A New Constructive Characterization of Stabilizing PID Controllers

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.