Stability Analysis And Stabilization Of Switched Positive Systems

By the development of times and the progress of science, the people’s understanding for the nature is deepening, and the models adopted for the design of control systems are more and more complicated and speci?c. In real world, many systems possess the features:(1) they cannot be described by a single model, and(2) for the systems that can be modeled by a single model, the expected performance standard of the systems cannot be obtained by using a single controller.Switched systems provide a possibility for solving the two problems. It motivates the researchers to pay their attention to studying switched systems over past several decades. During the process of investigating switched systems, the so-called switched positive systems attract much attention.Switched positive systems can be used to describe many practical systems, for instance, networks employing TCP in communication systems, formation ?ying in air traf?c systems, the treatment of HIV virus mutation in medical systems, etc. Although switched positive systems have extensive application, they are only concerned in very recent years. Many problems about switched positive systems are not yet solved. Thus, it is important and meaningful to study switched positive systems. This thesis focuses on the stability and stabilization of switched positive systems. The main contents and results obtained in this thesis are as follows:Chapter 1 is a survey. First, it sums up the recent development of switched positive systems and illustrates several hot topics of switched positive systems. Then, the switched positive systems considered and approaches used in this thesis are introduced with speci?c problems. Third, some basics of switched positive systems and the used mathematical tools are introduced. Finally, the major work of the paper is summarized.Chapter 2 investigates the problems of the stability and stabilization of switched positive systems. First, by using multiple linear copositive Lyapunov functions(MLCLF) integrated with average dwell time(ADT) switching, stability of switched positive systems is analyzed. A suf?cient condition of stability for the considered systems is established, and state-feedback and outputfeedback laws are proposed to stabilize switched positive systems. Second, by means of MLCLF integrated with the mode-dependent average dwell time(MDADT) switching, the stability and stabilization of switched positive systems are respectively addressed, a suf?cient condition of stability is presented, and state-feedback and output-feedback control laws are respectively constructed. By some simple comparisons, it is shown that MDADT is better than ADT when some subsystems of switched positive systems possess bad respond effect. Third, by employing linear programming(LP), stability and stabilization of switched positive systems are solved. A general feedback law is designed, for which the rank of gain matrix is not necessary 1. Finally, the LP approach is applied to the asynchronous switching design of switched positive systems, and stability and stabilization are solved.Chapter 3 deals with robust stability and stabilization of switched positive systems with uncertainty. First, robust problems of switched positive systems with interval and polytopic uncertainties is solved. By MLCLF and the properties of interval matrix, a suf?cient for robust stability of the systems is given, and robust state-feedback and output-feedback laws of the systems are respectively designed. Second, by using the MDADT switching approach, robust stability and stabilization of switched positive systems with interval and polytopic uncertainties are proposed.Finally, robust stability and stabilization of switched positive systems with exogenous disturbance are considered. By virtue of the MLCLF approach, a suf?cient condition for robust stability of switched positive systems is proposed, robust feedback control laws are designed, and an optimal L1 gain is obtained.Chapter 4 studies absolute stability and stabilization of switched nonlinear positive systems.The nonlinear functions in this chapter satisfy a sector ?eld condition. First, absolute stability and stabilization of the considered systems with nonlinear perturbation functions are solved, a suf?cient condition for the stability is given, and a control law guaranteeing the stabilization of the systems is constructed. Second, absolute stability and stabilization of switched nonlinear positive systems are solved. The obtained results reach to absolute exponential stability of the systems,which is better than absolute asymptotic stability in the literature. The present results are extended to more general systems.Chapter 5 summarizes the work presented in this thesis and provides several problems which are worth studying further.According to the speci?c kinds of the systems, the main contributions of this thesis can be summarized as the following three aspects.1. By using MLCLF integrated ADT and MDADT, respectively, the problems of stability and stabilization of switched positive systems are solved. The problems contain stability analysis, state-feedback design, and output-feedback design of switched positive systems. All present conditions can be solved in terms of LP, which is easier to implement than linear matrix inequalities. The obtained results develop the current results in the literature, perfect the stability theory of switched positive systems, and have a certain practical value.2. Based on the properties of interval and polytopic matrices, robust stability and stabilization of switched positive systems with uncertainties are solved. Several suf?cient conditions guaranteeing the robust stability of the systems are established, and the robust feedback control laws are constructed. By employing an L1-gain performance, robust stability and stabilization of switched positive systems with exogenous disturbance input are addressed.3. The problems of absolute stability and stabilization of switched nonlinear positive systems are considered. Two suf?cient conditions for the absolute stability are established, and feedback control laws are constructed. The obtained results are also developed to neural networks.