Control-Lyapunov function: Difference between revisions

Revision as of 11:52, 20 October 2010

Revision as of 16:02, 22 October 2010

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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==

Here is a characteristic example of applying a LyaponovLyapunov candidate function to a control problem.

Consider the non-linear system, which is a mass-spring-damper system with spring hardening and position dependantdependent mass described by

:<math>

m(1+q^2)\ddot{q}+b\dot{q}+K_0q+K_1q^3=u

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r=\dot{e}+\alpha e

</math>

A Control-LyaponovLyapunov candidate is then

:<math>

V=\frac{1}{2}r^2

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

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

:<math>

\dot{V}=r\dot{r}

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</math>

The goal is to get the time derivitivederivative to be

:<math>

\dot{V}=-\kappa V

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with <math>\kappa</math> and <math>\alpha</math>, both greater than zero, as tunable parameters

This control law will guarenteeguarantee global exponential stability since upon substitution into the time derivitivederivative yields, as expected

:<math>

\dot{V}=-\kappa V

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And hence the error and error rate, remembering that <math>V=\frac{1}{2}(\dot{e}+\alpha e)^2</math>, exponentially decay to zero.

If you wish to tune a particular response from this, it is necessary to subsititutesubstitute back into the solution we derived for for <math>V</math> and solve for <math>e</math>. This is left as an exercise for the reader but the first few steps at the solution are shown.

:<math>