Delay Robustness of PID Control of Second-Order Systems: Pseudoconcavity, Exact Delay Margin, and Performance Tradeoff

In this article, 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 pseudoconcave unimodal problem, i.e., a univariate optimization problem that admits a unique maximum and, hence, is a convex optimization problem in one variable. Finally, we demonstrate analytically the tradeoff 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.