All stabilizing PID controllers for time delay systems

This paper presents a method to compute the entire set of stabilizing PID controller parameters for an arbitrary (including unstable) linear time delay system. The main contribution is to handle the infinite number of stability boundaries in the ( k d , k i ) -plane for a fixed proportional gain k p . For retarded open loops, it is shown that the stable region in the ( k d , k i ) -plane consists of convex polygons. Concerning neutral loops, a new phenomenon is introduced. For certain systems and certain k p , the exact stable region in the ( k d , k i ) -plane can be described by the limit of a sequence of polygons with an infinite number of vertices. This sequence may be well approximated by convex polygons. Moreover, the paper describes a necessary condition for k p -intervals potentially having a stable region in the ( k d , k i ) -plane. Thus, the set of stabilizing controller parameters can be calculated after gridding k p in these intervals. A Matlab tool implementing the presented method is available.