances on the tracking error at zero initial conditions. In order to validate these theoretical results, a numerical example and two practical examples are given.

Fractional-Order Super-Twisting Sliding-Mode Procedure Design for a Class of Fractional-Order Nonlinear Dynamic Underwater Robots

The purpose of this study is to design a fractional-order super-twisting sliding-mode controller for a class of nonlinear fractional-order systems. The proposed method has the following advantages: (1) Lyapunov stability of the overall closed-loop system, (2) output tracking error’s convergence to zero, (3) robustness against external uncertainties and disturbances, and (4) reduction of the chattering phenomenon. To investigate the performance of the method, the proposed controller is applied to an autonomous underwater robot and Lorenz chaotic system. Finally, a simulation is performed to verify the potential of the proposed method.

Observer‐based modified repetitive control for fractional‐order non‐linear systems with unknown disturbances

This study investigates the problem of observer-based modified repetitive control for a class of fractional-order non-linear systems in the presence of unknown external disturbances. An equivalent-input-disturbance estimator is incorporated into the controller block to compensate the unknown external disturbances. The main objective of this study is to design a modified repetitive controller for the addressed system such that the system output can exactly trace the given periodic reference signal by attenuating both periodic and aperiodic disturbances. To be precise, by employing the Lyapunov stability theory together with the linear matrix inequality approach, sufficient conditions that ensure the existence of such a controller are obtained. It is to be noted that the advantage of using equivalent-input-disturbance estimator in the controller design is that it can reject both periodic and aperiodic disturbance signals effectively without requiring any prior knowledge of them. Finally, two numerical simulation examples are provided to illustrate the effectiveness and applicability of the proposed control technique.

Adaptive iterative learning control for a class of fractional-order nonlinear systems

Based on the discussions on the properties of fractional integral and Caputo fractional derivative, an adaptive iterative learning control approach is proposed for a class of fractional-order nonlinear system (FONS) with unknown time-varying parameter. With the design of learning controller and adaptive learning law for unknown parameter, the tracking error sequence converges to zero in the iteration domain while all the closed-loop signals remain bounded. Finally, a numerical example is given to verify the validity of the designed method.

A Fractional Adaptive Sliding Mode Control Configuration for Synchronizing Disturbed Fractional-Order Chaotic Systems

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.

A high-gain observer with Mittag–Leffler rate of convergence for a class of nonlinear fractional-order systems

In this paper, a class of nonlinear fractional systems of commensurate order is analyzed in order to solve the observation problem through a high-gain Mittag-Leffler observer. Some stability conditions are established to design Mittag-Leffler observers, and a high-gain observer is proposed to estimate the unknown states of a fractional-order nonlinear system expressed in the observer canonical form. It is shown that the designed algorithm is a Mittag-Leffler observer. Finally, some numerical simulations validate the proposed methodology.

Stabilization Conditions for a Class of Fractional-Order Nonlinear Systems

The stabilization problem of fractional-order nonlinear systems for 0<α<1 is studied in this paper. Based on Mittag-Leffler function and the Lyapunov stability theorem, two practical stability conditions that ensure the stabilization of a class of fractional-order nonlinear systems are proposed. These stability conditions are given in terms of linear matrix inequalities and are easy to implement. Moreover, based on these conditions, the method for the design of state feedback controllers is given, and the conditions that enable the fractional-order nonlinear closed-loop systems to assure stability are provided. Finally, a representative case is employed to confirm the validity of the designed scheme.

Global Stabilization for a class of nonlinear fractional-order systems

The problem of Mittag–Leffler stabilization (MLS) is studied for a class of nonlinear non-integer order systems. The stabilizer is constructed by using the Lyapunov function and backstepping algorithm. The continuous controller is designed to ensure that the state of the nonlinear fractional-order closed-loop system converges to the equilibrium. Two simulation examples are given to illustrate the effectiveness of the method.

Adaptive Sliding Mode Control of a Class of Fractional-Order Nonlinear Systems With Input Uncertainties

This paper presents a sliding mode control (SMC) strategy for a class of fractional-order nonlinear systems (FONSs) with unknown external disturbances and input uncertainties. A fractional-order integral sliding surface is introduced, and then, an adaptive sliding mode controller that can compensate the input uncertainties without any explicit fault detection is designed. The proposed controller together with two integer-order adaptation laws can guarantee that all signals in the closed-loop keep bounded, and the state variables converge to an arbitrarily small region eventually. Finally, a constructive example is given to verify the feasibility and superiority of the proposed method.

Fractional-Order Backstepping Sliding-Mode Control Based on Fractional-Order Nonlinear Disturbance Observer

In this paper, a novel fractional-order (FO) backstepping sliding-mode control is proposed for a class of FO nonlinear systems with mismatched disturbances. Here the matched/mismatched disturbances are estimated by an FO nonlinear disturbance observer (NDO). This FO NDO is proposed based on FO backstepping algorithm to estimate the mismatched disturbances. The stability of the closed-loop system is proved by the new extension of Lyapunov direct method for FO systems. Exponential reaching law considerably decreases the chattering and provides a high dynamic tracking performance. Finally, three simulation examples are presented to show the features and the effectiveness of the proposed method. Results show that this observer approximates the unknown mismatched disturbances successfully.

