Asymptotic stabilization of general nonlinear fractional-order systems with multiple time delays
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ORIGINAL PAPER
Asymptotic stabilization of general nonlinear fractionalorder systems with multiple time delays Zhang Zhe
. Zhang Jing
Received: 22 April 2020 / Accepted: 30 July 2020 Ó Springer Nature B.V. 2020
Abstract In this paper, two new control methods based on a Lyapunov-like function and a vector Lyapunov function separately were put forward to solve the asymptotic stabilization problem of general fractional-order nonlinear systems with multiple time delays. First, we deduced a new asymptotic stabilization control criterion based on a Lyapunov-like function after discussing the asymptotic stability criterion of general fractional-order nonlinear systems. Moreover, to address the problem of multiple time delays of the nonlinear system, we derived another asymptotic stabilization control criterion based on a vector Lyapunov function and an M-matrix via the new controller. Finally, the feasibility and effectiveness of the proposed controllers were verified by several common fractional-order nonlinear systems. Keywords Fractional-order nonlinear systems Asymptotic stabilization Stability analysis Time delays M-matrix
Z. Zhe (&) Z. Jing College of Electrical and Information Engineering, Hunan University, Changsha 410082, China e-mail: [email protected]; [email protected]
1 Introduction Recent years have witnessed the rapid development of the fractional derivative concept in the whole scientific community. An increasing number of researchers have devoted themselves to the study of fractional derivatives. Fractional derivatives have been applied in mathematics, physics, biology, electrical engineering, control and other fields, and achieved abundant research results [1–8]. Fractional-order derivatives are most widely used in fractional-order systems, which are worthy of research since many systems in nature are of non-integer order. There are often factors interfering with the study process of a variety of fractional-order systems, such as neural network systems [4, 5], gene regulation network systems [9, 10], HIV systems [11, 12], and Lorenz dynamical systems [8, 13], which necessitates the investigation of the stability and stabilization of fractional-order systems. Scholars have put forward different stability conditions and controllers of the fractional-order systems after years of research. These methods can be roughly divided into two types, with one based on a Lyapunov function and the other based on eigenvalue judgment. Since this paper focuses on the use of the Lyapunov-like function and the vector Lyapunov function to solve the problem, the method based on eigenvalue judgment is adopted [14, 15]. Li et al. derived the stability conditions for fractional-order systems via a Lyapunov function [16], and other
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stability conditions via the Lyapunov direct method [17]. Tuan et al. put forward a novel stability judgment method based on the fractional-order Lyapunov function [18], and investigated the asymptotic stability of non
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