Is it none of those?

>> None of those.

>> So I think it was one of those.

I was like, why are you saying that.

Okay, that makes much more sense.

Okay, it's none of those, and there's a word for that.

It's called indefinite, right?

Sometimes you have functions like that that just are going to be this.

Or you look at in the homework there's multi-dimensional stuff.

And some students always argue, well,

this is positive definite if I only consider positive xs.

But if we [INAUDIBLE] the stability,

we can't guarantee that people will only bump it to one side, not the other, right?

Remember, you always have to be able to draw a ball,

a finite neighborhood around the thing that you're studying.

And you can't say, well, along this trajectory, we would always be positive

here, we'd be negative, we will just never to perturb to the negative side.

Life doesn't work that way, that's not a stability argument.

So this one would be indefinite, right?

And the theorems we have wouldn't apply.

It doesn't mean it's unstable,

it just means we cannot say if it is stable or not.

We'll see, okay, Jordan.

>> So, for the last lecture I was watching it,

and you basically drew a problem below zero and above zero and you said that for

the above you could shift your coordinates system.

Can you also do that for if it's below?

You can do it both ways?

>> Well, to some point. This one, you could never shift up or

down to make it always positive or negative, right?

if your V function has the bowl shape you're looking for, but it's

not zero where it needs to be, typically what you do is a coordinate change.

And then you can talk about the stability, not about zero meters, but

the stability about one meters.

And that's where this function actually happens.

So really without loss and generality,

we're always going to be talking about driving either delta xs and

the tracking problem or just xs and a regulation problem to 0.

because we've assumed we've got a coordinate shift such that if

there's an equilibrium right at the origin but somewhere away,

you've made a new coordinate system there and you're driving things back.

Yep, so that can happen, so no good.

When you're looking at these functions and homeworks and trying to figure out what's

happening here, is it local, is it global, plot them out.

Go to math lab, go to mathematical something.

Plot these functions and you will quickly see visually, too, wait a minute.

Just think about it,

is definite about this point, so where is going to go to zero.

And what's happening there and

is there a finite region where it's guaranteed always positive.

That's something you have to consider, so

plots are actually very, very helpful in this stuff.

Especially when you only have one or two coordinates.

The F15, well, that's not a problem.

You have to look at the mathematics of it.

Okay, so these are definiteness, definition that we have,

basically making things positive.

There was also things about if you have a matrix, if you got a 3 by 3 matrix

if it's positive definite then this is the matrix version of it,

this will always be 0 unless x is all zeros right?

That's the origin.

If x is all zeroes, this is [INAUDIBLE] zero.

But anything else, this scalar answer will give you something positive.

And this works.

We've seen such functions.