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In this paper, the $\mathcal{H}_{2}$ optimal approximation of a$n_{y}\times{n_{u}}$ transfer function $\mathbf{G}(s)$ by a finite dimensionalsystem $\hat{\mathbf{H}}_{d}(s)$ including input/output delays, is addressed.The underlying $\mathcal{H}_{2}$ optimality conditions of the approximationproblem are firstly derived and established in the case of a poles/residuesdecomposition. These latter form an extension of the tangential interpolatoryconditions, presented in~\cite{gugercin2008h_2,dooren2007} for the delay-freecase, which is the main contribution of this paper. Secondly, a two stagealgorithm is proposed in order to practically obtain such an approximation.

Optimal $\mathcal{H}_{2}$ model approximation based on multiple input/output delays systems