Hurwitz-stability Boundary Crossing and Parameter Space Approach

Abstract

The basic idea of critical stability conditions is that a starting point in form of a stable characteristic polynomial p(s, q), q = qo is given. Assume that the (real) coefficients of p(s, q) are continuous in q. Then also the roots of p(s, q) are continuous in q, i.e. they cannot jump from the left half plane to the right half plane without crossing the imaginary axis. The stable neighborhood of go is bounded by the values of q, where for the first time one or more eigenvalues cross the imaginary axis under a continuous variation of q starting at go. Crossing of eigenvalues over the imaginary axis can occur in one of three ways: at s = 0, at s = ∞ and at s = ±jw.