Robust stability of polynomial families and robust strict positive realness of rational function families

Abstract

Dual to the interval model, the diamond model proves to be an alternative powerful device for taking into account the variation of parameters in prescribed ranges. This paper deals with the robust stability of certain diamond polynomial families and the strict positive realness of certain diamond rational function families. By exploiting the geometric characterizations of their value sets, we show that, for the strict positive realness of the rational function family with the coefficients of numerator and denominator each varying in diamonds, the same property of only 24 specially selected extreme rational functions will suffice. An interesting result concerning the stability of the so-called simplex polynomial family is provided, namely, the stability of 4 extreme points and 2 exposed edges in this family will ensure the stability of the whole family. We also extend this result to the strict positive realness of the simplex rational function family where 30 extreme points and 2 exposed edges need to be c...