Stability Of Fractional-order Nonlinear Systems Research Articles

Finite-time stability of fractional-order nonlinear systems.

The paper studies the finite-time stability (FTS) of equilibrium points (EPs) in fractional-order nonlinear systems (FONSs). Classifying the EPs into initial EP and finite-time EP innovatively, equations of the EPs in FONSs are investigated comprehensively, and a unified definition depicting the EPs in the FONSs is proposed. Furthermore, sufficient conditions of the FTS of EPs in FONSs are given. The proposed results are verified with an illustrative example. Thus, different from existing works that declare non-existence of finite-time stable equilibria in FONSs, the existence of finite-time stable EPs in FONSs is confirmed in this paper.

Robust Control of Nonlinear Fractional-Order Systems with Unknown Upper Bound of Uncertainties and External Disturbance

This paper investigates the asymptotic stabilization of nonlinear fractional-order systems under unknown upper bounds on external disturbances and uncertainties. The upper bound of uncertainties is a nonlinear function of the pseudostates norm with unknown coefficients. Based on the robust fractional-order sliding mode control, the states of the nonlinear fractional-order system under unknown upper bounds on external disturbances and uncertainties have been stabled. Adaptive control laws estimate the upper bound of external disturbances and uncertainties. In all cases, the stability proof is obtained using the Lyapunov theorem to show the convergence of the sliding surface to zero. Also, by introducing a suitable sliding surface, the chattering phenomenon is eliminated. Finally, the effectiveness of the proposed fractional-order controller is demonstrated using practical examples. The simulation results show the proposed controller has a proper performance in the presence of external disturbance and uncertainties.

On finite-time stability of nonlinear fractional-order systems with impulses and multi-state time delays

This work concentrates on the finite-time stability (FTS) analysis of fractional-order systems (FOS) with impulsive effects and multi-state time delays. The condition which give assurance for FTS of nonlinear FOS having impulsive behavior and multi-state time delay are derived by utilizing the generalization of Gronwall’s inequality (GI). At last, two numerical examples are given which provide the accuracy of the given result.

Stability analysis for a class of fractional-order nonlinear systems with time-varying delays

This paper presents the stability analysis problem of fractional-order nonlinear systems with time-varying delay. After formulating the problem and selecting the nonlinear model as the system under study, stability analysis and expression of the sufficient conditions for fractional-order nonlinear systems with time-varying delay are obtained using two different methods. In these methods, sufficient conditions for stability of fractional-order nonlinear systems are found in the form of satisfying some inequalities based on norms of nonlinear functions in the system and in terms of linear matrix inequality according fractional-order and nonlinear functions. In each case, despite the presence of time-varying delay, the system stability is ensured by meeting the stability sufficient conditions in terms of an inequality of functions and system parameters. Finally, numerical examples are given to determine the effectiveness of the proposed theorem.

Asymptotic Stability and Synchronization of Fractional-Order Neural Networks With Unbounded Time-Varying Delays

This article deals with the asymptotic stability and synchronization of fractional-order neural networks (FONNs) with unbounded time-varying delays, in which FONNs can be the general nonlinear system. First, the existence of the equilibrium point of the nonlinear system is obtained, and the μ-type stability of fractional-order nonlinear systems is proved by the analytical method. Second, several sufficient conditions are derived to guarantee the μ-type stability of FONNs by taking easily implemented parameters. Moreover, the μ-type synchronization of drive-response systems is established by designing the linear controller. Especially, μ-type stability is the general stability, including Mittag-Leffler stability and power-stability, which is the extension of the Mittag-Leffler stability of FONNs. Finally, two numerical examples are presented to show the effectiveness of the results.

Improved Approach for Finite-Time Stability of Nonlinear Fractional-Order Systems With Interval Time-Varying Delay

In this brief, we propose an approach based on the Laplace transform and “inf-sup” method for studying finite-time stability of fractional-order systems (FOS) with time-varying delay and nonlinear perturbation. Based on the proposed approach, we establish new delay-dependent conditions for finite-time stability of FOS with interval time-varying delay. The conditions are presented in terms of the Mittag-Leffler function and linear matrix inequalities, which are less conservative and easier to verify than the existing ones. The proposed method is also applicable for finite-time stability of linear uncertain time-delay FOS. Numerical example are given to show the validity and effectiveness of the proposed results.

Dynamical analysis of a new three-dimensional fractional chaotic system

In the present paper, a new fractional chaotic system proposed by the authors is discussed. Moreover, based on the stability theory of fractional-order systems, conditions for the stability of nonlinear fractional-order systems are presented, and the existence and uniqueness of the solutions of the resulting new fractional chaotic attractor are also studied. Next, the necessary conditions for the existence of chaotic attractors in new fractional chaotic system are reported, and at the end the stability analysis of the corresponding equilibria is given. Last but not the least, the presented numerical simulations confirm the validity of our analysis.

Stability of fractional‐order nonlinear systems by Lyapunov direct method

In this study, by using a characterisation of functions having a fractional derivative, the authors propose a rigorous fractional Lyapunov function candidate method to analyse the stability of fractional-order nonlinear systems. First, they prove an inequality concerning the fractional derivatives of convex Lyapunov functions without the assumption of the existence of the derivative of pseudo-states. Second, they establish fractional Lyapunov functions to fractional-order systems without the assumption of the global existence of solutions. Their theorems fill the gaps and strengthen results in some existing papers.

Synchronization of fractional-order chaotic systems with non-identical orders, unknown parameters and disturbances via sliding mode control

The scheme of synchronization between fractional-order chaotic systems with non-identical orders, unknown parameters and disturbances was investigated. A sliding surface was defined based on the theory of sliding mode control and a controller with adaptive laws was designed based on the stability of fractional-order nonlinear systems. The synchronization of two fractional-order hyperchaotic systems was simulated by using the fractional differential transform method to validate the effectiveness and the feasibility of the proposed scheme. All the theoretical analysis and simulation results showed the effectiveness of the proposed scheme.

Mittag-Leffler stabilization of fractional-order nonlinear systems with unknown control coefficients

In this paper, we consider the problem of Mittag-Leffler stabilization of fractional-order nonlinear systems with unknown control coefficients. With the help of backstepping design method, the stabilizing functions and tuning functions are constructed. The controller is designed to ensure that the pseudo-state of the fractional-order nonlinear system converges to the equilibrium. The effectiveness of the proposed method has been verified by some simulation examples.

On the robustness of linear and non-linear fractional-order systems with non-linear uncertain parameters

This paper presents the robust stabilization problem of linear and non-linear fractional-order systems with non-linear uncertain parameters. The uncertainty in the model appears in the form of the combination of ‘additive perturbation’ and ‘multiplicative perturbation’. Sufficient conditions for the robust asymptotical stabilization of linear fractional-order systems are presented in terms of linear matrix inequalities (LMIs) with the fractional-order |$0<\alpha <1$|⁠. Sufficient conditions for the robust asymptotical stabilization of non-linear fractional-order systems are then derived using a generalization of the Gronwall–Bellman approach. Finally, a numerical example is given to illustrate the effectiveness of the proposed results.