A case of mu-synthesis as a quadratic semidefinite program

TL;DR

This paper presents a new criterion for solving a specific spectral Nevanlinna-Pick problem, a special case of the μ-synthesis problem in $H^$ control, using quadratic semidefinite programming techniques.

Contribution

It introduces a novel quadratic semidefinite programming approach to determine the solvability of a structured spectral interpolation problem in control theory.

Findings

The criterion reduces the problem to a quadratic function minimization.

The minimum of the quadratic function is attained and can be zero.

The approach provides a new solvability condition for the spectral Nevanlinna-Pick problem.

Abstract

We analyse a special case of the robust stabilization problem under structured uncertainty. We obtain a new criterion for the solvability of the spectral Nevanlinna-Pick problem, which is a special case of the $\mu$-synthesis problem of $H^\infty$ control in which $\mu$ is the spectral radius. Given $n$ distinct points $\la_1,\dots,\la_n$ in the unit disc and $2\times 2$ nonscalar complex matrices $W_1,\dots,W_n$, the problem is to determine whether there is an analytic $2\times 2$ matrix function $F$ on the disc such that $F(\la_j)=W_j$ for each $j$ and the supremum of the spectral radius of $F(\la)$ is less than 1 for $\la$ in the disc. The condition is that the minimum of a quadratic function of pairs of positive $3n$-square matrices subject to certain linear matrix inequalities in the data be attained and be zero.