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

TL;DR

This paper develops an $ ext{H}_2$ optimal approximation method for systems with input/output delays, extending existing delay-free conditions and proposing a practical two-stage algorithm for implementation.

Contribution

It extends $ ext{H}_2$ optimal approximation conditions to delay systems and introduces a two-stage algorithm for practical realization.

Findings

Derived $ ext{H}_2$ optimality conditions for delay systems.

Extended tangential interpolatory conditions to include delays.

Proposed a practical two-stage approximation algorithm.

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.