Interpolatory Model Reduction of Parameterized Bilinear Dynamical Systems

TL;DR

This paper extends interpolatory projection methods to the reduction of parametric bilinear dynamical systems, ensuring accurate transfer function approximation and parameter sensitivity matching without computing derivatives.

Contribution

It introduces necessary conditions for projection subspaces to achieve parametric tangential interpolation of transfer functions in bilinear systems, including parameter sensitivities and Hessians.

Findings

Provides a framework for interpolatory model reduction of parametric bilinear systems.

Ensures transfer function and sensitivity matching without Jacobian or Hessian computations.

Enables efficient basis construction for reduced models.

Abstract

Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this is not the case for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function is matched tangentially by those of the corresponding reduced order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including extra vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian.