Stabilization of a class of fractional-order nonautonomous systems using quadratic Lyapunov functions

Quan Xu,Yankun Xia,Chang Che,Shengxian Zhuang,Xiaohui Xu

doi:10.1186/s13662-017-1459-9

Quan Xu, Yankun Xia + Show 3 more

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In this paper, we aim to solve the stabilization problem for a large class of fractional-order nonautonomous systems via linear state feedback control and adaptive control. By constructing quadratic Lyapunov functions and utilizing a new property for Caputo fractional derivative we derive some sufficient conditions for the global asymptotical stabilization of a class of fractional-order nonautonomous systems. We give two illustrative examples to validate the effectiveness of the theoretical results.

Highlights

1 Introduction Fractional calculus, as a mathematical tool dealing with derivatives and integrals of arbitrary orders, has played a central role in physics [1], differential and integral equations [2], signal processing [3], human relationships [4], image encryption [5], thermal conductivity [6], electrical circuits [7], dynamical models [8], nonlinear control systems [9], complex networks [10], and so on
Constructing quadratic Lyapunov functions and using a new property for the Caputo derivative, we respectively investigate the stabilization of a class of fractional-order nonautonomous systems via state feedback control and adaptive control
We further study the stabilization of system (3.1) via linear state feedback control and adaptive control

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Introduction

Fractional calculus, as a mathematical tool dealing with derivatives and integrals of arbitrary orders, has played a central role in physics [1], differential and integral equations [2], signal processing [3], human relationships [4], image encryption [5], thermal conductivity [6], electrical circuits [7], dynamical models [8], nonlinear control systems [9], complex networks [10], and so on. One of very important areas of application is nonlinear control systems. Stability analysis and stabilization are of theoretical and practical importance for control systems, certainly including fractional-order systems. The early studies on the stability of fractional-order systems mainly concentrated on the linear cases, and many well-known results have been obtained. The stability analysis of fractional-order nonlinear systems remains an open problem. More details on the development of stability of fractional-order systems can be found in [11,12,13,14,15]

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