Robust root-clustering of a matrix in intersections or unions of regions: Addendum

Abstract

This paper considers robust stability analysis for a matrix affected by LFT-based complex uncertainty (LFT for linear fractional transformation). A method is proposed to compute a bound on the amount of uncertainty ensuring robust root-clustering in a combination (intersection and/or union) of several possibly nonsymmetric half planes, discs, and exteriors of discs. In some cases to be detailed, this bound is not conservative. The conditions are expressed in terms of (linear matrix inequalities) LMIs and derived through Lyapunov's second method. As a distinctive feature of the approach, the Lyapunov matrices proving robust root-clustering (one per subregion) are not necessarily positive definite, but have prescribed inertias depending on the number of roots in the corresponding subregions. As a special case, when rootclustering in a single half plane, disc or exterior of a disc is concerned, the whole clustering region reduces to only one convex subregion and the corresponding unique Lyapunov matrix has to be positive definite as usual. The extension to polytopic LFT-based uncertainty is also addressed.