Analysis And Synthesis Of Switched Positive Linear Systems

Switched systems are a special class of hybrid systems, which consist of a familyof subsystems and a rule orchestrating the switching between them. Recently, switchedpositive linear systems （SPLSs） have drawn researchers’s great attention due to their im-portant applications in the control of network congestion, formation flying and infectiousdisease. Since SPLSs posses both the complex dynamics of switched systems and the dis-tinctive properties of positive systems, there are greater difculties in studying the analysisand synthesis problems of such systems. This dissertation will make deep discussions onSPLSs from the perspectives of stability analysis, controller synthesis, observer designand so on, and the main contributions are as follows:The stability problems for some kinds of SPLSs are investigated. Firstly, the stabilityunder arbitrary switching of SPLSs with commutable subsystems is analyzed by exploringthe existence of a common linear copositive Lyapunov function. The achieved resultsimply that such a function can always be constructed if only all subsystem matrices ofa SPLS are stable and pairwise commutable, whenever these subsystems are continuous-time, discrete-time or the both mixed, and hence it is concluded that the underlying systemis stable under arbitrary switching. Secondly, the stability of SPLSs under average dwelltime （ADT） switching is analyzed based on multiple linear copositive Lyapunov function（MLCLF） method, and sufcient stability criterions are obtained in the cases where allsubsystems are stable and where parts of the subsystems are stable, respectively. Theapplication of MLCLFs in the stability analysis guarantees that all the obtained stabilityconditions are given in terms of simple linear inequalities.The problems of D-stabilization and weighted H_∞control for SPLSs are studied bymaking use of multiple diagonal quadratic Lyapunov function approach and ADT switch-ing idea. Firstly, design methods of D-stabilizing state-feedback controllers are proposedfor continuous-and discrete-time SPLSs, respectively. The developed controllers canguarantee that the resultant closed-loop systems are not only positive but also globallyasymptotically stable under a set of corresponding ADT switching signals without theneed of positivity to original systems, and all their poles are assigned into the prescribeddisks. The derived results imply that the desired controller gain matrices and the corre-sponding minimal ADT can be easily obtained by solving the feasibility problem of a set of linear matrix inequalities （LMIs）. Secondly, design methods of weighted H_∞state-feedback controllers are given for continuous-and discrete-time SPLSs, respectively, andthe controller design problems are converted into convex optimization procedures subjectto LMIs. The jumping behavior of the used multiple functions in the framework of ADTswitching idea induces that the closed-loop systems achieve a weighted H_∞performanceindex rather than a standard one.On the basis of the obtained results in above part, the problem of weighted H_∞control is researched for both continuous-and discrete-time SPLSs with polytopic un-certainties under closed-loop D-stability constraints by employing multiple parameter-dependent quadratic Lyapunov function approach and ADT switching idea. According tothe online availability of uncertain parameters, design methods of robust and parameter-dependent weighted H_∞state-feedback controllers are proposed, respectively, and thecontroller design problems are formulated as the processes of solving convex optimizationprocedures based on the derived LMI conditions. The developed controllers can simul-taneously guarantee that the resultant closed-loop systems are positive, globally asymp-totically stable and have a desired weighted H_∞performance index under correspondingADT switching signals for all admissible uncertainties, with all subsystems D-stable.The utilization of multiple parameter-dependent quadratic Lyapunov functions leads tothe less conservativeness of the proposed controller design methods.The problem of weighted L₁-gain observation is investigated for both continuous-and discrete-time time-varying delay SPLSs by means of multiple linear copositive Lyapunov-Krasovskii functional approach and ADT idea. Firstly, the concept of weighted L₁-gainis proposed and an switched Luenberger-type observer is constructed for the underlyingsystem. Then, the stability and weighted L₁-gain under ADT switching is analyzed forthe corresponding positive error system, and a delay-dependent sufcient condition is de-rived. Based on this, a method of designing weighted L₁-gain observers for time-varyingdelay SPLSs is proposed, and the observer design problem is finally formulated as solv-ing a linear programming problem. The developed observer remains positive for all giventime-varying delays and the corresponding error system is not only globally asymptoti-cally stable but also has a desired weighted L₁-gain under any switching signal with ADTgreater than a certain constant.The issue of stabilization for discrete-time SPLSs with application to power con-trol in wireless communication networks is addressed. Firstly, the switching stabilization problem of autonomous SPLSs is investigated by using multiple linear copositive Lya-punov function approach. Two methods for designing time-and state-dependent switch-ing laws are proposed, respectively, and the design procedures are formulated as solvingcorresponding linear programming problems. The developed switching laws can guaran-tee that the underlying system is not only globally asymptotically stability but also has anupper bound of a cost function. Secondly, the feedback stabilization problem of intervalSPLSs is further researched. Two design methods of state-feedback stabilizing controllersare given under arbitrary and ADT switching cases, respectively. The achieved results im-ply that the desired controller parameters can be easily obtained by solving a set of linearinequalities. Lastly, the power control problem in wireless communication networks isanalyzed from the point of view of SPLSs, and the efectiveness and applicability of theproposed methods for designing stabilizing switching laws and controllers are verified inthis background.Finally, the achieved results of the dissertation are summarized and the direction offuture research is pointed out.