Optimal tuning rules of the fractional-order PID controllers with application to first-order plus time delay processes

The aim of this paper is to propose two sets of formulas for optimal tuning the PI^λD^μ controllers such that either the ISE or the ISTE performance index (corresponding to the tracking error of the unit step command) is minimized. The proposed formulas are parametric, simple, effective and easy-to-use, and moreover, they can be applied to all processes that can be modelled with a first-order plus time delay transfer function. The tuning rules are obtained by using the so-called dimensional analysis which avoids the complicated numerical optimizations. Two numerical examples are presented which confirm the effectiveness of the proposed tuning formulas.