Chapter 7: Model Reduction by Rational Interpolation

Abstract

The last two decades have seen major developments in interpolatory methods for model reduction of large-scale linear dynamical systems. Advances of note include the ability to produce (locally) optimal reduced models at modest cost; refined methods for deriving interpolatory reduced models directly from input/output measurements; and extensions for the reduction of parametrized systems. This chapter offers a survey of interpolatory model reduction methods starting from basic principles and ranging up through recent developments that include weighted model reduction and structure-preserving methods based on generalized coprime representations. Our discussion is supported by an assortment of numerical examples.Mathematics Subject Classification (2010). 41A05, 93A15, 93C05, 37M99.

System-Level Model and Simulation of a Frequency-Tunable Vibration Energy Harvester

“…The piezoelectric energy harvester model implemented in ANSYS ® Mechanical (V2019, R1) was thoroughly described in [34]. Considering its high computational cost for a transient simulation, Krylov subspace-based model order reduction (MOR) methods, also known as rational interpolation [38][39][40] were implemented to generate a highly compact but accurate reduced order model (ROM). Furthermore, based on this ROM, a circuit-device co-simulation of the piezoelectric energy harvester became feasible in the system-level simulation.…”

2 Model order reduction in mechanical engineering

“…This category is especially interesting for control engineering or for problems where inputs and outputs are defined. In this regard, approaches such as balanced truncation [78,44] and Krylov subspace methods [42,10] exploit the information contained in the input and output matrices B, C to obtain a reduced model which is tailored to approximate the input-output behavior.…”

“…Corresponding methods were intensively developed from the 1960s onwards ( [25,73,23,20] and Chapter 4 of Volume 1 of Model order reduction) and are applied in other areas of engineering as well, for instance in the reduction of power systems [64] and for the purpose of control design, e. g., [71,66,58,7]. With the advent of balanced truncation and of Krylov subspace methods ( [78,44], [42,45], overviews in [4,8,10], and Chapters 2 and 3 of Volume 1 of Model order reduction) the approximation quality and the applicability to high-and very high-order linear systems improved significantly and opened numerous fields of applications. Proper orthogonal decomposition (POD) methods based on snapshots [63,70,60] and hyper-reduction techniques [22,30,31] were developed for the reduction of nonlinear models.…”

Time‐limited pseudo‐optimal ‐model order reduction

“…Moment matching methods have been significantly advanced over the last two decades, and several generalisations and extensions exist which can preserve various system properties while ensuring good accuracy as well. Beattie and Gugercin [22] provided a detailed survey of moment matching methods.…”