Composite Trapezoidal Rule for Numerical Inverse Laplace Transform of Rational Functions in LTI Control Systems with C++ Programming

ชัยณรงค์ วิเศษศักดิ์วิชัย, ประเสรฐิ เผ่าชู


This paper presents a development of numerical methods for the inverse Laplace transform to analysis LTI (Linear Time Invariant) control systems by computer. The open source Standard C++ Library is used to create the programmatic objects by OOP (Object- Oriented Programming) for the numerical inverse Laplace transform object that uses numerical integration to the Bromwich integral. The composite trapezoidal rule is implemented on the simplified Bromwich integral which is real value integration that has only one cosine factor. The accuracy of numerical integration will be met by varying step size of the trapezoidal rule according to specified tolerance. The other numerical methods such that Laguerre’s method, Horner’s algorithm, synthetic division and nested multiplication will be included in the created programmatic objects for finding zeros of polynomial in order to avoid all singular points of integrand of Bromwich integral to make analytic integrand on contour of integration. The numerical inverse Laplace transform object of this article gives satisfactory results in both of the numerical inverse Laplace transform to the proper rational functions and unit step response of LTI control systems.


Object-oriented programming, Bromwich integral, Trapezoidal rule, Laguerre’s method, Horner’s algorithm, Synthetic division, Nested multiplication, Numerical inverse Laplace transform

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Trinchero, R., Stievano, I. S. and Canavero, F. G. Simulation of buck converters via numerical inverse Laplace transform. In: Signal and Power Integrity (SPI), 2017 IEEE 21st Workshopon. IEEE(2017), 1-4.

R-Smith, N. A. Z. and Brančík, L.

Comparative study on one-dimensional numerical inverse Laplace transform methods for electrical engineering. Elektrorevue-Electronic Journal, 18.1 (2016), 1-8.

Nuricumbo-Guillen, R., Gomez, P. and Espino-Cortes, F. P. Computation of transient voltage profiles along transmission lines by successive application of the numerical Laplace transform. In: North American Power Symposium (NAPS), 2013. IEEE(2013), 1- 6.

Mikulović, J. Č. and Šekara, T. B. The numerical method of inverse Laplace transform for calculation of over voltages in power transformers and test results. Serbian Journal of Electrical Engineering, 11.2(2014), 243-256.

Dubner, H. and Abate, J. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. Journal of the ACM (JACM), 15.1(1968), 115-123.

Durbin, F. Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate's method. The Computer Journal, 17.4(1974), 371-376.

Lin, F. F. Numerical inversion of Laplace transforms by the trapezoidal- type methods. 2003. PhD Thesis.

Cohen, A. M. Numerical methods for Laplace transform inversion. Springer Science & Business Media (2007), 26-28.

Gustafson, S. Computing inverse Laplace Transforms using convergence acceleration. In: Computation and Control II. Birkhäuser Boston (1991), 151-160.

Kincaid and W.Cheney, Numerical Analysis. 2d ed., California:Brooks/Cole Publishing Company(1996), 100-103

ชัยณรงค์ วิเศษศักดิ์วิชัย, สายชล ชุดเจือจีน และคณะ. วัตถุเชิงพหุนามของการโปรแกรม ซีพลัสพลัสสําหรับการประมวลผลฟังก์ชันถ่าย โอน. การประชุมวิชาการเครือข่ายวิศวกรรม ไฟฟ้ามหาวิทยาลัยเทคโนโลยีราชมงคล. ครั้งที่ 7. ชลบุรี; 2558. หน้า 186-189.

D. Xue, D., Chen, Y. and Atherton, D. P. Linear Feedback Control Analysis and Design with MATLAB, Springer-Verlag (2002), 28-29.