A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

• Autorzy: Oprzędkiewicz K. , Stanisławski R. , Gawin E. , Mitkowski W.

Identyfikatory

DOI

10.1515/bpasts-2017-0048

• Języki publikacji: EN

Abstrakty

EN

In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example confirms the usefulness of the method.

Słowa kluczowe

EN

fractional order calculus; discrete-time noninteger-order state equation; continuous fraction expansion; Al-Alaoui operator

PL

równanie dyskretne; aproksymacja Al-Alaoui; CFE

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2017
• Tom: Vol. 65, nr 4
• Strony: 429--437
• Opis fizyczny: Bibliogr. 23 poz., tab., wykr.

Twórcy

autor

Oprzędkiewicz K.

• AGH University of Science and Technology, Dept. of Automatics and Biomedical Engineering, 30 Mickiewicza Av., Krakow, Poland

autor

Stanisławski R.

• Opole University of Technology, Department of Electrical, Control and Computer Engineering, 76 Prószkowska St., 45-758 Opole, Poland

autor

Gawin E.

• State Higher Vocational School in Tarnow, Polytechnic Institute, 8 Mickiewicza St., Tarnow, Poland

autor

Mitkowski W.

• AGH University of Science and Technology, Dept. of Automatics and Biomedical Engineering, 30 Mickiewicza Av., Krakow, Poland

Bibliografia

• [1] R. Caponetto, G. Dongola, L. Fortuna, and I. Petras, (2010), Fractional Order Systems: Modeling and Control Applications, World Scientific Publishing, Singapore, 2010.
• [2] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, Berlin, 2008.
• [3] S. Das and I. Pan, Intelligent Fractional Order Systems andControl. An Introduction, Springer, Heidelberg, 2013.
• [4] A. Dzieliński, D. Sierociuk, and G. Sarwas, “Some applications of fractional order calculus”, Bull. Pol. Ac.: Tech. 58 (4), 583–592 (2010).
• [5] T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, Bialystok University of Technology, Bialystok, 2014.
• [6] K. Oprzędkiewicz and E. Gawin, “Non integer order, state space model for one dimensional heat transfer process”, Archives of Control Sciences 26 (2), 261–275 (2016).
• [7] R. Stanisławski, K.J. Latawiec, and M. Łukaniszyn, “A comparative analysis of Laguerre-based approximators to the Grünwald-Letnikov fractional-order difference”, Mathematical Problems in Engineering 2015, Article ID 512104 (2015).
• [8] T. Kaczorek, Selected Problems in Fractional Systems Theory, Springer-Verlag, Berlin, 2011.
• [9] P. Ostalczyk, “Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains”, International Journal of Applied Mathematics and Computer Science 22 (3), 533–538 (2012).
• [10] Y.Q. Chen and K.L. Moore, “Discretization schemes for fractional-order differentiators and integrators”, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49 (3), 363–367 (2002).
• [11] I. Petras, “Fractional order feedback control of a DC motor”, Journal of Electrical Engineering 60 (3), 117–128 (2009).
• [12] I. Petras, http: //people.tuke.sk/igor.podlubny/USU/matlab/petras/dfod2.m.
• [13] M.A. Al-Alaoui, “Novel digital integrator and differentiator”, Electronics Letters 29 (4), 376–378 (1993).
• [14] B.M. Vinagre, Y.Q. Chen, and I. Petras, “Two direct Tustin discretization methods for fractional-order differentiator-integrator”, Journal of the Franklin Institute 340, 349–362 (2003).
• [15] L. Dorcak, I. Petras, I. Kostial, and J. Terpak, “Fractional-order state space models”, Proc. International Carpathian Control Conference, ICCC 2002, 193–198 (2002).
• [16] J. Hammer, “Non-linear systems: Stability and rationality”, Int. J. of Control 39, 11–35 (1984).
• [17] Ü. Kotta, A.S.I. Zinober, and P. Liu, “Transfer equivalence and realization of nonlinear higher order input–output difference equations”, Automatica 37 (11), 1771–1778 (2001).
• [18] Ü. Kotta, E. Pawłuszewicz, and S. Nomm, “Generalization of transfer equivalence for discrete-time non-linear systems: Comparison of two definitions”, Int. J. of Control 77 (8), 741–747 (2004).
• [19] R.K. Pearson and Ü. Kotta, “Nonlinear discrete-time models: state-space vs. I/O representations”, Journal of Process Control 14 (5), 533–538 (2004).
• [20] P. Ostalczyk, Discrete Fractional Calculus. Applications in Control and Image Processing, World Scientific Publishing, Singapore, 2016.
• [21] M. Siami, M.S. Tavazoei, and M. Haeri, “Stability preservation analysis in direct discretization of fractional order transfer functions”, Signal Processing 9 (2011), 508–512 (2011).
• [22] R. Stanisławski and K.J. Latawiec, “Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for asymptotic stability”, Bull. Pol. Ac.: Tech 61 (2), 353–361 (2013).
• [23] R. Stanisławski and K.J. Latawiec, “Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: New stability criterion for FD-based systems”, Bull. Pol. Ac.: Tech 61 (2), 362–370 (2013).

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).