Abstract
Nowadays, a realization of non-integer (fractional) order elements on a digital platform is a widely researched problem. The theory of such dynamic components is relatively well grounded. However, many problems of implementation on a digital platform are still open. Popular methods of implementation completely fail when used in real-time control applications. A need for efficient, numerically robust and stable implementation is obvious. These types of controllers and filters can be used in areas like telemedicine, biomedical engineering, signal processing, control, and many others. In this paper, the authors present the basic level of preliminary implementation of Matlab library for a realization of fractional order dynamic elements.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bania, P., Baranowski, J.: Laguerre polynomial approximation of fractional order linear systems. In: Mitkowski, W., Kacprzyk, J., Baranowski, J. (eds.) Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, Cracow, Poland, pp. 171–182. Springer (2013)
Bania, P., Baranowski, J., Zagórowska, M.: Convergence of Laguerre impulse response approximation for non-integer order systems. Math. Prob. Eng. 2016, 13 (2016). https://doi.org/10.1155/2016/9258437. Article ID 9258437
Baranowski, J.: Quadrature based approximations of non-integer order integrator on finite integration interval. In: Babiarz, A., Czornik, A., Klamka, J., Niezabitowski, M. (eds.) Theory and Applications of Non-integer Order Systems, Lecture Notes in Electrical Engineering, vol. 407, pp. 11–20. Springer International Publishing (2017). https://doi.org/10.1007/978-3-319-45474-0_2
Baranowski, J., Bauer, W., Zagórowska, M.: Stability properties of discrete time-domain oustaloup approximation. In: Domek, S., Dworak, P. (eds.) Theoretical Developments and Applications of Non-Integer Order Systems, Lecture Notes in Electrical Engineering, vol. 357, pp. 93–103. Springer International Publishing (2016). https://doi.org/10.1007/978-3-319-23039-9_8
Baranowski, J., Bauer, W., Zagórowska, M., Dziwiński, T., Piątek, P.: Time-domain Oustaloup approximation. In: 2015 20th International Conference On Methods and Models in Automation and Robotics (MMAR), pp. 116–120. IEEE (2015)
Baranowski, J., Zagórowska, M.: Quadrature based approximations of non-integer order integrator on infinite integration interval. In: 2016 21st International Conference On Methods and Models in Automation and Robotics (MMAR) (2016)
Bauer, W., Baranowski, J., Dziwiński, T., Piątek, P., Zagórowska, M.: Stabilisation of magnetic levitation with a PI\(^{\lambda }\)D\(^{\mu }\) controller. In: 2015 20th International Conference On Methods and Models in Automation and Robotics (MMAR), pp. 638–642. IEEE (2015)
De Keyser, R., Muresan, C., Ionescu, C.: An efficient algorithm for low-order direct discrete-time implementation of fractional order transfer functions. ISA Trans. 74, 229–238 (2018)
Kapoulea, S., Psychalinos, C., Elwakil, A.: Single active element implementation of fractional-order differentiators and integrators. AEU - Int. J. Electron. Commun. 97, 6–15 (2018)
Kawala-Janik, A., Bauer, W., Al-Bakri, A., Haddix, C., Yuvaraj, R., Cichon, K., Podraza, W.: Implementation of low-pass fractional filtering for the purpose of analysis of electroencephalographic signals. Lect. Notes Electr. Eng. 496, 63–73 (2019)
Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-order systems and controls: Fundamentals and applications. Advances in Industrial Control. Springer-Verlag, London (2010)
Monteghetti, F., Matignon, D., Piot, E.: Time-local discretization of fractional and related diffusive operators using gaussian quadrature with applications. Appl. Numer. Math. (2018)
Mozyrska, D., Wyrwas, M.: Stability of linear systems with Caputo fractional-, variable-order difference operator of convolution type (2018)
Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 47(1), 25–39 (2000)
Petráš, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Nonlinear Physical Science. Springer (2011)
Piątek, P., Zagórowska, M., Baranowski, J., Bauer, W., Dziwiński, T.: Discretisation of different non-integer order system approximations. In: 2014 19th International Conference on Methods and Models in Automation and Robotics (MMAR), pp. 429–433. IEEE (2014)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. Elsevier Science (1998)
Rydel, M., Stanisławski, R., Latawiec, K., Gałek, M.: Model order reduction of commensurate linear discrete-time fractional-order systems. IFAC-PapersOnLine 51(1), 536–541 (2018)
Stanisławski, R., Latawiec, K.J., Gałek, M., Łukaniszyn, M.: Modeling and identification of fractional-order discrete-time laguerre-based feedback-nonlinear systems. In: Latawiec, K.J., Łukaniszyn, M., Stanisławski, R. (eds.) Advances in Modelling and Control of Non-integer-Order Systems, Lecture Notes in Electrical Engineering, vol. 320, pp. 101–112. Springer International Publishing (2015)
Tepljakov, A., Petlenkov, E., Belikov, J.: FOMCON: a MATLAB toolbox for fractional-order system identification and control. Int. J. Microelectron. Comput. Sci. 2, 51–62 (2011)
Trigeassou, J., Maamri, N., Sabatier, J., Oustaloup, A.: State variables andtransients of fractional order differential systems. Comput. Math. Appl. 64(10), 3117–3140 (2012). https://doi.org/10.1016/j.camwa.2012.03.099. http://www.sciencedirect.com/science/article/pii/S0898122112003173. Advances in FDE, III
Acknowledgment
Work partially realized in the project “Development of efficient computing software for simulation and application of non-integer order systems”, financed by National Centre for Research and Development with TANGO programme.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Bauer, W., Baranowski, J., Piątek, P., Grobler-Dębska, K., Kucharska, E. (2020). SoftFRAC - Matlab Library for Realization of Fractional Order Dynamic Elements. In: Bartoszewicz, A., Kabziński, J., Kacprzyk, J. (eds) Advanced, Contemporary Control. Advances in Intelligent Systems and Computing, vol 1196. Springer, Cham. https://doi.org/10.1007/978-3-030-50936-1_99
Download citation
DOI: https://doi.org/10.1007/978-3-030-50936-1_99
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-50935-4
Online ISBN: 978-3-030-50936-1
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)