0:07

Lyapunov stability, this is the one if you just hear somebody talk about

Â the non-linear system and say, is that stable?

Â They're implying Lyapunov stability.

Â And that's a very common assumption.

Â There's many, many control papers, all vigorous math and

Â they just talk stability.

Â It's implied to be Lyapunov stability.

Â So what Lyapunov stability mean?

Â Here we think of this in state space.

Â If you have this ball that can move around the reference, it generates this tube.

Â This tube size was not a function of the initial conditions.

Â Now with the Lyapunov stability, what we're saying is,

Â we're getting in there, but you can pick an epsilon.

Â That means you have a size of a ball at the end,

Â if I want to be within one degree of the attitude.

Â There is a neighborhood you have tp be within ten degrees to begin with and

Â then I can guarantee at some point you enter one degree and

Â stay in one degree all right.

Â 1:07

So you get to pick epsilon Lyapunov stability you don't get to pick a bound

Â it just comes form the physics of the problem independent of initial conditions.

Â Here you get to pick the bound, and say hey, if I want to be within one,

Â this is what you have to do.

Â A simple example is actually the spring mass system.

Â 1:25

It's an oscillatory response.

Â If I want to guarantee my wiggles are less than one-degree and

Â I'm letting it go with zero rates just make sure you deflect that pendulum less

Â than one degree and you're guaranteed you're staying in there.

Â Their epsilon delta happens to be the same.

Â Don't always have to be the same.

Â But that's kind of the stability argument that we have here.

Â 2:00

Three's no guarantee it'll converge, all right?

Â So while we can make epsilon arbitrary small, we can't make epsilon zero.

Â because that would imply convergence that you have, right?

Â There's just a bound that I can guarantee that once I get within it,

Â and you could make that aperture very small, so

Â it gives you the kind of neighborhood, within what neighborhood would I converge

Â to within some tolerance of your pointing accuracy as an example.

Â 2:31

Lagrange, think bounded doesn't depend on initial conditions.

Â The Lyapunov stability, the final tube that you have, that ball

Â that moves through around the reference, depends on initial conditions, right?

Â That's why we have to stay within some accuracy, but you can pick these now.

Â So, let's talk about an example and apply this.

Â We've seen Robert was talking earlier about spins, about intermediate access.

Â If you remember that video I showed from the astronaut taking that one little key,

Â spinning it, right.

Â 3:19

No, it kept flipping back again, right?

Â And from the pole hold plots, we understand why it's doing that.

Â And as we get closer to separate trigs that's pure spin.

Â 3:29

It hangs out a long time there, and the pole plot time slows down, so to speak.

Â And then it goes quickly across.

Â So that's a situation where if you looked at a short period of time,

Â you might go well, wait a minute.

Â It wobbled maybe within five degrees.

Â That would be the stable.

Â But that is not true.

Â Because while it was here for a while, at some point and with these panes we know

Â we wait long enough it will eventually leave again, it leaves as to begin.

Â That's why the separate x motion when you're doing this looks kind of stable for

Â short period of time that's not stable.

Â It's still unstable because at some point it's going to leave this

Â neighbourhood again.

Â All these definitions, all these stability definitions mean once you enter

Â this desired stuff, you're proving that it will remain in there forever.

Â So the spring amper system with gravity.

Â Once you let go and it wiggles, it stays here.

Â It is there, it's never going to all of a sudden have a huge oscillation again.

Â That's not in this dynamical system.

Â But the seperate nix motion does do that.

Â So that's why you have to be careful if you use numerical tools, because if we try

Â to publish and go well I ran six million simulations and I ran them all for five

Â seconds and within five seconds everything kind of wiggled and stayed close.

Â Must be stable they're going to be yelling at you going wait you need to sim it way

Â longer and really prove this.

Â And if you can do it analytically you avoid all those arguments, right?

Â So these definitions are nice.

Â You need to be familiar with the basics, I like the visual part of it.

Â The wordings you going to remember great but

Â just, these are the concepts you have, proving this is a realm pain.

Â 5:07

We 're not going to do that, that's where the Lyapunov theory comes in.

Â Lyapunov theory gives you a nice elegant mathematical tool to prove these

Â properties without having to actually solve it.

Â because otherwise, in a complicated system you have to look at the full response.

Â In the spring mass system, I can get an analytic answer,

Â what those oscillations are going to be and

Â you can come up with bounding functions, and for every epsilon I can do this.

Â But for nonlinear systems, an attitude problem,

Â you're not going to find analytic answers.

Â So those approaches can be very difficult.

Â So, this is just a definition, boundedness.

Â Here we have stability but not convergence, right?

Â On the linear system,

Â stability would imply convergence not with a non-linear system.

Â