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- Publication history
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Received January 3, 2026
Revised February 25, 2026
Accepted February 27, 2026
Available online April 21, 2026
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This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/bync/3.0) which permits
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반 무한 구조에서의 확산 공정을 위한 몇몇 전달함수의 유리 근사
Rational Approximations of Some Transfer Functions for Diffusion Processes in the Semi-infinite Geometry
https://doi.org/10.9713/kcer.2026.64.2.105158
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Abstract
반 무한 구조물 표면으로부터 열 혹은 물질이 확산하는 공정의 동특성을 나타내는 전달함수에는 exp(-√s)와 √s 같은 Padé 근사가 불가능한 요소들이 있을 수 있다 . 이 가운데 exp(-√s)의 경우 먼저 Padé 근사가 가능한 1/cosh(√s) 함수의 급수로 근사한 후, 이 급수에 Padé 근사를 적용하여 최종적으로 Padé 근사와 유사한 s의 유리 전달함수를 얻는 방법 이 제안되어 있다 . 여기서는 이 1/cosh(√s) 함수로 표현된 급수의 수렴을 촉진하는 방법을 제안하고 , 여러 Padé 근사 가 불가능한 전달함수 요소들의 근사로 확장하는 방법을 다루고자 한다 . 이들을 s의 유리 전달함수로 근사할 수 있으 면, 편미분방정식으로 표현되는 공정 동특성 모델을 상미분 형태의 상태방정식으로 근사할 수 있는데 , 이는 공정의 모 사, 해석 및 제어시스템 설계에 큰 편리를 가져다준다 . 본 연구에서는 다양한 형태의 전달함수를 근사하여 계단응답을 각각 모사하였고 , 4차 이하의 근사에서 우수한 정확도를 보였다 . 이는 특정 주파수 대역에서의 근사 정확도에 초점을 둔 기존 방법과 달리 , 시간영역 응답과 정상상태 특성을 동시에 만족시키는 근사라는 점에서 차별성을 갖는다 .
Transfer functions describing the dynamics of heat or mass diffusion from the surface of a semi-infinite medium may contain terms such as exp(-√s ) and √s, which cannot be directly approximated using Padé approximation.
Forexp(-√s ), an approach has been proposed in which it is first expressed as a series involving the function 1/cosh(√s ), to which Padé approximation can be applied. Then, Padé approximation is applied to this series to obtain a
rational transfer function in s. In this study, a method to accelerate the convergence of the series representation based on 1/cosh(√s ) is proposed and extended to the approximation of various transfer function elements that are not amenable to Padé approximation. Approximating these terms by rational transfer functions enables dynamic models originally formulated as partial differential equations to be represented in ordinary differential equation-based state-space form, facilitating simulation, analysis, and control system design. Step-response simulations demonstrate that satisfactory accuracy can be achieved with approximation orders of four or less while preserving steady-state characteristics and time-domain response accuracy.
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