Control-Lyapunov function: Difference between revisions

Revision as of 05:37, 3 August 2022

Revision as of 02:38, 24 October 2022

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==Theorems==

E. D. Sontag showed that for a given control system, there exists a continuous CLF if and only if the origin is asymptotic stabilizable.<ref>{{cite journal |first=E.D. |last=Sontag |title=A Lyapunov-like characterization of asymptotic controllability|journal=SIAM J. Control Optim.|volume=21 |issue=3 |year=1983 |pages=462–471|doi=10.1137/0321028 |s2cid=450209 }}</ref> It was later shown by [[Francis Clarke (mathematician)|Francis H. Clarke]] that every [[Controllability|asymptotically controllable]] system can be stabilized by a (generally discontinuous) feedback.<ref>{{cite journal |firstfirst1=F.H.|lastlast1=Clarke |first2=Y.S.|last2=Ledyaev |first3=E.D.|last3=Sontag |first4=A.I.|last4=Subbotin |title=Asymptotic controllability implies feedback stabilization |journal=IEEE Trans. Autom. Control|volume=42 |issue=10 |year=1997 |pages=1394–1407|doi=10.1109/9.633828 }}</ref>

Artstein proved that the dynamical system ({{EquationNote|2}}) has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback ''u''(''x'').

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*{{cite book|last=Freeman|first=Randy A.|author2=Petar V. Kokotović|title=Robust Nonlinear Control Design|publisher=Birkhäuser|year=2008|edition=illustrated, reprint|pages=257|isbn=978-0-8176-4758-92|url=https://books.google.com/books?id=_eTb4Yl0SOEC|accessdate=2009-03-04}}

*{{cite book | last = Khalil | first = Hassan | year = 2015 | title = Nonlinear Control| publisher = Pearson | isbn = 9780133499261}}

*{{cite book | last = Sontag | first = Eduardo | author-link = Eduardo D. Sontag | year = 1998 | title = Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition | publisher = Springer | url = http://www.sontaglab.org/FTPDIR/sontag_mathematical_control_theory_springer98.pdf | isbn = 978-0-387-98489-6 }}