Research On Approximation Method Of Fractional Calculus Operator And Its Applications

Fractional calculus is a generalized calculus, the order of which is extended from integer domain to the field of real numbers, even to the complex field. With the rapid development of rational approximation method, numerical computation method, integrated circuit and computer technology, fractional calculus theory has been greatly researched and permeated to other important application fields. In control area, since some control plants have certain properties of non-integer order and it has existed a fact that fractional order controller has better control performance than that of integer controller, in recent decades, more and more attention are paid to the research of fractional calculus theory in control field. The research on fractional calculus theory in control area mainly covers fractional system modeling,identification, analog operational circuit, fractional controller design and fractional filter.The research on approximation and numerical implementation of fractional calculus operation is the premise on the issue of fractional order control. Therefore, the paper firstly studies the direct discretization and indirect discretization method of fractional operator in z domain, and approximation method in s domain. Then it constructs the approximation function of fractional order operator based on approximation theory in s domain to propose a practical method, which can adjust approximation error and approximation region.The paper also realizes the actual fractional physical device and circuit. It includes the realization of fractance, fractional integrator and differentiator. The existed design circuit defects of fractional calculus have been analyzed and the improved circuit has been proposed and designed. It adopts a method of constructing the rational approximation function in the form of continued product to design fractance and fractional integrator and differentiator. By simulation tool of Multisim 10, it illustrates the rationality of the proposed fractional circuit.Based on the design method of constructing the rational approximation function in the form of continued product, it indirectly discretizes the fractional operator and then the implementation of fractional order PID controller is achieved. In the process of designing fractional order PID controller, it will be involved in tuning parameter problem. This paper gives two kinds of tuning method. The first method is based on vector method to tuning the controller parameters for simplifying the solving process and computation. Another method is an parameter tuning method of enhanced robust controller. The simulations show that the approximation method for fractional calculus operator is effective and the fractional order PID controller system based on the two tuning method has great control performance.The innovation of the paper is as follows. Firstly, it gives the rational approximation algorithm and applies the approximation algorithm to design fractional order PID controller.It also gives two kinds of new tuning method. Those are tuning the controller parameters for simplifying the solving process and computation based on vector method and an parameter tuning method of enhanced robust fractional order controller.