H-? Norm Based Robustly Stabilizing PI-PD Controller Design for the Unstructured Uncertainty Modelling

Abstract

The design of effective control systems in real-world applications is frequently complicated by system uncertainties and the limitations of conventional controllers. While widely adopted, proportional-integral-derivative controllers often struggle with complex processes such as unstable, integrating, or oscillating systems, particularly in the presence of uncertainties. The proportional integral-proportional derivative (PI-PD) controller structure offers a more robust four-parameter framework that provides superior control capabilities for these challenging systems. However, determining the optimal values or regions for these four tuning parameters remains a significant and complex challenge, especially when systems include various forms of uncertainty. This paper addresses the identification of the region containing all PI-PD controllers that robustly stabilize systems subjected to unstructured uncertainty, using H-? norm robust stability conditions. The proposed study is based on the graphical representation of the nominal stability boundary locus and robust stability boundary locus in the (kd–kf)-plane for the inner loop and the (kp–ki)-plane for the outer loop. The methodology is systematically outlined and illustrated with an example. Furthermore, stability analyses are conducted visually using Nyquist envelopes and uncertainty discs. © King Fahd University of Petroleum & Minerals 2025.