Control-Lyapunov function: Difference between revisions

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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>.

who[[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:

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>

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

:<math>

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

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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 andis luckfound, 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]]

:<math>