Approximating arbitrary impulse response functions with Prony basis functions

Herber, Daniel Ronald; Allison, James T.

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https://hdl.handle.net/2142/106010 Copy

Description

Title

Approximating arbitrary impulse response functions with Prony basis functions

Author(s)

• Herber, Daniel Ronald
• Allison, James T.

Issue Date

2019-10-21

Keyword(s)

• convolution integral
• dynamics
• fitting
• impulse response
• modeling
• Prony
• state-space model
• transfer function

Abstract

In this report, we are concerned with approximating the input-to-output behavior of a type of scalar convolution integral given its so-called impulse response function by constructing an appropriate linear time-invariant state-space model. Such integrals frequently appear in the modeling of hydrodynamic forces, viscoelastic materials, among other applications. First, linear systems theory is reviewed. Next, Prony basis functions, which are exponentially decaying cosine waves with phase delay and variable amplitude, are described as potential objects to be used to approximate a given impulse response function. Then it is shown how a superposition of Prony basis functions can be directly mapped back to an equivalent linear state-space model. Also, it is directly shown that both the Golla-Hughes-McTavish model and Prony series (generalized Maxwell model) are special cases of the considered Prony basis function. Several nonlinear optimization (fitting) problems are then described to determine the value of the model parameters that result in the desired approximation. Finally, a few numerical examples are presented to demonstrate that Prony basis functions can approximation a diverse set of impulse response behaviors.

Series/Report Name or Number

UIUC-ESDL-2019-01

Type of Resource

text

Language

en

Permalink

http://hdl.handle.net/2142/106010

Copyright and License Information

Copyright 2019 Daniel R. Herber

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