Have a personal or library account? Click to login
Normalized finite fractional differences: Computational and accuracy breakthroughs Cover

Normalized finite fractional differences: Computational and accuracy breakthroughs

International Journal of Applied Mathematics and Computer Science
Volume 22 (2012): Issue 4 (December 2012)
By:
Open Access
|Dec 2012

References

  1. Bandrowski, B., Karczewska, A. and Rozmej, P. (2010). Numerical solutions to integral equations equivalent to differential equations with fractional time, <em>International Journal of Applied Mathematics and Computer Science </em><bold>20</bold>(2): 261-269, DOI: <a href="https://doi.org/10.2478/v10006-010-0019-1." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10006-010-0019-1.</a><dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.2478/v10006-010-0019-1" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10006-010-0019-1</a></dgdoi:pub-id>
    Search in Google Scholar
  2. Barbosa, R. and Machado, J. (2006). Implementation of discrete-time fractional-order controllers based on LS approximations, <em>Acta Polytechnica Hungarica </em><bold>3</bold>(4): 5-22.
    Search in Google Scholar
  3. Busłowicz, M. and Kaczorek, T. (2009). Simple conditions for practical stability of positive fractional discrete-time linear systems, <em>International Journal of Applied Mathematics and Computer Science </em><bold>19</bold>(2): 263-269, DOI: <a href="https://doi.org/10.2478/v10006-009-0022-6." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10006-009-0022-6.</a><dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.2478/v10006-009-0022-6" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10006-009-0022-6</a></dgdoi:pub-id>
    Search in Google Scholar
  4. Chen, Y., Vinagre, B. and Podlubny, I. (2003). A new discretization method for fractional order differentiators via continued fraction expansion, <em>Proceedings of DETC’2003, ASME Design Engineering Technical Conferences, Chicago, IL, USA, </em>Vol. 340, pp. 349-362.
    Search in Google Scholar
  5. Debeljković, D.L., Aleksendric´, M., Yi-Yong, N. and Zhang, Q. (2002). Lyapunov and nonlyapunov stability of linear discrete time delay systems, <em>Facta Universitatis Mechanical Engineering </em><bold>14</bold>(9-10): 1147-1160.
    Search in Google Scholar
  6. Delavari, H., Ranjbar, A., Ghaderi, R. and Momani, S. (2010). Fractional order control of a coupled tank, <em>Nonlinear Dynamics </em><bold>61</bold>(3): 383-397.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1007/s11071-010-9656-z" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1007/s11071-010-9656-z</a></dgdoi:pub-id>
    Search in Google Scholar
  7. Dzieliński, A. and Sierociuk, D. (2008). Stability of discrete fractional order state-space systems, <em>Journal of Vibration and Control </em><bold>14</bold>(9-10): 1543-1556.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1177/1077546307087431" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1177/1077546307087431</a></dgdoi:pub-id>
    Search in Google Scholar
  8. Guermah, S., Djennoune, S. and Bettayeb, M. (2010). A new approach for stability analysis of linear discrete-time fractional-order systems, <em>in </em>D. Baleanu, Z.B. Güvenç and J.A.T. Machado (Eds.), <em>New Trends in Nanotechnology and Fractional Calculus Applications</em>, Springer, Dordrecht, pp. 151-162.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1007/978-90-481-3293-5_11" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1007/978-90-481-3293-5_11</a></dgdoi:pub-id>
    Search in Google Scholar
  9. Hunek, W.P. and Latawiec, K.J. (2011). A study on new right/left inverses of nonsquare polynomial matrices, <em>International Journal of Applied Mathematics and Computer Science </em><bold>21</bold>(2): 331-348, DOI: <a href="https://doi.org/10.2478/v10006-011-0025-y." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10006-011-0025-y.</a><dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.2478/v10006-011-0025-y" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10006-011-0025-y</a></dgdoi:pub-id>
    Search in Google Scholar
  10. Kaczorek, T. (2008). Practical stability of positive fractional discrete-time linear systems, <em>Bulletin of the Polish Academy of Sciences: Technical Sciences </em><bold>56</bold>(4): 313-317.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.2478/v10175-010-0143-y" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10175-010-0143-y</a></dgdoi:pub-id>
    Search in Google Scholar
