In this paper, the design of fractional-order PID controller is considered in order to minimize certain performance indices such as integral absolute error, integral square error and integral time absolute error. The design-construction leads to a high-dimensional, multi-modal, complex optimization problem which is difficult to solve analytically. We show that it can be solved heuristically using an artificial bee colony (ABC) algorithm, which is a recently emerged 'stochastic' technique inspired from the intelligent foraging behavior of honey bee swarm. For the numerical examples under consideration, we further compare the performance of ABC with a 'deterministic' Nelder-Mead simplex algorithm.

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In this paper, a constrained optimization problem is formulated to tune the limit cycle minimizing controllers meeting additional loop-shaping performances such as phase margin and gain crossover frequency. A graphical approach is proposed so as to determine the superior controller in terms of better limit-cycle suppression. The framework is illustrated with a suitable case of elementary servo plant which has separable static backlash nonlinearity in its model. For this plant, integer-order controllers and their fractional counterparts (PI and PI α , [PI] α ; PID and PI α D β ) are designed and compared. Interestingly, it is found that the fractional controllers produce better limit-cycle responses than their integer counterparts while both meeting the rest of the specifications. Correspondingly, the better sustained oscillations in the plant output response are obtained with fractional controllers. Such a 'fractional superiority' is further verified with the closed-loop nonlinear simulation.

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