Lets go through quickly the stability definitions. Maurice,long gross table, what does that mean? >> Long gross table just means once you enter a certain neighborhood, region, it's that you'll stay within that neighborhood. >> True, and the key thing is what about this neighborhood? What was your name again? >> Brian. >> Brian. With log on stability. Because we also have the neighborhood with the optimum stability, right. Both of this final neighborhood, what's the key stability difference between them? >> Does it depend on the initial state? >> Exactly, so if I want stability, think of a system and you've taken it and tilted it 90 degrees where gravity now is an offset. It's always going to settle at the steady state. It doesn't matter how close you were to the origin. It's always going to settle there. It's always one meter, that's your balance. That's not going to stability. The next stronger statement is stability. So, fine. What makes something enough stable then? >> Where you can pick what that neighborhood is depending on the initial conditions. >> Yes, what's the mechanical analog for this? What's the system that's clearly up and up stable but it's not asymptotically stable, right? That's the next one we're going to. Mariel do you remember mechanical animals? >> [INAUDIBLE] >> Well yeah we're not quite to asymptotic yet. So what, Chip. >> Spring mass system. >> Spring mass system and even for a linear system spring mass is kind of just marginal, they call it marginal stable. There are roots of the characteristic equations, imaginary axis. It's just going to oscillate. So with stability we always talk about small departures, you can say well my domain of attraction is zero, have to be on it. Now you just have an equilibrium, that's not a stability. Stability is always you've departed, there's some epsilon, no matter how small but there's some find the epsilon that you bump it what's the use of that? And that's kind of the challenge. So, as soon as you take the spring master system and you detour it ever so slightly it's just going to keep on wiggling. You can make them wiggle as small as you wish. So if your mission says I have to be within one art session you can just bump it very, very lightly. If any mechanical thing move the fuels wiggles a little bit. We might be exceeding it already. So, that now dictates other things in your, what are the disturbance torques acting on the spacecraft? If your good to within one degree, well now you can actually, and you have the optimal stability. You can bump it quite a bit more, obviously. And stay within one degree then one arc second. So there's that kind of stability. So, Andrew, what's the next level of stability, beyond Lyapunov stable? >> Asymptotic? >> Asymptotic stability, right? So now, that final neighborhood has gone to zero. Does it do this for any initial conditions, Tony, or only for local initial conditions, if you say it's asymptotically stable? >> Local >> If you just see asymptotically stable, you'll have to be somewhat imprecise it's going to be a trick question. [LAUGH] All right so it's what often sometimes you see in it's maybe within the context it's implied to be local I mean that's typically the case. If people think it's global, they're very proud of having a global answer and they typically mention that very explicitly. This is globally asymptotically stable that's their conclusion out of all the math. That means it's good for any initial conditions. Then the other thing called Exponential Stability which we don't cover in class but it's one that I mention. Now we have a performance guarantee. He doesn't just say, I'm going to get to zero, which may take 6 billion years. But I'm going to get to zero within an exponentially decaying envelope, essentially. Which is what you would have with linear control. Every linear system, once it's stable, it is asymptotically, globally, exponentially steep. You get all that stuff for free for modeling your systems? No.
