Model reduction with pole-zero placement and matching of derivatives.

In this paper we consider the model reduction of a large, minimal, linear, time-invariant system of order $n$ using moment matching techniques. Our goal is to compute an approximation of order $\nu \ll n$ that matches $\nu$ moments of the transfer function, has $\ell$ poles and $k$ zeros fixed and also matches a number of moments of its derivative. Assuming the original model is known, using a moment matching-based parameterization of the reduced model, we derive explicit linear algebraic constraints to place the desired poles and zeros and to match some moments of the derivative of the transfer function. The corresponding constraints are given by linear systems with the free parameters as unknowns together with solving low order Sylvester equations. Furthermore, since in practice data sets are available rather than the explicit model, we extend these results to the framework of data-driven model reduction. We generalize the Loewner matrices to include the measured data and the imposed pole and derivative constraints as well and use them to compute the approximation that satisfies all the imposed constraints simultaneously through solving again a linear system in the free parameters.