Extreme Point Results for Robust Stabilization of Interval Plants with First Order Compensators

It has recently been shown that a first order compensator robustly stabilizes an interval plant family if and only if it stabilizes all of the extreme plants. That is, if the plant is described by m-th order numerator and monic n-th order denominator with coefficients lying in prescribed intervals, it is necessary and sufficient to stabilize the set of 2^m+n+l extreme plants. These extreme plants are obtained by considering all possible combinations for the extreme values of the numerator and denominator coefficients. In this paper, we prove a stronger result. Namely, it is necessary and sufficient to stabilize only sixteen of the extreme plants. These sixteen plants are generated using the Kharitonov polynomials associated with the numerator and denominator. Furthermore, when additional apriori information about the compensator is specified (sign of the gain and signs and relative magnitudes of the pole and zero), then in some cases, it is necessary and sufficient to stabilize eight critical plants while in other cases, it is necessary and sufficient to stabilize twelve critical plants.