This course trains you in the skills needed to program specific orientation and achieve precise aiming goals for spacecraft moving through three dimensional space. First, we cover stability definitions of nonlinear dynamical systems, covering the difference between local and global stability. We then analyze and apply Lyapunov's Direct Method to prove these stability properties, and develop a nonlinear 3-axis attitude pointing control law using Lyapunov theory. Finally, we look at alternate feedback control laws and closed loop dynamics. After this course, you will be able to... * Differentiate between a range of nonlinear stability concepts * Apply Lyapunov’s direct method to argue stability and convergence on a range of dynamical systems * Develop rate and attitude error measures for a 3-axis attitude control using Lyapunov theory * Analyze rigid body control convergence with unmodeled torque
5: Definitions: Asymptotic Stability
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Nonlinear Stability Definitions
Discusses stability definitions of nonlinear dynamical systems, and compares to the classical linear stability definitions. The difference between local and global stability is covered.
教学方
Hanspeter Schaub
Glenn L. Murphy Chair of Engineering, Professor
So now, if we are going to talk Asymptotic Stability, it's very easy with these tubes
because it means you still get to pick an Epsilon,
but in fact, now with Asymptotic Stability that Epsilon goes to zero.
That means there's no final tube, that tube is gonna shrink infinitesimally.
And whatever bound you come up with,
at some point that tube is gonna shrink less than that.
So it converges, it may not reach zero but it's asymptotically converging.
You can never find a finite shape and then it never gets less than that.
So, that's what asymptotic stability means.
And so, as T goes to infinity your state now becomes your reference.
And of course, from a control perspective, this is what we love.
We love to have convergence always.
Don't always get it, but that's what we're looking for.
But with a nonlinear system, you still have to argue about local and global stability.
So, there's a neighborhood for which this is true,
and we can do that and we've talked about this pendulum problem for example, right?
If this thing is oscillating,
and I'm only oscillating within 10 degrees with some damping,
it will actually converge to a steady state position.
But it's not global because some joker could put it up here perfectly balanced
and it would not converge all of the sudden, right?
So, convergence is not global.
The stability... So, let's argue this other one.
On this example, is stability global, Lyapunov stability.
Supply that, Martha.
If we, our initial condition is less than 180, then?
Or if it is 180, can you come up with a bound within which the attitude air will remain?
If somebody gives you this condition, what's your bound then?
Any small perturbation it will go back to it.
I know but I don't have a small perturbation.
I have-- So, this is the perturbation, yeah?
This is where we wanna be.
I've given you a 180 degree.
Theta is 180 degrees.
Theta dot is zero precisely.
So, what is the bound that we have to pick targets to go with.
I'm confused.
Ok, sure.
With that case, why isn't it just an equilibrium but not stabilized, it's stable?
It's not stable.
All right.
We're looking at the stability of this point, but I'm allowing very large deflections,
and with nonlinear systems you might have multiple equilibrias in the system.
So, you have to account for, well,
if you're saying it converges back for any set of states,
you have to include also this state, right?
So, we are talking about stability of this one right now,
we're just throwing in a 180 degree perturbation with zero rate.
What do you think [inaudible]?
I have a question, is global stability only possible if there's one equilibrium?
Oh, that's a good question.
The question is, is global stability only possible if there's one equilibrium?
I would lean towards yes, but live right now, teaching, I wouldn't bet my life on it
that some smarter mathematician hasn't come up with a weird degenerate case
where something could asymptotically go crazy and never actually reach that equilibrium.
But it's probably a good sign.
With reference trajectory tracking though,
it changes a lot of that because you're creating artificial equilibria, right?
I wanna follow this path and things get more complicated.
So, there'll be some rigorous tools that we will have to argue it.
But we said, ok, so, we know this pen cannot be asymptotically globally stable
'cause I found a set of states that don't make it converge.
But, is it still stable, right?
For stable it means, can you pick an Epsilon and a Delta,
such that once it enters that condition, it stays within that set of conditions.
What would your-- Daniel, David.
Basically, yeah.
So, if you just make your bound 180, which you can, you're gonna stay in that bound.
Of course, that attitude's almost cheating because we can't get worse than 180,
but some people go to 200 and higher.
But that's right, right?
With 180 plus with zero rates.
That's gonna be the kind of, there's a full state spacing.
That's gonna be the bound that you're going to have.
So, you could say that this system is bounded.
You're never gonna have-- With friction,
you're never gonna have a set of initial conditions where the attitude angle keeps--
Even if you went beyond 180.
So, let's forget that it's, you know, the same thing.
360 is same thing as zero.
There is no way the system would, you know,
asymptotically spin out of control because with damping you would always be losing energy,
and it eventually it would settle there and there would be a bound that you can have.
If you do the clipping of multi revolutions,
it's 180, otherwise you can come up with something slightly different.
But that's an example quickly that hopefully starts to illustrate what we're doing, right?
Our controls.
We will design them to be asymptotic.
But then once we throw in other issues, you will see where it comes up.
So, asymptotic stability, the Epsilon now can go to zero.
We converge.
Andrew.
You know...
Tough demo.
I've had two hours of sleep last night, sorry.
So, just the pendulum problem.
It's that what you're talking about.
It's both.
You said stable as both Lagrange and Lyaponuv stable.
Not asymptotically but.
Yes.
Even though it's for all Epsilon, but that Epsilon would-- You wouldn't say Lagrange
and Lyapunov stable typically, cause if it's Lyapunov stable, it's a stronger argument.
Yeah, yeah.
That already overrides Lagrange stability.
But it's not--the key is, it's not just locally stable,
it's actually globally as Lyapunov stable.
But it's not globally asymptotically stable
because I could give it initial conditions where if you get that error,
you're stuck, right?
And this happens a lot then in stuff.
And what if you happen to be just exactly upside down?
You're gonna stay there.
And if you're off a little bit, you might recover, but very often what happens then,
kind of like with the separate tricks,
it might take a really long time to get there as well.
