International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2007 / Article

Research Article | Open Access

Volume 2007 |Article ID 069093 | https://doi.org/10.1155/2007/69093

Sergey Nikitin, "Generalized Persistency of Excitation", International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 069093, 11 pages, 2007. https://doi.org/10.1155/2007/69093

Generalized Persistency of Excitation

Academic Editor: Irena Lasiecka
Received08 May 2007
Revised15 Sep 2007
Accepted10 Oct 2007
Published30 Dec 2007

Abstract

The paper presents the generalized persistency of excitation conditions. Not only they are valid for much broader range of applications than their classical counterparts but also they elegantly prove the validity of the latter. The novelty and the significance of the approach presented in this publication is due to employing the time averaging technique.

References

  1. S. Sastry and M. Bodson, Adaptive Control. Stability, Convergence, and Robustness, Prentice Hall, Englewood Cliffs, NJ, USA, 1989. View at: Google Scholar | Zentralblatt MATH
  2. B. D. O. Anderson, “Exponential stability on linear equations arising in adaptive identification,” IEEE Transactions on Automatic Control, vol. 22, no. 1, pp. 83–88, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. D. Janecki, “Persistency of excitation for continuous-time systems---time-domain approach,” Systems & Control Letters, vol. 8, no. 4, pp. 333–344, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. A. Loría and E. Panteley, “Uniform exponential stability of linear time-varying systems: revisited,” Systems & Control Letters, vol. 47, no. 1, pp. 13–24, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  5. A. P. Morgan and K. S. Narendra, “On the uniform asymptotic stability of certain linear nonautonomous differential equations,” SIAM Journal on Control and Optimization, vol. 15, no. 1, pp. 5–24, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. A. P. Morgan and K. S. Narendra, “On the stability of nonautonomous differential equations x˙=[A+B(t)]x, with skew symmetric matrix B(t),” SIAM Journal on Control and Optimization, vol. 15, no. 1, pp. 163–176, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. G. Kreisselmeier, “Adaptive observers with exponential rate of convergence,” IEEE Transactions on Automatic Control, vol. 22, no. 1, pp. 2–8, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. M. M. Sondhi and D. Mitra, “New results on the performance of a well-known class of adaptive filters,” Proceedings of the IEEE, vol. 64, no. 11, pp. 1583–1597, 1976. View at: Google Scholar | MathSciNet
  9. D. Aeyels, R. Sepulchre, and J. Peuteman, “Asymptotic stability for time-variant systems and observability: uniform and nonuniform criteria,” Mathematics of Control, Signals, and Systems, vol. 11, no. 1, pp. 1–27, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. E. A. Coddington and N. Levinson, “Theory of Ordinary Differential Equations,” McGraw-Hill, New York, NY, USA, 1955. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  11. G. Sansone, “Equazioni differenziali nel compo reale,” Nicola Zanichelli, Bologna, Italy, 1949. View at: Google Scholar | MathSciNet
  12. N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov's Direct Method, Springer, New York, NY, USA, 1977. View at: Google Scholar
  13. F. R. Gantmacher, “The Theory of Matrices,” Chelsea, New York, NY, USA, 1959. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  14. S. Gershgorin, “Über die Abgrenzung der Eigenwerte einer Matrix,” Izvestiya Akademii Nauk USSR, pp. 749–754, 1931. View at: Google Scholar

Copyright © 2007 Sergey Nikitin. 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.


More related articles

 PDF Download Citation Citation
 Order printed copiesOrder
Views215
Downloads551
Citations

Related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.