Improving integral square error performance with implementable fractional-order PI controllers

In this paper, an algebraic rule for tuning the integer realizations of fractional‐order PI controllers is developed, with an integral square error performance index, which outperforms that of an optimal ordinary PI controller. To this end, the PI^λ control structure is used in conjunction with a third‐order integer approximating filter to provide a three parameter fixed‐structure extension of the ordinary PI controller. Next, the extra degree of freedom in setting the order of integration λ is leveraged to introduce a steepest descent direction in the extended controller parameter space. It is then stated that shifting the parameters of an ordinary PI controller along the proposed descent direction will result in a fractional‐based three parameter controller with a performance index, which is superior to that of the original PI controller. The stability of the controller parameters derived in this manner is then analyzed, and examples and simulation results are offered to verify the theoretical expectations and analyses.