For a general time-varying system, we prove that existence of an “Output Robust Control Lyapunov Function” implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well known result due to Coron and Rosier [J. Math. Syst. Estim. Control 4 (1994) 67-84] concerning stabilization of autonomous systems by means of time-varying periodic feedback.
Classification: 93C10, 93B52, 93D09
Keywords: control Lyapunov function, feedback stabilization, time-varying systems
@article{COCV_2009__15_3_599_0,
author = {Karafyllis, Iasson and Tsinias, John},
title = {Control Lyapunov functions and stabilization by means of continuous time-varying feedback},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {15},
number = {3},
year = {2009},
pages = {599-625},
doi = {10.1051/cocv:2008046},
zbl = {1167.93021},
mrnumber = {2542575},
language = {en},
url = {http://www.numdam.org/item/COCV_2009__15_3_599_0}
}
Karafyllis, Iasson; Tsinias, John. Control Lyapunov functions and stabilization by means of continuous time-varying feedback. ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 3, pp. 599-625. doi : 10.1051/cocv:2008046. http://www.numdam.org/item/COCV_2009__15_3_599_0/
[1] and , Continuous control-Lyapunov functions for asymptotic controllable time-varying systems. Int. J. Control 72 (1990) 1630-1641. | MR 1733874 | Zbl 0949.93035
[2] , Stabilization with relaxed controls. Nonlinear Anal. Theory Methods Appl. 7 (1983) 1163-1173. | MR 721403 | Zbl 0525.93053
[3] and , Liapunov Functions and Stability in Control Theory, Lecture Notes in Control and Information Sciences 267. Springer-Verlag, London (2001). | MR 1826052 | Zbl 0968.93004
[4] and , State constrained feedback stabilization. SIAM J. Contr. Opt. 42 (2003) 422-441. | MR 1982277 | Zbl 1037.93068
[5] , , and , Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Contr. 42 (1997) 1394-1407. | MR 1472857 | Zbl 0892.93053
[6] , , and , Feedback stabilization and Lyapunov functions. SIAM J. Contr. Opt. 39 (2000) 25-48. | MR 1780907 | Zbl 0961.93047
[7] and , A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Syst. Estim. Control 4 (1994) 67-84. | MR 1298548 | Zbl 0925.93827
[8] , Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publishers (1988). | Zbl 0664.34001
[9] and , Robust Nonlinear Control Design- State Space and Lyapunov Techniques. Birkhauser, Boston (1996). | MR 1396307 | Zbl 0857.93001
[10] and , Topology. Dover Editions (1988). | MR 1016814 | Zbl 0718.55001
[11] , Necessary and sufficient conditions for the existence of stabilizing feedback for control systems. IMA J. Math. Control Inf. 20 (2003) 37-64. | MR 1963063 | Zbl 1109.93362
[12] , Non-uniform in time robust global asymptotic output stability. Systems Control Lett. 54 (2005) 181-193. | MR 2115536 | Zbl 1129.93480
[13] and , Robust output feedback stabilization and nonlinear observer design. Systems Control Lett. 54 (2005) 925-938. | MR 2166284 | Zbl 1129.93523
[14] and , A converse Lyapunov theorem for non-uniform in time global asymptotic stability and its application to feedback stabilization. SIAM J. Contr. Opt. 42 (2003) 936-965. | MR 2002141 | Zbl 1049.93073
[15] , A Lyapunov approach to detectability of nonlinear systems. Dissertation thesis, Rutgers University, Department of Mathematics, USA (2000).
[16] and , A Lyapunov characterization of robust stabilization. Nonlinear Anal. Theory Methods Appl. 37 (1999) 813-840. | MR 1695080 | Zbl 0947.34054
[17] , and , A smooth converse Lyapunov theorem for robust stability. SIAM J. Contr. Opt. 34 (1996) 124-160. | MR 1372908 | Zbl 0856.93070
[18] and , Averaging results and the study of uniform asymptotic stability of homogeneous differential equations that are not fast time-varying. SIAM J. Contr. Opt. 37 (1999) 997-1010. | MR 1680944 | Zbl 0924.34047
[19] , Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Contr. Opt. 39 (2000) 1043-1064. | MR 1814266 | Zbl 0982.93068
[20] , On the existence of nonsmooth control-Lyapunov function in the sense of generalized gradients. ESAIM: COCV 6 (2001) 593-612. | Numdam | MR 1872388 | Zbl 1002.93058
[21] , A universal construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett. 13 (1989) 117-123. | MR 1014237 | Zbl 0684.93063
[22] , Clocks and insensitivity to small measurement errors. ESAIM: COCV 4 (1999) 537-557. | Numdam | MR 1746166 | Zbl 0984.93068
[23] and , Notions of input to output stability. Systems Control Lett. 38 (1999) 235-248. | MR 1754906 | Zbl 0985.93051
[24] and , Lyapunov characterizations of input-to-output stability. SIAM J. Contr. Opt. 39 (2001) 226-249. | MR 1780918 | Zbl 0968.93076
[25] and , A smooth Lyapunov function from a class- estimate involving two positive semidefinite functions. ESAIM: COCV 5 (2000) 313-367. | Numdam | MR 1765429 | Zbl 0953.34042
[26] , A general notion of global asymptotic controllability for time-varying systems and its Lyapunov characterization. Int. J. Control 78 (2005) 264-276. | MR 2141864 | Zbl pre02176073
[27] , Partial Stability and Control. Birkhauser, Boston (1998). | MR 1487080 | Zbl 0891.93004
