>> Well, if we could make it, yes.

But unfortunately, for these kind of second order systems it's very tough.

I haven't seen one that actually does it successfully.

>> We're looking then at the second order derivative.

>> Exactly, so there's other methods.

If you do nominal control, you may have heard of invariance principle.

It's kind of related.

It's a different approach.

You find the largest invariant set where this thing manages and

then there's some extra arguments.

But the chain theorem I like very much,

because it's a nice approach especially for engineers.

You know how to take derivatives, you do it and

then we have to look at that, right?

So Daniel look at your chin, could you quickly outline that one for me?

>> Well, we keep on deriving to the Lyapunov function until we

get to something that's not 0.

>> Right, so

we take an extra time derivative of that second derivative, third derivative.

And you expect these things to be 0 for the even ones and you're looking for

an odd derivative that's non 0.

On which set, Kevin, do we actually evaluate these higher derivatives?

>> The set where the V dot function is going to 0.

>> Right, which in this kind of formulation is always tends to be

the rates of 0.

With integral feedback we had something where it was the measure of the rates and

integral term combined goes to 0.

But it's something of that form, exactly.

So good, we went through that.

Global stability we argued if you treat these as just going to infinity,

which we can do.

But of course there we exclude the case that the spacecraft tumbles 360 degrees,

because then our measure goes to infinity.

So you can talk about global stability, but

you still can't do a completely continuous tumbling body.

To do that, we kind of had to start talking about switch Lyapunov functions.

And what saves us there was that as we switch at sigma is equal to 1,

the Lyapunov function itself actually remains continuous.

Not smooth, but continuous.

And in those arguments you can do.

So we do have a globally stabilizing one and

here's the outline of what we had with asymptotic stability.

You take the second derivatives.

Why could we always assume that delta omega dots and so forth or

delta omega double dots are finite?

What was that argument?

>> [INAUDIBLE] >> Exactly, all right, sometimes students.

because you could plug in and in the spring mass damper system I show you,

you have an x double dot.

You can take the dynamics, plug it in, put it in x dot equal to 0, and

then show okay, if this is all that it vanishes.

But if you can argue that this is stable, this is finite.

0 times finite has to be 0 and

you save yourself some steps in algebra so, that's kind of a nice approach.