Stability Analysis For Fractional Order Singular Systems

Fractional order calculus as an extension of ordinary calculus has more than 300 years history.Fractional-order systems are represented by differential equations with non-integer order.It has been found that the behavior of many physical systems can be more properly described by fractional-order systems.This paper considers three classes of systems in the fractional-order system theory: the fractional-order singular systems,the fractional-order singular systems with time-varying and the fractional-order singular systems with time-delay.For fractional-order linear and nonlinear singular systems,we discuss the stability of the systems with the method of Lyapunov function;for fractional-order systems with time-varying,we use inequalities and give a sufficient condition of stability without pulse,and use Riccati equation to research its stability;for fractional-order systems with time-delay,we design a state-feedback control law by using Lyapunov-Krasovskii function,we also discuss its stability.For the study of the above several systems,we all give examples by Simulink.The main contributions and achievements of the paper are as follows:Firstly,we investigate the stability of fractional order singular systems.Sufficient conditions of asymptotical stability for fractional order singular linear systems are obtained by Lyapunov function method.We design the state feedback controller of the fractional order singular systems and give the criteria of asymptotical stability.We also give criterion of nonlinear systems.Secondly,we study the problems of stability for fractional order linear singular systems and nonlinear singular systems with time-varying.A sufficient condition is derived based on matrix inequality,in which linear singular system is found to be impulse-free and stable.Due to fact that the slow sub-systems can be determined by the fast sub-systems with appropriate conditions,some concise criteria of stability for nonlinear singular systems with time-varying are obtained by using Riccati equation.Finally,the problems of Lyapunov uniform stability and stabilization for fractional order positive singular systems with time delay are studied.A co-positive Lyapunov-Krasovskii functional is constructed for fractional order positive time delay singular systems,then a sufficient condition for the Lyapunov uniform stability of fractional order positive singular systems with time delay is established in terms of linear programming.By designing a state-feedback control law,we solve the problem of Lyapunov uniform stabilization for the closed-loop systems.