Maximization of Gain/Phase Margins by PID Control

Proportional-integral-derivative (PID) control has been the workhorse of control technology for about a century. Yet to this day, designing and tuning PID controllers relies mostly on either tabulated rules (Ziegler–Nichols) or on classical graphical techniques (Bode). Our goal in this article is to take a fresh look on PID control in the context of optimizing stability margins for low-order (first- and second-order) linear time-invariant systems. Specifically, we seek to derive explicit expressions for gain and phase margins that are achievable using PID control, and thereby gain insights on the role of unstable poles and nonminimum-phase zeros in attaining robust stability. In particular, stability margins attained by PID control for minimum-phase systems match those obtained by more general control, while for nonminimum-phase systems, PID control achieves margins that are no worse than those of general control modulo to a predetermined factor. Furthermore, integral action does not contribute to robust stabilization beyond what can be achieved by PD control alone.

I. Introduction

The primary goal of feedback regulation is to maintain stability and performance in the presence of modeling uncertainty and external disturbances. Amongst the metrics that are commonly used to quantify robustness against such factors, traditionally, the most important have been gain and phase margins, various types of induced norms (, ), the gap metric, the structured singular value, and so forth. Each of these, and a few others, have been the subject of respective chapters in modern robust control literature. Herein, we focus on gain and phase margins that historically have been the first to be considered. Interestingly but perhaps unsurprisingly, these same metrics have also been the first to be tackled in the waning years of the 1970s with the modern tools of analytic function theory that gave rise to optimal designs [1], [2], [3]; see also [4], [5], [6] as well as [7, Ch. 11].