Abstract
We consider the simplest design problem for nonlinear systems: the problem of rendering asymptotically stable a given equilibrium by means of feedback. For such a problem, we provide a necessary condition, known as Brockett condition, and a sufficient condition, which relies upon the definition of a class of functions, known as control Lyapunov functions. The theory is illustrated by means of a few examples. In addition we discuss a nonlinear enhancement of the so-called separation principle for stabilization by means of partial state information.
This is a preview of subscription content, log in via an institution to check access.
Bibliography
Artstein Z (1983) Stabilization with relaxed controls. Nonlinear Anal Theory Methods Appl 7:1163–1173
Astolfi A, Karagiannis D, Ortega R (2008) Nonlinear and adaptive control with applications. London: Springer.
Astolfi A, Praly L (2006) Global complete observability and output-to-state stability imply the existence of a globally convergent observer. Math Control Signals Syst 18:32–65
Bacciotti A (1992) Local stabilizability of nonlinear control systems. World Scientific, Singapore
Brockett RW (1983) Asymptotic stability and feedback stabilization. In: Brockett RW, Millman RS, Sussmann HJ (eds) Differential geometry control theory. Birkhauser, Boston, pp 181–191
Gauthier J-P, Kupka I (2001) Deterministic observation theory and applications. Cambridge University Press, Cambridge
Isidori A (1995) Nonlinear control systems, 3rd edn. Springer, Berlin
Isidori A (1999) Nonlinear control systems II. Springer, London,
Jurdjevic V, Quinn JP (1978) Controllability and stability. J Differ Equ 28:381–389
Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice-Hall, Upper Saddle River
Krstić M, Kanellakopoulos I, Kokotović P (1995) Nonlinear and adaptive control design. John Wiley and Sons, New York
Marino R, Tomei P (1995) Nonlinear control design: geometric, adaptive and robust. Prentice-Hall, London
Mazenc F, Praly L (1996) Adding integrations, saturated controls, and stabilization for feedforward systems. IEEE Trans Autom Control 41:1559–1578
Mazenc F, Praly L, Dayawansa WP (1994) Global stabilization by output feedback: examples and counterexamples. Syst Control Lett 23:119–125
Ortega R, Loría A, Nicklasson PJ, Sira-Ramírez H (1998) Passivity-based control of Euler-Lagrange systems. Springer, London
Ryan EP (1994) On Brockett’s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J Control Optim 32:1597–1604
Sepulchre R, Janković M, and Kokotović P (1996) Constructive nonlinear control. Springer, Berlin
Sontag ED (1989) A “universal” construction of Artstein’s theorem on nonlinear stabilization. Syst Control Lett 13:117–123
Teel A, Praly L (1995) Tools for semiglobal stabilization by partial state and output feedback. SIAM J Control Optim 33(5):1443–1488
van der Schaft A (2000) L2-gain and passivity techniques in nonlinear control, 2nd edn. Springer, London
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2020 Springer-Verlag London Ltd., part of Springer Nature
About this entry
Cite this entry
Astolfi, A. (2020). Feedback Stabilization of Nonlinear Systems. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_85-2
Download citation
DOI: https://doi.org/10.1007/978-1-4471-5102-9_85-2
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-5102-9
Online ISBN: 978-1-4471-5102-9
eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering
Publish with us
Chapter history
-
Latest
Feedback Stabilization of Nonlinear Systems- Published:
- 03 January 2020
DOI: https://doi.org/10.1007/978-1-4471-5102-9_85-2
-
Original
Feedback Stabilization of Nonlinear Systems- Published:
- 15 March 2014
DOI: https://doi.org/10.1007/978-1-4471-5102-9_85-1