Robust Control And Stability Analysis Of Singular Fractional Order Systems

Singular system is a kind of dynamical system with wide application background.It is an advantageous tool to describe and characterize the actual system.Many practical system models need to be described by singular systems,such as power systems,economic systems,electronic networks and aerospace systems,and large-scale interconnected systems.Therefore,the singular system has more extensive significance than the normal system,and its research has more important theory and application value.Because the generalized system is more complex and novel in structure than the normal system,it is more difficult and challenging in theoretical research.Because of its many practical backgrounds and engineering requirements,fractional order system has been widely valued in control field in recent years,and has gradually become a research hotspot.Therefore,fractional order singular system has also become the focus of scholars at home and abroad.Great progress has been made in the theoretical research of singular systems.Many research results of normal systems based on state space are successfully extended to singular systems,which further enrich and improve singular systems and even linear singular systems.However,even for linear systems,the control of fractional singular systems has huge space,and there are few research results for nonlinear systems.The synchronization of fractional singular systems and the stability analysis and robust control are studied using the fractional-order calculus theory,fractional-order stability theory and property and Lyapunov stability theory.The main contents of this paper are as follows:1.singular systems,fractional-order differential theory and fractional-order singular systems are introduced in this paper.In order to solve the problem of the stability analysis of the linear singular fractional-order systems,the continuous frequency distributed equivalent model and the indirect Lyapunov approach are applied in the research of the linear singular fractional-order systems,an PD(Proportional-Derivative)controller is proposed to normalize the singular fractional order system.A novel sufficient condition for asymptotic stability of the singular fractional-order system with the order ?(0(27)?(27)1)is presented.Next using LMI(linear matrix inequalities)techniques and matrix's SVD(Singular Value Decomposition),in oder to obtain the results.A numerical example is provided to demonstrate the effectiveness of the proposed methods.2.Based on the fractional-order singular system theory and robust control theory,we study the controller design method for singular fractional order nonlinear dynamic systems.The continuous frequency distributed equivalent model and the indirect Lyapunovapproach are applied in the research of the singular fractional-order systems,the fractional order derivative(D)and fractional order proportional-derivative(PD)controllers are proposed to normalize the singular fractional order system.A novel sufficient condition for asymptotic stability of the singular fractional-order system with the order ?(0(27)?(27)1)is presented.In order to obtain the results,we use the LMI techniques.Then,we use these methods to solve the problem of the synchronization for singular fractional-order complex dynamic networks.Some numerical examples are provided to demonstrate the effectiveness of the proposed methods.3.On the basis of the passive control theory of an integer order singular system,this paper investigates the passivity analysis and passivity-based control for fractional nonlinear singular systems with fractional-order: 0(27)?(27)1.By using Lyapunov direct approach and linear matrix inequalities,the passivity condition and sufficient conditions for the asymptotic stability of this class of system are derived.Then,the state feedback controller based on passivity is designed.Finally,the validity of the proposed methods are demonstrated by three simple examples.