Calculation of all robust PID type controllers for unstable high-order time delay systems based on gain and phase margin specifications

This paper proposes a novel alternative method to graphically calculate all feasible gain and phase margin specifications-oriented robust PID controllers to stabilize open-loop unstable high-order time delay (OLUHOPTD) systems. Distinguished from other proposed methods, in addition to the conventional positive gain margin (GM) and phase margin (PM), the negative GM is particularly considered in this study. Thus, the closed-loop control system guarantees robustness either subject to positive or negative gain variations. To guarantee considered system with pre-specified safety margins, a virtual gain-phase margin tester compensator is incorporated in the forward path of the concerned control system. Combining the stability equation method and the parameter plane method allows us to establish the conditions for stability. Accordingly, the stability boundary, the constant positive GM boundary, the constant negative GM boundary as well as the constant phase margin (PM) boundary can be portrayed. A gain and phase margin specifications-oriented region (GPMSOR) is successfully located in the parameter plane. The GPMSOR characterizes all the feasible robust stabilizing PID controllers for the considered OLUHOPTD systems. Finally, an illustrative example is provided to demonstrate the effectiveness of this proposed method.