  11. Latawiec, K.J. (2004). <em>The Power of Inverse Systems in Linear and Nonlinear Modeling and Control</em>, Opole University of Technology Press, Opole.
    Search in Google Scholar
  12. Liavas, A.P. and Regalia, P. (1999). On the numerical stability and accuracy of the conventional recursive least squares algorithm, <em>IEEE Transactions on Signal Processing </em><bold>47</bold>(1): 88-96.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1109/78.738242" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1109/78.738242</a></dgdoi:pub-id>
    Search in Google Scholar
  13. Lubich, C.H. (1986). Discretized fractional calculus, <em>SIAM Journal on Mathematical Analysis </em><bold>17</bold>(3): 704-719.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1137/0517050" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1137/0517050</a></dgdoi:pub-id>
    Search in Google Scholar
  14. Maione, G. (2006). A digital, noninteger order, differentiator using laguerre orthogonal sequences, <em>International Journal of Intelligent Control and Systems </em><bold>11</bold>(2): 77-81.
    Search in Google Scholar
  15. Miller, K. and Ross, B. (1993). <em>An Introduction to the Fractional Calculus and Fractional Differential Equations</em>, Willey, New York, NY.
    Search in Google Scholar
  16. Momani, S. and Odibat, Z. (2007). Numerical approach to differential equations of fractional order, <em>Journal of Computational and Applied Mathematics </em><bold>207</bold>(1): 96 - 110.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1016/j.cam.2006.07.015" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1016/j.cam.2006.07.015</a></dgdoi:pub-id>
    Search in Google Scholar
  17. Monje, C., Chen, Y., Vinagre, B., Xue, D. and Feliu, V. (2010). <em>Fractional-order Systems and Controls</em>, Springer-Verlag, London.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1007/978-1-84996-335-0" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1007/978-1-84996-335-0</a></dgdoi:pub-id>
    Search in Google Scholar
  18. Oldham, K. and Spanier, J. (1974). <em>The Fractional Calculus</em>, Academic Press, Orlando, FL.
    Search in Google Scholar
  19. Ortigueira, M.D. (2000). Introduction to fractional linear systems, II: Discrete-time case, <em>IEE Proceedings on Vision, Image and Signal Processing </em><bold>147</bold>(1): 71-78.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1049/ip-vis:20000273" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1049/ip-vis:20000273</a></dgdoi:pub-id>
    Search in Google Scholar
  20. Ostalczyk, P. (2000). The non-integer difference of the discrete-time function and its application to the control system synthesis, <em>International Journal of Systems Science </em><bold>31</bold>(12): 1551-1561.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1080/00207720050217322" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1080/00207720050217322</a></dgdoi:pub-id>
    Search in Google Scholar
  21. Ostalczyk, P. (2010). Stability analysis of a discrete-time system with a variable-fractional-order controller, <em>Bulletin of the Polish Academy of Sciences: Technical Sciences </em><bold>58</bold>(4): 613-619.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.2478/v10175-010-0063-x" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10175-010-0063-x</a></dgdoi:pub-id>
    Search in Google Scholar
  22. Petráš, I., Dorčák, L. and Koštial, I. (2000). The modelling and analysis of fractional-order control systems in discrete domain, <em>Proceedings of the International Carpa-tian Control Conference, High Tatras, Slovak Republic</em>, pp. 257-260.
    Search in Google Scholar
  23. Petráš, I. and Vinagre, B. (2002). Practical application of digital fractional-order controller to temperature control, <em>Acta Montanistica Slovaca </em><bold>7</bold>(2): 131-137.
    Search in Google Scholar
  24. Podlubny, I. (1999). <em>Fractional Differential Equations</em>, Academic Press, Orlando, FL.
    Search in Google Scholar
  25. Riu, D., Retiére, N. and Ivanes, M. (2001). Turbine generator modeling by non-integer order systems, <em>IEEE International Conference on Electric Machines and Drives, Cambridge, MA, USA</em>, pp. 185-187.
    Search in Google Scholar
  26. Saeedi, H., Mollahasani, N., Moghadam, M.M. and Chuev, G.N. (2011). An operational Haar wavelet method for solving fractional Volterra integral equations, <em>International Journal of Applied Mathematics and Computer Science </em><bold>21</bold>(3): 535-547, DOI: <a href="https://doi.org/10.2478/v10006-011-0042-x." target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10006-011-0042-x.</a><dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.2478/v10006-011-0042-x" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2478/v10006-011-0042-x</a></dgdoi:pub-id>
