About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 346041, 7 pages
http://dx.doi.org/10.1155/2013/346041
Review Article

Observability of Nonlinear Fractional Dynamical Systems

1Department of Mathematics, Bharathiar University, Coimbatore 641 046, India
2Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, 4200-072 Porto, Portugal
3Departamento de Matemática Fundamental, Universidad de La Laguna, La Laguna, 38271 Tenerife, Spain
4Departamento de Análisis Matemático, Universidad de La Laguna, La Laguna, 38271 Tenerife, Spain

Received 19 March 2013; Accepted 7 June 2013

Academic Editor: Hossein Jafari

Copyright © 2013 K. Balachandran et al. 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. J. A. T. Machado, “And i say to myself: what a fractional world!,” Fractional Calculus and Applied Analysis, vol. 14, no. 4, pp. 635–654, 2011. View at Publisher · View at Google Scholar · View at Scopus
  2. R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, Mass, USA, 2006.
  3. I. Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, Heidelberg, Germany, 2011.
  4. V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Nonlinear Physical Science, Springer, Heidelberg, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  5. D. Valério and J. S. da Costa, An Introduction to Fractional Control, IET, Stevenage, UK, 2012.
  6. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2008. View at MathSciNet
  7. B. Bonilla, M. Rivero, L. Rodríguez-Germá, and J. J. Trujillo, “Fractional differential equations as alternative models to nonlinear differential equations,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 79–88, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, Czestochowa University of Technology, Czestochowa, Poland, 2009.
  9. V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Volume 1. Back-Ground and Theory; Volume II. Applications, Springer Jointly published with Higher Education Press, 2013.
  10. J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. J. Tenreiro Machado, A. M. Galhano, and J. J. Trujillo, “Science metrics on fractional calculus development since 1966,” Fractional Calculus and Applied Analysis, vol. 16, no. 2, pp. 479–500, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  12. M. Bettayeb and S. Djennoune, “New results on the controllability and observability of fractional dynamical systems,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1531–1541, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  13. D. Matignon and B. d’Andréa-Novel, “Some results on controllability and observability of finite dimensional fractional differential systems,” in Proceedings of the IAMCS, IEEE Conference on Systems, Man and Cybernetics, pp. 952–956, Lille, France, 1996.
  14. J. Sabatier, M. Merveillaut, L. Fenetau, and A. Oustaloup, “On observability of fractional order systems,” in Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (DETC '09), pp. 253–260, September 2009. View at Scopus
  15. A. B. Shamardan and M. R. A. Moubarak, “Controllability and observability for fractional control systems,” Journal of Fractional Calculus, vol. 15, pp. 25–34, 1999. View at MathSciNet
  16. B. M. Vinagre, C. A. Monje, and A. J. Calderon, “Fractional order systems and fractional order control actions,” Fractional Calculus Applications in Automatic Control and Robotics, 2002.
  17. C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls, Springer, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  18. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, vol. 3, World Scientific Publishing, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  20. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at MathSciNet
  21. F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 283–299, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  22. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons Inc., New York, NY, USA, 1993. View at MathSciNet
  23. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, London, UK, 1974. View at MathSciNet
  24. M. Caputo, “Linear model of dissipation whose Q is almost frequency independent,” Geophysical Journal of Royal Astronomical Society, vol. 13, pp. 529–539, 1967.
  25. S. D. Éidel'man and A. A. Chikriĭ, “Dynamic game-theoretic approach problems for fractional-order equations,” Ukrainian Mathematical Journal, vol. 52, no. 11, pp. 1787–1806, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  26. T. Kaczorek, Selected Problems of Fractional Systems Theory, vol. 411, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  27. X.-J. Wen, Z.-M. Wu, and J.-G. Lu, “Stability analysis of a class of nonlinear fractional-order systems,” IEEE Transactions on Circuits and Systems II, vol. 55, no. 11, pp. 1178–1182, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser, Berlin, Germany, 1992. View at MathSciNet