Model reduction with pole-zero placement and high order moment matching

In this paper, we compute a low order approximation of a system of large order $n$ that matches $ν$ moments of order $j_i$ of the transfer function, at $ν$ interpolation points, has $\ell$ poles and $k$ zeros fixed and also matches $ν-(\ell +k)$ moments of order $j_i+1$, where $j_i+1$ is the multiplicity of the $i$-th interpolation point. We derive explicit linear systems in the free parameters to simultaneously achieve the required pole-zero placement and match the desired high order moments. We compute the closed form of the free parameters that meet the constraints, as the solution of a $ν$ order linear system. Furthermore, for data-driven model reduction, we generalize the construction of the Loewner matrices to include the data and the imposed pole and higher order moment constraints. The resulting approximations achieve a trade-off between the good norm approximation and the preservation of the dynamics of the original system in a region of interest.

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