Fractional-order Switching Research Articles

Multi-scroll Attractors and Multi-stability Generated Via Fractional-order Switching Linear Systems

This work presents a method for fractional order dynamical systems design that exhibits multi-scrolls by means of a fractional-order switching piecewise linear system. This method is based on the equilibrium points positioning and the region sizes in which the system switches. Numerical results of the proposed method are also presented and the multi-stability phenomenon is exhibited.

Enhanced fractional order sliding mode control for a class of fractional order uncertain systems with multiple mismatched disturbances

To improve the control performance of the fractional order uncertain systems with multiple mismatched disturbances, an enhanced fractional order sliding mode control (FOSMC) method is developed in this paper. The multiple disturbances and uncertainties are estimated by the finite-time disturbance observers (FTDO) and a fractional order extended state observer (FOESO), respectively. A fractional order switching law is designed to provide a fast convergence mode for the system states. Then a novel FOSMC law is developed by incorporating the feedforward compensation, the fractional order switching law, and the auxiliary state for input saturation. The proposed method is applied to numerical examples and to a motor speed control problem. The effectiveness of the proposed method is demonstrated by the performance comparisons with some existing control methods.

Robust H∞ Control of Fractional-Order Switched Systems with Order 0 < α < 1 and Uncertainty

In this paper, robust H∞ control for fractional-order switched systems (FOSSs) with uncertainty is studied. Firstly, the fractional-order switching law for FOSSs is proposed. Then, H∞ control for FOSSs is proven based on the switching law and linear matrix inequalities (LMIs). Moreover, H∞ control for FOSSs with a state feedback controller is extended. Furthermore, the LMI-based condition of robust H∞ control for FOSSs with uncertainty is proven. Furthermore, the condition of robust H∞ control is proposed to design the state feedback controller. Finally, four simulation examples verified the effectiveness of the proposed methods.

Updating [formula omitted] is significant to Caputo fractional order switching systems: A reply to Hu’s comments

This is to reply the concerns about our NAHS paper “Lyapunov and external stability of Caputo fractional order switching systems” pointed out by Hu in “Comments on ‘Lyapunov and external stability of Caputo fractional order switching systems”’ recently.

Comments on “Lyapunov and external stability of Caputo fractional order switching systems”

The stability of Caputo fractional order switching systems is studied in the article by Wu C. etc (Wu and Liu (2019)). The authors claim that the lower bound of the Caputo fractional order derivative needs to be updated at each switching instant. However, the lower bound is relevant to the initial condition and reflects the historical information of a fractional system. No historical information can be changed by subsequent control input as all physical systems are causal systems. The model in Wu and Liu (2019) is physically unattainable and the theoretical achievements cannot be applied in engineering.

The Asymptotic Stability of Caputo Fractional Order Switching Systems With Only Continuous Vector Field Functions

In this paper, a theorem with very weak conditions on vector field functions for asymptotic stability of Caputo fractional order switching systems is proposed. Based on Vainikko’s lemmas, the Caputo fractional derivative of a continuously differentiable and convex Lyapunov function, along trajectories of any Caputo fractional order switching systems with vector field function being only continuous, proves to be continuous and have a very useful estimation. This weakens the smoothness requirement on the vector field functions for fractional stability analysis from differentiability as in existing results to only continuity. Finally, the numerical implementation of a Caputo fractional order switching system with only continuous vector field function illustrates the effectiveness of the proposed theorem.

Lyapunov and external stability of Caputo fractional order switching systems

This paper investigates Lyapunov and external stability of Caputo fractional order switching systems. It proves an estimation for the Caputo fractional order derivative of an arbitrary quadratic function whose variable is a function of time with the same differential properties as those given by the most recent smoothness result, and then establishes criteria on Lyapunov stability that bridge over existing difficulties. On the other hand, the equivalence between Caputo fractional order switching control systems and their diffusive realizations is shown. This together with a Lyapunov-like function proven to be well defined shows their advantages in solving the problem of external stability.

The Stability of a Class of Fractional Order Switching System with Time-Delay Actuator

Since switching systems are important in research and industry, the article is concerned about the stabilization of fractional order switching systems with the order of 1 < q < 2 and a time delay actuator. To this end, the so-called system was initially converted to a system with no delay using a trick, such that the impact of delay was considered in the state matrix of the system in form of a coefficient. In the following, the switching rule was obtained based on the variable structure control with the sliding section. The necessary stability condition for the fractional order switching system with the order of 1 < q < 2 and time delay actuator is presented and approved based on the convex analysis and linear matrix inequalities. Then, a Lyapunov function was introduced with its negative derivative. By defining the Lyapunov function, the system that can be chosen at any time by the switching rule would be stable. Finally, the simulation results were expressed to show the impact of the proposed method.

