Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 887-928.

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.

DOI: 10.1051/cocv/2009029
Classification: 93D15, 93D30
Keywords: 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://archive.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://archive.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://archive.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, Volume 16 (2010) no. 4, pp. 887-928. doi : 10.1051/cocv/2009029. http://archive.numdam.org/articles/10.1051/cocv/2009029/

[1] D. Aeyels and J. Peuteman, A new asymptotic stability criterion for nonlinear time-variant differential equations. IEEE Trans. Automat. Contr. 43 (1998) 968-971. | Zbl

[2] Z. Artstein, Stabilization with relaxed controls. Nonlinear Anal. Theory Methods Appl. 7 (1983) 1163-1173. | Zbl

[3] J.P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhauser, Boston, USA (1990). | Zbl

[4] F.H. Clarke, Y.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Contr. 42 (1997) 1394-1407. | Zbl

[5] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs 136. AMS, USA (2007). | Zbl

[6] J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Syst. Estim. Contr. 4 (1994) 67-84. | Zbl

[7] R.A. Freeman and P.V. Kokotovic, Robust Nonlinear Control Design-State Space and Lyapunov Techniques. Birkhauser, Boston, USA (1996). | Zbl

[8] J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations. Springer-Verlag, New York, USA (1993). | Zbl

[9] C. Hua, G. Feng and X. Guan, Robust controller design of a class of nonlinear time delay systems via backstepping methods. Automatica 44 (2008) 567-573.

[10] M. Jankovic, Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems. IEEE Trans. Automat. Contr. 46 (2001) 1048-1060. | Zbl

[11] M. Jankovic, 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] Z.-P. Jiang, Y. Lin and Y. Wang, 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.

[13] I. Karafyllis, 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] I. Karafyllis, Non-uniform in time robust global asymptotic output stability. Syst. Contr. Lett. 54 (2005) 181-193. | Zbl

[15] I. Karafyllis, Lyapunov theorems for systems described by retarded functional differential equations. Nonlinear Anal. Theory Methods Appl. 64 (2006) 590-617. | Zbl

[16] I. Karafyllis, 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] I. Karafyllis and C. Kravaris, Robust output feedback stabilization and nonlinear observer design. Syst. Contr. Lett. 54 (2005) 925-938. | Zbl

[18] I. Karafyllis and J. Tsinias, 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

[19] I. Karafyllis and J. Tsinias, Control Lyapunov functions and stabilization by means of continuous time-varying feedback. ESAIM: COCV 15 (2009) 599-625. | Numdam | Zbl

[20] I. Karafyllis, P. Pepe and Z.-P. Jiang, Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations. Eur. J. Contr. 14 (2008) 516-536.

[21] M. Krstic, I. Kanellakopoulos and P.V. Kokotovic, Nonlinear and Adaptive Control Design. John Wiley (1995).

[22] Y. Lin, E.D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability. SIAM J. Contr. Optim. 34 (1996) 124-160. | Zbl

[23] F. Mazenc and P.-A. Bliman, Backstepping design for time-delay nonlinear systems. IEEE Trans. Automat. Contr. 51 (2006) 149-154.

[24] F. Mazenc, M. Malisoff and Z. Lin, 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.

[25] E. Moulay and W. Perruquetti, Stabilization of non-affine systems: A constructive method for polynomial systems. IEEE Trans. Automat. Contr. 50 (2005) 520-526.

[26] S.K. Nguang, Robust stabilization of a class of time-delay nonlinear systems. IEEE Trans. Automat. Contr. 45 (2000) 756-762. | Zbl

[27] J. Peuteman and D. Aeyels, Exponential stability of nonlinear time-varying differential equations and partial averaging. Math. Contr. Signals Syst. 15 (2002) 42-70. | Zbl

[28] J. Peuteman and D. Aeyels, Exponential stability of slowly time-varying nonlinear systems. Math. Contr. Signals Syst. 15 (2002) 202-228. | Zbl

[29] L. Praly, G. Bastin, J.-B. Pomet and Z.P. Jiang, Adaptive stabilization of nonlinear systems, in Foundations of Adaptive Control, P.V. Kokotovic Ed., Springer-Verlag (1991) 374-433. | Zbl

[30] E.D. Sontag, A universal construction of Artstein's theorem on nonlinear stabilization. Syst. Contr. Lett. 13 (1989) 117-123. | Zbl

[31] E.D. Sontag and Y. Wang, Notions of input to output stability. Syst. Contr. Lett. 38 (1999) 235-248. | Zbl

[32] E.D. Sontag, and Y. Wang, Lyapunov characterizations of input-to-output stability. SIAM J. Contr. Optim. 39 (2001) 226-249. | Zbl

[33] J. Tsinias, Sufficient Lyapunov-like conditions for stabilization. Math. Contr. Signals Syst. 2 (1989) 343-357. | Zbl

[34] J. Tsinias and N. Kalouptsidis, Output feedback stabilization. IEEE Trans. Automat. Contr. 35 (1990) 951-954. | Zbl

[35] S. Zhou, G. Feng and S.K. Nguang, Comments on robust stabilization of a class of time-delay nonlinear systems. IEEE Trans. Automat. Contr. 47 (2002) 1586-1586. | Zbl

Cited by Sources: