PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2012 | 22 | 4 | 907-919
Tytuł artykułu

Normalized finite fractional differences: Computational and accuracy breakthroughs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This paper presents a series of new results in finite and infinite-memory modeling of discrete-time fractional differences. The introduced normalized finite fractional difference is shown to properly approximate its fractional difference original, in particular in terms of the steady-state properties. A stability analysis is also presented and a recursive computation algorithm is offered for finite fractional differences. A thorough analysis of computational and accuracy aspects is culminated with the introduction of a perfect finite fractional difference and, in particular, a powerful adaptive finite fractional difference, whose excellent performance is illustrated in simulation examples.
Rocznik
Tom
22
Numer
4
Strony
907-919
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-06-16
poprawiono
2011-11-18
poprawiono
2012-04-16
Twórcy
  • Institute of Control and Computer Engineering, Opole University of Technology, ul. Sosnkowskiego 31, 45-272 Opole, Poland
  • Institute of Control and Computer Engineering, Opole University of Technology, ul. Sosnkowskiego 31, 45-272 Opole, Poland
Bibliografia
  • Bandrowski, B., Karczewska, A. and Rozmej, P. (2010). Numerical solutions to integral equations equivalent to differential equations with fractional time, International Journal of Applied Mathematics and Computer Science 20(2): 261-269, DOI: 10.2478/v10006-010-0019-1.
  • Barbosa, R. and Machado, J. (2006). Implementation of discrete-time fractional-order controllers based on LS approximations, Acta Polytechnica Hungarica 3(4): 5-22.
  • Busłowicz, M. and Kaczorek, T. (2009). Simple conditions for practical stability of positive fractional discrete-time linear systems, International Journal of Applied Mathematics and Computer Science 19(2): 263-269, DOI: 10.2478/v10006-009-0022-6.
  • Chen, Y., Vinagre, B. and Podlubny, I. (2003). A new discretization method for fractional order differentiators via continued fraction expansion, Proceedings of DETC'2003, ASME Design Engineering Technical Conferences, Chicago, IL, USA, Vol. 340, pp. 349-362.
  • Debeljković, D.L., Aleksendrić, M., Yi-Yong, N. and Zhang, Q. (2002). Lyapunov and nonlyapunov stability of linear discrete time delay systems, Facta Universitatis Mechanical Engineering 14(9-10): 1147-1160.
  • Delavari, H., Ranjbar, A., Ghaderi, R. and Momani, S. (2010). Fractional order control of a coupled tank, Nonlinear Dynamics 61(3): 383-397.
  • Dzieliński, A. and Sierociuk, D. (2008). Stability of discrete fractional order state-space systems, Journal of Vibration and Control 14(9-10): 1543-1556.
  • Guermah, S., Djennoune, S. and Bettayeb, M. (2010). A new approach for stability analysis of linear discrete-time fractional-order systems, in D. Baleanu, Z.B. Güvenç and J.A.T. Machado (Eds.), New Trends in Nanotechnology and Fractional Calculus Applications, Springer, Dordrecht, pp. 151-162.
  • Hunek, W.P. and Latawiec, K.J. (2011). A study on new right/left inverses of nonsquare polynomial matrices, International Journal of Applied Mathematics and Computer Science 21(2): 331-348, DOI: 10.2478/v10006-011-0025-y.
  • Kaczorek, T. (2008). Practical stability of positive fractional discrete-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 313-317.
  • Latawiec, K.J. (2004). The Power of Inverse Systems in Linear and Nonlinear Modeling and Control, Opole University of Technology Press, Opole.
  • Liavas, A.P. and Regalia, P. (1999). On the numerical stability and accuracy of the conventional recursive least squares algorithm, IEEE Transactions on Signal Processing 47(1): 88-96.
  • Lubich, C.H. (1986). Discretized fractional calculus, SIAM Journal on Mathematical Analysis 17(3): 704-719.
  • Maione, G. (2006). A digital, noninteger order, differentiator using laguerre orthogonal sequences, International Journal of Intelligent Control and Systems 11(2): 77-81.