    Search in Google Scholar
  27. Sierociuk, D. and Dzielin´ski, A. (2006). Fractional Kalman filter algorithm for states, parameters and order of fractional system estimation, <em>International Journal of Applied Mathematics and Computer Science </em><bold>16</bold>(1): 129-140.
    Search in Google Scholar
  28. Stanisławski, R. (2009). Identification of open-loop stable linear systems using fractional orthonormal basis functions, <em>Proceedings of the 14th International Conference on Methods and Models in Automation and Robotics, Mie˛dzyzdroje, Poland</em>, pp. 935-985.
    Search in Google Scholar
  29. 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, <em>Proceedings of the 15th International Conference on Methods and Models in Automation and Robotics, Mie˛dzyz-droje, Poland</em>, pp. 411-414.
    Search in Google Scholar
  30. Stanisławski, R. and Latawiec, K.J. (2011). Finite approximations of a discrete-time fractional derivative, <em>16th International Conference on Methods and Models in Automation and Robotics, Mie˛dzyzdroje, Poland</em>, pp. 142-145.
    Search in Google Scholar
  31. Stojanovic, S.B. and Debeljkovic, D.L. (2010). Simple stability conditions of linear discrete time systems with multiple delay, <em>Serbian Journal of Electrical Engineering </em><bold>7</bold>(1): 69-79.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.2298/SJEE1001069S" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.2298/SJEE1001069S</a></dgdoi:pub-id>
    Search in Google Scholar
  32. Sun, H., Chen, W. and Chen, Y. (2009). Variable-order fractional differential operators in anomalous diffusion modeling, <em>Physica A: Statistical Mechanics and Its Applications </em><bold>388</bold>(21): 4586-4592.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1016/j.physa.2009.07.024" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1016/j.physa.2009.07.024</a></dgdoi:pub-id>
    Search in Google Scholar
  33. Tseng, C., Pei, S. and Hsia, S. (2000). Computation of fractional derivatives using Fourier transform and digital FIR diferentiator, <em>Signal Processing </em><bold>80</bold>(1): 151-159.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1016/S0165-1684(99)00118-8" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1016/S0165-1684(99)00118-8</a></dgdoi:pub-id>
    Search in Google Scholar
  34. Valério, D. and Sá da Costa, J. (2011). Variable-order fractional derivatives and their numerical approximations, <em>Signal Processing </em><bold>91</bold>(3): 470-483.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1016/j.sigpro.2010.04.006" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1016/j.sigpro.2010.04.006</a></dgdoi:pub-id>
    Search in Google Scholar
  35. Verhaegen, M. H. (1989). Round-off error propagation in four generally-applicable, recursive, least-squares estimation schemes, <em>Automatica </em><bold>25</bold>(3): 437-444.<dgdoi:pub-id xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" pub-id-type="doi"><a href="https://doi.org/10.1016/0005-1098(89)90013-7" target="_blank" rel="noopener noreferrer" class="text-signal-blue hover:underline">10.1016/0005-1098(89)90013-7</a></dgdoi:pub-id>
    Search in Google Scholar
  36. Vinagre, B., Podlubny, I., Hernandez, A. and Feliu, V. (2000). Some approximations of fractional order operators used in control theory and applications, <em>Fractional Calculus &amp; Applied Analysis </em><bold>3</bold>(3): 945-950.
    Search in Google Scholar
  37. Zaborowsky, V. and Meylaov, R. (2001). Informational network traffic model based on fractional calculus, <em>Proceedings of the International Conference on Info-tech and Info-net, ICII 2001, Beijing, China, </em>Vol. 1, pp. 58-63.
    Search in Google Scholar
DOI: https://doi.org/10.2478/v10006-012-0067-9 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
Journal RSS Feed
Language: English
Page range: 907 - 919
Published on: Dec 28, 2012
Published by: Sciendo
In partnership with: Paradigm Publishing Services
Publication frequency: 4 times per year
Keywords:
fractional difference,
Grünwald-Letnikov difference,
stability analysis,
recursive computation,
adaptive systems.
Related subjects:
Mathematics,
Applied mathematics

© 2012 Rafał Stanisławski, Krzysztof J. Latawiec, published by Sciendo
This work is licensed under the Creative Commons License.

Previous articleVolume 22 (2012): Issue 4 (December 2012)Next article