Stability Analysis And Controller Design Of Fractional Order T-S Fuzzy Systems

Since its birth in 1695,fractional order calculus,as a important branch of mathematics,has been continuously studied.Because a lot of realistic systems can be modelled by the fractional order systems more accurately.In recent years,fractional order calculus is also more and more applied in many different fields of science and engineering,such as fluid mechanics,image processing,seismic analysis of viscoelastic damper,fractional order PID controller of power network,the fractal signal processing and control on different fields.Due to the mathematical properties of the fractional order calculus itself,the fractional order derivative equation of numerical computation complexity is relatively high,but with the improvement of computer computing ability,fractional order systems research are getting more and more widely and deeply.At present,the study of the fractional order systems involves many aspects,but the theory of fractional order T-S fuzzy systems research is still relatively small,therefore,in this article stability and stabilization problems of fractional order T-S fuzzy systems are studied and discussed.In this paper,study of fractional order systems is based on a new linear matrix inequality(LMI)criterion,with fractional order T-S fuzzy systems as the object,mainly includes the fractional order T-S fuzzy systems stability problem and state feedback controller and output feedback controller design problems,gives stability and stabilization conditions of the related models.In addition,based on the conditions of stability and stabilization,strictly feasible solutions can be obtained,state feedback controllers and output feedback controllers can be solved.Simulation examples are given to verify the validity and feasibility of the proposed conditions and controllers.The main contents of this paper are introduced as follows:Firstly,this paper introduces development history and research status of the fractional order calculus and fuzzy systems.The basis of relevant knowledge of fractional order calculus are introduced in detail.The main theories of fractional order calculus are introduced.Briefly,the relationships between the different definitions of fractional order calculus are illustrated.Three generalized functions are introduced:Mittag-Leffler function,Beta function and Gamma function.These three functions are needed in the operation of fractional calculus.Secondly,the stability problems of fractional order T-S fuzzy systems are studied.Mainly using linear matrix inequality criterion of the fractional order systems,the criterion is extended to the fractional order T-S fuzzy systems to study the stability problems of fractional order T-S fuzzy systems.Asymptotic stability conditions of fractional order T-S fuzzy systems are given which are sufficient conditions.Then stability conditions of fractional order T-S fuzzy systems with nonlinear uncertain parameters are given.Third,stabilization problems of fractional order T-S fuzzy systems are studied.The stabilization conditions of fractional order T-S fuzzy systems and fractional order T-S fuzzy system with nonlinear uncertainties are given.Using the method of parallel distributed compensation(PDC),according to the stabilization conditions of fractional order T-S fuzzy systems and uncertain fractional order T-S fuzzy system,the fuzzy state feedback controllers are solved with linear matrix inequality method.If feasible solutions are obtained,it indicates that state feedback controllers solved by stabilization conditions can asymptotically stabilize the unstable systems.Fourth,output feedback control problems of the fractional order T-S fuzzy system and uncertain fractional order T-S fuzzy system are studied.Using the method of parallel distributed compensation,fuzzy static output feedback controllers and dynamic output feedback controllers are designed for the fuzzy subsystems and output stabilization conditions of fractional order T-S fuzzy system and uncertain fractional order T-S fuzzy system are given respectively.Based on these stabilization conditions,coefficient matrix inequalities are solved.If feasible solutions are obtained,it indicates that output feedback controllers solved by stabilization conditions can asymptotically stabilize the unstable systems.Simulation platform based on MATLAB/SIMULINK is built.The detailed simulation results are provided,and the results are analyzed from the theoretical foundation.The simulation results prove the availability of the given criteria.