In this work, we propose a methodology for the expression of necessary and sufficient Lyapunov-like conditions for the existence of stabilizing feedback laws. The methodology is an extension of the well-known Control Lyapunov Function (CLF) method and can be applied to very general nonlinear time-varying systems with disturbance and control inputs, including both finite and infinite-dimensional systems. The generality of the proposed methodology is also reflected upon by the fact that partial stability with respect to output variables is addressed. In addition, it is shown that the generalized CLF method can lead to a novel tool for the explicit design of robust nonlinear controllers for a class of time-delay nonlinear systems with a triangular structure.
Mots clés : control Lyapunov function, stabilization, time-varying systems, nonlinear control
@article{COCV_2010__16_4_887_0, author = {Karafyllis, Iasson and Jiang, Zhong-Ping}, title = {Necessary and sufficient {Lyapunov-like} conditions for robust nonlinear stabilization}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {887--928}, publisher = {EDP-Sciences}, volume = {16}, number = {4}, year = {2010}, doi = {10.1051/cocv/2009029}, mrnumber = {2744155}, zbl = {1202.93117}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2009029/} }
TY - JOUR AU - Karafyllis, Iasson AU - Jiang, Zhong-Ping TI - Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 887 EP - 928 VL - 16 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009029/ DO - 10.1051/cocv/2009029 LA - en ID - COCV_2010__16_4_887_0 ER -
%0 Journal Article %A Karafyllis, Iasson %A Jiang, Zhong-Ping %T Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 887-928 %V 16 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2009029/ %R 10.1051/cocv/2009029 %G en %F COCV_2010__16_4_887_0
Karafyllis, Iasson; Jiang, Zhong-Ping. Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 887-928. doi : 10.1051/cocv/2009029. http://www.numdam.org/articles/10.1051/cocv/2009029/
[1] A new asymptotic stability criterion for nonlinear time-variant differential equations. IEEE Trans. Automat. Contr. 43 (1998) 968-971. | Zbl
and ,[2] Stabilization with relaxed controls. Nonlinear Anal. Theory Methods Appl. 7 (1983) 1163-1173. | Zbl
,[3] Set-Valued Analysis. Birkhauser, Boston, USA (1990). | Zbl
and ,[4] Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Contr. 42 (1997) 1394-1407. | Zbl
, , and ,[5] Control and Nonlinearity, Mathematical Surveys and Monographs 136. AMS, USA (2007). | Zbl
,[6] A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Syst. Estim. Contr. 4 (1994) 67-84. | Zbl
and ,[7] Robust Nonlinear Control Design-State Space and Lyapunov Techniques. Birkhauser, Boston, USA (1996). | Zbl
and ,[8] Introduction to Functional Differential Equations. Springer-Verlag, New York, USA (1993). | Zbl
and ,[9] Robust controller design of a class of nonlinear time delay systems via backstepping methods. Automatica 44 (2008) 567-573.
, and ,[10] Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems. IEEE Trans. Automat. Contr. 46 (2001) 1048-1060. | Zbl
,[11] Stabilization of Nonlinear Time Delay Systems with Delay Independent Feedback, in Proceedings of the 2005 American Control Conference, Portland, OR, USA (2005) 4253-4258.
,[12] Stabilization of time-varying nonlinear systems: A control Lyapunov function approach, in Proceedings of IEEE International Conference on Control and Automation 2007, Guangzhou, China (2007) 404-409.
, and ,[13] The non-uniform in time small-gain theorem for a wide class of control systems with outputs. Eur. J. Contr. 10 (2004) 307-323.
,[14] Non-uniform in time robust global asymptotic output stability. Syst. Contr. Lett. 54 (2005) 181-193. | Zbl
,[15] Lyapunov theorems for systems described by retarded functional differential equations. Nonlinear Anal. Theory Methods Appl. 64 (2006) 590-617. | Zbl
,[16] A system-theoretic framework for a wide class of systems I: Applications to numerical analysis. J. Math. Anal. Appl. 328 (2007) 876-899. | Zbl
,[17] Robust output feedback stabilization and nonlinear observer design. Syst. Contr. Lett. 54 (2005) 925-938. | Zbl
and ,[18] A converse Lyapunov theorem for non-uniform in time global asymptotic stability and its application to feedback stabilization. SIAM J. Contr. Optim. 42 (2003) 936-965. | Zbl
and ,[19] Control Lyapunov functions and stabilization by means of continuous time-varying feedback. ESAIM: COCV 15 (2009) 599-625. | Numdam | Zbl
and ,[20] Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations. Eur. J. Contr. 14 (2008) 516-536.
, and ,[21] Nonlinear and Adaptive Control Design. John Wiley (1995).
, and ,[22] A smooth converse Lyapunov theorem for robust stability. SIAM J. Contr. Optim. 34 (1996) 124-160. | Zbl
, and ,[23] Backstepping design for time-delay nonlinear systems. IEEE Trans. Automat. Contr. 51 (2006) 149-154.
and ,[24] On input-to-state stability for nonlinear systems with delayed feedbacks, in Proceedings of the American Control Conference (2007), New York, USA (2007) 4804-4809.
, and ,[25] Stabilization of non-affine systems: A constructive method for polynomial systems. IEEE Trans. Automat. Contr. 50 (2005) 520-526.
and ,[26] Robust stabilization of a class of time-delay nonlinear systems. IEEE Trans. Automat. Contr. 45 (2000) 756-762. | Zbl
,[27] Exponential stability of nonlinear time-varying differential equations and partial averaging. Math. Contr. Signals Syst. 15 (2002) 42-70. | Zbl
and ,[28] Exponential stability of slowly time-varying nonlinear systems. Math. Contr. Signals Syst. 15 (2002) 202-228. | Zbl
and ,[29] Adaptive stabilization of nonlinear systems, in Foundations of Adaptive Control, P.V. Kokotovic Ed., Springer-Verlag (1991) 374-433. | Zbl
, , and ,[30] A universal construction of Artstein's theorem on nonlinear stabilization. Syst. Contr. Lett. 13 (1989) 117-123. | Zbl
,[31] Notions of input to output stability. Syst. Contr. Lett. 38 (1999) 235-248. | Zbl
and ,[32] Lyapunov characterizations of input-to-output stability. SIAM J. Contr. Optim. 39 (2001) 226-249. | Zbl
, and ,[33] Sufficient Lyapunov-like conditions for stabilization. Math. Contr. Signals Syst. 2 (1989) 343-357. | Zbl
,[34] Output feedback stabilization. IEEE Trans. Automat. Contr. 35 (1990) 951-954. | Zbl
and ,[35] Comments on robust stabilization of a class of time-delay nonlinear systems. IEEE Trans. Automat. Contr. 47 (2002) 1586-1586. | Zbl
, and ,Cité par Sources :