Control-Lyapunov function: Difference between revisions

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In [[control theory]], a '''control-Lyapunov function (cLf)'''<ref>Isidori</ref><ref>Freeman (46)</ref><ref>Khalil</ref><ref>Sontag</ref> is an extension of the idea of [[Lyapunov function]] <math>V(x)</math> to systems with control inputs. 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 ''asymptotically controllable'', 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 asymptotically by applying the control ''u''.

More formally, suppose we are given an autonomous dynamical system with inputs

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This notion was introduced by [[Eduardo D. Sontag|E. D. Sontag]] in

<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}}</ref>

<ref>sontag83</ref>

who showed that the existence of a continuous cLf is equivalent to asymptotic stabilizability. It was later shown that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback

<ref>{{cite journal |first=F.H.|last=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. Automat. Control|volume=42 |issue=10 |year=1997 |pages=1394-1407}}</ref>. One may also ask when there is a continuous feedback stabilizer. For systems affine on controls, and differentiable cLf's, the definition translates as follows:

<ref>Clarke et al</ref>. One may also ask when there is a continuous feedback stabilizer. For systems affine on controls, and differentiable cLf's, the definition translates as follows:

'''Definition.''' A control-Lyapunov function is a function <math>V:D\rightarrow\mathbb{R}</math> that is continuously differentiable, positive-definite (that is <math>V(x)</math> is positive except at <math>x=0</math> where it is zero), and such that

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'''Artstein's theorem.''' The dynamical system has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback ''u''(''x'').

It may not be easy to find a control-Lyapunov function for a given system, but if we can find one thanks to some ingenuity and luck, then the feedback stabilization problem simplifies considerably, in fact it reduces to solving a static non-linear [[optimization (mathematics)|programming problem]].

The ''Sontag's universal formula'' writes the feedback law directly in terms of the derivatives of the cLf<ref>Isidori</ref><ref>Khalil</ref>. An alternative is to solve a static non-linear [[optimization (mathematics)|programming problem]]

:<math>

u^*(x) = \underset{u}{\operatorname{arg\,min}} \nabla V(x) \cdot f(x,u)

</math>

for each state ''x''.

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. A ''universal formula'' can be given for the feedback that stabilizes the system, provided that a differentiable cLf is known.

==Example==

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which can then be solved using any linear differential equation methods.

Relations between different types of stabilization (continuous and discontinuous feedback, robustness) are studied in <ref>notions</ref>.

==Notes==

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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=0-8176-4758-9|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 }}

{{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}}

==See also==