Stable polyhedra in parameter space

Abstract

A typical uncertainty structure of a characteristic polynomial is P(s) = A(s) * Q(s) + B(s) with A(s) and B(s) fixed and Q(s) uncertain. In robust controller design Q(s) may be a controller numerator or denominator polynomial; an example is the PID controller with Q(s) = KI + KP(s) + KD (s*s) . In robustness analysis Q(s) may describe a plant uncertainty. For fixed imaginary part of Q(j*omega), it is shown that Hurwitz stability boundaries in the parameter space of the even part of Q(j*omega) are hyperplanes and the stability regions are convex polyhedra. A dual result holds for fixed real part of Q(j*omega) . Also sigma -stability with the real parts of all roots of P(s) smaller than sigma is treated.