Pole Placement Based On Fractional Order PI~λD~μ Controllers

Fractional order calculus is an extended theory of integer order calculus, Fractional order calculus equation can more accurately describe many processes in engineering. Currently, the research of the parameters setting for fractional order controller achieves many results, which laid the theoretical foundation to apply in engineering. But the theoretical research on pole placement for fractional order systems is not studied completely, which has the theoretical and practical significance. The method of pole placement with fractional order controller to realize the controller parameter setting is a new hot topic.This dissertation studies the pole placement with fractional order PI~λ controller mainly. Based on the elementary fractional transfer functions, the pole placement with fractional order PI~λ controller is investigated, as well as the pole placement with fractional order PI~λ controller for time-delay systems. The main works of this dissertation are as follows:(1) Characteristic polynomial P(s) of fractional order control system is a fractional order, which is not easy to determine the stability in classical control theory. Through a transformation from s plane to w plane, stability criterion of characteristic polynomial P(w) can be used to judge the stability. The influences of the fractional order and the simulation step size on the system performances are studied and analyzed for fractional order PI~λD~μ controlled system, and the research provide a reliable basis for parameters setting.(2) The simple pole placement method with fractioned PI~λ controllers is researched based on the elementary fractional transfer functions. The elementary fractional transfer functions of the first kind is taken as the controlled process and the elementary fractional transfer functions of the second kind as the desired closed transfer function. Select the parameter appropriately of kp and ki, so as to obtain the closed-loop poles in the stability area of the complex plane. Simulation results shows that when the fractional order v taking any non integer in range 0< v< 2, the step response using fractional PI;’controller is superior to that using fractional I’P controller in the fractional case.(3) This chapter discusses pole placement using parameter space approach for time-delay systems. The damping ratio sector region and the relative stability region in s-plane, which form a trapezoid region in the left-half of s-plane, are mapped into the controller parameters space. Thus, the corresponding controller parameters can place all closed-loop poles in a specified trapezoid region, and guarantee the performances of the closed-loop systems. The simulation example shows that fractional order controller can obtain better attenuation characteristic and the damping characteristics than integer order controller.The first innovation of this dissertation is the simple pole placement method with fractioned PI~λ controllers based on the elementary fractional transfer functions. The second innovation of this dissertation is pole placement using parameter space approach for time-delay systems. In the latter case, from the simulation examples, the choice of parameters of fractional order PI~λ controller has much flexibility, and the method does not have any constraints for fractional order.