Stability Analysis Of Nonlinear Systems Based On Polynomial Fuzzy Model

Fuzzy control is a hot topic in the mathematical and control community,and has been received extensive attention of researchers all over the world as soon as it was put forward,and has a big applications in the practice,among which the related research of the Takagi-Sugeno(T-S)fuzzy model have been fruitful and influential deeply.The diversity and complexity of dynamic systems put forward a higher request to the related research.This dissertation based on the method of Sum of Squares(SOS),studies the related problems of the finite-time stability and stabilizing control of a class of polynomial fuzzy system mainly,considering the finite-time stable and stabilizing conditions of both continuous time polynomial fuzzy dynamic systems and discrete time dynamic systems.The main contents of this dissertation are as follows:As the research background and practical significance depicted in the first chapter,it mainly introduces the historical background and the current research progress of the stability and stabilization of a class of nonlinear fuzzy system,focus deeply on continuous time dynamic systems,discrete time dynamic systems,finite time stability and the method of sum of squares to deal with polynomials.In the second chapter,the related problems of the finite-time stability and corresponding controller design of continuous-time polynomial fuzzy systems are studied.In this chapter,the concept of finite-time stability and stabilization of continuous-time polynomial fuzzy systems is given,and two situations without controlled input and with controller are discussed respectively.Combining Sum of squares optimization method(SOS),polynomial Lyapunov function and the differential equation stability theory to obtain the sufficient conditions of finite-time stability and stabilization,which include some existing classical results as special cases.Two numerical examples and a coaxial counter rotating micro-helicopter dynamics instance are introduced to verify the effectiveness of our proposed theoretical method and the application in actual operation with the help of the MATLAB toolbox SOSTOOLS and semi-definite-program solver(SDP).Chapter 3 discusses the finite-time stability and stabilizing controller design of discrete-time polynomial fuzzy systems.Combining Schur Complement theorem,difference equation,polynomial Lyapunov function and Sum of Squares method,to extend relevant results in the second chapter further for the discrete-time dynamic systems.The concept of finite-time stability and stabilization is given for the discrete-time polynomial fuzzy system.At the same time,this Chapter theoretically analysis the sufficient conditions of the nonlinear dynamic systems without control input under the polynomial fuzzy model structure reaching the finite-time stability.Aiming at the discrete-time nature,we design the finite-timestabilization controller in the form of Sum of Squares.Numerical simulation results verify the validity and feasibility of the theoretical results,and discuss the finite-time stable conditions of the nonlinear dynamic systems without control input.Finally,the stabilizing effect of the control input is further discussed for the discrete-time polynomial fuzzy system.In the fourth Chapter,the main contents and the innovation of this dissertation are summarized.Meanwhile,future studies are also given.At the end of this dissertation,all references are listed.All conclusions are numerically simulated by Matlab to ensure correctness and effectiveness.The simulation results bring into correspondence with the theoretical analysis.