Control Of Nonlinear Positive Systems Based On T-S Fuzzy Model

On a bounded compact set,fuzzy system can approximate a complex nonlinear system,which makes it an effective way to handle complex nonlinear systems.In particular,the recently proposed Takagi-Sugeno?T-S?fuzzy system,which consists of a set of linear systems and approximates a complex nonlinear system with fuzzy inference and membership functions on a bounded compact set.Studies have shown that it is esier to control a T-S fuzzy system which is in an easy form than a complex nonlinear system.Therefore,linear system theory can be comblined with the study of the nonlinear system based on T-S fuzzy model and some classical linear system theory can be used for the study of complex nonlinear system,which has attracted a lot of attention in the past decades.In this paper,according to the Lyapunov stability theorem,linear programming technique,L₁ gain control theory,parallel distributed compensation?PDC?algorithm,switched systems theory,four hot issues in nonlinear positive system are studied.The main work is summarized as following:Firstly,the stability of T-S positive system is investigated by building T-S fuzzy system to approximate the nonlinear positive system.Based on ₁L gain control theory and the use of linear Lyapunov function,a sufficient condition in LP for the ₁L gain performance stability is obtained.Output feedback controller is designed by PDC algorithm,which makes the T-S fuzzy system controlled positive and asymptotically stable with the ₁L gain performance indexg.Then,baesd on the results,we go one step farther and take into account the optimal problem,and obtain the optimal ₁L gain performance index and the optimal control gain matrices with LP.The problem of stabilization with optimal ₁L gain for positive T-S fuzzy system is studied and the obtained conditions can be solved by LP.Compared to the conditions in LMI,the computation is more than the conditions in LP.Finally,an instance example is introduced to show the design of the ₁L gain stability controller for nonlinear system and.the effectiveness of the proposed methods are proved.Secondly,by building discrete time T-S fuzzy positive model and PDC algorithm to design controller,the condition of ₁l gain stability is obtained,and the optimal ₁l gain performance indexgand the optimal control gain matrices are solved.Then,respectively construct nonnegative constrained controller and non-symmetrically constrained controller to obtain the sufficient condition of ₁l gain stability and guarantee the state bounded and the controller satisfying given prescribed boundaries,and obtain the optimal ₁l gain performance indexsgand the optimal control gain matrices.The condition of initial state and controller are designed to satisfy any given prescribed boundaries,which can be proved by two examples.Thirdly,considering the effects of delay on the stability and control peformance of the system,build T-S fuzzy mixed delay positive system for nonlinear mixed delay positive system to study the stability of the T-S fuzzy mixed delay positive system.With the use of Lyapunov stability theorem,delay-dependent exponential stability condition with ₁L gain in LP is obtained.Based on PDC algorithm,the controller is built to guarantee the T-S fuzzy mixed delay system controlled positive and exponential stable with the ₁L gain performance indexg.And the optimal ₁L gain performance index and the optimal control gain matrices are finally obtained.Compared with the existing results,the delay-dependent exponential stabilization and exponential stability conditions in LP are obtained.Fourthly,considering the switch problem of parameters of the system,the stability and stabilization for nonlinear switched mixes delay positive system is considered in this chapter.A T-S fuzzy switched mixed delay positive system is built to approximate the nonlinear system.With the Mode-dependent average dwell time?MDADT?approach and multiple co-positive type Lyapunov–Krasovskii functional,the delay-dependent exponential stability condition is obtained.Then,with the fuzzy controller designed,the sufficient condition in LP of delay-dependent exponential stabilization and positivity for the T-S fuzzy switched mixed delay system is obtained.Finally,some summaries for the main results of this paper and some further reseach works are given.