Delay Robustness of PID Control of Second-Order Systems : Pseudo-Concavity, Exact Delay Margin, and Performance Trade-Off

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Original languageEnglish
Journal / PublicationIEEE Transactions on Automatic Control
Online published12 Feb 2021
Publication statusOnline published - 12 Feb 2021

Abstract

In this paper we study delay robustness of PID controllers in stabilizing systems containing uncertain delays. We consider second-order systems and seek analytical characterization and exact computation of the PID delay margin, where by PID delay margin we mean the maximal range of delay values within which the system can be robustly stabilized by a PID controller. Our primary contribution is threefold. First, we show that the delay margin achieved by PID control coincides with that by PD controllers. Second, we show that the PID delay margin can be computed efficiently by solving a pseudo-concave unimodal problem, that is, a univariate optimization problem that admits a unique maximum and hence is a convex optimization problem in one variable. Finally, we demonstrate analytically the trade-off between achieving delay margin and tracking performance, showing that for several canonical performance criteria, integral control reduces the delay margin. These results lend useful insights into the PID control of delay systems, and useful guidelines in the tuning and analytical design of PID controllers.

Research Area(s)

  • Delay margin, nonlinear programming, PID controller, pseudo-concavity, robust stabilization