| Title:
|
Stabilization of homogeneous polynomial systems in the plane (English) |
| Author:
|
Jerbi, Hamadi |
| Author:
|
Kharrat, Thouraya |
| Author:
|
Sioud, Khaled |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
52 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
131-152 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we study the problem of stabilization via homogeneous feedback of single-input homogeneous polynomial systems in the plane. We give a complete classification of systems for which there exists a homogeneous stabilizing feedback that is smooth on $\mathbb{R}^2 \setminus\{ (0,0)\}$ and preserve the homogeneity of the closed loop system. Our results are essentially based on Theorem of Hahn in which the author gives necessary and sufficient conditions of stability of homogeneous systems in the plane. (English) |
| Keyword:
|
polynomial system |
| Keyword:
|
control system |
| Keyword:
|
homogeneous feedback |
| Keyword:
|
stabilization |
| MSC:
|
93D15 |
| idZBL:
|
Zbl 1374.93301 |
| idMR:
|
MR3482615 |
| DOI:
|
10.14736/kyb-2016-1-0131 |
| . |
| Date available:
|
2016-03-21T17:56:56Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144867 |
| . |
| Reference:
|
[1] Artstein, Z.: Stabilization with relaxed controls..Nonlinear Analysis, TMA, 7 (1983), 1163-1173. Zbl 0525.93053, MR 0721403, 10.1016/0362-546x(83)90049-4 |
| Reference:
|
[2] Coron, J. M., Praly, L.: Adding an integrator for the stabilization problem..Systems Control Lett. 17 (1991), 89-104. Zbl 0747.93072, MR 1120754, 10.1016/0167-6911(91)90034-c |
| Reference:
|
[3] Hahn, W.: Stability of Motion..Springer-Verlag 1967. Zbl 0189.38503, MR 0223668, 10.1007/978-3-642-50085-5 |
| Reference:
|
[4] Hermes, H.: Homogeneous feedback controls for homogeneous systems..Systems Control Lett. 24 (1995), 7-11. Zbl 0877.93088, MR 1307121, 10.1016/0167-6911(94)00035-t |
| Reference:
|
[5] Jerbi, H., Maaloum, A. Ould: Feedback stabilization of homogeneous polynomial systems of odd degree in the plane..Systems Control Lett. 56 (2007), 611-617. MR 2344649, 10.1016/j.sysconle.2007.04.002 |
| Reference:
|
[6] Jerbi, H., Kharrat, T.: Asymptotic stabilizability of homogeneous polynomial systems of odd degree..Systems Control Lett. 48 (2003), 87-99. Zbl 1097.93033, MR 2011869, 10.1016/s0167-6911(02)00248-7 |
| Reference:
|
[7] Kawski, M.: Homogeneous stabilizing feedback laws..Control-Theory Advanced Technol. 6 (1990), 497-516. MR 1092775 |
| Reference:
|
[8] Rosier, L.: Homogeneous Lyapunov function for homogeneous continuous vector field..Systems Control Lett. 19 (1992), 467-473. Zbl 0762.34032, MR 1195304, 10.1016/0167-6911(92)90078-7 |
| Reference:
|
[9] Massera, J. L.: Contibutions to stability theory..Ann. Math. 64 (1956), 182-206. MR 0079179, 10.2307/1969955 |
| Reference:
|
[10] Sontag, E. D.: A "universal" construction of Artstein's theorem on nonlinear stabilization..Systems Control Lett. 13 (1989), 117-123. Zbl 0684.93063, MR 1014237, 10.1016/0167-6911(89)90028-5 |
| Reference:
|
[11] Tsinias, J.: Stabilization of affine in control nonlinear systems..Nonlinear Analysis, TMA 12 (1988), 1283-1296. Zbl 0662.93055, MR 0969506, 10.1016/0362-546x(88)90060-0 |
| Reference:
|
[12] Tsinias, J.: Sufficient Lyapunov like conditions for stabilization..Math. Control, Signals Systems 2 (1989), 343-357. Zbl 0688.93048, MR 1015672, 10.1007/bf02551276 |
| . |
