Tradeoff between Delay Margin and Asymptotic Tracking by PID Control

This paper studies the tradeoff between delay robustness and tracking performance of PID controllers in stabilizing systems containing uncertain, variable delays. We consider the first-order unstable system in one of which PID controllers are most commonly used, and then study the corresponding delay margin which means the maximal range of delay within which the system can be robustly stabilized by a PID controller. On the other hand, to characterize the tradeoff concerned, we impose a tracking performance assumption simultaneously of PID controllers. How to seek the analytical characterization and exact computation of the PID delay margin under tracking performance consideration? We answered this question. Our contribution is fourfold. First, we show that the derivative control provides an active effect to enlarge the delay margin. Second, opposite to the derivative control, we show that the integral control which plays a dominated role in maintaining tracking accuracy contributes a negative action to increase the delay margin. Third, we show that the exact computation of delay margin is equivalent to solving a unimodal problem, that is, a univariate convex optimization problem that admits a unique maximum, which hence can be computed efficiently. Finally, we show that the higher the tracking accuracy is, the less the delay margin can be achieved, explaining well the tradeoff between delay robustness and tracking accuracy of PID controllers. Except for answering the above fundamental question and demonstrating the intrinsic tradeoff concerned, moreover, these results lend useful guidelines in the tuning and analytical design of PID controllers in practical implementations.