Analysis and Design of Strongly Stabilizing PID Controllers for Time-Delay Systems

TL;DR

This paper analyzes the stability of PID controllers for systems with time delays, introduces a robust stability concept called strong stability, and provides a computational method for designing controllers that ensure this enhanced stability.

Contribution

It introduces the concept of strong stability for time-delay systems and offers a computational procedure for designing PID controllers that achieve this robustness.

Findings

Strong stability can be achieved by adding a low-pass filter.

Regions of stability and strong stability differ in parameter space.

Theoretical results are supported by analytical examples.

Abstract

This paper presents the analysis of the stability properties of PID controllers for dynamical systems with multiple state delays, focusing on the mathematical characterization of the potential sensitivity of stability with respect to infinitesimal parametric perturbations. These perturbations originate for instance from neglecting feedback delay, a finite difference approximation of the derivative action, or neglecting fast dynamics. The analysis of these potential sensitivity problems leads us to the introduction of a `robustified' notion of stability called \emph{strong stability}, inspired by the corresponding notion for neutral functional differential equations. We prove that strong stability can be achieved by adding a low-pass filter with a sufficiently large cut-off frequency to the control loop, on the condition that the filter itself does not destabilize the nominal closed-loop system. Throughout the paper, the theoretical results are illustrated by examples that can be analyzed analytically, including, among others, a third-order unstable system where both proportional and derivative control action are necessary for achieving stability, while the regions in the gain parameter-space for stability and strong stability are not identical. Besides the analysis of strong stability, a computational procedure is provided for designing strongly stabilizing PID controllers. Computational case-studies illustrating this design procedure complete the presentation.