2
$\begingroup$

I was reading an article on Euler-Lagrange systems. It is stated there that since M(q) and C(q,q') depend on q, it is not autonomous. As a result, we cannot use LaSalle's theorem. I have uploaded that page of the article and highlighted the sentence. (ren.pdf)

Then, I read Spong's book on robotics, and he had used LaSalle's theorem. I am confused. (spong.pdf)

I did some research, and found out that non-autonomous means it should not explicitly depend on the independent variable. Isn't independent variable time in these systems? So, shouldn't they be considered autonomous?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Autonomous systems are also called time invariant. They don't depend on the time variable. $\endgroup$ – CroCo Mar 27 '16 at 17:00
  • $\begingroup$ Dear @CroCo, So you belive Euler-Lagrange systems are autonomous? $\endgroup$ – Has Mar 27 '16 at 21:15
1
$\begingroup$

If you structure the system dynamics equations to not depend on the independent variable, an Euler-Lagrange system is autonomous. If, however, you recast the equations to depend on the state variables (and their derivatives), as Ren did, it becomes non-autonomous. That clever manipulation allowed him to prove asymptotic stability for the consensus algorithm $\tau_i$. It might be possible to prove without taking that mathematical direction, but I cannot think of a better way to do it.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.