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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 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 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▼
|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
▲
The theory and application of control-Lyapunov functions were developed by
More formally, suppose we are given an autonomous dynamical system with inputs▼
where <math>x\in\mathbb{R}^n</math> is the state vector and <math>u\in\mathbb{R}^m</math> is the control vector, and we want to drive states to an equilibrium, let us <math>x=0</math>, from every initial state in some ___domain <math>D\subset\mathbb{R}^n</math>.▼
==Definition==
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. Autom. 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:▼
▲
'''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▼
{{NumBlk|:|<math>\dot{x}=f(x,u)</math>|{{EquationRef|1}}}}
▲where <math>x\in\mathbb{R}^n</math> is the state vector and <math>u\in\mathbb{R}^m</math> is the control vector
▲'''Definition.'''
:<math>
</math>
where <math>\langle u, v\rangle</math> denotes the [[inner product]] of <math>u, v \in\mathbb{R}^n</math>.
The last condition is the key condition; in words it says that for each state ''x'' we can find a control ''u'' that will reduce the "energy" ''V''. Intuitively, if in each state we can always find a way to reduce the energy, we should eventually be able to bring the energy asymptotically to zero, that is to bring the system to a stop. This is made rigorous by [[Artstein's theorem]]
Some results apply only to control-affine systems—i.e., control systems in the following form:
{{NumBlk|:|<math>\dot x = f(x) + \sum_{i=1}^m g_i(x)u_i</math>|{{EquationRef|2}}}}
where <math>f : \mathbb{R}^n \to \mathbb{R}^n</math> and <math>g_i : \mathbb{R}^n \to \mathbb{R}^{n}</math> for <math>i = 1, \dots, m</math>.
==Theorems==
▲
▲The last condition is the key condition; in words it says that for each state ''x'' we can find a control ''u'' that will reduce the "energy" ''V''. Intuitively, if in each state we can always find a way to reduce the energy, we should eventually be able to bring the energy asymptotically to zero, that is to bring the system to a stop. This is made rigorous by [[Artstein's theorem]], repeated here:
=== Constructing the Stabilizing Input ===
▲'''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 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. 5.56
}} 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>
where <math>L_f V(x) := \langle \nabla V(x), f(x)\rangle</math> and <math>L_g V(x) := \langle \nabla V(x), g(x)\rangle</math> are the [[Lie derivative|Lie derivatives]] of <math>V</math> along <math>f</math> and <math>g</math>, respectively.
For the general nonlinear system ({{EquationNote|1}}), the input <math>u</math> can be found by solving 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.
==Example==
Line 42 ⟶ 74:
A Control-Lyapunov candidate is then
:<math>
r \mapsto V(r) :=\frac{1}{2}r^2
</math>
which is positive
Now taking the time derivative of <math>V</math>
Line 105 ⟶ 137:
which can then be solved using any linear differential equation methods.
==
{{Reflist}}
▲*{{cite book
▲ | author = Isidori, A.
▲ | year = 1995
▲ | title = Nonlinear Control Systems
▲ | publisher = Springer
▲ | isbn = 978-3-540-19916-8
==See also==
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