Five-Full-Block Structured Singular Values of Real Matrices Equal Their Upper Bounds

TL;DR

This paper proves that for real matrices with five full complex uncertainty blocks, the structured singular value equals its convex upper bound, using SDP feasibility and low-rank solutions, with a counterexample for six blocks.

Contribution

It establishes the equality of structured singular value and its upper bound for five blocks and introduces a new SDP-based approach for analyzing such matrices.

Findings

Equality holds for five blocks, not six.

SDP feasibility conditions characterize the equality.

Revisits known results with the new approach.

Abstract

We show that the structured singular value of a real matrix with respect to five full complex uncertainty blocks equals its convex upper bound. This is done by formulating the equality conditions as a feasibility SDP and invoking a result on the existence of a low-rank solution. A counterexample is given for the case of six uncertainty blocks. Known results are also revisited using the proposed approach.