Stability Analysis And Control Synthesis For Polynomial Fuzzy Positive Systems

Positive system is a special kind of system with unique positivity.It has attracted many attentions of researchers because of its wide application in many fields such as biomedicine,chemistry,ecology and so on.But due to the positivity,many research results on general systems cannot be applied to positive systems.How to fully use the unique properties of the positive systems requires further research and analysis.Therefore,in terms of the polynomial fuzzy theory,linear copositive Lyapunov stability theory and membership function-dependent technology,this paper mainly focuses on the output feedback control,filter design,L₁ performance,stability analysis and relaxed conservatism of positive nonlinear systems.The main research contents are as follows:(1)Based on the polynomial fuzzy model,a polynomial static output feedback controller is designed for a positive nonlinear system.Meanwhile,the conservativeness of the stability analysis results is relaxed by introducing the membership function-dependent technology.Firstly,a polynomial fuzzy model is used to express the positive nonlinear system so that the polynomials can exist in the sub-fuzzy systems,thereby,the research scope of the positive nonlinear systems can be expanded.Secondly,in view of the fact that the imperfect premise matching technique allows the fuzzy controller to have completely different membership functions from the fuzzy system,a polynomial fuzzy static output feedback controller is designed for the positive polynomial fuzzy system,hence,the difficulty of design and implementation of the fuzzy controller can be reduced.In addition,by introducing a non-singular transformation matrix,combined with matrix processing technology,the non-convex stable conditions and positive conditions can be transformed into convex conditions so that the feasible solutions can be found.Finally,by introducing the membership function-dependent technique,the relaxed stability conditions and positivity conditions are derived.(2)In terms of the superiority of polynomial fuzzy theory,the design of output feedback controller and the analysis of asymptotic stability are studied for positive polynomial fuzzy systems.At the same time,the polynomial membership function approximation method is used to relax the stability analysis results of the closed-loop system.Firstly,by using the non-parallel distributed compensation technology,a polynomial fuzzy output feedback controller is designed for the polynomial fuzzy positive systems.Secondly,the augmented vector method is used to construct the augmented system of the closed-loop control system.By selecting the appropriate linear copositive Lyapunov function,using advanced sum-of-squares(SOS)processing technology,and combining a non-zero constant vector,a new method to solve the non-convex problem is proposed and the SOS-based stability conditions as well as positivity conditions are obtained.Finally,the advanced polynomial membership function approximation method is used to introduce the shape information and approximate error information of the membership functions into the stability analysis so that the conservativeness of the stability analysis can be reduced.(3)A L₁-gain fuzzy filter is designed for the positive T-S fuzzy system with constant time delay and external disturbance.Meanwhile,the introduction of membership function information helps to relax the stability analysis results of the positive T-S fuzzy filter error system.Firstly,an auxiliary variable is used to construct the augmented system of the positive T-S fuzzy filtering error system to promote the transformation from the non-convex stability conditions to the convex ones.Then,in view of the existence of the time-delay term and the capture of the positivity,a linear copositive Lyapunov-Krasovskii functional is constructed.By combining with the L₁performance,the sufficient conditions which not only can guarantee the stability and positivity,but also can guarantee the system performance index are derived.Finally,in order to improve the conservitiveness of the results,the piecewise linear membership function(PLMF)approximation method is applied to the stability analysis of the positive T-S fuzzy filtering error system so as to obtain the relaxed stability and positivity analysis results.(4)For positive polynomial fuzzy systems with external disturbances in different situations,different L₁-gain output feedback fuzzy controllers are designed.Based on the Lyapunov stability theory and the L₁ performance index,the stability and positivity of the positive polynomial fuzzy L₁-gain output feedback control system are analyzed,and the stability and positivity conditions under the L₁ performance are deduced.For solving the tricky non-convex problem,the augmented vector method,matrix transformation technology,and constraint conditions are used to successfully transform the non-convex stability and positivity conditions into convex conditions.Finally,based on the membership function-dependent technology,a high-order polynomial membership function approximation method is proposed,which helps to achieve better relaxing effect.