Construction of Robust Root Loci for Linear Systems with Ellipsoidal Uncertainty of Parameters

In this paper we consider the construction of the robust root locus (RRL) for the systems with ellipsoidally parametric uncertainties. By characterizing the principal points of ellipsoidal parameter set $\bQ$ associated with the root mapping $s(\bq): \bR^m \rightarrow \bC$, we present a necessary condition for the point $(s,\bq) \in \bC\times \bQ$ to satisfy $p(s;\bq)=0$ and $s\in\partial Z(p,\bQ)$, the boundary of the RRL $Z(p,\bQ)$. This condition renders analytic manifolds of dimension one in the domain $\bC\times\bQ$. Hence, the boundary of each section of the RLL $Z(p,\bQ)$ can be accurately constructed via tracing the manifolds by a path-following algorithm. This approach to constructing the RRL provides an alternative way of verifying the robust stability of uncertain systems with ellipsoidal perturbations.