Research On Fractional Order Control Systems And Its Application

Fractional calculus is the calculus whose integration or differentiation order is not conventional integer number but real or even complex one. It is extensiton of integer calculus. Farctional order control, which is established on the idea of fractional order operators and the theory of fractional order dieffrential equations, is now a quite new research direction. Practice has proved that better results could be obtained by introduction of fractional calculus in control theory. Fractional calculus provides a powerful support for the expansion of the classic research methods in control theory and a better explaination of the current results.With the need of development of control theory, we take fractional-order calculus as a new tool. Four aspects of research work including fractional-order system analysis, fractional-order system identification, fractional order controller design and fractional order operator approximation are developed in this dissertation. Main contributions are given as follows:(1) Generally, the transfer function of fractional order system is nolonger a rational function of complex variable s, thus the stability analysis of the fractional order system is more complicated than the integer order system's. Two kinds of fractional order systems are analysised. Firstly, for a class of fractional order system which has similar transfer function structure to the traditional first order system, the scope of parameters which can ensure stability of such fractional order system is given and the relationship between time response and fractional order is analyzed. Secondly, for a class of fractional order system which has similar transfer function structure to the traditional second order system, the range of damping ratio is derived for closed-loop stability(time domain). Finally, complements to the existing fractional nyquist criterion and logarithm frequency stability criterion are given and then stability and relative stability is analysised for the second kind of fractional order system using the new complementary stability criterions(frequency domain).(2) The fractional order sysem is not easy to be identified though it can describe physical phenomena more accurately than the integer order one. For a class of fractional order system with known transfer function structure, a method of simultaneously identifying parameters and factional order using PSO (Particle Swarm Optimization) is given. Then, a new fractional transfer function is used to describe the dynamics of main steam temperature of a circulating fluidized bed boiler under the disturbance of spray water. PSO is used to estimate the parameters of the fractional transfer function. Simulation results show that the proposed scheme offers a higher accuracy and adaptiveness to describe the dynamics of the main stea temperature compared with integral models obtained with the same method.(3) Fractional order PIλDμcontroller is the generalization and development of traditional PID controller. The introduction of fractional differential and integral operators makes the PIλDμcontroller more flexible and robust. Aiming at a triple-input triple-output nonlinear boiler-turbine system which has uncertainties and input constraints, a fractional order control system with PIλDμcontrollers is designed. The tuning algorithm of the controller parameters is PSO. Simulation results show that more robustness and adaptability can be obtained compared with traditional PID control.(4) The famous Oustaloup's method is a typical global approximation method which can approximate ideal fractional operator on a given frequency interval by continuous integer order zero/pole transfer function. However, the quality of the Oustaloup's approximation method may not be satisfactory in high and low frequency bands. An evolutionary based approximation method with the same expression as the Oustaloup's method but different coefficients is presented. The coefficients are optimized with PSO. A set of experiments demonstrate that, compared with the Oustaloup's method, significant improvement can be obtained especially near the interested frequency bands.