Estimate of the farthest possible non-dominant pole locations with PID controllers

It is important to know how far non-dominant poles can be located away from the dominant poles since the closed-loop system transient response may not be as expected, if the non-dominant poles are not placed far enough from the dominant pole pair. Even if the arbitrary pole assignment is possible with a full state feedback or an output feedback controller with the order of plant order minus one, low order controllers and especially PID controllers are used in most practical applications. Therefore, the farthest possible non-dominant pole locations with PID controllers are found for all pole systems in this study. Proposed calculation method is demonstrated on two different examples and the closed-loop system performance is investigated with the designed PID controllers.