Stability Analysis And Synthesis Of Switched Positive Systems

Switched positive systems have recently drawn researcher's great attentions due to their important applications in the control of networks employing TCP in communication sys-tems,formation flying in air traffic systems,and the treatment of HIV virus mutation in medical systems,etc.Since switched positive systems possess both the distinctive proper-ties of positive systems and the complex dynamics induced by switching signals,there are greater difficulties in studying the stability analysis and synthesis problems of such systems.Thus,it is of important theoretical significance and widespread application value to study switched positive systems,however,the research work is still in its infancy at present,and many research problems remain to be solved.Based on the basic concepts and methods of the control theory,by using multiple Lyapunov functions,Lyapunov-Krasovskii functional technique and average dwell time method,the stability analysis and synthesis of switched positive systems are deeply discussed in this thesis from perspectives of stability analysis,stabilization and observer design.Some novel analysis methods and design schemes are pro-posed,and a series of well-established,systematical and important results are obtained.The main results of this thesis are summarized as follows:1.The problems of stability analysis and L1 stabilization for a class of switched positive systems with mixed time-varying delays are investigated,and the mixed delays are presented in the forms of discrete delay and distributed delay.Firstly,by constructing an appropriate multiple linear co-positive type Lyapunov-Krasovskii functional and using the average dwell time approach,a sufficient condition is proposed to ensure the exponential stability with an L1-gain performance for the system.Then,on the basis of the obtained results,an effec-tive state-feedback controller design scheme is proposed such that the resulting closed-loop system is positive and exponentially stable with an prescribed L1-gain performance level.2.The problem of positive L1 observer design for a class of switched positive systems with observable and unobservable subsystems is investigated.Firstly,a switched Luenberger-type observer is constructed for the considered system,and the concept of positive L1 ob-server is proposed based on the definition of L1-gain.Then,based on the multiple linear co-positive type Lyapunov function method and the average dwell time approach,a suffi-cient condition is proposed for delay-free switched positive systems to ensure the existence of a positive L1 observer,and an effective design scheme is also presented correspondingly.Finally,the proposed positive L1 observer design scheme is extended to a class of switched positive systems with mixed time-varying delays,where the mixed delays are presented in the forms of discrete delay and distributed delay.3.The problems of L_fault detection observer design and muti-objective L_/L1 fault detection observer design for a class of switched positive systems with time-varying delay are investigated,respectively.A new fault sensitivity measuret,called L_index,is proposed.Firstlyr,based on the average dwell time approach and the multiple linear co-positive type Lyapunov-Krasovskii functional method,a sufficient condition is proposed to ensure the ex-ponential stability of the error switched system.Then,a sufficient L1 robustness condition and a sufficient L_fault sensitivity condition are developed for the error switched system,respectively.Finally,based on the obtained results,sufficient conditions for the existence of L_and muti-objective L_/L1 fault detection observers are given respectively,and the ef-fective design schemes are presented correspondingly,such that the designed fault detection systems are sensitive to faults and simultaneously robust to unknown disturbances.4.The problems of stochastic stability analysis and L1 stabilization for a class of posi-tive Markovian jump systems with time-varying delays are investigated.Firstly,by introduc-ing an "equivalent",deterministic linear positive system,a necessary and sufficient condition of stochastic stability with a prescribed L1-gain performance level is proposed for the un-derlying system.Then,by considering the uncertainties in system matrices and transition probabilities,some robust L1-gain performance analysis results are obtained.Finally,based on the obtained results,an effective scheme for designing a desired state-feedback controller is proposed such that the resulting closed-loop system is positive and stochastically stable with an prescribed L1-gain performance level.5.The stabilization problem for a class of nonlinear switched positive systems un-der asynchronous switching is investigated.The continuous-time and discrete-time cases are considered respectively.Asynchronous switching means that the switches between the candidate controllers and the system modes are not synchronous,and it normally exits in practical switching.Firstly,for continuous-time nonlinear switched positive systems,based on the average dwell time approach,an effective asynchronous switching control law is first-ly provided to guarantee the resulting closed-loop system to be positive and exponentially stable,and the corresponding admissible switching signal is presented.Then,an effective asynchronous switching control law is further provided for discrete-time nonlinear switched positive systems.As special cases,the stabilization results for linear switched positive sys-tems under asynchronous switching are provided accordingly.