Control-Lyapunov function: Difference between revisions

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In [[control theory]], a '''control-Lyapunov function (cLfCLF)'''<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''). ThatLyapunov is,stability whethermeans that if the system startingstarts in a state <math>x \ne 0</math> in some ___domain ''D'', then the state will remain in ''D'', or for all time. For ''asymptotic stability'', willthe eventuallystate returnis also required to converge to <math>x = 0</math>. TheA 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 Z.[[Zvi Artstein]] and [[Eduardo D. Sontag|E. D. Sontag]] in the 1980s and 1990s.

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

:<math>

\dot{x}=f(x,u)

</math>

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==

[[Eduardo D. Sontag|E. D. Sontag]] showed that the existence of a continuous cLf is equivalent to asymptotic stabilizability.<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> 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:

MoreConsider formally,an suppose[[Autonomous we are given ansystem (mathematics)|autonomous dynamical]] system with inputs

'''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,. Suppose andour wegoal wantis to drive statesthe system to an equilibrium, let us <math>x=0x_* \in \mathbb{R}^n</math>, from every initial state in some ___domain <math>D\subset\mathbb{R}^n</math>. Without loss of generality, suppose the equilibrium is at <math>x_*=0</math> (for an equilibrium <math>x_*\neq 0</math>, it can be translated to the origin by a change of variables).

'''Definition.''' A control-Lyapunov function (CLF) is a function <math>V : D \rightarrowto \mathbb{R}</math> that is [[Differentiable function#continuously differentiable|continuously differentiable]], positive-definite (that is, <math>V(x)</math> is positive for all <math>x\in D</math> except at <math>x=0</math> where it is zero), and such that for all <math>x \in \mathbb{R}^n (x \neq 0),</math> there exists <math>u\in \mathbb{R}^m</math> such that

:<math>

\forall x \ne 0, \exists u \qquad \dot{V}(x, u) := \langle \nabla V(x) \cdot, f(x,u)\rangle < 0.,

</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]], repeated here:.

Some results apply only to control-affine systems—i.e., control systems in the following form:

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

{{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==

It may not be easy to find a control-Lyapunov function for a given system, but if one is found, then the feedback stabilization problem simplifies considerably.

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]]

[[Eduardo D. Sontag|E. D. Sontag]] showed that thefor existencea ofgiven control system, there exists a continuous cLfCLF isif equivalentand only if the toorigin is asymptotic stabilizabilitystabilizable.<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> 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 |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:

'''Artstein's theorem.'''proved that Thethe 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>Sontag (1998). ''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>

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==