Conference article

Fractional-Order Modelling in Modelica

Alexander Pollok
Institute of System Dynamics and Control, German Aerospace Center (DLR), Germany

Dirk Zimmer
Institute of System Dynamics and Control, German Aerospace Center (DLR), Germany

Francesco Casella
Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italy

Download articlehttp://dx.doi.org/10.3384/ecp15118109

Published in: Proceedings of the 11th International Modelica Conference, Versailles, France, September 21-23, 2015

Linköping Electronic Conference Proceedings 118:11, s. 109-115

Show more +

Published: 2015-09-18

ISBN: 978-91-7685-955-1

ISSN: 1650-3686 (print), 1650-3740 (online)

Abstract

Most dynamic systems with a basis in nature can be described using Differential-Algebraic Equations (DAE), and hence be modelled using the modelling language Modelica. However, the concept of DAEs can still be generalised, when differential operators of non-integer order are considered. These so called fractional order systems have counterparts in naturally occuring systems, for instance in electrochemistry and viscoelasticity. This paper presents an implementation of approximate fractional-order differential operators in Modelica, increasing the scope of systems that can be described in a meaningful way. Properties of fractional-order systems are discussed and some approximation methods are presented. An implementation in Modelica is proposed for the first time. Several testing procedures and their results are displayed. The work is then illustrated by the application of the model to several physically motivated examples. A possible usability-enhancement using the concept of "Calling Blocks as functions" is suggested.

Keywords

Fractional Order Systems; fractional calculus; Integer-Order Approximations

References

Michele Caputo. Linear models of dissipation whose q is almost frequency independent part 2. Geophysical Journal International, 13(5):529–539, 1967.

G Carlson and C Halijak. Approximation of fractional capacitors (1/s)ˆ(1/n) by a regular newton process. Circuit Theory, IEEE Transactions on, 11(2):210–213, 1964.

Lokenath Debnath. Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences, 2003(54):3413–3442, 2003.

Bill Goodwine and Kevin Leyden. Recent results in fractionalorder modeling in multi-agent systems and linear friction welding. IFAC-PapersOnLine, 48(1):380–381, 2015.

Andreas Klöckner, Andreas Knoblach, and Andreas Heckmann. How to shape noise spectra for continuous system simulation. In Proceedings of the 11th International Modelica Conference, 2015.

RC Koeller. Applications of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics, 51(2):299–307, 1984.

VV Kulish and JL Lage. Fractional-diffusion solutions for transient local temperature and heat flux. Transactions-American Society of Mechanical Engineers Journal of Heat Transfer, 122(2):372–375, 2000.

Richard L Magin. Fractional calculus in bioengineering. Critical Reviews in Biomedical Engineering, 32(1), 2004.

Keith B Oldham. The fractional calculus. Elsevier, 1974.

Alain Oustaloup, Francois Levron, Benoit Mathieu, and FlorenceMNanot. Frequency-band complex noninteger differentiator: characterization and synthesis. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, 47(1):25–39, 2000.

Indranil Pan and Saptarshi Das. Intelligent fractional order systems and control: an introduction. Springer Publishing Company, Incorporated, 2012.

Yu Z Povstenko. Fractional heat conduction equation and associated thermal stress. Journal of Thermal Stresses, 28(1): 83–102, 2004.

J Sabatier, Om P Agrawal, and JA Tenreiro Machado. Advances in fractional calculus, volume 4. Springer, 2007.

Michael Stiassnie. On the application of fractional calculus for the formulation of viscoelastic models. Applied Mathematical Modelling, 3(4):300–302, 1979.

BM Vinagre, I Podlubny, A Hernandez, and V Feliu. Some approximations of fractional order operators used in control theory and applications. Fractional calculus and applied analysis, 3(3):231–248, 2000.

Dingyu Xue, Chunna Zhao, and Yang Quan Chen. A modified approximation method of fractional order system. In Mechatronics and Automation, Proceedings of the 2006 IEEE International Conference on, pages 1043–1048. IEEE, 2006.

Dirk Zimmer. Equation-based modeling of variable-structure systems. PhD thesis, Swiss Federal Institute of Technology, Zürich, 2010.

Citations in Crossref