Anyway, no excuse, I was wrong.

This would be basically what we just did.

It would be theta and theta dot.

Those are the two things.

If you have N degrees of freedom, we have three attitudes,

three positions, it's a second order differential equation.

I also have to keep track of their velocities.

So this becomes a 2 times N here would be 2 times the number of degrees

of freedom, yes.

That would make more sense, good.

So we've got that and that's basically so if we go here,

if we just wrap this up and you save this, then you're saying,

okay, x dot is going to be theta dot and theta double dot.

And theta dot, let's call this x1, x2.

Then theta dot, the x1 dot is going to be nothing but x2.

And x2 dot is going to be -omega and

sine of x1 that you would have, right?

So you'd just introduce these forms.

So you could always take a set of second, third, fourth order differential equations

and introduce, as you're talking about, these other derivatives.

That's what we're doing, right?

So, this can always be done, so this form here is very general.

And we have our first order differential equations.

Now we want to talk about control.

So we don't just have x double dot plus something x equal to 0,

but it's equal to some control effort.

There's some server, some thruster, something you're controlling, right?

So you have an authority over this.

How much thrust is produced?

How much torque are you creating with wheels?

And this has some function, so depending on the states that happen here,

there's some function that could be going in here.

This function is written as g(x) so it could be non-linear.

But you can also, as you will see in several problems we do,

we have some elegant controls where we could just throw in -k times sigma,

sigma being my attitude error.

And it's still going to be stable, but

we can also throw in non-linear feedback controls.

We'll see some of them. So very general.

It's just our control formulation is a non-linear, encompasses linear.

And then when you apply this into the dynamics,

you get what's called the Closed-Loop System.

So that's when you had the natural dynamics, which were unstable,

you apply some control, hey, I'm going to take this attitude angle and

feedbacks on it, and that's going to stabilize it.

And then once you apply that control, that's the Closed-Loop System.

So you just plug in that u into it, and

that's the stability that we're going to be looking at.

Not the original system, but

we really care about the new system that we've created this way.

Equilibrium states, we discussed this.

We've seen this with the rigidi body spinners,

we've seen it with dual spinners.

It's when, if you have always this form, x dot = f, for

what set of states x is this derivative across all the states going to go to zero?

That means in an example we had earlier with the theta double dot + omega

n sin theta, that vector was theta and theta dot.

So I have to figure out for what set of states is that all going to vanish?

And typically, you also need theta dots to be zero.

You're really just looking for the states in the end.

And when you find that, that's your equilibrium point.