H2-optimal model reduction with higher-order poles

We revisit the problem of approximating a multiple-input multiple-output $p imes m$ rational transfer function $H(s)$ of high degree by another $p imes m$ rational transfer function $widehat{H}(s)$ of much smaller degree, so that the $mathcal{H}_2$-norm of the approximation error is minimized. We show that in the general case of higher-order poles in the reduced-order model, called the defective case, the stationary points of the $mathcal{H}_2$-norm of the approximation error can still be characterized by tangential interpolation conditions. We also indicate that the sensitivity of the solution of this problem depends on the parameterization used