0:51

What's the dimension of my state vector x here?

Â Lewis? Two, why two?

Â Talk me through that.

Â >> Make it two-by-one where you can have a system of first order.

Â >> Yes, so you have maybe x is equal to theta and theta dot, right?

Â This is a scalar equation but it's a second order differential equation.

Â So you're going to need two initial conditions really.

Â That's another way to look at it.

Â What's the dimension or state vector?

Â It has to be the same as the number of initial conditions.

Â because initial conditions should define this is a fully deterministic problem.

Â Once you have all initial conditions needed,

Â you should be able to fully propagate.

Â So, here, you need two so the state vector has to be of dimension two.

Â Let's say we have this though.

Â This is you classic orbit one, your unpatterebed orbit two problem.

Â Here, what's the dimension of the state vector that we had?

Â Andre, what do you think?

Â >> Six.

Â >> Six right, because here, in MATLAB, this in make vector form right,

Â you have your r and then you need your r dot.

Â 2:17

>> 12?

Â >> How did you get to 12?

Â >> I just caught the end of your question.

Â >> [LAUGH] So, if you want to write your attitude problem in this form, right?

Â That's if you're doing integration, which you're doing also in this last problem.

Â You're doing in this next problem, especially now,

Â where you're getting the full six stuff.

Â So, how many estates do you need in x if you have the complete attitude motion?

Â >> You have the attitude, you need three degrees of [INAUDIBLE] for the attitude,

Â so- >> Okay.

Â >> And then don't you need three for the position?

Â >> Well, we're not doing translation.

Â >> Okay.

Â >> We're just doing rotation.

Â 2:58

All the code you're writing right now really is just rotation.

Â If you're adding translation, you're doing extra work.

Â You don't get credit for that.

Â >> So for the attitude alone, wouldn't you need six?

Â >> Six then, right?

Â because you need positions and rates but the positions are easy because it's r and

Â r dot or x and x dot, whatever you called that position vector and its derivative.

Â That's typically how we write it.

Â It's kind of easy, it's simple.

Â The attitude, though, we use different coordinates.

Â We use different attitude coordinates specific for

Â the orientation description and then we don't use yaw, pitch, roll rates.

Â We use actually omegas.

Â That's always angular velocity measure, so

Â that's a six-dimensional vector that you'd have.

Â Now, is it always going to be six dimensional?

Â [SOUND] What else?

Â How would you increase this dimension?

Â >> Increase the dimension?

Â >> Yeah, how would you have more?

Â >> If we interested in higher dimension?

Â >> No, 3D attitude.

Â 4:00

>> Well, yes, if your differential equations need it,

Â you don't just have omega dot that you're solving or omega double dot.

Â Then, you might have to go to higher order and

Â you're looking at those kinds of responses.

Â But what's the easy one?

Â >> If you're using something like Oiler parameters?

Â >> If you're using quartarians, all of a sudden, we moved to four.

Â Did it make my problem now, instead of six degree or a three-degree freedom problem,

Â second order differential equation times two is six.

Â Now, we have seven states.

Â >> [COUGH] >> What must be happening, right?

Â because we didn't just increase the degrees of freedom.

Â >> Because of constraints.

Â >> You have to have constraints, right?

Â And that means in your code after you integrate from one step to another,

Â you better normalize those quartanians again because otherwise,

Â little numerical errors creep in.

Â If you're using a DCM, you're adding nine coordinates and

Â you'd have re orthogonise a DCM every time, which you could do.

Â And if you use MRP, we can avoid.

Â There is no constraints, but

Â we have to deal with the switching just to avoid singularities.

Â So there is still an if statement, right?

Â So good, but that's kind of a quick run down.

Â This is what you should be doing.

Â Now, let's talk about neighborhoods.

Â What is a neighborhood?

Â >> Is it Daniel?

Â No, David.

Â >> David, in what context?

Â A neighborhood, it's like a region around an equilibrium point where

Â stability can be proven or shown.

Â >> Okay, so this is typically what we are writing a lot.

Â What do I mean by this?

Â Does this xr have to be an equilibrium?

Â >> No.

Â >> We also be writing it around references, right?

Â So the reference problem is typically the distraction problem.

Â If it's in equilibrium, it's more like a regulation problem,

Â just drive everything into a steady state, right?

