Stability Analysis And Observer Design For Uncertain Fractional Order Systems

In recent years,fractional order systems have attracted increasing interests.The reason is that many real world physical objects can be characterized more precisely by fractional order state equations than by integer order state equations,such as in servo motor systems,viscoelastic materials and signal processing etc.Therefore more and more researchers pay their attention on the research of fractional order systems.Fractional order systems have already become a hot topic of control field.Descriptor fractional order systems have special advantages:descriptor systems can describe real world physical objects better and more directly.However,descriptor fractional order systems are much more complicated than normal systems,because not only stability but also regularity and impulse free need to be taken in account.Furthermore,the controlled object in reality has nonlinear characteristics,T-S fuzzy system is a universal approximator for nonlinear systems so that fractional order T-S fuzzy systems are very important in studying fractional nonlinear problems.This paper focuses on the observer design of uncertain fractional order systems.This paper is organized as follows:(1)The origin and the research status of fractional order calculus are systematically introduced.The common functions and basic definitions of fractional order systems,descriptor fractional order systems and fractional order T-S fuzzy systems are introduced.The lemmas used in this paper and the definitions of mathematical symbols are given;(2)Three equivalent necessary and sufficient generalized quadratically stable conditions for the descriptor fractional order system are given.Under the existing stable criterion,the state feedback problem of uncertain descriptor fractional order systems is solved.The observer design problems for uncertain descriptor fractional order systems are studied under two cases of fractional derivatives with 0<?<1 and 1<?<2,respectively;(3)An uncertain fractional order nonlinear system is described by using the T-S fuzzy model.The observer designs for uncertain fractional order T-S fuzzy systems are presented under two cases of fractional derivatives with 0<?<1 and 1<?<2,respectively;(4)Using T-S fuzzy model to describe uncertain descriptor fractional order nonlinear systems,the observer design for uncertain descriptor fractional order T-S fuzzy systems with fractional derivatives under 1<?<2 is designed.