Calculation of robustly relatively stabilizing PID controllers for linear time-invariant systems with unstructured uncertainty

This article deals with the calculation of all robustly relatively stabilizing (or robustly stabilizing as a special case) Proportional–Integral–Derivative (PID) controllers for Linear Time-Invariant (LTI) systems with unstructured uncertainty. The presented method is based on plotting the envelope that corresponds to the trios of P–I–D parameters marginally complying with given robust stability or robust relative stability condition formulated by means of the H∞ norm. Thus, this approach enables obtaining the region of robustly stabilizing or robustly relatively stabilizing controllers in a P–I–D space. The applicability of the technique is demonstrated in the illustrative examples, in which the regions of robustly stabilizing and robustly relatively stabilizing PID controllers are obtained for a controlled plant model with unstructured multiplicative uncertainty and unstructured additive uncertainty. Moreover, the method is also verified on the real laboratory model of a hot-air tunnel, for which two representative controllers from the robust relative stability region are selected and implemented. [Display omitted] •Calculation of robustly relatively stabilizing PID controllers is introduced.•Controlled plants are in the form of LTI systems with unstructured uncertainty.•Boundaries of H∞ norm-based robust relative stability conditions are utilized.•The regions of robust relative stability are plotted in a P–I–D space.•Robustly (absolutely) stable controllers can be obtained as a special case.