Research On Robust Control Of Nonlinear Fractional Order Systems

Fractional calculus is a generalization of the classical integration and differentiation operators to non-integer order.Because of the unique characteristics of historical memory,the dynamic process of a real problem can be described more accurately by fractional order systems.Meanwhile,most systems in reality are highly coupled nonlinear systems and there are always dynamic uncertainties in the modeling.These uncertainties reduce the performance of controllers designed based on the accurate mathematical models.Therefore,it is necessary to study the robust control of the uncertain nonlinear fractional order systems.In this thesis,the robust control of a nonlinear fractional order system by designing a controller for fractional T-S fuzzy system is investigated.Firstly,a fractional T-S fuzzy system is obtained by using the fuzzy modeling method which is composed by several linear fractional order subsystems.The state feedback controller is designed for each linear fractional order subsystem to make it stable.Then,based on the thought of fuzzy weighting,the stability conditions and the controller design method of an uncertain fractional order T-S fuzzy system with incomplete measurable state are given.The main contents of the thesis are as follows:1)The modeling problem for a class of nonlinear systems is investigated.Based on sector nonlinearity,a T-S fuzzy modeling algorithm is proposed.Firstly,the maximum and minimum values of the nonlinear term in the system are determined.Secondly,according to the extreme values,the membership functions are derived.Finally,a T-S fuzzy model is established.The algorithm is simple and easy to implement.Meanwhile,simulation results show that the proposed algorithm can effectively achieve the approximation of the original system.2)For a nonlinear uncertain fractional order system with 0<?<1,the introduction of robust control theory of linear fractional order system is introduced.Then,associated with bounded input and bounded output stability theory and the properties of Hermite matrix,the stability criterion of the system and the observer-based state feedback controller design method are given for the fractional order T-S fuzzy system with incomplete measurable state.Finally,the effectiveness of the approach is verified by some numerical examples.3)For a nonlinear uncertain fractional order system with 1??<2,because of the stability region is different from the system with 0<?<1(the former is convex region),the T-S fuzzy model of nonlinear fractional order system is established.Then,considering the incomplete state of the system,the Luenberger-type observer is constructed for each subsystem.Based on the related theories of linear fractional order systems and using parallel distributed compensation theory,a robust controller based on the observer is proposed.Then,a sufficient condition for the stability of fractional T-S fuzzy systems is given in the form of linear matrix inequalities.By means of singular value decomposition,the observer and controller's parameters are obtained.Finally,numerical simulation examples are given to illustrate the effectiveness of the results.4)Using GUI tool of MATLAB,a visual interface is designed to show the main contents of the thesis.