×

Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization. (English) Zbl 1202.93117

Summary: 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 of a class of time-delay nonlinear systems with a triangular structure.

MSC:

93D15 Stabilization of systems by feedback
93D30 Lyapunov and storage functions
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML

References:

[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 0982.34044 · doi:10.1109/9.701102
[2] Z. Artstein, Stabilization with relaxed controls. Nonlinear Anal. Theory Methods Appl.7 (1983) 1163-1173. Zbl0525.93053 · Zbl 0525.93053 · doi:10.1016/0362-546X(83)90049-4
[3] J.P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhauser, Boston, USA (1990). · Zbl 0713.49021
[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. Zbl0892.93053 · Zbl 0892.93053 · doi:10.1109/9.633828
[5] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs136. AMS, USA (2007).
[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 0925.93827
[7] R.A. Freeman and P.V. Kokotovic, Robust Nonlinear Control Design-State Space and Lyapunov Techniques. Birkhauser, Boston, USA (1996). · Zbl 0863.93075
[8] J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations. Springer-Verlag, New York, USA (1993). · Zbl 0787.34002
[9] C. Hua, G. Feng and X. Guan, Robust controller design of a class of nonlinear time delay systems via backstepping methods. Automatica44 (2008) 567-573. · Zbl 1283.93099
[10] M. Jankovic, Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems. IEEE Trans. Automat. Contr. 46 (2001) 1048-1060. Zbl1023.93056 · Zbl 1023.93056 · doi:10.1109/9.935057
[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. · Zbl 1293.93629
[14] I. Karafyllis, Non-uniform in time robust global asymptotic output stability. Syst. Contr. Lett.54 (2005) 181-193. Zbl1129.93480 · Zbl 1129.93480 · doi:10.1016/j.sysconle.2004.08.004
[15] I. Karafyllis, Lyapunov theorems for systems described by retarded functional differential equations. Nonlinear Anal. Theory Methods Appl.64 (2006) 590-617. · Zbl 1163.34389 · doi:10.1016/j.na.2005.04.045
[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. Zbl1120.93029 · Zbl 1120.93029 · doi:10.1016/j.jmaa.2006.05.059
[17] I. Karafyllis and C. Kravaris, Robust output feedback stabilization and nonlinear observer design. Syst. Contr. Lett.54 (2005) 925-938. · Zbl 1129.93523 · doi:10.1016/j.sysconle.2005.02.002
[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 1049.93073 · doi:10.1137/S0363012901392967
[19] I. Karafyllis and J. Tsinias, Control Lyapunov functions and stabilization by means of continuous time-varying feedback. ESAIM: COCV15 (2009) 599-625. · Zbl 1167.93021 · doi:10.1051/cocv:2008046
[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. · Zbl 1293.93667
[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. Zbl0856.93070 · Zbl 0856.93070 · doi:10.1137/S0363012993259981
[23] F. Mazenc and P.-A. Bliman, Backstepping design for time-delay nonlinear systems. IEEE Trans. Automat. Contr.51 (2006) 149-154. Zbl1158.93337 · Zbl 1158.93337 · doi:10.1007/s11434-006-2052-x
[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. · Zbl 1365.93379
[26] S.K. Nguang, Robust stabilization of a class of time-delay nonlinear systems. IEEE Trans. Automat. Contr.45 (2000) 756-762. · Zbl 0978.93067 · doi:10.1109/9.847117
[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 1010.34035 · doi:10.1007/s004980200002
[28] J. Peuteman and D. Aeyels, Exponential stability of slowly time-varying nonlinear systems. Math. Contr. Signals Syst.15 (2002) 202-228. · Zbl 1019.93050 · doi:10.1007/s004980200008
[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 0787.93083
[30] E.D. Sontag, A universal construction of Artstein’s theorem on nonlinear stabilization. Syst. Contr. Lett.13 (1989) 117-123. Zbl0684.93063 · Zbl 0684.93063 · doi:10.1016/0167-6911(89)90028-5
[31] E.D. Sontag and Y. Wang, Notions of input to output stability. Syst. Contr. Lett.38 (1999) 235-248. Zbl0985.93051 · Zbl 0985.93051 · doi:10.1016/S0167-6911(99)00070-5
[32] E.D. Sontag, and Y. Wang, Lyapunov characterizations of input-to-output stability. SIAM J. Contr. Optim.39 (2001) 226-249. Zbl0968.93076 · Zbl 0968.93076 · doi:10.1137/S0363012999350213
[33] J. Tsinias, Sufficient Lyapunov-like conditions for stabilization. Math. Contr. Signals Syst.2 (1989) 343-357. Zbl0688.93048 · Zbl 0688.93048 · doi:10.1007/BF02551276
[34] J. Tsinias and N. Kalouptsidis, Output feedback stabilization. IEEE Trans. Automat. Contr.35 (1990) 951-954. · Zbl 0723.93054 · doi:10.1109/9.58511
[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. Zbl1106.93339 · Zbl 1106.93339 · doi:10.1016/S0167-6911(02)00203-7
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.