Solution of a non-convex optimization arising in PI/PID control design

As shown by Astrom et al. (Automatica 34(5) (1998) 585), the problem of designing a stabilizing PI controller based on minimizing integral of error associated with step load disturbance while subjecting to constraints on maximum sensitivity and/or complementary sensitivity amounts to that of finding the maximum allowable integral gain. The latter problem is a non-convex optimization problem whose true solution cannot be obtained with a guarantee by a gradient-based search algorithm. In this paper, we present a novel and effective approach to solve such a non-convex optimization problem. Our approach is based on regarding an equality constraint set on controller gain parameters as a two-dimensional value set in the complex plane and using the notion of principal points to characterize its boundary. With this treatment, we are able to derive analytical expressions for describing the boundary of an equality constraint set in the controller gain plane. These expressions allow one to trace the boundaries of equality constraint sets using an existing path-following algorithm. Hence, by constructing the boundary of the feasible domain in the controller gain space, the maximum allowable integral gain can be obtained. In addition to having the ability to obtain global optimal solution, our approach can handle sensitivity and complementary sensitivity constraints simultaneously without using an iterative procedure. (C) 2002 Elsevier Science Ltd. All rights reserved.