0:04

So now, if we are going to talk Asymptotic Stability, it's very easy with these tubes

Â because it means you still get to pick an Epsilon,

Â but in fact, now with Asymptotic Stability that Epsilon goes to zero.

Â That means there's no final tube, that tube is gonna shrink infinitesimally.

Â And whatever bound you come up with,

Â at some point that tube is gonna shrink less than that.

Â So it converges, it may not reach zero but it's asymptotically converging.

Â You can never find a finite shape and then it never gets less than that.

Â So, that's what asymptotic stability means.

Â And so, as T goes to infinity your state now becomes your reference.

Â And of course, from a control perspective, this is what we love.

Â We love to have convergence always.

Â Don't always get it, but that's what we're looking for.

Â But with a nonlinear system, you still have to argue about local and global stability.

Â So, there's a neighborhood for which this is true,

Â and we can do that and we've talked about this pendulum problem for example, right?

Â If this thing is oscillating,

Â and I'm only oscillating within 10 degrees with some damping,

Â it will actually converge to a steady state position.

Â But it's not global because some joker could put it up here perfectly balanced

Â and it would not converge all of the sudden, right?

Â So, convergence is not global.

Â The stability... So, let's argue this other one.

Â On this example, is stability global, Lyapunov stability.

Â Supply that, Martha.

Â If we, our initial condition is less than 180, then?

Â Or if it is 180, can you come up with a bound within which the attitude air will remain?

Â If somebody gives you this condition, what's your bound then?

Â Any small perturbation it will go back to it.

Â I know but I don't have a small perturbation.

Â I have-- So, this is the perturbation, yeah?

Â This is where we wanna be.

Â I've given you a 180 degree.

Â Theta is 180 degrees.

Â Theta dot is zero precisely.

Â So, what is the bound that we have to pick targets to go with.

Â I'm confused.

Â Ok, sure.

Â With that case, why isn't it just an equilibrium but not stabilized, it's stable?

Â It's not stable.

Â All right.

Â We're looking at the stability of this point, but I'm allowing very large deflections,

Â and with nonlinear systems you might have multiple equilibrias in the system.

Â So, you have to account for, well,

Â if you're saying it converges back for any set of states,

Â you have to include also this state, right?

Â So, we are talking about stability of this one right now,

Â we're just throwing in a 180 degree perturbation with zero rate.

Â What do you think [inaudible]?

Â I have a question, is global stability only possible if there's one equilibrium?

Â Oh, that's a good question.

Â 3:03

The question is, is global stability only possible if there's one equilibrium?

Â I would lean towards yes, but live right now, teaching, I wouldn't bet my life on it

Â that some smarter mathematician hasn't come up with a weird degenerate case

Â where something could asymptotically go crazy and never actually reach that equilibrium.

Â But it's probably a good sign.

Â With reference trajectory tracking though,

Â it changes a lot of that because you're creating artificial equilibria, right?

Â I wanna follow this path and things get more complicated.

Â So, there'll be some rigorous tools that we will have to argue it.

Â But we said, ok, so, we know this pen cannot be asymptotically globally stable

Â 'cause I found a set of states that don't make it converge.

Â But, is it still stable, right?

Â For stable it means, can you pick an Epsilon and a Delta,

Â such that once it enters that condition, it stays within that set of conditions.

Â What would your-- Daniel, David.

Â Basically, yeah.

Â So, if you just make your bound 180, which you can, you're gonna stay in that bound.

Â Of course, that attitude's almost cheating because we can't get worse than 180,

Â but some people go to 200 and higher.

Â But that's right, right?

Â With 180 plus with zero rates.

Â That's gonna be the kind of, there's a full state spacing.

Â That's gonna be the bound that you're going to have.

Â So, you could say that this system is bounded.

Â You're never gonna have-- With friction,

Â you're never gonna have a set of initial conditions where the attitude angle keeps--

Â Even if you went beyond 180.

Â So, let's forget that it's, you know, the same thing.

Â 360 is same thing as zero.

Â There is no way the system would, you know,

Â asymptotically spin out of control because with damping you would always be losing energy,

Â and it eventually it would settle there and there would be a bound that you can have.

Â If you do the clipping of multi revolutions,

Â it's 180, otherwise you can come up with something slightly different.

Â But that's an example quickly that hopefully starts to illustrate what we're doing, right?

Â Our controls.

Â We will design them to be asymptotic.

Â But then once we throw in other issues, you will see where it comes up.

Â So, asymptotic stability, the Epsilon now can go to zero.

Â We converge.

Â Andrew.

Â You know...

Â Tough demo.

Â I've had two hours of sleep last night, sorry.

Â So, just the pendulum problem.

Â It's that what you're talking about.

Â It's both.

Â You said stable as both Lagrange and Lyaponuv stable.

Â Not asymptotically but.

Â Yes.

Â Even though it's for all Epsilon, but that Epsilon would-- You wouldn't say Lagrange

Â and Lyapunov stable typically, cause if it's Lyapunov stable, it's a stronger argument.

Â Yeah, yeah.

Â That already overrides Lagrange stability.

Â But it's not--the key is, it's not just locally stable,

Â it's actually globally as Lyapunov stable.

Â But it's not globally asymptotically stable

Â because I could give it initial conditions where if you get that error,

Â you're stuck, right?

Â And this happens a lot then in stuff.

Â And what if you happen to be just exactly upside down?

Â You're gonna stay there.

Â And if you're off a little bit, you might recover, but very often what happens then,

Â kind of like with the separate tricks,

Â it might take a really long time to get there as well.

Â