Which is a nice result.

So now with unmodeled torques, I can actually argue that I will have asymptotic

error as long as the torque is constant.

That was one of the requirement that we have.

If it's not constant, then doesn't quite work.

But, the z will not go to zero with unmodeled torques.

So, following the earlier steps, you can do this,

plug it in into the exclusive dynamics.

Evaluated on these sets.

And we know del omega's going to go to zero.

In the end, you can predict now what the z variable is going to be.

If this is some unmodeled torque,

I know what the inertia is, I know what this gain is.

This is what that value will convert to.

So your integral term, as it builds up, it actually, in fact,

learns what that external torque is.

And it brings you to zero because at zero it has to compensate.

Something's pushing me one direction one Newton meter.

This integral term has to push back one Newton meter,

otherwise you wouldn't stay in place.

So it's a way of doing a simple form of doing adaptation.

It's learning with an external forces.

Which is nice, so if we wanted the same simulation we had.

That's the same external force.

We want to derive this to zero.

Before the rates went to zero, but the attitudes didn't go to zero.

Now by using this new control with the z interval term in there,

this large departure tumble and recovers, comes down.

You can see our attitudes do go to zero, which is nice.

And from the analysis we're predicting here that.

I'm using a single inertia here, so delta l over iki should give me these z values.

That's where this integral term should converge to.

And I believe yup there we go.

There's the z's plotted out and they actually match the 1.6, 3.3 and minus 3.3.

So the numerical results match up very well with what we analytical predicted,

this is going to be the off set.

And the control as expected doesn't converge to zero here.

It can't.

because when its holding the heading,

there's always a torque that's pushing you out of that heading.

So, it has to converge to a non-zero torque and that can be a challenge.

If you do this for example with reaction wheels.

What do you think Brian?

If I'm using reaction wheels to create this torque and

you have to have a non-zero torque all the time just to hold the heading,

any problems you can envision?

>> Couple.

>> A couple?

[LAUGH] Give me one.

>> I know your reaction wheel is always going to be changing or