Stable polyhedra in parameter space

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)=K_I+K_Ps+K_Ds². In robustness analysis Q(s) may describe a plant uncertainty. For fixed imaginary part of Q(jω), it is shown that Hurwitz stability boundaries in the parameter space of the even part of Q(jω) are hyperplanes and the stability regions are convex polyhedra. A dual result holds for fixed real part of Q(jω). Also σ-stability with the real parts of all roots of P(s) smaller than σ is treated.Under the above conditions, the roots of P(s) can cross the imaginary axis only at a finite number of discrete “singular” frequencies. Each singular frequency generates a hyperplane as stability boundary. An application is robust controller design by simultaneous stabilization of several representatives of A(s) and B(s) by a PID controller. Geometrically, the intersection of convex polygons must be calculated and represented tomographically for a grid on K_P.