Computation of stabilizing PI and PID controllers

In this paper, a simple method for the calculation of stabilizing PI controllers is given. The proposed method is based on plotting the stability boundary locus in the (k_p, k_i)-plane and then computing 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 is 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. Furthermore, computation of stabilizing PI controllers, which achieve user specified gain and phase margins is studied. It is also shown via an example that the stabilizing region in the (k_p, k_i)-plane is not always a convex set. The proposed method is also used to design PID controllers. The limiting values of the PID controller, which stabilize a given system are obtained in the (k_p, k_d)-plane and (k_i, k_d)-plane. Examples are given to show the benefit of the method presented.