Control-Lyapunov function: Difference between revisions

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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>

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>{{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:

'''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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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.

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)