Stability margin gap of LTI and PID control capped at two in lower-order systems

This paper examines second-order non-minimum phase systems with a single unstable pole, focusing on the maximal stability margins under unknown gain and phase variations. These margins are optimized using both proportional–integral–derivative (PID) control and general linear time-invariant (LTI) control. We first demonstrate that LTI control can offer, at most, a twofold improvement over PID control, meaning the maximal stability margins achievable under LTI controllers are always within a factor of two of those attainable with PID controllers. Interestingly, introducing a suitably placed stable pole enables a PID controller to deliver the same margins as an LTI controller. The results further clarify how plant characteristics constrain tolerable gain and phase variations, providing key insights for PID design and tuning.