Christopher A. Beattie

Christopher A. Beattie, Serkan Gugercin, Volker Mehrmann

We consider the model reduction problem for linear time-invariant dynamical systems having nonzero (but otherwise indeterminate) initial conditions. Building upon the observation that the full system response is decomposable as a superposition of the response map for an unforced system having nontrivial initial conditions and the response map for a forced system having null initial conditions, we develop a new approach that involves reducing these component responses independently and then combining the reduced responses into an aggregate reduced system response. This approach allows greater flexibility and offers better approximation properties than other comparable methods.

Christopher A. Beattie, Serkan Gugercin, Sarah Wyatt

We investigate the use of inexact solves for interpolatory model reduction and consider associated perturbation effects on the underlying model reduction problem. We give bounds on system perturbations induced by inexact solves and relate this to termination criteria for iterative solution methods. We show that when a Petrov-Galerkin framework is employed for the inexact solves, the associated reduced order model is an exact interpolatory model for a nearby full-order system; thus demonstrating backward stability. We also give evidence that for $\h2$-optimal interpolation points, interpolatory model reduction is robust with respect to perturbations due to inexact solves. Finally, we demonstrate the effecitveness of direct use of inexact solves in optimal ${\mathcal H}_2$ approximation. The result is an effective model reduction strategy that is applicable in realistically large-scale settings.

Branimir Anic, Christopher A. Beattie, Serkan Gugercin et al.

This paper introduces an interpolation framework for the weighted-H2 model reduction problem. We obtain a new representation of the weighted-H2 norm of SISO systems that provides new interpolatory first order necessary conditions for an optimal reduced-order model. The H2 norm representation also provides an error expression that motivates a new weighted-H2 model reduction algorithm. Several numerical examples illustrate the effectiveness of the proposed approach.

Caleb C. Magruder, Serkan Gugercin, Christopher A. Beattie

Linear time-periodic (LTP) dynamical systems frequently appear in the modeling of phenomena related to fluid dynamics, electronic circuits, and structural mechanics via linearization centered around known periodic orbits of nonlinear models. Such LTP systems can reach orders that make repeated simulation or other necessary analysis prohibitive, motivating the need for model reduction. We develop here an algorithmic framework for constructing reduced models that retains the linear time-periodic structure of the original LTP system. Our approach generalizes optimal approaches that have been established previously for linear time-invariant (LTI) model reduction problems. We employ an extension of the usual H2 Hardy space defined for the LTI setting to time-periodic systems and within this broader framework develop an a posteriori error bound expressible in terms of related LTI systems. Optimization of this bound motivates our algorithm. We illustrate the success of our method on two numerical examples.