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.
S. Sastry and M. Bodson, Adaptive Control. Stability, Convergence, and Robustness, Prentice Hall, Englewood Cliffs, NJ, USA, 1989.
View at: Google Scholar | Zentralblatt MATHB. 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 | MathSciNetD. 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 | MathSciNetA. 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 | MathSciNetA. 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 | MathSciNetA. P. Morgan and K. S. Narendra, “On the stability of nonautonomous differential equations , with skew symmetric matrix ,” SIAM Journal on Control and Optimization, vol. 15, no. 1, pp. 163–176, 1977.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetG. 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 | MathSciNetM. 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 | MathSciNetD. 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 | MathSciNetE. A. Coddington and N. Levinson, “Theory of Ordinary Differential Equations,” McGraw-Hill, New York, NY, USA, 1955.
View at: Google Scholar | Zentralblatt MATH | MathSciNetG. Sansone, “Equazioni differenziali nel compo reale,” Nicola Zanichelli, Bologna, Italy, 1949.
View at: Google Scholar | MathSciNetN. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov's Direct Method, Springer, New York, NY, USA, 1977.
View at: Google ScholarF. R. Gantmacher, “The Theory of Matrices,” Chelsea, New York, NY, USA, 1959.
View at: Google Scholar | Zentralblatt MATH | MathSciNetS. Gershgorin, “Über die Abgrenzung der Eigenwerte einer Matrix,” Izvestiya Akademii Nauk USSR, pp. 749–754, 1931.
View at: Google Scholar