On frequency- and time-limited H2-optimal model order reduction

TL;DR

This paper investigates frequency- and time-limited H2-optimal model reduction for linear systems, revealing inherent limitations, establishing key equivalences, and proposing iterative algorithms with improved optimality satisfaction as model order increases.

Contribution

It introduces new iterative algorithms for frequency- and time-limited H2-optimal model reduction, addressing the challenge of satisfying optimality conditions within the oblique projection framework.

Findings

Proposed algorithms nearly satisfy optimality conditions with increasing model order.

Deviation from optimality conditions decreases as model order increases.

Validated algorithms on benchmark high-order models.

Abstract

In this paper, the problems of frequency-limited and time-limited H2-optimal model order reduction of linear time-invariant systems are considered within the oblique projection framework. It is shown that it is inherently not possible to satisfy all the necessary conditions for the local minimizer in the oblique projection framework. The conditions for exact satisfaction of the optimality conditions are also discussed. Further, the equivalence between the tangential interpolation conditions and the gramians-based necessary condition for the local optimum is established. Based on this equivalence, iterative algorithms that nearly satisfy these interpolation-based necessary conditions are proposed. The deviation in satisfaction of the optimality conditions decay as the order of the reduced-model is increased in the proposed algorithms. Moreover, stationary point iteration algorithms that satisfy two out of three necessary conditions for the local minimizer are also proposed. There also, the deviation in satisfaction of the third optimality conditions decay as the order of the reduced-model is increased in the proposed algorithms. The efficacy of the proposed algorithms is validated by considering one illustrative and three high-order models that are considered a benchmark for testing model order reduction algorithms.