Talk:Lyapunov stability

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[edit] Application

You who wrote this page: are you sure that we cannot apply the Lyapunov stability to forced systems????

[edit] Overall coments

Hi,

I'm not in the mood to put it all correctly right now, but I'd like at least to comment some details.

First, I've already altered the definition here (I didn't have a login at that time, sorry... when I find out how, I promise to put my IP here), and basically that's the one present now, when it comes to &epsillon; and δ.

But the article is, actually, wrong in the following sense: we're dealing with Lyapunov Stability of Equilibrium Points. There's no need to specify if the system is unforced.

The concept of Lyapunov Stability is, actually, original from the study of trajectories from ODEs. Then, you should define when a solution of a ODE is Lyapunov Stable. It's a rather similar definition to the one on the page, but since the solution need not to be a equilibrium point anymore, you should compare the norm of the difference between y(t) (the solution you're studing) and x(t) as in the article.

As you can see (gotta find the reference to put it in the article too) in LaSalle's book on Stability, if you do make a change of variables based on a "translation by y(t)", then y(t) is the origin and a equilibrium point for the new system, and the definition just given in the article is applied.

Although we do not need to specify the kind of system for the definitions, it's very important to realize that a great part of classic results and theorems applies only to unforced systems (actually, I'd use "conservative" systems, since that's the context I do work with them everyday...)

Well, sorry for anything! This is my first writing here... so I'll try to learn it better how to contribute!

--Rfreire 22:40, May 5, 2005 (UTC)

Hi, in the definition don't we need to mention the system dynamics being considered are ordinary differential equations of the form xdot=f(x) (i.e. not time varying or other dynamics), and that they must be "well behaved" (e.g. f satisfies a Lipschitz condition within a ball of radius epsilon of the origin)? Also you might want to mention the norms involved.

[edit] forced systems can be stable ITSOL

Can't Lyapunov stability apply to forced systems, like closed-loop systems for example?

dx/dt - 4*x = f(t)

is unstable when f(t) = 0, but stable when f(t) = -5*x. The closed-loop system in this case is

dx/dt + x = 0

It's true that the closed-loop system is unforced mathematically, but from the perspective of the original system, we have a system that was stabilized using the control force.

[edit] Equilibrium point

In the 9th line the 0 is not the EP.--Pokipsy76 18:57, 11 February 2006 (UTC)

[edit] The Van der Pol oscillator example is Incorrect.

In the Van der Pol oscillator example, we clearly have the origin as an equilibrium point to the system. The problem stems from the "claim" that \dot{V}(x) is negative definite. Since \dot{V}(x) has no x1 contribution, this equation is negative semi-definite, i.e. \dot{V}(x) = 0 when x2 = 0 and \forall x_1. To prove asymptotic stability, either a new Lyapunov candidate function needs to be considered, or LaSalle's invariance principal needs to be applied.

I agree. To use a Lyapunov function itself to prove asymptotic stability (without resorting to the invariance principle or Barbalat's Lemma), you have to choose a different function. I suggest changing the section to use a simpler first order system. S280Z28 06:06, 26 April 2007 (UTC)

[edit] lack of references

well a short bibliography would be useful, wouldn't it?


[edit] Lyapunov second theorem on stability

Isn't it also required that V(0) = 0? The definition given allows V(x) = 1 / (1 + | x | ) to prove that x(t) = t is asymptotically stable.

Also, my understanding was that such a V(x) is called a Lyapunov function, and that the term 'candidate' is used to distinguish functions which have yet to be proven to be Lyapunov functions (candidate Lyapunov functions) from those which have been proven. LachlanA 19:00, 31 January 2007 (UTC)


[edit] Barbalat's lemma

This looks like a really cute lemma, but does it belong on this page? LachlanA 22:21, 24 February 2007 (UTC)

[edit] 'Iterated' systems

I never heard the term 'iterated' to refer to 'discrete-time' or just 'discrete' dynamics in the control literature. Though correct, I would suggest that anyone having a sounder background than me should consider replacing 'iterated' with 'discrete-time'. -- Biscay 12:10, 31 August 2007 (UTC).