Stability And Stabilization Of Discrete Switched Positive Systems

The stability and stabilization of discrete switched positive system-s(DSPSs)are investigated in this thesis.Firstly,the exponential stability and sta-bilization of DSPSs are studied.Secondly,when the switching instants of the con-trollers lag behind those of the subsystems,we further discuss the asynchronous stabilization and finite-time control of DSPSs.Finally,the exponential stability of discrete positive singular time-delay systems(DPSDSs)and finite-time stabili-ty(FTS)of discrete switched positive singular systems(DSPSSs)are also discussed in this thesis.It is organized as follows:The first chapter introduces some basic konwledge of switched systems,switched positive systems,Lyapunov stability and FTS.In order to give the explicit solution of DPSSs,we also present the relevant theory about generalized inverse.In the second chapter,the stability and stabilization of DSPSs are studied.First,by constructing multiple-linear copositive Lyapunov functions,a sufficient condition is obtained for exponential stability of DSPSs by making use of the for-ward mode-dependent average dwell time(MDADT)apprpach.And we also point out,when the switching signal satisfies some certain assumptions,some results in previous literature can be seen as corollaries of the result.Second,by virtue of multiple-sample Lyapunov-like functions variation,another sufficient criterion for exponential stability of DSPSs is derived.Then,a set of mode-dependent state feedback controllers are designed to guarantee the resultant closed-loop system is positive and exponentially stable under the switching signal with forward MDADT.Finally,two illustrative examples are given to show the correctness of the obtained theoretical results.The asynchronous stabilization of DSPSs with time-varying delays is discussed in third chapter.Firstly,by constructing an appropriate Lyapunov-Krasovskii func-tional,a sufficient condition is derived for the exponential stability of discrete switched positive time-delay systems under the switching signal with MDADT.Sec-ondly,by allowing the selected Lyapunov-Krasovskii functional to increase during the mismatched period between the activated subsystems and the controllers,with the aid of MDADT swiitching signal,a sufficient condition is obtained for the ex-istence of state feedback controllers such that the resulting closed-loop system is positive and exponentially stable under asynchronous switching.Finally,two nu-merical examples are given to illustrate the validity of the proposed methods.In the fourth chapter,the finite-time control of discrete impulsive switched positive time-delay systems(DISPDSs)under asynchronous switching is addressed.First,by constructing an appropriate Lyapunov function,a sufficient condition is derived for FTS of DISPDSs under the MDADT switching signal.Second,by con-structing another Lyapunov function,several sufficient conditions are obtained for the existence of state feedback controllers such that the obtained closed-loop sys-tem is positive and finite-time stable under asynchronous switching,and the specific form of controller gain is given.Finally,a numerical example is given to show the obtained results is feasible and valid.The fifth chapter investigates the exponential stability of DPSDSs.Firstly,by using the singular value decomposition and monomial coordinate transformation,a necessary and sufficient condition is established to guarantee the discrete singu-lar time-delay systems is positive.Secondly,by constructing an appropriate linear copositive Lyapunov function,a sufficient condition is obtained such that the DPS-DSs are exponentially stable.Moreover,the obtained results are formulated in terms of algebraic matrix inequalities which can be numerically tractable by virtue of LP toolbox in Matlab.Finally,a numerical simulation is given to show the effectiveness of the proposed techniques.The FTS of DSPSSs is addressed in sixth chapter.Firstly,the concept of FTS for DSPSSs is proposed,and an assumption is given to guarantee DSPSSs possess consistent switching.Secondly,with the aid of the state transition matrix,a nec-essary and sufficient condition is obtained for the FTS of DSPSSs with consistent switching under arbitrary switching.Then,by constructing a quasi-linear Lyapunov function,another sufficient condition is derived for the FTS of DSPSSs with consis-tent switching under MDADT switching signal in terms of a set of algebraic matrix inequalities.Finally,a numerical example is given to show the proposed techniques is valid.