Research On Fuzzy Approximation Of Nonlinear Singular Systems

With the development of science and technology,many practical control systems are becoming more and more complex,and these systems are always nonlinear.Therefore,the research of control problems for nonlinear singular systems is of great theoretical significance and wide application value.There are many methods for solving nonlinear singular systems.In the past few decades,fuzzy linearization of nonlinear singular systems has been extensively studied by using fuzzy logic,and a lot of research results have been achieved.However,the approximation of fuzzy models for nonlinear singular systems with impulses has not been effectively solved.In this thesis,the T-S fuzzy approximation method and the polynomial fuzzy approximation method are used as the theoretical framework.The nonlinear singular systems with singular induction bifurcation and nonlinear singular systems with input derivatives are approximated by the T-S fuzzy model.The algorithms of T-S fuzzy approximation for several classes of nonlinear singular systems with impulses and the algorithm of polynomial fuzzy approximation for nonlinear singular systems without impulses are summarized.The main research works are as follows:1.The problem of impulse analysis in the T-S fuzzy singular system is considered by using the Kronecker index or the generalized Kronecker index in this thesis.Firstly,the T-S singular system is divided into two cases:the uniformly regular T-S fuzzy singular system and the non-uniformly regular T-S fuzzy singular system.A new definition of the generalized Kronecker form is introduced,and the algorithm of finding the generalized Kronecker form is given.Then,the theorem is proved,which the T-S fuzzy singular system is impulsive if and only if the(generalized)Kronecker index is greater than one.Finally,two numerical simulations and the population model with stage structure are carried out to show the consistency with theoretical analysis and illustrate the effectiveness of the algorithm.2.The present thesis solves the approximation problem of T-S fuzzy linear singular system for a class nonlinear singular system with impulses.Considering a special nonlinear singular bio-economic system with impulses,the T-S fuzzy linear singular system of the original system has been calculated.For this particular case,it is extended to more generally nonlinear singular system.For a class nonlinear singular system that is Bounded Impulsefree Item and Separable Impulse Item with singularity induced bifurcation,we proved that it can be approximated by T-S fuzzy singular system with arbitrary accuracy.Finally,a numerical simulation is carried out to show the consistency with theoretical analysis and illustrate the effectiveness of approximation.3.This thesis solves the approximation problem of T-S fuzzy linear singular system for a class of nonlinear singular system with derivative of input and the nonlinear singular system exists impulses.Considering a numerical example and a two-wheel drive robot,the T-S fuzzy singular systems are calculated for original system with derivative of input.According to solvability and solving steps of the two examples,the results are extended to more generally nonlinear singular system with derivative of input.The theorem and algorithm are given,which if input-state system is Bounded Impulse-free Item and Separable Impulse Item,it can be approximated by T-S fuzzy singular system with arbitrary accuracy.Finally,a numerical simulation is carried out to show the consistency with theoretical analysis and illustrate the effectiveness of approximation.4.In this thesis,the problem that using the Sum-of-square(SOS)fuzzy linear singular system to approximate a class of nonlinear singular system without impulses is solved.Firstly,considering two numerical examples,the SOS fuzzy singular systems of the nonlinear singular system are calculated.According to solvability and the steps of two examples,the results are extended to more generally nonlinear singular system and the nonlinear singular system can be divided into two types of cases,one is nonlinear model with compound function type and the other is nonlinear model with simple function type.Then,the theorem and algorithms are given that the nonlinear singular system without impulse can be approximated by SOS fuzzy singular system with arbitrary accuracy and it can effectively reduce the number of fuzzy rules.Finally,a human posture inverted pendulum model example and a numerical simulation are carried out to show the consistency with theoretical analysis and illustrate the effectiveness of algorithms.