Structured singular values with nondiagonal structures. I. Characterizations

The purpose of this two-part series is to provide a robustness analysis framework for a class of problems with highly structured modeling uncertainties. This framework is more general than that of the usual block, diagonally structured uncertainties, and it corresponds to a structure consisting of block-by-block matrix perturbations. We study the structured singular value with respect to this structure, and we establish a number of novel results for this notion. This paper contains a study on the properties of the structured singular value. We give an alternative characterization of this notion as the solution of a smooth optimization problem. Furthermore, we show that under a certain circumstance the structured singular value reduces to a vector-induced matrix norm.