Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2015, Article ID 638420, 7 pages
http://dx.doi.org/10.1155/2015/638420
Research Article

Constrained Controllability of the -Difference Fractional Control Systems with Caputo Type Operator

Bialystok University of Technology, 15-351 Bialystok, Poland

Received 7 April 2015; Revised 18 October 2015; Accepted 20 October 2015

Academic Editor: Alicia Cordero

Copyright © 2015 Ewa Pawluszewicz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Z. Bartosiewicz and E. Pawłuszewicz, “Realization of linear control systems on time scales,” Control & Cybernetics, vol. 35, no. 4, pp. 769–786, 2006. View at Google Scholar · View at Scopus
  2. J. M. Davis, I. A. Gravagne, B. J. Jackson, and I. Marks, “Controllability, observability, realizability, and stability of dynamic linear systems,” Electronic Journal of Differential Equations, vol. 37, pp. 1–32, 2009. View at Google Scholar · View at MathSciNet
  3. T. Kaczorek, Selected Problems of Fractional Systems Theory, vol. 411 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar
  4. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  5. T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. K. S. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, pp. 139–152, Nihon University, Kōriyama, Japan, 1988.
  7. M. D. Ortigueira, “Fractional discrete-time linear systems,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '97), vol. 3, pp. 2241–2244, Munich, Germany, April 1997.
  8. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  9. D. S. Karanjkar, S. Chatterij, and P. R. Vankateswam, “Trends in fractional order controller,” International Journal of Emerging Technology and Advanced Engineering, vol. 2, no. 3, pp. 383–389, 2012. View at Google Scholar
  10. D. Sierociuk, A. Dzielinski, G. Sarwas, I. Petras, I. Podlubny, and T. Skovranek, “Modelling heat transfer in heterogeneous media using fractional calculus,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 371, no. 1990, 2013. View at Publisher · View at Google Scholar · View at Scopus
  11. K. Balachandran and J. Kokila, “On the controllability of fractional dynamical systems,” International Journal of Applied Mathematics and Computer Science, vol. 22, no. 3, pp. 523–531, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. M. Bettayeb and S. Djennoune, “A note on the controllability and the obseravbility of fractional dyanmical systems,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Application, pp. 493–498, Porto, Portugal, July 2006.
  13. J. Klamka, “Local controllability of fractional discrete-time nonlinear systems with delay in control,” in Advances in Control Theory, M. Buslowicz and K. Malinowski, Eds., pp. 25–34, Committe on Automatic Control and Robotics, Polish Academy of Sciences, Białystok, Poland, 2012. View at Google Scholar
  14. D. Mozyrska and E. Pawluszewicz, “Fractional discrete-time linear control systems with initialisation,” International Journal of Control, vol. 85, no. 2, pp. 213–219, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. D. Mozyrska and E. Pawluszewicz, “Controllability of h-difference linear control systems with two fractional orders,” International Journal of Systems Science, vol. 46, no. 4, pp. 662–669, 2015. View at Publisher · View at Google Scholar
  16. D. Mozyrska, E. Pawłuszewicz, and M. Wyrwas, “The h-difference approach to controllability and observability of fractional linear systems with Caputo type opeartor,” Asian Journal of Control, vol. 17, no. 4, pp. 1163–1173, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  17. D. L. Abel, “Constrains vs controls,” The Open Cybernetics & Systemics Journal, vol. 4, pp. 14–27, 2010. View at Google Scholar
  18. B. R. Barmish and W. E. Schmitendorf, “A necessary and sufficient condition for local constrained controllability of a linear system,” IEEE Transactions on Automatic Control, vol. 25, no. 1, pp. 97–100, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  19. W. E. Schmitendorf and W. G. Hwang, “Global reachability results for systems with constrained controllers,” Journal of Optimization Theory and Applications, vol. 46, no. 4, pp. 581–590, 1985. View at Publisher · View at Google Scholar
  20. R. P. Van Til and W. E. Schmitendorf, “Constrained controllability of discrete-time systems,” International Journal of Control, vol. 43, no. 3, pp. 941–956, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. K. Janglajew and E. Pawłuszewicz, “Constrained local controllability of dynamic systems on time scales,” Advances in Difference Equations, vol. 2015, article 89, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  22. E. Pawłuszewicz and D. Mozyrska, “Constrained controllability of h-difference linear systems with two fractional orders,” in Advances in the Theory and Applications of Non-Integer Order Systems, W. Mitkowski, J. Kacprzyk, and J. Baranowski, Eds., vol. 257 of Lecture Notes in Electrical Engineering, pp. 67–75, Springer, Berlin, Germany, 2013. View at Publisher · View at Google Scholar
  23. K. Balachandran and J. Kokila, “Constrained controllability of fractional dynamical systems,” Numerical Functional Analysis and Optimization, vol. 34, no. 11, pp. 1187–1205, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. D. Mozyrska and M. Wyrwas, “The Z-transform method and delta type fractional difference operators,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 852734, 12 pages, 2015. View at Publisher · View at Google Scholar
  25. S. Rolewicz, Functional Theory and Control Theory, WNT, 1997 (Polish).
  26. D. Mozyrska and E. Girejko, “Overview of the fractional h-difference operators,” in Advances in Harmonic Analysis and Operator Theory: The Stefan Samko Anniversary Volume, vol. 229 of Operator Theory: Advances and Applications, pp. 253–267, Springer, 2013. View at Google Scholar
  27. D. Mozyrska and E. Pawluszewicz, “Local controllability of nonlinear discrete-time fractional order systems,” Bulletin of the Polish Academy of Sciences: Technical Sciences, vol. 61, no. 1, pp. 251–256, 2013. View at Publisher · View at Google Scholar · View at Scopus
  28. E. D. Sontag, Mathematical Control Theory, Springer, 1998.
  29. D. Mozyrska, E. Girejko, and M. Wyrwas, “Fractional nonlinear systems with sequential operators,” Central European Journal of Physics, vol. 11, no. 10, pp. 1295–1303, 2013. View at Publisher · View at Google Scholar · View at Scopus
  30. J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhuser, Boston, Mass, USA, 1992. View at MathSciNet
  31. E. Pawłuszewicz, “Null-controllability of linear systems on time scales,” Acta Mechanica et Automatica, vol. 6, no. 4, pp. 50–55, 2012. View at Google Scholar · View at Scopus