Control-Lyapunov function: Difference between revisions

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In [[control theory]], a '''control-Lyapunov function (CLF)'''<ref name="Isidori">{{cite book

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 [[Control system|systems with control inputs]]. The ordinary Lyapunov function is used to test whether a [[dynamical system]] is [[Lyapunov stability|''(Lyapunov) stable'']] or (more restrictively) ''asymptotically stable''. Lyapunov stability means that if the system starts in a state <math>x \ne 0</math> in some ___domain ''D'', then the state will remain in ''D'' for all time. For ''asymptotic stability'', the state is also required to converge to <math>x = 0</math>. A control-Lyapunov function is used to test whether a system is [[Controllability#Stabilizability|''asymptotically 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 asymptotically by applying the control ''u''.

| author = Isidori, A.

| year = 1995

| title = Nonlinear Control Systems

| publisher = Springer

| isbn = 978-3-540-19916-8

*}}</ref><ref>{{cite book

|last=Freeman

|first=Randy A.

|author2=Petar V. Kokotović

|title=Robust Nonlinear Control Design

|chapter=Robust Control Lyapunov Functions

|chapter-url=https://link.springer.com/chapter/10.1007/978-0-8176-4759-9_3

|publisher=Birkhäuser

|year=2008|pages=33–63

|doi=10.1007/978-0-8176-4759-9_3

|edition=illustrated, reprint

|isbn=978-0-8176-4758-2|

url=https://books.google.com/books?id=_eTb4Yl0SOEC|

accessdate=2009-03-04}}</ref><ref>{{cite book

| last = Khalil | first = Hassan

| year = 2015 | title = Nonlinear Control

In| [[controlpublisher theory]],= aPearson '''control-Lyapunov| functionisbn (CLF)'''<ref>Isidori= 9780133499261}}</ref><ref>Freeman name="Sontag (461998)</ref><ref>Khalil</ref><ref">{{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 }}</ref> is an extension of the idea of [[Lyapunov function]] <math>V(x)</math> to [[Control system|systems with control inputs]]. The ordinary Lyapunov function is used to test whether a [[dynamical system]] is [[Lyapunov stability|''(Lyapunov) stable'']] or (more restrictively) ''asymptotically stable''. Lyapunov stability means that if the system starts in a state <math>x \ne 0</math> in some ___domain ''D'', then the state will remain in ''D'' for all time. For ''asymptotic stability'', the state is also required to converge to <math>x = 0</math>. A control-Lyapunov function is used to test whether a system is [[Controllability#Stabilizability|''asymptotically 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 asymptotically by applying the control ''u''.

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

Line 23 ⟶ 45:

==Theorems==

E.[[ D.Eduardo 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]], Yuri Ledyaev, [[Eduardo Sontag]], and A.I. Subbotin 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'').

=== Constructing the Stabilizing Input ===

It is often difficult to find a control-Lyapunov function for a given system, but if one is found, then the feedback stabilization problem simplifies considerably. For the control affine system ({{EquationNote|2}}), ''Sontag's formula'' (or ''Sontag's universal formula'') gives the feedback law <math>k : \mathbb{R}^n \to \mathbb{R}^m</math> directly in terms of the derivatives of the CLF.<ref> name="Sontag (1998)"/>{{rp|Eq. ''Mathematical Control Theory'', Equation 5.56</ref>
}} In the special case of a single input system <math>(m=1)</math>, Sontag's formula is written as

:<math>k(x) = \begin{cases} \displaystyle -\frac{L_{f} V(x)+\sqrt{\left[L_{f} V(x)\right]^{2}+\left[L_{g} V(x)\right]^{4}}}{L_{g} V(x)} & \text { if } L_{g} V(x) \neq 0 \\

0 & \text { if } L_{g} V(x)=0 \end{cases} </math>

Line 51 ⟶ 74:

A Control-Lyapunov candidate is then

:<math>

r \mapsto V(r) :=\frac{1}{2}r^2

</math>

which is positive definite for all <math> q \ne 0</math>, <math>\dot{q}r \ne 0</math>.

Now taking the time derivative of <math>V</math>

Line 114 ⟶ 137:

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

==NotesReferences==

{{Reflist}}

==References==

*{{cite book

| author = Isidori, A.

| year = 1995

| title = Nonlinear Control Systems

| publisher = Springer

| isbn = 978-3-540-19916-8

}}

*{{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 }}

==See also==