Fractional Order PD~? Controller And Analysis Of Robustness For Fractional Order System

The fractional calculus theory makes the traditional integer order calculus theory popularized and generalized in the real field.Fractional order control theory is a new research direction based on fractional calculus theory and fractional order differential equations(FODEs).The importance of the fractional order control theory is to generalize the traditional integer order control theory,and help to establish the mathematical model more completely.Compared with the integer order control system,the fractional order control system can describe the real physical system more accurately.So,the fractional order model is called a mathematical model for describing natural phenomena.This thesis firstly introduces the development and the current situation of the fractional calculus theory.The design of the fractional order PD~? controller and the stability of the fractional order systems are discussed.Also the robustness problems related to the fractional order control systems are discussed simultaneously.The main contents of this thesis are as follows:For the fractional order plant with time delay,according the three robustness specifications,the parameters of the fractional order PD~? controller are calculated.So the fractional order PD~? controller has been obtained to satisfy the design requirement.And then,the Oustaloup recursive filter is employed to estimate the fractional order plant and the fractional order PD~? controller which has be obtained already.Simulation experiments show that the fractional order PD~? controller has better control performance and stability than the integer PD controller for the fractional order system,and the fractional order control system fulfills the robustness conditions.For the fractional order parameter uncertainties system with time delay,a method for solving the parameter stability regions of the fractional order PD~? controllers is proposed.Firstly,the parameters uncertain system is divided into a number of subsystems with certain parameters by using the Kharitonov theory.Secondly,D-decomposition method was applied to solve stability region of each subsystem.Thirdly,the value of ? is chosen according the biggest stability region which the fractional order PD~? controller obtain for each subsystem.The new fractional order PD~? controller was constructed by the value of ?.Finally,the intersection of stability region of each subsystem is stability region of parameter uncertain system with time delay.Through the Matlab simulation example,it is obviously obtained that the stability region of the fractional order PD~? controller is much bigger than the integer one.The step response curves of the control system corresponding fractional order PD~? controller which selects the parameters in the stability region,is convergent with a small overshoot and has a good stability.The fractional order PD~? controller for the fractional order parameter uncertain system with time delay has a strong robustness.For the fractional order plant with time delay,by the relationship of the sensitivity function with the amplitude margin and phase margin,the method for parameter tuning of the fractional order PD~? controller is studied.The maximum sensitivity function index is a designed requirement for all frequencies,rather than only for a certain frequency range.So,it is more applicable than the robustness constraints of amplitude margin and phase margin.The Matlab simulation shows that the fractional order PD~? controller can obtain a better dynamic performance for the fractional control system than the integer order PD controller under the same maximum sensitivity constraint.Finally,the main work of the thesis is summarized,and the main innovation points of this work are stated clearly.And the future research work is prospected.