Padé-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

Abstract

Summary.A standard approach to reduced-order modeling of higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for reduced-order modeling of first-order systems. While this approach results in reduced-order models that are characterized as Padé-type or even true Padé approximants of the system's transfer function, in general, these models do not preserve the form of the original higher-order system. In this paper, we present a new approach to reduced-order modeling of higher-order systems based on projections onto suitably partitioned Krylov basis matrices that are obtained by applying Krylov-subspace techniques to an equivalent first-order system. We show that the resulting reduced-order models preserve the form of the original higher-order system. While the resulting reduced-order models are no longer optimal in the Padé sense, we show that they still satisfy a Padé-type approximation property. We also introduce the notion of Hermitian higher-order linear dynamical systems, and we establish an enhanced Padé-type approximation property in the Hermitian case.

Structure-Preserving Model Order Reduction of RCL Circuit Equations

“…In terms of transfer functions, the problem of order reduction of the original system (16) is equivalent to the problem of constructing the matricesM ,D, K, andF in (18) such that the transfer function (21),H(s), is a 'sufficiently accurate' approximation to the original transfer function (20), H(s).…”

“…Therefore, the reduced-order model (18) should be constructed such that it corresponds to an actual RCL circuit. This is the problem of structure-preserving model order reduction of RCL circuits: Generate matricesM ,D,K, andF such that the reduced-order model (18) can be synthesized as an RCL circuit.…”

“…Therefore, the reduced-order model (18) should be constructed such that it corresponds to an actual RCL circuit. This is the problem of structure-preserving model order reduction of RCL circuits: Generate matricesM ,D,K, andF such that the reduced-order model (18) can be synthesized as an RCL circuit. Obviously, (18) corresponds to an actual RCL circuit if the matricesM ,D,K, andF are constructed such that they have analogous structures as the matrices M , D, K, and F of the original given RCL circuit.…”

“…This is the problem of structure-preserving model order reduction of RCL circuits: Generate matricesM ,D,K, andF such that the reduced-order model (18) can be synthesized as an RCL circuit. Obviously, (18) corresponds to an actual RCL circuit if the matricesM ,D,K, andF are constructed such that they have analogous structures as the matrices M , D, K, and F of the original given RCL circuit. Unfortunately, for general RCL circuits, no order reduction method is known that is guaranteed to preserve all these 'RCL structures'.…”

“…While PRIMA generates provably passive reduced-order models, it does not preserve other structures, such as reciprocity or the block structure of the circuit matrices, inherent to RCL circuits. This has motivated the development of the reduction technique SPRIM [17,18], which overcomes these disadvantages of PRIMA. In particular, SPRIM generates provably passive and reciprocal macromodels of multi-port RCL circuits.…”