Tradeoff between Delay Robustness and Tracking Performance by PID Control: Second-Order Unstable Systems

In this paper we study delay robustness and tracking performance of PID controllers in stabilizing systems containing uncertain delays. In order to verify the potential tradeoff between the delay robustness and the tracking performance clear, we first propose a tracking performance assumption of concerned PID controllers. Next, we examine second-order unstable time-delay systems and seek the analytical characterization and exact computation of the delay margin. The delay margin defines the maximal range of delay within which the system can be robustly stabilized by a PID controller. In this paper, our contribution is twofold. First, we develop the analytical expressions, the efficient computation, and the explicit upper bounds of the PID delay margin achievable. The results also provide a well interpretation of the roles of three components of PID control on achieving the delay margin. Second, we prove that the higher the tracking accuracy is, the smaller a delay margin can be achieved, explaining well the tradeoff between delay robustness and tracking accuracy of PID controllers.