Stability Analysis Of Switched Singular Systems Via A Multiple Discontinuous Lyapunov Function Approach

Switched systems,as special hybrid systems,consist of a finite number of continuous-time(or discrete-time)subsystems and a logical rule orchestrating the switching between them.Singular systems are also known as descriptor variable systems,implicit system-s,differential algebraic systems,which could describe a larger class of systems than the normal system model,so they can provide convenient and natural representation-s in the description of dynamic systems.The switched systems that contain singular subsystems are called switched singular systems.In recent years,due to their broad applications foregrounds in numerous practical areas such as power systems,economic systems,switched singular systems have aroused researcher s general concern.Stabil-ity analysis is very crucial and fundamental problems in the area of switched singular systems.However,because these kind of systems have the characteristics of switched system and singular system,the research on switched singular systems is more complex and challenging than normal switched systems.In this dissertation,the stability problems of linear switched singular systems are investigated by using a novel multiple discontinuous Lyapunov function(MDLF)and mode-dependent average dwell time(MDADT)switching signal approach.Compared with the traditional multiple Lyapunov function(MLF),the advantage of adopted mul-tiple discontinuous Lyapunov function is that each Lyapunov function is required only to be piecewise continuous during the operation time on each activated subsystems instead of continuously differentiable,which can enhance the flexibility in our actual applica-tions.In addition,the mode-dependent average dwell time switching signal overcomes the limitations of mode-independent of average dwell time(ADT)switching signal,and reduces the conservatism of all subsystems must share a common dwell time.The main reason is that each mode not only has its own ADT but also has its own control strategyunder MDADT switching.Furthermore,tighter bounds on dwell time can be obtained than existing results.The main contents of this.dissertation include:Chapter 1 is the preface,the research background and research states of this dis-sertation are introduced.Firstly,the research background and research meaning of switched singular systems are summarized,the actual applications of switched singular systems are introduced.Then,the research states of stability and control problems for the switched normal and switched singular systems are presented.Finally,the main work of this dissertation is briefly introduced.The exponential stability of linear switched singular systems in continuous-time case are studied in Chapter 2.First,the stability problems of linear switched singular systems with stable and unstable subsystems are considered under the assumption of regularity and impulse free.Then,by constructing a novel multiple discontinuous Lya-punov function and utilizing mode-dependent average dwell time switching signal,and adopting the switching strategy where fast switching and slowing switching are respec-tively applied to unstable and stable subsystems,the new sufficient stability conditions for linear switched singular systems are presented in the form of linear matrix inequal-ities.Moreover,compared with the existing results,tighter bounds on the dwell time can be obtained based on above approaches.Finally,a numerical example is provided to illustrate the effectiveness and feasibility of the proposed criteria.The stability of linear switched singular systems in discrete-time case are considered in Chapter 3.We extend the continuous results to discrete case for the linear switched singular systems composed of stable and unstable subsystems.First,an equivalence lemma between the E-exponential stability and exponential stability is derived based on regularity and causality.Next,by constructing an appropriate multiple discontinuous Lyapunov function and utilizing mode-dependent average dwell time switching signal,and adopting the fast and slow switching strategy,the sufficient stability conditions for considered systems are given in terms of linear matrix inequalities.Moreover,comparedwith the existing results,the proposed method can obtain tighter bounds on dwell time that stable and unstable subsystems should respectively satisfy.Finally,a numerical example is provided to illustrate the effectiveness and feasibility of the proposed results.The stability and L2-gain problems of linear switched singular systems contain no unstable subsystems in discrete-time case are investigated in Chapter 4.Along the above multiple discontinuous Lyapunov function and mode-dependent average dwell time switching signal approach,new sufficient stability conditions for considered systems are proposed in terms of linear matrix inequalities.In addition,to meet the practical engineering needs,the controlled systems with external disturbance are usually requested to robust.Therefore,by using a multiple discontinuous Lyapunov function approach and adopting the mode-dependent average dwell time switching signal,the L2-gain problems of switched singular systems are discussed and the weighted L2-gain performance index are also derived.At last,a numerical example is given to show the effectiveness and feasibility of the results.Chapter 5 summarizes the main results and contributions of this dissertation and point out the further research.