Research On Several Control Problems Of Switched Positive Systems

In real life,there are a large number of systems that can be described by switched positive systems,such as economy,biology,transportation and other fields.A switched positive system is a special kind of switched system,the switching characteristics between each subsystem and non-negative characteristics of the system state make the study of such system complex and challenging.Although some progress has been made in the study of switched positive systems.However,it is still in the preliminary research stage,and a lot of analysis and comprehensive problems need to be solved urgently.Therefore,it is of theoretical significance and practical value to study the problems related to the switched positive systems.This dissertation considers the problems of stability,controller design and observer design for switched positive systems.The main work of this dissertation consists of the following aspects:In order to find the smaller bound of dwell time for switched positive systems under slow switching,a novel multiple discontinuous linear co-positive Lyapunov function approach is pro-posed.Different from the classical Multiple linear co-positive Lyapunov function method,which only requires that each co-positive Lyapunov function is piecewise continuous during operating time on each activated subsystem.Based on this method,the stability problems are addressed for discrete-time switched positive linear systems,discrete-time switched positive linear time-delay systems and continuous-time switched positive linear time-delay systems.Based on the switch-ing signals of average dwell time and mode-dependent average dwell time,by constructing a suitable multiple discontinuous linear co-positive Lyapunov-Krasovskii functional,the stability criteria of the above systems are established by using the linear programming method.Moreover,the traditional multiple linear co-positive Lyapunov-Krasovskii functional method is used to ob-tain the stability for switched positive linear time-delay systems in continuous time and discrete time and the simulation comparison result is given,thus the superiority of the proposed method can be validated.Furthermore,this method is further extended to solve the stability problem of switched positive linear systems with multiple delays.Furthermore,the finite time stability of switched positive linear time-delay systems un-der state constraints is studied.By developing an multiple discontinuous linear co-positive Lyapunov-Krasovskii functional,finite-time stability criterion is established for such systems under the switching signals of average dwell time and mode-dependent average dwell time in the form of linear programming.Then,by proposing a combinatorial linear co-positive Lyapunov-Krasovskii functional approach,a numerical construction of the controller is designed.And a state-feedback control scheme is developed.The finite-time stabilization is achieved for switched positive linear time-delay systems under the switching signals of average dwell time and mode-dependent average dwell time.This method overcomes the shortcoming that multiple discontin-uous linear co-positive Lyapunov-Krasovskii functional cannot be directly used to design con-troller.An observer design method is proposed for state immeasurability in engineering applica-tions.This method transforms the observer gain setting problem into a linear programming problem and reduces the computational complexity of the observer.Based on this method,the problem of observer design is studied in continuous-time domain for switched positive linear sys-tems,uncertain switched positive linear systems and switched positive linear time-delay systems.Based on average dwell time switching signal,the existence conditions of multi-modal positive observer,robust multi-modal interval positive observer and multi-modal positive time-delay ob-server are given by means of linear programming method.At the same time,the observer design method is extended to solve the controller design problem of switched positive linear systems,two control schemes such as static output-feedback controller and state-feedback controller are designed such that the closed-loop system is globally uniformly exponentially stable.Considering the effects of switched transition probability and time-varying delay on nonlin-ear switched positive systems,the stability and L1 performance of a class of nonlinear positive Markov jump systems are analyzed.First,the stability of nonlinear positive markov jump sys-tem is studied based on mode-dependent average dwell time switching signal.A sector field condition is used to analyze the nonlinear behavior of such system,and bounded time-varying delay is considered.By constructing a nonlinear stochastic co-positive Lyapunov-Krasovskii functional,an exponential stability criterion is established for nonlinear positive Markov jump systems under the switching signals of average dwell time and mode-dependent average dwell time based on linear programming.Then,for the effect of random external perturbations,the L1-gain performance analysis is given for nonlinear positive Markov jump systems.