A New Constructive Characterization of Stabilizing PID Controllers

Abstract

In this paper, we present a new complete and constructive characterization of all the stabilizing regions in the PID gain space. It is based on D-partition theory and the fact that for a given proportional gain $k_{p}$ the stability boundaries in the plane of integral and derivative gains are all straight lines. Actually, we present the notion of the critical $k_{p}$ point, which is defined in the sense that the topology of the D-partition of $(k_{i},k_{d})$ plane can change when the $k_{p}$ crosses a critical $k_{p}$ point. Moreover, we identify seven types of critical $k_{p}$ points and provide formulas for their computation. With the availability of all critical $k_{p}$ points, the stabilizing $k_{p}$ intervals can be exhaustively and exactly determined. By sweeping the $k_{p}$ parameter over the stabilizing $k_{p}$ intervals, the whole set of stabilizing PID controllers for a given plant is created. In addition, it allows for an efficient test of the existence of a stabilizing PID controller set without sweeping the $k_{p}$ parameter. To validate the newly presented analytical and constructive characterization of stabilizing PID controller sets and demonstrate the existence of seven types of critical $k_{p}$ points, five examples are provided.

Keywords

• Stabilizing PID controller set
• D-partition theory
• parametric-space method
• stability domain