Â Whatever that orientation is good.

Â And then there's a neighborhood.

Â How do we define these neighborhoods typically?

Â 6:10

coordinates could be different sizes, couldn't they?

Â So- >> No, it's one delta.

Â >> Really, okay so- >> So,

Â however you define this system if you're mixing positions and angles,

Â it's going to give you weird norms and that's part of the issue with this stuff.

Â Sometimes, you can rescale it and prove the stability for a bigger neighborhood.

Â And we talked about this too like if we look at the doffing

Â equation, there was an equilibrium here and an equilibrium here.

Â These were unstable.

Â This one was stable, the phase based plots looked at this.

Â But, if you fit the nearest sphere inside,

Â this is the region that we can prove stability.

Â Here, it's actually stable as well.

Â It's just you may not get out of this result.

Â So as we're doing stability arguments, we often have hey if this is the case,

Â then it's stable.

Â 7:21

could be awake if the lights are not on but if they're on, you're awake for sure.

Â If and only if means you're only awake if the lights are on.

Â If you're awake, lights must be on, right?

Â Now, you can go both ways with those arguments right,

Â and that's a thing we use a lot in control.

Â So we'll be looking for that a little bit.

Â This coming up in today's lecture as well.

Â That's an if statement, or an if and only if part is mathematical proof.

Â So we'll be covering that.

Â So neighbourhoods, we have this L2 normsthat we're dealing with, good.

Â Then we have different types of stability that we discussed.

Â What was the simplest type of stability?

Â >> Lagrange.

Â >> Lagrange, what does Lagrange mean in basic words?

Â >> Bounded input, bounded output?

Â >> Yeah, essentially, it's a bounded response.

Â That means there is a neighborhood delta somewhere.

Â This is your x r, such that at some point,

Â you enter this tube, and that's typically set as t naught, just out of convenience.

Â At sometime, you've entered it, and

Â then you remain within this boundedness forever, right?

Â That's great.

Â Now, the key question though is does this neighborhood delta,

Â this bound that you have, is it a function of the initial conditions?

Â 8:39

Andrew?

Â >> No, it's not. >> No, is not,

Â that's the key thing, right?

Â And the example we had was, think of a spring mass system,

Â a spring master amp system.

Â Just floating in space, it would oscillate and settle down as zero equilibrium.

Â So now, you're studying this equilibrium subject to a disturbance and

Â you're treating gravity as a disturbance, it's going to settle to this deflection.

Â But it settles to the same deflection no matter how big you bumped it.

Â It's always going to come to the same one.

Â So it doesn't matter on initial conditions, right?

Â It's just a quick visual way to think of this.

Â What's the next stronger level of stability?

Â We have Lagrange, Downtanis, then we have the Lyapunov, good.

Â So Matt, talk me through the Lyapunov now.

Â All right, we have some initial delta, [SOUND].

Â >> So for the Lyapunov, for any initial delta, there's some final epsilon that if

Â you start in the b sub delta, you will end up and in the b selector.

Â >> And stay within it, right?

Â We talked about the separatrix motion.

Â The astronaut spinning that key.

Â It kind of stable for

Â short period but then it flipped around then it will stable here.

Â And then it flips again and it continues this.

Â So that would never fit the stability requirements because it might be there for

Â a short period of time.

Â But at some point, it leaves again and then it comes back.

Â And that's a whole different kind of a thing, okay, CK.

Â >> I had a question on that because when it's flipping around,

Â it's sort of flipping within these two like defined equilibria.

Â So could you bound around that and call that-

Â >> You could call it bounded.

Â You could come up with boundedness arguments for that one, and say that for

Â the system, the rates aren't spinning up like crazy.

Â Something could be unstable like if you look at Europe effects,

Â people keep studying on asteroids and debris.

Â There, the spin's going to get bigger, and bigger, and bigger, and

Â bigger, and bigger, and the cements are going to break apart.

Â This spinning thing is not going to do that.

Â It is bounded in its response.

Â And you could argue some types of Lagrange stability around it,

Â that I know it's not going to 6 billion RPMs all of a sudden, right?

Â Exactly, but now, what's the analog we use here for the Lyapunov stability?

Â because we say, we can pick any epsilon, any epsilon, really, really small.

Â Now, the corresponding delta might be really, really small too, but

Â you can find it, right?

