Implementation-oriented filtered PID control: Optimization of robustness margins

In this paper, we investigate the gain and phase robustness of PID control in stabilizing linear time-invariant (LTI) systems subject to gain and phase variations. We consider specifically filtered PID controllers, out of the necessity in practical implementation of PID controllers. We examine first-order unstable systems and seek to find analytical expressions of the maximal gain and phase margins achievable by filtered PID control, where the maximal gain and phase margins are referred to as the largest ranges of gain and phase variations within which the system is guaranteed to be stabilizable. Our results show that the maximal gain margin can be determined explicitly, while the maximal phase margin can be computed by solving a third-order polynomial, which in turn can be solved explicitly. We also investigate the gain and phase maximization problems subject to a performance constraint, which has direct linkages to the steady-state and transient performances of a system. 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 also shed light into the tradeoff between performance and robustness of PID controllers. © 2023 Elsevier Ltd.