Control-Lyapunov function: Difference between revisions

Revision as of 18:33, 24 November 2008

Revision as of 02:14, 5 March 2009

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In [[control theory]], a '''control-Lyapunov function''' <math>V(x,u)</math> <ref>Freeman (46)</ref>is a generalization of the notion of [[Lyapunov function]] <math>V(x)</math> used in [[Lyapunov stability|stability]] analysis. The ordinary Lyapunov function is used to test whether a [[dynamical system]] is ''stable'' (more restrictively, ''asymptotically stable''). That is, whether the system starting in a state <math>x \ne 0</math> in some ___domain ''D'' will remain in ''D'', or for ''asymptotic stability'' will eventually return to <math>x = 0</math>. The control-Lyapunov function is used to test whether a system is ''feedback stabilizable'', that is whether for any state ''x'' there exists a control <math> u(x,t)</math> such that the system can be brought to the zero state by applying the control ''u''.

More formally, suppose we are given a dynamical system

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The theory and application of control-Lyapunov functions were developed by Z. Artstein and [[Eduardo D. Sontag|E. D. Sontag]] in the 1980s and 1990s.

==Notes==

{{unreferenced|date=August 2006}}

{{Reflist}}

==References==

*{{cite book|last=Freeman|first=Randy A.|coauthors=Petar V. Kokotović|title=Robust Nonlinear Control Design|publisher=Birkhäuser|date=2008|edition=illustrated, reprint|pages=257|isbn=0817647589|url=http://books.google.com/books?id=_eTb4Yl0SOEC|accessdate=2009-3-4|language=English}}

==See also==

* [[Artstein's theorem]]

[[Category:Stability theory]]