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In this paper we are concerned with the design of a Fractional-order PID controller which involves noninteger-order integrator and differentiator. The three controller gains and two real orders of the fractional-order PID controller are determined to minimize an integral square error (ISE)performance index while satisfying the specified gain and phase margins. We formulate the design problem as a constrained parametric optimization and apply a differential evolution algorithm to search globally the optimal controller parameters. Due to the problem lack of analytic time-domain analysis methods for fractional-order systems, and effective numerical method is utilized to compute the ISE performance index as well as to check the stability in the frequency domain. Design examples are given to compare the performance of the optimal fractional-order PID controller with the optimal integer-order PID controller in controlling integer-order as well as fractional-order processes.
Leu et al. (Fri,) studied this question.