The thesis contributes to robustness studies of PID controllers in stabilizing linear time-invariant (LTI) systems subject to uncertain gain and phase variations. 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 PID controllers, in the sense that the maximal gain margins achievable by general LTI controllers are within a factor of two of those by PID controllers, whereas the gain margins are measured on the logarithmic scale. Moreover, 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. Finally, we demonstrate that the maximal gain and phase margins can be computed efficiently by solving a univariate convex optimization problem that admits a unique maximum. In this regard, from a computational perspective, the PID gain and phase margin problems can be considered resolved. The results show analytically the dependence of the maximal margins on the pole and zero, indicating how a plant’s unstable pole and nonminimum phase zero may limit the ranges of gain and phase values over which a system is robustly stabilized by PID control.

We next consider specifically filtered PID controllers, out of the necessity in practical implementation of PID controllers. We examine first-order unstable systems and attempt to find analytical expressions and exact computations of the maximal gain and phase margins achievable by filtered PID control, where the maximal gain and phase margins are also referred to as the largest ranges of gain and phase variations within which the system is guaranteed to be stabilizable. The results show, from a perspective of practical implementation, the effect of filtered PID control on the gain and phase margins. In yet another interesting observation, we find further that it is possible for filtered PID controllers to achieve the maximally possible gain and phase margins, i.e., those by LTI controllers.

In an additional effort, we examine the gain and phase maximization problems subject to steady-state tracking performance constraint, which sheds light into the tradeoff between performance and robustness of PID controllers. Our results show how in a practically implementable PID control scheme the gain and phase margins may be confined by the plant characteristics, and how in the limit it can perform equally well as other optimal controllers.

The results given in this thesis, as alluded to above, establish the strong robustness properties of PID controllers when appropriately designed, and from the perspective of gain and phase margins, lend theoretical justifications for the successes of PID control, leading to a theory of "explainable" analytical design and tuning of PID controllers.