Computation of stabilizing PI and PID controllers using the stability boundary locus

In this paper, a new method for the calculation of all stabilizing PI controllers is given. The proposed method is based on plotting the stability boundary locus in the (kp, ki)-plane and then computing the stabilizing values of the parameters of a PI controller. The technique presented does not require sweeping over the parameters and also does not need linear programming to solve a set of inequalities. Thus it offers several important advantages over existing results obtained in this direction. Beyond stabilization, the method is used to shift all poles to a shifted half plane that guarantees a specified settling time of response. Computation of stabilizing PI controllers which achieve user specified gain and phase margins is studied. It is shown via an example that the stabilizing region in the (kp, ki)-plane is not always a convex set. The proposed method is also used to design PID controllers. The limiting values of a PID controller which stabilize a given system are obtained in the (kp, ki)-plane, (kp, kd)-plane and (ki, kd)-plane. Furthermore, the proposed method is used to compute all the parameters of a PI controller which stabilize a control system with an interval plant family. Examples are given to show the benefits of the method presented.