Exact Computation of Delay Margin by PID Control: It Suffices to Solve a Unimodal Problem!

In this paper we study delay robustness of PID controllers in stabilizing systems containing uncertain, variable delays. We consider second-order unstable 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 contribution is threefold. First, we show that the delay margin achieved by PID control coincides with that by PD controllers. Second, we show that other than helping stabilize the delay-free part of a plant, the proportional control contributes no action to increase the delay margin. Finally, we show that the PID delay margin can be computed efficiently by solving a unimodal problem, that is, a univariate optimization problem that admits a unique maximum and hence is a convex optimization problem in one variable. This unimodal problem is one of pseudo-concave optimization and hence can be solved using standard convex optimization or gradient-based methods. As such, from a computational perspective, the PID delay margin problem is completely resolved in this paper! The results not only insure that the PID delay margin problem be readily solvable, but also provide fundamental conceptual insights into the PID control of delay systems, and analytical justifications to long-held engineering intuitions and heuristics, thus lending useful guidelines in the tuning and analytical design of PID controllers.