zbMATH — the first resource for mathematics

Stabilization of nonlinear systems: A bilinear approach. (English) Zbl 0797.93038
Summary: This paper gives a necessary and sufficient condition for the stabilization of planar bilinear systems. Local stabilization results are obtained for other systems.

93D15 Stabilization of systems by feedback
93C15 Control/observation systems governed by ordinary differential equations
93C10 Nonlinear systems in control theory
Full Text: DOI
[1] D. Aeyels, Local and global stabilizability for nonlinear systems, inTheory and Applications of Nonlinear Control Systems (C. I. Byrnes and A. Lindquist, eds.), Elsevier, Amsterdam, 1986, pp. 93-105.
[2] Z. Artstein, Stabilization with relaxed controls,Nonlinear Anal.,7 (1983), 1163-1173. · Zbl 0525.93053 · doi:10.1016/0362-546X(83)90049-4
[3] A. Bacciotti and P. Boieri, Linear stabilizability of planar nonlinear systems,Math. Control Signals Systems,3 (1990), 183-193. · Zbl 0694.93082 · doi:10.1007/BF02551367
[4] A Bacciotti and P. Boieri, A characterization of single input planar bilinear systems which admit a smooth stabilizer,Systems Control Lett.,16 (1991), 139-143. · Zbl 0732.93067 · doi:10.1016/0167-6911(91)90008-3
[5] W. Boothby and R. Marino, Feedback stabilization of planar nonlinear systems,Systems Control Lett.,12 (1989), 87-92. · Zbl 0684.93062 · doi:10.1016/0167-6911(89)90100-X
[6] R. W. Brockett. Asymptotic stability and feedback stabilization, inDifferential Geometric Control Theory (R. W. Brockett, R. S. Millman, and H. J. Sussmann, eds.), Birhäuser, Boston, 1983, pp. 181-191. · Zbl 0528.93051
[7] C. Byrnes and A. Isidori, Local stabilization of minimum phase nonlinear systems,Systems Control Lett.,11 (1988), 9-17. · Zbl 0649.93030 · doi:10.1016/0167-6911(88)90105-3
[8] C. Byrnes, A. Isidori, and J. C. Willems, Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems,IEEE Trans. Automat. Control,36 (1991), 1228-1240. · Zbl 0758.93007 · doi:10.1109/9.100932
[9] R. Chabour and J. C. Vivalda, Stabilisation des systèmes bilinéaires dans le plan par une commande non régulière,Proceedings of the European Control Conference, Hermes, 1991, pp. 485-487.
[10] W. P. Dayawansa and C. F. Martin, Asymptotic stabilization of two-dimensional real analytic systems,Systems Control Lett,12 (1989), 205-211. · Zbl 0673.93064 · doi:10.1016/0167-6911(89)90051-0
[11] W. P. Dayawansa, C. F. Martin, and G. Knowles, Asymptotic stabilization of a class of smooth two-dimensional systems,SIAM J. Control Optim.,28 (1990), 1321-1349. · Zbl 0731.93076 · doi:10.1137/0328070
[12] J. P. Gauthier and G. Bornard, Stabilisation des systèmes non linéaires, inOutils et Modeles Mathematiques pour l’Automatique, l’Analyse de Systemes et le Traitement du Signal, Editions du CNRS, Paris, 1981, pp. 307-324.
[13] H. Hermes, Homogeneous coordinates and continuous stabilizing feedback controls, inDifferential Equations: Stability and Control (S. Elaydi, ed.), Marcel Dekker, New York, 1991. · Zbl 0736.93069
[14] M. W. Hirsch and S. Smale,Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974. · Zbl 0309.34001
[15] V. Jurdjevic and J. P. Quinn, Controllability and stability,J. Differential Equations,28 (1978), 381-389. · Zbl 0417.93012 · doi:10.1016/0022-0396(78)90135-3
[16] M. Kawski, Homogeneous stabilizing feedback laws,Control Theory Adv. Tech.,6 (1990), 497-516.
[17] M. Kawski, Stabilization of nonlinear systems in the plane,Systems Control Lett.,12 (1990), 169-175. · Zbl 0666.93103 · doi:10.1016/0167-6911(89)90010-8
[18] J. L. Massera, Contribution to stability theory,Ann of Math.,64 (1956), 182-206. · Zbl 0070.31003 · doi:10.2307/1969955
[19] L. Praly, B. d’Andréa-Novel, and J.-M. Coron, Lyapunov design of stabilizing controllers,Proceedings of the 28th IEEE Conference on Decision and Control, Tampa, FL, 1989, pp. 1047-1052.
[20] N. Rouche and J. Mawhin,Equations Différentielles Ordinaires, vol. 2, Masson, Paris, 1973. · Zbl 0289.34001
[21] A. Saberi, P. V. Kokotovi?, and H. J. Sussmann. Global stabilization of partially linear composed systems.SIAM J. Control Optim.,28 (1990), 1491-1503. · Zbl 0719.93071 · doi:10.1137/0328079
[22] E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability,SIAM J. Control Optim.,21 (1983), 462-471. · Zbl 0513.93047 · doi:10.1137/0321028
[23] E. D. Sontag and H. J. Sussmann, Further comments on the stabilizability of the angular velocity of the rigid body,Systems Control Lett.,12 (1988), 213-217. · Zbl 0675.93064 · doi:10.1016/0167-6911(89)90052-2
[24] J. Tsinias, Sufficient Lyapunov-like conditions for stabilization,Math. Control Signals Systems,2 (1989), 343-357. · Zbl 0688.93048 · doi:10.1007/BF02551276
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.