Lyapunov stability, this is the one if you just hear somebody talk about the non-linear system and say, is that stable? They're implying Lyapunov stability. And that's a very common assumption. There's many, many control papers, all vigorous math and they just talk stability. It's implied to be Lyapunov stability. So what Lyapunov stability mean? Here we think of this in state space. If you have this ball that can move around the reference, it generates this tube. This tube size was not a function of the initial conditions. Now with the Lyapunov stability, what we're saying is, we're getting in there, but you can pick an epsilon. That means you have a size of a ball at the end, if I want to be within one degree of the attitude. There is a neighborhood you have tp be within ten degrees to begin with and then I can guarantee at some point you enter one degree and stay in one degree all right. So you get to pick epsilon Lyapunov stability you don't get to pick a bound it just comes form the physics of the problem independent of initial conditions. Here you get to pick the bound, and say hey, if I want to be within one, this is what you have to do. A simple example is actually the spring mass system. It's an oscillatory response. If I want to guarantee my wiggles are less than one-degree and I'm letting it go with zero rates just make sure you deflect that pendulum less than one degree and you're guaranteed you're staying in there. Their epsilon delta happens to be the same. Don't always have to be the same. But that's kind of the stability argument that we have here. There's no guarantee you could make epsilon as small as you want. I can make it a millionth of a degree, great. And it's going to stay within a millionth of a degree. Three's no guarantee it'll converge, all right? So while we can make epsilon arbitrary small, we can't make epsilon zero. because that would imply convergence that you have, right? There's just a bound that I can guarantee that once I get within it, and you could make that aperture very small, so it gives you the kind of neighborhood, within what neighborhood would I converge to within some tolerance of your pointing accuracy as an example. Good, these are three definitions, any questions on the neighborhood? Lagrange, think bounded doesn't depend on initial conditions. The Lyapunov stability, the final tube that you have, that ball that moves through around the reference, depends on initial conditions, right? That's why we have to stay within some accuracy, but you can pick these now. So, let's talk about an example and apply this. We've seen Robert was talking earlier about spins, about intermediate access. If you remember that video I showed from the astronaut taking that one little key, spinning it, right. It looked stable. In simple layman's terms. It was kind of wobbling, but it was there. But then one happened? Trevor. >> Flipped. >> It flipped, okay, then it looked stable. Did it stay there? No, it kept flipping back again, right? And from the pole hold plots, we understand why it's doing that. And as we get closer to separate trigs that's pure spin. It hangs out a long time there, and the pole plot time slows down, so to speak. And then it goes quickly across. So that's a situation where if you looked at a short period of time, you might go well, wait a minute. It wobbled maybe within five degrees. That would be the stable. But that is not true. Because while it was here for a while, at some point and with these panes we know we wait long enough it will eventually leave again, it leaves as to begin. That's why the separate x motion when you're doing this looks kind of stable for short period of time that's not stable. It's still unstable because at some point it's going to leave this neighbourhood again. All these definitions, all these stability definitions mean once you enter this desired stuff, you're proving that it will remain in there forever. So the spring amper system with gravity. Once you let go and it wiggles, it stays here. It is there, it's never going to all of a sudden have a huge oscillation again. That's not in this dynamical system. But the seperate nix motion does do that. So that's why you have to be careful if you use numerical tools, because if we try to publish and go well I ran six million simulations and I ran them all for five seconds and within five seconds everything kind of wiggled and stayed close. Must be stable they're going to be yelling at you going wait you need to sim it way longer and really prove this. And if you can do it analytically you avoid all those arguments, right? So these definitions are nice. You need to be familiar with the basics, I like the visual part of it. The wordings you going to remember great but just, these are the concepts you have, proving this is a realm pain. We 're not going to do that, that's where the Lyapunov theory comes in. Lyapunov theory gives you a nice elegant mathematical tool to prove these properties without having to actually solve it. because otherwise, in a complicated system you have to look at the full response. In the spring mass system, I can get an analytic answer, what those oscillations are going to be and you can come up with bounding functions, and for every epsilon I can do this. But for nonlinear systems, an attitude problem, you're not going to find analytic answers. So those approaches can be very difficult. So, this is just a definition, boundedness. Here we have stability but not convergence, right? On the linear system, stability would imply convergence not with a non-linear system.