Â What kind of a mechanical system can you pick?

Â I want to be oscillating within one degree, 0.1 degree,

Â 1 arc second and all this find an initial condition that puts you there.

Â >> [INAUDIBLE] >> No temper, you're close.

Â It's just a spring mass system.

Â If you look at a classic spring mass system, you can deflect it, right?

Â And the stability, we are always talking about protobation.

Â You can't go while otherwise, you don't deflect it.

Â You can do anything but nothing.

Â [LAUGH] Right?

Â You can't do nothing, you have to do something.

Â And so you bump it infinitesimally and whatever infinitesimal bump you gave it,

Â it's just going to wiggle and there's no damping, right?

Â It's just going to wiggle.

Â So that's why we can never set epsilon here to zero.

Â 12:05

So stability is good.

Â That get's you there and

Â we can get some now more control over how much deflection we'll have at the end.

Â Boundedness, you don't have such control.

Â What's the next level of stability,

Â the stronger stability beyond the Lyapunov stable?

Â >> Asymptotic.

Â >> Asymptotic, okay?

Â Now, asymptotic basically means epsilon is going to go to zero.

Â That will be a spring mass damper system essentially, right?

Â Everything is going to converge to zero.

Â But here's also a challenge.

Â If you use two linear systems, if things are stable, not marginally stable, stable,

Â all the roots on the left-hand side, you have an exponentially decaying response,

Â in all those different modes in a linear system.

Â That means with the next exponential decay, and if we do this on a log scale

Â 12:56

This gives you a performance guarantee.

Â I can come up with a half life and say, okay, in 30 seconds,

Â I want my errors to to decay by half.

Â Another 30 seconds, they're another half.

Â Another 30 seconds, they're another half, right?

Â That's great.

Â Every linear system acts that way.

Â Non-linear systems, that's not true.

Â Asymptotic stability just means this error will do something, and eventually,

Â it's going to go to zero, if that's your reference that you have, right?

Â It gives no guarantee on performance and that's a big tricky thing.

Â So non-linear control papers,

Â you often see people arguing forever about stability.

Â And they got it's a wonderful, great, crazy, control and it's stable.

Â Wonderful but is it worth anything?

Â Because this control may take 6 million years to converge.

Â You've proven it converges after much effort, but are you that patient, right?

Â And you can come up with weird nonlinear systems that you guaranteed

Â will get there.

Â But man, the convergence rate is just atrocious, right?

Â So there's actually a extra level of stability, people argue.

Â Asymptotic stability, we'll show you today, and we'll use in class.

Â There's an extra even stronger argument in non-linear control called exponential

Â stability.

Â And that basically means, if you look in the book, you'll find the definitions,

Â that you can upper and

Â lower bound the response by an exponentially decaying function.

Â So whatever the craziness happens in-between may not follow exactly a linear

Â response, but it's decaying fast enough that I can bound it by an exponential.

Â So, that gives you a performance guarantee that hey, with this, I know it will be at

Â least in 30 seconds, it's at least half as big the error, maybe even more.

Â We're not going to deal much with exponential stability in this class, but

Â if you're curious, I just wanted to highlight that.

Â So you can see, different levels of strength and as we go from Lagrange,

Â to stable, to asymptotically stable, to exponentially stable,

Â there's always more and more and more things you have to argue.

Â Whereas linear systems, once it's stable, you're done.

Â Linear systems, are they locally stable or globally stable?

Â 15:03

You guys are way too shy.

Â >> Global.

Â >> Global, thank you, yes.

Â It's good. Spring mass system, the math,

Â nothing says you can only defect 1 meter, 1.5, you're really pushing it.

Â Come on, what are you thinking?

Â Nothing in there about that.

Â You can throw in any number you wish.

Â And it'll oscillate.

Â Whereas non-linear systems and

Â we've seen that with equation highlighted there is actually multiple equilibrium.

Â Some of them are stable, some of them aren't.

Â Your arguments may only be good locally.

Â We've also talked about this planear pendulum.

Â This is one that you look at the homework as well, right?

Â And what you're going to find is this system is actually stable.

Â It's globally stable.

Â So even in this one, I can come up with a bound as CK was talking about, and

Â saying hey, if you're good within 180 degrees, this system is stable.

Â