Genuine feasible stability margin region for integrating unstable processes with PD control

PID control is widely used to control stable processes; however, its application to integrating unstable processes is less common. Integrating unstable process is often approximated by an integrating-unstable-first-order-plus-dead-time model based on which the simple analytical tuning formulae of PD control are derived to meet gain and phase margin (GPM) specifications. The previous tuning approach based on GPM only takes the increasing gain margin into account; however, the decreasing gain margin is commonly more critical. Using Routh stability criterion, the decreasing gain margin is obtained by Pade approximation of the delay based on the established tuning rule. The numerical polynomial solving approach is employed to seek the feasible stability margin region, which is explicitly plotted in the 2-D plane. At the same time, a novel tuning rule based on the decreasing gain margin is presented. The results demonstrate that the feasible GPM region is a small portion of the original one. Finally, some numerical examples are provided to validate the analysis result.