On the Kalman---Yakubovich---Popov lemma and the multidimensional models
Multidimensional Systems and Signal Processing
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This paper considers robust stability analysis for a matrix affected by unstructured complex uncertainty. 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 outsides 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 root-clustering in a single half plane, disc, or outside 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.