  • Miller, K. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Willey, New York, NY.
  • Momani, S. and Odibat, Z. (2007). Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics 207(1): 96-110.
  • Monje, C., Chen, Y., Vinagre, B., Xue, D. and Feliu, V. (2010). Fractional-order Systems and Controls, Springer-Verlag, London.
  • Oldham, K. and Spanier, J. (1974). The Fractional Calculus, Academic Press, Orlando, FL.
  • Ortigueira, M.D. (2000). Introduction to fractional linear systems, II: Discrete-time case, IEE Proceedings on Vision, Image and Signal Processing 147(1): 71-78.
  • Ostalczyk, P. (2000). The non-integer difference of the discrete-time function and its application to the control system synthesis, International Journal of Systems Science 31(12): 1551-1561.
  • Ostalczyk, P. (2010). Stability analysis of a discrete-time system with a variable-fractional-order controller, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(4): 613-619.
  • Petráš, I., Dorčák, L. and Koštial, I. (2000). The modelling and analysis of fractional-order control systems in discrete domain, Proceedings of the International Carpatian Control Conference, High Tatras, Slovak Republic, pp. 257-260.
  • Petráš, I. and Vinagre, B. (2002). Practical application of digital fractional-order controller to temperature control, Acta Montanistica Slovaca 7(2): 131-137.
  • Podlubny, I. (1999). Fractional Differential Equations, Academic Press, Orlando, FL.
  • Riu, D., Retiére, N. and Ivanes, M. (2001). Turbine generator modeling by non-integer order systems, IEEE International Conference on Electric Machines and Drives, Cambridge, MA, USA, pp. 185-187.
  • Saeedi, H., Mollahasani, N., Moghadam, M.M. and Chuev, G.N. (2011). An operational Haar wavelet method for solving fractional Volterra integral equations, International Journal of Applied Mathematics and Computer Science 21(3): 535-547, DOI: 10.2478/v10006-011-0042-x.
  • Sierociuk, D. and Dzieliński, A. (2006). Fractional Kalman filter algorithm for states, parameters and order of fractional system estimation, International Journal of Applied Mathematics and Computer Science 16(1): 129-140.
  • Stanisławski, R. (2009). Identification of open-loop stable linear systems using fractional orthonormal basis functions, Proceedings of the 14th International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 935-985.
  • Stanisławski, R. and Latawiec, K.J. (2010). Modeling of open-loop stable linear systems using a combination of a finite fractional derivative and orthonormal basis functions, Proceedings of the 15th International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 411-414.
  • Stanisławski, R. and Latawiec, K.J. (2011). Finite approximations of a discrete-time fractional derivative, 16th International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 142-145.
  • Stojanovic, S.B. and Debeljkovic, D.L. (2010). Simple stability conditions of linear discrete time systems with multiple delay, Serbian Journal of Electrical Engineering 7(1): 69-79.
  • Sun, H., Chen, W. and Chen, Y. (2009). Variable-order fractional differential operators in anomalous diffusion modeling, Physica A: Statistical Mechanics and Its Applications 388(21): 4586-4592.
  • Tseng, C., Pei, S. and Hsia, S. (2000). Computation of fractional derivatives using Fourier transform and digital FIR diferentiator, Signal Processing 80(1): 151-159.
  • Valério, D. and Sá da Costa, J. (2011). Variable-order fractional derivatives and their numerical approximations, Signal Processing 91(3): 470-483.
  • Verhaegen, M. H. (1989). Round-off error propagation in four generally-applicable, recursive, least-squares estimation schemes, Automatica 25(3): 437-444.
  • Vinagre, B., Podlubny, I., Hernandez, A. and Feliu, V. (2000). Some approximations of fractional order operators used in control theory and applications, Fractional Calculus & Applied Analysis 3(3): 945-950.
  • Zaborowsky, V. and Meylaov, R. (2001). Informational network traffic model based on fractional calculus, Proceedings of the International Conference on Info-tech and Info-net, ICII 2001, Beijing, China, Vol. 1, pp. 58-63.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv22z4p907bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.