Parameterization of All Differential-Algebraic Moment Matching Interpolants

In this article, we extend the notion of time-domain moment to a class of general systems of nonlinear differential-algebraic equations. In addition, a new family of systems achieving moment matching is constructed. Provided that mild conditions hold, and for any given dimension at least as large as that of the signal generator, it is proven that this family of systems parameterizes all systems achieving moment matching. Four illustrative examples are provided. In the first and second example, a family of reduced-order models interpolating the response of a nonlinear circuit driven by a Van der Pol oscillator is constructed and higher order families with additional degrees of freedom are presented; in the third example a structure-preserving reduced-order model of a transmission line is constructed by considering a family of interpolants having one more state than the minimal order; and in the final example a differential-algebraic reduced-order model is constructed for a system of semidiscretized Oseen equations to obtain a better high-frequency approximation than what is achievable by a differential reduced-order model.