This would be something that is now, for example, globally positive semi-definite.

I'm always positive or I'm zero.

And semi-definite is away from the zero part.

Yes, Matt?

>> Does the ball have to be [INAUDIBLE] so like the x cubed, for example?

Instead of the ball or it has to be definite?

>> No, it has to be finite, yeah,

because otherwise you're talking about the equilibrium.

You've just discovered that page.

Hey this is good if I have zero disturbances.

That is what the ball being zero would mean.

And for stability we have to look at finite epsilon disturbances.

Yep, it can be really, really small but it has to be finite.

Very good question.

Okay, good.

So you could see, there's things we could play, but

there's nice visual representations.

That's always positive, this could be positive or zero, right?

And they could be local or it could be global.

Then simple examples could be things like these where often have these kinds of

functions.

Where if this were mass, this would be like potential energy of a spring, k

times x squared over 2 would be potential energy, this could be kinetic energy.

So potential and kinetic energy, what happens there, that's an example?

So with more than just one, you can create these functions.

This function v is only 0 if x is 0 and x square is 0.

If either of them is perturbed, this function is non-zero, all right?

But it is positive.

So this would actually be a globally positive definite function.

This function also never goes to zero away from the origin.

It doesn't do that bend down and then go up again and stuff.

So it's not semi definite, it's actually globally positive definite in a sense.

So you can do it on functions.

Yes, Marta?

>> In all of these examples, xr is origin, like the equilibrium is origin.

What if it's not 0,0?

What if it's not- >> Then you do a coordinate change.

And xr would be, for example, assume we had the linearization we said.

We took x minus xr and then the new coordinates were delta u or

delta x actually in that case.

That is the departure motion about the reference.

If you have a tracking problem, that's why I always have these xrs,

that's my reference.

Reference could be an equilibrium and then you could define your frame to be

at that equilibrium so everything drives to zero.

So you can always do that.

But for the tracking problem, your reference keeps moving,

I want to be here then I have to be here.

So now, you define your state relative to this time varying things.

So, that is a delta x.

So, if you did tracking problem, for example, down here,

all you would have to do is replace x with delta x.

And now you have a positive definite function in terms of your tracking errors,

my tracking position errors and tracking rate errors.

If I had delta x squared and delta x dot squared.