Fractional Order PID Controllers And Analysis Of Stability Region For Fractional Order Systems With Uncertain Parameters

Fractional-order calculus is a generalized form of traditional integer order calculus, which is birth with the same time as the integer order calculus. It has same conception and analysis tools of mathematics with the traditional integer calculus. Fractional order control system has different expression methods in time domain, frequency domain and complex domain. The basic analysis methods of fractional order control system have also included the stability condition, observability, controllability and son on. All these analysis methods are same as the integer order control system. With the deeply studying the fractional order calculus in the control theory, the fractional order controllers and fractional order plants are proposed. The fractional order PIλDμcontrollers are extend of traditional integer order PID controllers. It is very important that the fractional order PIλDμcontroller has excess two parametersλandμthan the integer order PID controller, which can provide better dynamic performance and robustness than the older one. However, due to the immaturity of fractional order control theory, there are still many aspects and key points need to be improved for the fractional order controllers and the stability of fractional order control systems. In this dissertation, the design of fractional order PIλDμcontrollers and analysis of the stability for the fractional order control systems with parameters uncertainties are discussed. The main research work of this dissertation can be described as follows:1. This chapter illustrates an overview of the development, foundation theory and characteristic of fractional order calculus, and summarizes the application of fractional order calculus in the control system, the design methods of fractional order PIλDμcontrollers and the analysis methods of stability for fractional order interval plants using fractional order PIλDμcontrollers.2. In this chapter, the performance of control systems were analyzed by change the order parameters and simulation step for fractional order systems and fractional order PIλDμcontrollers. Firstly, the change of simulation step and differential order were produce the effect for control system by numerical solution of fractional order differential equation. Secondly, when the integral orderλand differential orderμwere changed in the range (0,2) for the fractional order PIλDμcontroller, it will influence frequency characteristic and step response performance for the fractional order control system. The analysis of frequency characteristic are accord with the result of practical step response, which shown the integral orderλand differential orderμof fractional order controller have the better range. 3. Using the robust stability conditions, the fractional order PIλDμcontrollers are presented for plant uncertainties with time delay. The fractional order PIλDμcontroller with five parameters is present for the time constant of plant changed in the interval, when the control system fulfills the robust conditions of phase margin and amplitude margin. At the same time, the fractional order PIλDμcontroller is also design for the gain constant of plant changed in the interval, when the control system fulfills the robust conditions of phase margin and the phase forced to be flat at cut off frequency in the Bode plot.4. A new algorithm was presented to solve the stability region of the parameter uncertainties system with time delay using fractional order PIλDμcontroller. Firstly, it is use the Kharitonov theorem to decompose parameter uncertainties system into several subsystems. Secondly, D-decomposition method was applied to solve stability region of each subsystem. The values ofλandμare chosen respectively according the biggest stability region which the fraction order PIλD and PλDμcontrollers obtain for each subsystem. The new fraction order PIλDμcontroller was constructed by the values ofλandμ. Thirdly, the stability region of each subsystem was plotted by using new fractional order PIλDμcontroller. Finally, the intersection of stability region of each subsystem is stability region of parameter uncertainties system with time delay.5. The design method of the robust fractional order PIλcontroller was pre-sented by computing the stability region of fractional order parameter uncertain system. Firstly, the Kharitonov theorem is adopted to decompose the fractional order param-eter uncertain system into several subsystems of parameter certainties. Secondly, the D-decomposition technique is applied to compute the stability region of each subsystem and obtain the A value which can take the bigger stability region for each subsystem. Thirdly, the fractional order PIλAcontroller is constructed by the A value. The stability region of fractional order PIλcontroller for each subsystem is computed in the plane of (kp, kI). The intersection of stability region of each subsystem is parameters set of fractional order PIλcontroller. By this way, all parameters of fractional order PIλcontroller can be obtained by selecting in this stability region.6. The method of computing the robust stability region of fractional order interval plant with time delay by using fractional order PIλDμcontroller was presented. First, the edge theorem is adopted to decompose interval quasi-polynomial of fractional order close loop control system to several vertices quasi-polynomials. The exposed edges of polytope were constructed by vertices quasi-polynomial. Secondly, D-decomposition technique is applied to solve stability region of each vertices quasi-polynomial. The values ofλandμare chosen respectively according the biggest stability region which the fraction order and controllers obtain for each vertices quasi-polynomials. Third, the stability region of each vertices quasi-polynomial is plotted in the plane of (kp, kI) by using fractional order PIλDμcontroller. Furthermore, the overlap of stability region of each vertices quasi-polynomial is stability region of fractional order interval plant with time delay. The parameters of fractional order PIλDμcontroller are chose arbitrary in the intersection of stability. The robust stability can be checked by value set theorem and zero exclusion principle.7. The fuzzy self-adaptive fractional order PIλDμcontroller was presented for the fractional order plant by combine the fractional order control theory and fuzzy control principle. The operators of derivative and integral of fractional order equation replaces the units of derivative and integral in the classical PID controller, which construct the fractional order PIλDμcontroller. The numerical evaluation of fuzzy self-adaptive fractional order PIλDμcontroller was realized by fuzzy logic and numerical solution of fractional differential equation.In final, the main research work is summarized. As a whole, the conclusions derived from this dissertation is illustrated. The innovation points and the future outlook of the research work is given.