Strange attractors generated by a fractional order switching system and its topological horseshoe

Chaos generation in a new fractional order unstable dissipative system with only two equilibrium points is reported. Based on the integer version of an unstable dissipative system (UDS) and using the same system’s parameters, chaos behavior is observed with an order less than three, i.e., 2.85. The fractional order can be decreased as low as 2.4 varying the eigenvalues of the fractional UDS in accordance with a switching law that fulfills the asymptotic stability theorem for fractional systems. The largest Lyapunov exponent is computed from the numerical time series in order to prove the chaotic regime. Besides, the presence of chaos is also verified obtaining the topological horseshoe. That topological proof guarantees the chaos generation in the proposed fractional order switching system avoiding the possible numerical bias of Lyapunov exponents. Finally, an electronic circuit is designed to synthesize this fractional order chaotic system.

Circuit simulation on control and synchronization of fractional order switching chaotic system

In view of obtaining a larger key space and thus improve the level of security attainable in chaotic communication, this paper examines the dynamics of a switched, fractional-order chaotic system. Taking the fractional-order Chen system as a starting point, the dynamics of a modified version of that system is examined, and the composed system is hereafter realized by allowing switching between the two different subsystems. Our work first demonstrates how fractional-order chaotic systems can be realized physically. For the fractional-order switching chaotic system, a linear feedback controller and a number of synchronization schemes are designed. Our simulations show that control and synchronization of the individual subsystems can be achieved with the same feedback and synchronization controller.

Output Feedback Fractional-Order Nonsingular Terminal Sliding Mode Control of Underwater Remotely Operated Vehicles

For the 4-DOF (degrees of freedom) trajectory tracking control problem of underwater remotely operated vehicles (ROVs) in the presence of model uncertainties and external disturbances, a novel output feedback fractional-order nonsingular terminal sliding mode control (FO-NTSMC) technique is introduced in light of the equivalent output injection sliding mode observer (SMO) and TSMC principle and fractional calculus technology. The equivalent output injection SMO is applied to reconstruct the full states in finite time. Meanwhile, the FO-NTSMC algorithm, based on a new proposed fractional-order switching manifold, is designed to stabilize the tracking error to equilibrium points in finite time. The corresponding stability analysis of the closed-loop system is presented using the fractional-order version of the Lyapunov stability theory. Comparative numerical simulation results are presented and analyzed to demonstrate the effectiveness of the proposed method. Finally, it is noteworthy that the proposed output feedback FO-NTSMC technique can be used to control a broad range of nonlinear second-order dynamical systems in finite time.

Stability of fractional order switching systems

This paper studies the stability problem for fractional order switching systems. The conditions for the stability of such systems are presented in terms of common Lyapunov theory, generalized to fractional order systems, and in frequency domain, an approach equivalent to the previous one. The effectiveness of the developed theory is shown through some illustrative examples.

Synchronization of uncertain fractional-order chaotic systems with disturbance based on a fractional terminal sliding mode controller

This paper provides a novel method to synchronize uncertain fractional-order chaotic systems with external disturbance via fractional terminal sliding mode control. Based on Lyapunov stability theory, a new fractional-order switching manifold is proposed, and in order to ensure the occurrence of sliding motion in finite time, a corresponding sliding mode control law is designed. The proposed control scheme is applied to synchronize the fractional-order Lorenz chaotic system and fractional-order Chen chaotic system with uncertainty and external disturbance parameters. The simulation results show the applicability and efficiency of the proposed scheme.

Study on the Nonsingular Problem of Fractional-Order Terminal Sliding Mode Control

An improved type of control strategy combining the fractional calculus with nonsingular terminal sliding mode control named non-singular fractional terminal sliding mode control (NFOTSM) is proposed for the nonlinear tire-road friction control system of vehicle in this paper. A fractional-order switching manifold is proposed, and the corresponding control law is formulated based on the Lyapunov stability theory to guarantee the sliding condition. The proposed controller ensures the finite time stability of the closed-loop system. Then, a terminal attractor is introduced to solve the singularity problem of fractional terminal sliding mode control (FOTSM). Finally, the performance of the NFOTSM is fully investigated compared with other related algorithms to find the effectiveness for the tire-road friction system. The results show that the NFOTSM has better performance than other related algorithms.

Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty

A novel type of control strategy combining the fractional calculus with terminal sliding mode control called fractional terminal sliding mode control is introduced for a class of dynamical systems subject to uncertainties. A fractional-order switching manifold is proposed and the corresponding control law is formulated based on the Lyapunov stability theory to guarantee the sliding condition. The proposed fractional-order terminal sliding mode controller ensures the finite time stability of the closed-loop system. Finally, numerical simulation results are presented and compared to illustrate the effectiveness of the proposed method.