Non-smooth convex Lyapunov functions for stability analysis of fractional-order systems

Based on proximal subdifferentials and subgradients, and instrumented with an extended Caputo differintegral operator, the stability analysis of a general class of fractional-order nonlinear systems is considered by means of non-smooth but convex Lyapunov functions. This facilitates concluding the Mittag–Leffler stability for fractional-order systems whose solutions are not necessarily differentiable in any integer-order sense. As a solution to the problem of robust command of fractional-order systems subject to unknown but Lebesgue-measurable and bounded disturbances, a unit-vector-like integral sliding mode controller is proposed. Numerical simulations are conducted to highlight the reliability of the proposed method in the analysis and design of fractional-order systems closed by non-smooth robust controllers.

A novel Mittag–Leffler stable estimator for nonlinear fractional-order systems: a linear quadratic regulator approach

In this article, the estimation problem in a class of nonlinear fractional-order systems is solved by a stable estimator that generalizes the classical exponential observers, establishing a new class called Mittag–Leffler observers. The solution to the linear quadratic regulator problem is proposed to design optimal control laws for fractional-order linear systems and its applications. All the results are based on the Caputo derivative for commensurate fractional-order systems. Numerical simulations validate the proposed theory.

Quadratic Lyapunov functions for stability analysis in fractional-order systems with not necessarily differentiable solutions

Solutions of fractional-order differintegral equations are generally not necessarily integer-order differentiable, neither in the strong nor in the weak sense, thus limiting the stability analysis in systems based on the most conventional fractional-order operators. In this paper, a consistent and well-posed definition for fractional-order systems is performed based on the study of alternative fractional-order operators that preserve the most interesting and useful properties of differintegrals, even in the case of not necessarily integer-order (weakly) differentiable functions. In addition, it is shown that these operators comply to a recently verified well-known inequality, which allows us to demonstrate Mittag-Leffler stability in a more general class of fractional-order systems, considering quadratic Lyapunov functions, by demonstrating a generalization of the Lyapunov direct method for a class of fractional-order nonlinear systems. Illustrative examples are given to highlight the feasibility of the proposed method, and a multivariable fractional integral sliding mode control application is presented.

Adaptive sliding mode synchronisation for fractional‐order non‐linear systems in the presence of time‐varying actuator faults

This study presents an adaptive sliding mode synchronisation scheme for a class of fractional-order non-linear systems in the presence of unknown parameters, actuator faults, and disturbances. The designed adaptive controller compensates a general class of actuator faults without the need for any explicit fault detection. The parameters, times, and patterns of the considered faults are completely unknown. The proposed adaptive sliding mode synchronisation scheme can guarantee the asymptotic convergence of the synchronisation error to the origin despite the presence of actuator faults and uncertainties. The simulation results show the correctness and effectiveness of the proposed adaptive sliding mode fault compensation approach.

Stability and Stabilization of a Class of Fractional-Order Nonlinear Systems for 1 < α < 2

This study is interested in the stability and stabilization of a class of fractional-order nonlinear systems with Caputo derivatives. Based on the properties of the Laplace transform, Mittag-Leffler function, Jordan decomposition, and Grönwall's inequality, some sufficient conditions that ensure local stability and stabilization of a class of fractional-order nonlinear systems under the Caputo derivative with 1<α<2 are presented. Finally, typical instances, including the fractional-order three-dimensional (3D) nonlinear system and the fractional-order four-dimensional (4D) nonlinear hyperchaos, are implemented to demonstrate the feasibility and validity of the proposed method.

Fuzzy generalised predictive control for a class of fractional‐order non‐linear systems

Fractional-order fuzzy generalised predictive control (FFGPC) for a class of non-linear systems is studied. Based on the Grunwald-Letnikov fractional calculus, Laplace transform, and discretisation, a class of fractional-order non-linear systems are transformed to fit the standard controlled autoregressive integrating moving average (CARMA) model. With the help of the Takagi-Sugeno fuzzy linearisation theory, a linear CARMA model for the non-linear systems is represented. Then, a new FFGPC method is proposed for the fractional-order non-linear systems based on the obtained CARMA predictive model and generalised predictive control theory. Numerical simulations are implemented to verify the effectiveness and superiority of the new method. This work also provides a reference for the control of relevant fractional-order systems.

An Improved Numerical Algorithm for the Fractional Differential Equations and Its Application in the Fractional-Order Nonlinear Systems

In this paper, an improved numerical algorithm for the fractional differential equations is proposed based on the variational iteration method. Using the improved numerical scheme, we investigate a new fractional-order hyperchaotic system, and find that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4. The lowest order we find to yield hyperchaos is 3.46 in this new fractional-order system. The existence of two positive Lyapunov exponents further verifies our results. Numerical results show that the proposed method has a faster speed and more accurate comparing with the traditional predictor-corrector algorithm. Based on the stability theory of the fractional-order system, a nonlinear controller is constructed to achieve synchronization fora class of nonlinear fractional-order systems using the pole placement technique. The nonlinear control method can synchronize two different fractional-order hyperchaotic systems. This synchronization scheme which is simple and theoretically rigorous enables synchronization of fractional-order hyperchaotic systems to be achieved in a systematic way and does not need to compute the conditional Lyapunov exponents. Simulation results are given to validate the effectiveness of the proposed synchronization method.

A novel fractional sliding mode control configuration for synchronizing disturbed fractional-order chaotic systems

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.