| Fractional order calculus is the promotion of traditional integer order calculus. It promotes the order of calculus from the integer field to the real field, even the complex field. The theory of fractional order calculus and integer order calculus were proposed at almost the same time for300years. With the rapid development of the numerical calculation method, rational approximation method and computer technology, the research of fractional calculus theory has greatly promoted. Nowadays, the fractional calculus also is gradually applied into more and more research fields. In control field, application researches of the fractional calculus theory in control systems are getting more attention, because some control plants are with non-integer order characteristics, and fractional order controller has different performance with conventional controller in the last decades. In control fields some researcher subjects on fractional order calculus theory involve fractional order system modeling, fractional order controller designing and implementing, fractional order system identification, and fractional order filter, etc.Approximation and numerical implementation methods of the fractional calculus operator are preconditions of some research issues related to fractional order control. Therefore usual approximation methods of fractional calculus operator are reviewed in greater detail firstly, and two methods of fractional calculus operators, best rational approximation and joint optimal rational approximation, are proposed in this paper. In this basis, this dissertation makes a thorough research about design and digital implementation methods of the fractional order controller. Finally, some control experiments with different fractional order controllers were conducted on the AC motor control platform based on LabView, and the performance of fractional order controller in suppressing the motor vibration is analyzed from different aspects.The main researches of this paper are as follows:1. Research on the approximation methods of common fractional order calculus operators. Because that the fractional order operator is an irrational function of complex variable s and it can’t be directly implemented in the numerical simulation and practical application, so fractional order operator should be rational approximated. The process, that fractional order operator was transferred from frequency domain to discrete time domain and to be rationalized, is called operator discretization. There are two discretization methods:direct discretization and indirect discretization. This paper systematically described the direct discretization process of fractional order calculus operator with Euler generating function, Tustin generating function, Al-Alaoui generating function and combined with continued fraction expansion, and plotted the corresponding frequency domain approximation curve and analyzed the approximating characteristics. With Oustaluop method this paper shows the thinking of indirect discretization method, and with the sample of fractional order filter studied the selection of approximation order.2. Put forward the best rational approximation and the joint optimal rational approximation methods of fractional-order operator. Based on the theory of rational approximation and logarithmic amplitude-frequency characteristic asymptotes ideological, definition of two approximation concept of fractional-order calculus operators which named the best rational approximation and the joint optimal rational approximation are given in S-domain. Then proposed a construction method of rational approximation function of the fractional-order calculus operators and validated the effectiveness of rational approximation function of constructed fractional order calculus based on the two approximation method. The analysis of the amplitude frequency characteristics and phase frequency characteristics show that the proposed method is effective; the simulation is simple and convenient. And can be analyzed with the related analysis method of linear control theory, the methods can be easily and effectively applied to the fractional-order controller design, analysis and implementation.3. The design and implementation of fractional order PIαDβ controller. The fraction order PIαDβ controller has two extra adjustable parameters and valued in the interval [0,1] then the integer order PID controller, so the turning of fractional order PIαDβ controller is much flexible and convenient, and can be better adapted to controlled objects with different characteristics to get better control effect. Direct discretization method (Al-Alaoui+CFE) and indirect discrete method (joint optimal rational approximation method) based on fractional-order calculus operators are used to design and implement the fractional order PIαDβ controller, and neural network method was adopted to adjust the parameters of fractional order PIαDβ controller for the AC motor speed control system.4. Analysis of vibration spectrum characteristics of AC motor under VVVF. The harmonics contained in the output voltage of the inverter will affect the output torque of AC motor. So the analysis of the harmonic of the output voltage of the inverter, analysis of the harmonic torque of asynchronous motor and the comparative analysis of vibration spectrum of the motor driven by inverter and regulator, are helpful for the comprehensive understanding of the vibration characteristics of motors, and provide a reference for the more effective vibration control strategies and algorithms seeking.5. Analysis of vibration suppression performance of fractional order PIαDβ controller. New methods for vibration suppression of motor were investigated from the perspective of the control law, and control experiment was executed on the VVVF System of AC motor control platform based on LabView with the designed fractional controller. In the AC motor speed control system controlled by integer order PID controller and fractional order PIαDβ controller, the spectrum of stator current and vibration data sampled at different speed were analyzed. By observing the spectrum characteristics of stator current and vibration, the difference result of vibration suppression to the AC motor of these two controllers is analyzed and the mechanism of vibration suppression that the fractional order controller did to the AC motor is revealed. |
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