Abstract
ABSTRACTWe investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [1] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation, we show a connection to a rational Krylov subspace, and thus to moment matching. This theoretical link between the MOR techniques is illustrated by three numerical examples. For linear time-invariant systems, the link also motivates that the time-domain approach can be at best as accurate as moment matching, since the expansion points are fixed by the choice of the polynomial basis, while in moment matching they can be adapted to the system.
Highlights
IntroductionVarious mathematical and physical processes can be modeled as linear time-invariant (LTI) input-output systems
Various mathematical and physical processes can be modeled as linear time-invariant (LTI) input-output systemsEx (t) = Ax(t) + Bu(t), y(t) = Cx(t), (1)where E, A ∈ Rn×n are sparse matrices, B ∈ Rn×p and C ∈ Rq×n are input and output matrices, respectively, x(t) ∈ Rn is the state vector, u(t) ∈ Rp is the input vector, y(t) ∈ Rq is the output vector and t ∈ R represents time.Since the order of the LTI system (1) is often huge n 103, a numerical simulation might be too expensive or even impossible, caused by immense computational time and memory requirements
We investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [1] and extend their idea by exploiting the structure of the corresponding linear system of equations
Summary
Various mathematical and physical processes can be modeled as linear time-invariant (LTI) input-output systems. In this paper we only consider single-input single-output (SISO) systems, i.e. p = q = 1 in (1) to simplify the notation Drawbacks of this method are the dependence of the reduced order model (ROM) on the input u(t) and the initial state x(t0) = x0. The framework in [1] uses W = V in (5) and obtains the projection matrix V ∈ Rn×r from the vector valued coefficients in series expansions of the state and input, sampling their time dependence via orthogonal polynomials [5, Chapter 22]. The matrix H has a certain block-structure We exploit this structure to derive an equivalent formulation and a more well-posed variation of this MOR method
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