Controller Design And Stability Analysis For Nonlinear Systems Based On Polynomial Fuzzy Models

With the development of the science and technology,industrial process control systems studied by the engineers become more and more complex.Due to the nonlinearity,uncertainty and some other causes of the practical systems,it's difficult to provide a precisely description for them.The research and application of the nonlinear system is hindered greatly.The fuzzy control theory solves the problem of these complex systems properly.Due to the high ability on modeling nonlinear systems,the T-S fuzzy model,by use of linear matrix inequalities,has developed plentiful results.As an extension of the T-S fuzzy model,the polynomial model drew the extensive attention of scholars in recent 10 years due to the good description of the nonlinearity.This model inherits the advantages of the T-S fuzzy model.Due to the polynomial model in the consequent part,the polynomial fuzzy model simplifies the modeling process and has fewer rules.It also reduces the conservativeness of stability conditions to some extent.The system matrix and input matrix of each sub-system could be polynomial matrices.In view of the cases whether the state could be measured or not,the study and design of the controller for it becomes more complicated than the T-S fuzzy model.And the LMI method could not be used to solve the problem of the polynomial fuzzy systems.The control criteria based on the polynomial fuzzy model could be written in sum of squares(SOS).By use of the SOSTOOLS,the problem could be resolved smoothly.The stability analysis and the controller design are studied in this dissertation.First,a polynomial fuzzy system is modeled for the considered nonlinear system.Then,a candidate Lyapunov function is proposed.And by use of the SOS method,some stability conditions are obtained.At last,some simulations are provided to illustrate the effectiveness of the theoretical results.The main contents and contributions could be summarized as below:(1)The control problem of some discrete-time nonlinear systems is investigated.Based on polynomial fuzzy models,for several cases of the system matrix and the input matrix,some novel polynomial controllers are designed.By use of the Lyapunov method,the SOS conditions are provided to make the equilibrium of the closed-loop system be asymptotically stable.Compared with the LMI approaches to T-S model ones,some simulations are provided to illustrate the advantages of the SOS-based approaches.For some cases,the polynomial fuzzy controller and observer can be separately designed.Therefore,the design flexibility could be increased,and the computation complexity reduced.(2)The control problem of a class of continuous nonlinear system is investigated.For the polynomial fuzzy model-based systems with the common input matrix,based on a mathematical transformation,for three cases of the system matrix and the input matrix,different polynomial controllers are designed.By use of the SOS and Lyapuno-v method,the SOS conditions are provided to make the equilibrium of the closed-loop system be asymptotically stable.According to some published results,the computational burden might be reduced.(3)The stability analysis and controller design of discrete-time polynomial fuzzy time-varying delay systems are investigated.Unlike the most existing literature,the delayed state is also considered in the controller design.By construct-ing a new parameter-dependent polynomial Lyapunov-Krasovskii function and introducing some free weighting matrices,some novel delay-dependent stability and stabilization conditions are obtained.Then,the results are ex-tended to discrete time-delay systems with norm-bounded uncertainties.The conditions proposed are all derived in terms of SOS.At the same time,the LMIs are provided for the according T-S fuzzy models.(4)The problem of designing a robust controller for a class of nonlinear systems which could be modeled by polynomial fuzzy ones is investigated.The local input matrices of the considered system could be not equal to one another.Every column vector of the input matrix could be rewritten by use of the ones of the basis matrix.The novel sliding mode surface could also be designed by use of the basis matrix.And the sliding mode surface's parameter matrix could be characterized in terms of the solution of the proposed SOS conditions.Then,based on a lemma provided,a sliding mode controller is proposed to stabilize the considered system.At the same time,the LMIs are provided for the according T-S fuzzy models.