Maximal Gain and Phase Margins Attainable by PID Control

In this paper, we study the gain and phase margins achievable by PID controllers in stabilizing linear time-invariant (LTI) systems. The problem under consideration amounts to determining the largest ranges of gain and phase variations such that there exists a single PID controller capable of stabilizing all the plants within the variation ranges. We consider low-order systems, notably the first- and second-order systems. For each class of these systems, we derive explicit expressions of the maximal gain and phase margins achievable. The results demonstrate analytically how a plant's unstable poles and nonminimum phase zeros may confine the maximal gain and phase margins attainable by PID control, which lead to a number of useful observations. First, for minimum phase systems, we show that the maximal gain and phase margins achievable by PID controllers coincide with those by general LTI controllers. Second, for nonminimum phase systems, we show that LTI controllers perform no better than twice than PID controllers, in the sense that the maximal gain and phase margins achievable by general LTI controllers are within a factor of two of those by PID controllers, whereas the former is measured on a logarithmic scale and latter on a linear scale. Finally, we show that PID and PD controllers achieve the same maximal margins, indicating that integral control is immaterial in improving a system's robustness in feedback stabilization. These results thus provide useful insights into PID control, and from a system robustness perspective, offer an interpretation on the effectiveness of PID controllers.