The Optimal Control Of Fractional Order System

Although fractional calculus have been proposed more than300years,but it was not became a research hotspot and been used in various fields until recent decades.Fractional order systems is described by fractional differential equations system.It has been proved that some systems be modeled by fractional calculus are more accurate and reliable.With the development of fractional order system theory,fractional order system will be applied to wider areas to meet higher and more accurate requirements.Optimization problem of the integer order system will also expand to fractional fileds and become a new hot spot.In this paper,we used the variational method to solve several specific fractional optimal control problem,The main contributions of the thesis can be summarized as follows:First,the existence and uniqueness of the solution of fractional differential equations in the case of fixed initial conditions was discussed in the chapter2.In chapter3,the variational method will be applied to solve the extreme value of the fractional functional,In the basis of solving the extreme value of fractional functionals withe fixed end point,The Euler equation which used to solve the extreme value of the fractional funcional with constraint case is derived.The boundary conditions and the transversality conditions which used to solve the extreme value of the fractional functional with free endpoints was deduced and proved.Second,the fractional functionals variational theory and knowledge which has been mentioned above will be used in the actual control systems to solve the optimal control problem which its fractional system indicator function with fixed endpoints,free terminal endpoints and function constrained end-point.The state equation,adjoint equation,control equation,transversality conditions and terminal constraint equation is obtained and proved in this chapter.In the last,he knowledge of the fractional linear quadratic optimal control was introduced in the beginning.The method which used to solve the optimal control of fractional linear quadratic was gived, Then we use the Oustaloup’s recursve filtering method to derive the Riccati equations for the LQR of fractional order system. the method of solving the optimal control problem for integer-order systems is extended into fractional-order in this paper. Several solving fractional optimal control theory and methods was obtained in this article.The process of proof was very detailed and sufficient.It was a very useful exploration of the optimal control of ractional order systems.