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

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