Exact Solution for the Rank-One Structured Singular Value with Repeated Complex Full-Block Uncertainty

TL;DR

This paper provides an exact solution for the structured singular value of rank-one complex matrices with repeated complex full-block uncertainty, using Von Neumman's trace inequality, with applications in fluid flow analysis.

Contribution

It introduces a novel exact solution for rank-one SSV with repeated complex full-block uncertainty, extending previous results and applicable to fluid dynamics problems.

Findings

Exact SSV solution for rank-one matrices with repeated complex full-blocks.

Application to turbulent channel flow model.

Enhanced understanding of flow physics in incompressible fluids.

Abstract

In this note, we present an exact solution for the structured singular value (SSV) of rank-one complex matrices with repeated complex full-block uncertainty. A key step in the proof is the use of Von Neumman's trace inequality. Previous works provided exact solutions for rank-one SSV when the uncertainty contains repeated (real or complex) scalars and/or non-repeated complex full-block uncertainties. Our result with repeated complex full-blocks contains, as special cases, the previous results for repeated complex scalars and/or non-repeated complex full-block uncertainties. The repeated complex full-block uncertainty has recently gained attention in the context of incompressible fluid flows. Specifically, it has been used to analyze the effect of the convective nonlinearity in the incompressible Navier-Stokes equation (NSE). SSV analysis with repeated full-block uncertainty has led to an improved understanding of the underlying flow physics. We demonstrate our method on a turbulent channel flow model as an example.