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Control Lyapunov functions and stabilization by meansof continuous time-varying feedback

Published online by Cambridge University Press:  19 July 2008

Iasson Karafyllis  and
John Tsinias
Iasson Karafyllis
Affiliation:
Department of Environmental Engineering, Technical University of Crete, 73100, Chania, Greece. [email protected]
John Tsinias
Affiliation:
Department of Mathematics, National Technical University of Athens, Zografou Campus 15780, Athens, Greece. [email protected]
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Abstract

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. Control4 (1994) 67–84] concerning stabilization of autonomous systems by means of time-varying periodic feedback.

Keywords

Control Lyapunov Functionfeedback stabilizationtime-varying systems
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 3 , July 2009 , pp. 599 - 625
Copyright
© EDP Sciences, SMAI, 2008

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References

Albertini, F. and Sontag, E.D., Continuous control-Lyapunov functions for asymptotic controllable time-varying systems. Int. J. Control 72 (1990) 1630–1641. CrossRef
Artstein, Z., Stabilization with relaxed controls. Nonlinear Anal. Theory Methods Appl. 7 (1983) 1163–1173. CrossRef
A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, Lecture Notes in Control and Information Sciences 267. Springer-Verlag, London (2001).
Clarke, F.H. and Stern, R.J., State constrained feedback stabilization. SIAM J. Contr. Opt. 42 (2003) 422–441. CrossRef
Clarke, F.H., Ledyaev, Y.S., Sontag, E.D. and Subbotin, A.I., Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Contr. 42 (1997) 1394–1407. CrossRef
Clarke, F.H., Ledyaev, Y.S., Rifford, L. and Stern, R.J., Feedback stabilization and Lyapunov functions. SIAM J. Contr. Opt. 39 (2000) 25–48. CrossRef
Coron, J.-M. and Rosier, L., A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Syst. Estim. Control 4 (1994) 67–84.
A.V. Fillipov, Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publishers (1988).
R.A. Freeman and P.V. Kokotovic, Robust Nonlinear Control Design- State Space and Lyapunov Techniques. Birkhauser, Boston (1996).
J.G. Hocking and G.S. Young, Topology. Dover Editions (1988).
Karafyllis, I., Necessary and sufficient conditions for the existence of stabilizing feedback for control systems. IMA J. Math. Control Inf. 20 (2003) 37–64. CrossRef
Karafyllis, I., Non-uniform in time robust global asymptotic output stability. Systems Control Lett. 54 (2005) 181–193. CrossRef
Karafyllis, I. and Kravaris, C., Robust output feedback stabilization and nonlinear observer design. Systems Control Lett. 54 (2005) 925–938. CrossRef
Karafyllis, I. and Tsinias, J., 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. CrossRef
M. Krichman, A Lyapunov approach to detectability of nonlinear systems. Dissertation thesis, Rutgers University, Department of Mathematics, USA (2000).
Ledyaev, Y.S. and Sontag, E.D., Lyapunov, A characterization of robust stabilization. Nonlinear Anal. Theory Methods Appl. 37 (1999) 813–840. CrossRef
Lin, Y., Sontag, E.D. and Wang, Y., A smooth converse Lyapunov theorem for robust stability. SIAM J. Contr. Opt. 34 (1996) 124–160. CrossRef
Peuteman, J. and Aeyels, D., 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. CrossRef
Rifford, L., Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Contr. Opt. 39 (2000) 1043–1064. CrossRef
Rifford, L., On the existence of nonsmooth control-Lyapunov function in the sense of generalized gradients. ESAIM: COCV 6 (2001) 593–612. CrossRef
Sontag, E.D., A universal construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett. 13 (1989) 117–123. CrossRef
Sontag, E.D., Clocks and insensitivity to small measurement errors. ESAIM: COCV 4 (1999) 537–557. CrossRef
Sontag, E.D. and Wang, Y., Notions of input to output stability. Systems Control Lett. 38 (1999) 235–248. CrossRef
Sontag, E.D. and Wang, Y., Lyapunov characterizations of input-to-output stability. SIAM J. Contr. Opt. 39 (2001) 226–249. CrossRef
Teel, A.R. and Praly, L., A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions. ESAIM: COCV 5 (2000) 313–367. CrossRef
Tsinias, J., A general notion of global asymptotic controllability for time-varying systems and its Lyapunov characterization. Int. J. Control 78 (2005) 264–276. CrossRef
V.I. Vorotnikov, Partial Stability and Control. Birkhauser, Boston (1998).