This paper presents a research effort focused on the problem of robust stability of the closed-loop adaptive system. It is aimed at providing a general framework for the investigation of continuous-time, state-space systems required to track a (stable) reference model. This is motivated by the model reference adaptive control (MRAC) scheme, traditionally considered in such a setting. The application of differential inequlities results to the analysis of the Lyapunov stability for a class of nonlinear systems is investigated and it is shown how the problem of model following control may be tackled using this methodology.
@article{bwmeta1.element.bwnjournal-article-amcv15i4p455bwm,
author = {Dzielinski, Andrzej},
title = {Stability of a class of adaptive nonlinear systems},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {15},
year = {2005},
pages = {455-462},
zbl = {1127.93044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv15i4p455bwm}
}
Dzielinski, Andrzej. Stability of a class of adaptive nonlinear systems. International Journal of Applied Mathematics and Computer Science, Tome 15 (2005) pp. 455-462. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv15i4p455bwm/
[000] Corduneanu C. (1960): Application of differential inequalities to stability theory.- Analele Stiinctifice ale Universitatii 'Al. I. Cuza' din Iasi (Serie Noua). Sectiunea I (Matematica, Fizica, Chimie), Vol. VI, No. 1, pp. 47-58, (in Russian).
[001] Corduneanu C. (1961): Addendum to the paper Application of differential inequalities to stability theory. - Analele Stiinctifice ale Universitatii 'Al. I. Cuza' din Iasi (Serie Noua). Sectiunea I (Matematica, Fizica, Chimie), Vol. VII(2), pp. 247-252, (in Russian).
[002] Corduneanu C. (1964): On partial stability. - Revue Roumaine de Mathematiques Pures et Appliquees, Vol. IX(3), pp. 229-236, (in French).
[003] Dzieliński A. (2002a): Neural networks based NARX models in nonlinear adaptive control. - App. Math. Comput. Sci., Vol. 12, No. 2, pp. 101-106. | Zbl 1004.93500
[004] Dzieliński A. (2002b): Difference inequalities and BIBO stability of approximate NARX models. - Bull. Polish Acad. Sci., Techn. Sci., Vol. 50, No. 4, pp. 295-311. | Zbl 1175.93202
[005] Hahn W.(1963): Theory and Application of Liapunov's Direct Method. - Englewood Cliffs, NJ: Prentice-Hall. | Zbl 0119.07403
[006] Hahn W. (1967): Stability of Motion. - New York: Springer. | Zbl 0189.38503
[007] Hatvany L.(1975): On the application of differential inequalities to stability theory.- Vestnik Moskovskogo Universiteta, Vol. I30, No. 3, pp. 83-89, (in Russian).
[008] Lakshmikantham V.(1962a): Differential systems and extension of Lyapunov's method. - Michigan Math. J., Vol. 9, No. 4, pp. 311-320. | Zbl 0109.31301
[009] Lakshmikantham V. (1962b): Notes on variety of problems of differential systems. - Arch. Rat. Mech. Anal., Vol. 10, No. 2, pp. 119-126. | Zbl 0106.06103
[010] Lakshmikantham V. and Leela S. (1969a): Differential and Integral Inequalities. Theory and Applications, Vol. I: Ordinary Differential Equations. - New York: Academic Press. | Zbl 0177.12403
[011] Lakshmikantham V. and Leela S. (1969b): Differential and Integral Inequalities. Theory and Applications, Vol. II: Functional, Partial, Abstract, and Complex Differential Equations. - New York: Academic Press. | Zbl 0177.12403
[012] Liu X. and Siegel D., editors. (1994): Comparison Method in Stability Theory.- Amsterdam: Marcel Dekker.
[013] Luzin N. N. (1951): On the method of approximate integration due to academician S. A. Chaplygin. - Uspekhi matematicheskikh nauk, Vol. 6, No. 6, pp. 3-27, (in Russian).
[014] Makarov S. M. (1938): A generalisation of fundamental Lyapunov's theorems on stability of motion. - Izvestiya fiziko-matematicheskogo obshchestva pri Kazanskom gosudarstvennom universitete (Seriya 3), Vol. 10, No. 3, pp. 139-159, (in Russian).
[015] Narendra K. S. and Annaswamy A. M. (1989): Stable Adaptive Systems. - Eglewood Cliffs, NJ: Prentice-Hall.
[016] Pachpatte B. G. (1971): Finite-difference inequalities and an extension of Lyapunov's method. - Michigan Math. J., Vol. 18, No. 4, pp. 385-391. | Zbl 0237.39001
[017] Rabczuk R. (1976): Elements of Differential Inequalities. - Warsaw: Polish Scientific Publishers, (in Polish). | Zbl 0351.34006
[018] Szarski J. (1967): Differential Inequalities, 2nd Ed.. - Warsaw: Polish Scientific Publishers, (in Polish). | Zbl 0171.01502
[019] Walter W. (1970): Differential and Integral Inequalities. - Berlin: Springer.