2020/09/07 by Pieter Appeltans, Appeltans, Pieter, Silviu‐Iulian Niculescu +3 · 1 citation
Engineering · #93B35 #93B52 #93C23 #93D15 #93D22 #Advanced Control Systems Design #Advanced Control Systems Optimization #Extremum Seeking Control Systems #FOS: Mathematics #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2009.02924
openalex publication_date 2020/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper presents the analysis of the stability properties of PID\ncontrollers for dynamical systems with multiple state delays, focusing on the\nmathematical characterization of the potential sensitivity of stability with\nrespect to infinitesimal parametric perturbations. These perturbations\noriginate for instance from neglecting feedback delay, a finite difference\napproximation of the derivative action, or neglecting fast dynamics. The\nanalysis of these potential sensitivity problems leads us to the introduction\nof a `robustified' notion of stability called \strong stability, inspired\nby the corresponding notion for neutral functional differential equations. We\nprove that strong stability can be achieved by adding a low-pass filter with a\nsufficiently large cut-off frequency to the control loop, on the condition that\nthe filter itself does not destabilize the nominal closed-loop system.\nThroughout the paper, the theoretical results are illustrated by examples that\ncan be analyzed analytically, including, among others, a third-order unstable\nsystem where both proportional and derivative control action are necessary for\nachieving stability, while the regions in the gain parameter-space for\nstability and strong stability are not identical. Besides the analysis of\nstrong stability, a computational procedure is provided for designing strongly\nstabilizing PID controllers. Computational case-studies illustrating this\ndesign procedure complete the presentation.\n