So now, let's go back to these definitions.

With linear control, If you have a spring-mass time per system,

and the gain is positive, so all of my roots are particularly on the left hand side,

does it matter if I pull that spring mass a little bit or a huge amount?

Is the stability argument going to be different?

CK, what do you think?

No.

So, for linear systems once it's stable,

it's regardless of initial conditions, you have to have this nice linear math form,

linear matrix math, and then you can throw in any initial conditions.

And, big oscillations will just be big wiggles

but they will go to zero; small oscillations will be small wiggles but they go to zero.

It's always stable.

Right?

Nonlinear systems, not always the case.

So we have to define neighborhoods.

We will say, "Hey, the system is stable", but not only within, you know,

if you don't depart more than 10 degrees;

if you do 20 degrees, sometimes we just don't know,

or sometimes we can even prove instability as well.

So, that's where things get a bit more complicated.

But if you think of a simple nonlinear system, just pendulum problem, right?

This point here is stable.

So, is it globally stable?

Can I give it any initial conditions and it will always converge back to this point?

I have friction, you know, and this thing can wobble.

Andrew, you're shaking your head.

It will?

Yeah, always.

Always?

Always.

If you're not applying control to it, yeah.

What if I put it here?

You're holding it at the bottom?

No, it's...Pretend I perfectly balance this thing.

Right?

Because mathematically, this is an equilibrium.

What if somebody gave you this initial condition?

Are you going to converge to here?

Oh, ok.

That's right?

So, even in the simple example that's one way you can see.

We actually have local stability.

It's almost global, and there's all these papers talking about almost global stability.

And I think that's rubbish.

Either it's like, "I'm almost alive", that means you're dead, you know?

So, that's a local stability actually with this pendulum system.

And that's one of the homeworks you go through,

it's actually to look at this and argue, and look at the math,

and you can find all of a sudden, great.

And, what does this mean?

Right?

So, there's neighborhood that we have to define,

and the way to do this here is we use the classic L2-norm, you know?

Where you take one state

representation minus another, and then do the classic L2-norm, you know,

to some square, the components set this square root.

What that means geometrically,

is really you taking a ball in this multi-dimensional space,

so in 3D, it would be a ball;

if it's 2D, it's a disk; if it's 5D, it's a hyperball, right?

Anything more than three will be called hyper.

So, that's really your neighborhood.

So, we're talking about neighborhoods, we define it through assigned Delta.

Hey, if I'm within five degrees here, I can guarantee I'm stable.

And if I'm within 90 degrees, I'm guaranteed I'm going to converge and be down here.

But if I include 180 degrees,

I'm not guaranteed that it necessarily would converge back down again, right?

So, that's what the Deltas mean.

And the B of Delta just means - this is the neighborhood of a ball, size, Delta,

and it's around some reference.

This reference could be equilibria or it could be that path that you decided to follow

or, you know, hey you want to scan first London, then,

some other city and then the next city, and off you go.

Right?

So, you have some time during introductory.

Doesn't have to be a negative.

So, that's a neighborhood.