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A proposal for optimal tuning of fractional order proportional integral-proportional derivative PIα-PDß controllers

In this paper we propose a fractional PI<sup>α</sup>-PD<sup>ß</sup> controller that tuned with integral performance criterion. The orders of the integral and derivative parts, α and ß, respectively, are fractional. This controller is a generalization of a conventional PI-PD controller. This expansion could provide much more flexibility than a conventional PI-PD controller design. The tuning method is based on the solution of an optimization problem. For setting the six parameters of the fractional PI<sup>α</sup>-PD<sup>ß</sup> controller, the proposed method is based on the minimum integral squared error (ISE) criterion with a minimum control effort. The integral criterion is calculated by using Hall-Sartaurius method. Two cases of study are given; the results demonstrate that gains and orders optimization leads to better transient performance of the proposed fractional PI<sup>α</sup>-PD<sup>ß</sup> control structure.