A case of mu-synthesis as a quadratic semidefinite program

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.