0:05

Now Lyapunov's Direct Method, why do we use this?

Â [LAUGH] To apply these definitions directly,

Â you really have to solve these equations of motion.

Â In a very general form, for any initial conditions, so you come up with these

Â bounds, and many equations of motion are not analytically solvable.

Â So you could use numerical stuff but then how do you prove that for

Â any inital condition these wiggles are always going to stay within that bound and

Â they don't eventually leave again like they do in the.

Â It would require infinite integration time to do it completely rigorously.

Â So this method is going to save you tons of headache and issues.

Â 0:42

What we're doing now is, this is going to be an energy-based method.

Â We're going to write these energy like functions.

Â And energy's nice it's a scalar.

Â If you're not moving, it's zero.

Â And that's typically our equilibrium.

Â If I am moving, I'm looking at the partial motions.

Â So I can look at it in an energy kind of a sense, I might always losing energy.

Â And if I'm losing energy, where do I settle?

Â To what minimum energy state do I settle?

Â And it works for linear systems, but in particular, it works also for

Â very non-linear systems as well.

Â So this becomes a huge building block for many non-linear control theories.

Â The question is always what are these functions?

Â Now, more definitions, you staying awake?

Â I know, have a little sleep too, more definitions.

Â So let's look at these, because these are things that are going to,

Â everyone of these lectures are going to be dense with this stuff.

Â We're going to start using it, after this lecture, please go back before Thursday,

Â go over these definitions again.

Â Come back Thursday, bring your review as we always do, we will be asking you,

Â what is Lagrange stability, what is the Oppenhoff stability,

Â what is other one, which ones depend on initial states, which ones don't?

Â Practice beforehand, it's going to sink in,

Â you're going to fly through this material much better.

Â So, here's a definition; positive definite functions.

Â And the negative definite is almost the same except for one little bracketed term,

Â that's why I'm just, instead of having two statements, I'm making it one statement.

Â But if it's negative definite, then instead of being locally positive,

Â you replace positive with negative and you're fine.

Â And this argument instead of v being bigger than 0,

Â it has to be v negative than 0.

Â So I've got two definitions in one.

Â But essentially what positive definite function means is you're locally positive

Â definite about xr if as the reference we have to be 0.

Â So you think of energy, if this is my equilibrium, everything's at rest,

Â my energy would be 0.

Â If I'm up here, actually it has potential energy.

Â So that would be non zero right?

Â I have a moving, I have kinetic energy that is non zero.

Â It doesn't literally have to be energy but

Â energy is a nice analog in this mathematics.

Â So at our desired state, x is equal to xr.

Â We're tracking our path, I need the function to be 0.

Â 4:26

Matt, what do you think?

Â >> Yes.

Â >> So it's 0 where it has to be 0.

Â I missed it by a smidgen, but let's pretend I hit it, right?

Â So up to what point is this, what's the neighborhood?

Â >> That's other place where 0 is noninclusive.

Â >> Noninclusive perfectly, right?

Â That was my next question, then.

Â So here, it's up to these points inside of that, at the boundary you're 0 again and

Â that require you to be bigger than zero not zero anywhere else, good.

Â 6:15

Well, essentially, at this point here, this is there.

Â But this reference instead of calling this zero,

Â you can always do a change of coordinates that makes that your reference, right?

Â If you have a function right here, here my energy is a zero.

Â But everything works well from that.

Â I can always redefine your height to be one and

Â a half meters plus whatever you do.

Â So a shift in coordinate systems is often done, and

Â that gives you the properties you're looking for.

Â And then at the end,

Â you unshift it again to get the overall arguments you're doing, right?

Â So you could make it but as it is, you're absolutely correct.

Â It doesn't satisfy that first criteria that it's zero.

Â But you're onto something because then it grows everywhere else.

Â So let's see, what other colors do we have?

Â Purple, there we go.

Â Let's say I have this function,

Â 7:18

Andre.

Â >> I would say for the right half of it, it is.

Â >> Okay, good, I'm glad you said that.

Â Completely wrong, but I'm glad you said it because many students do this.

Â I mean researchers do this.

Â If you go back to the definition, we talked about balls.

Â So around that reference, so what is the circle that you

Â can draw around the origin that guarantees every point in that circle.

Â 7:52

And you can't do that, right?

Â So there some homeworks you're getting into.

Â And it hey, for these things argue, is locally pause a definite, negative

Â definite, globally, all these kind of stuff, you have to think about those.

Â And some students always go well x cubed,

Â well, it's positive definitely if I only consider right handed disturbances.

Â But all over a sudden, if you bump it negative, then you're unstable, right?

Â So any perturbations around them,

Â you can't just say look I'm only going to fall to the left, never to the right.

Â So therefore, the system would be unstable as a whole.

Â But you have regained some extra insights with this kind of function.

Â You go actually maybe if I do fall to the right, it might recover,

Â maybe I can take advantage of that and other stuff.

Â But as a system, it is unstable.

Â because you can't draw a ball within which this function is there.

Â And you can't just say only consider disturbances along this plane of

Â my state space, right?

Â You have to be able to draw a continuous sphere,

Â that's how we're defining these things, okay?

Â So lots of wiggles.

Â But pause it definite is basically a function.

Â It's like being strictly positive away from zero.

Â And that's what we're looking at.

Â Yes.

Â >> A cubic function like that, could you draw a reference shift like you did for

Â the yellow one?

Â >> How would I do that?

Â >> And shift your origin down so that most of your cubic is above zero?

Â 9:13

>> But you have to be zero, at zero, right?

Â And that would be the hard part.

Â Now, so you're saying to shift it down to here where everything is positive and

Â then this is just a neighborhood that you could come up within a ball.

Â In this case it has to be a symmetric x-axis neighborhood.

Â But no, but then you don't have this behavior being zero.

Â And that wouldn't work either.

Â Yes, Nathaniel.

Â >> Just out of curiosity, can you also do rotation?

Â So you could draw a diagonal line.

Â >> Uh-huh.

Â >> That would, your cubic would always be- >> Yeah, I can.

Â Because this axis, let me just label the axis.

Â This is my states phase.

Â I'm just using a single variable right now.

Â And then this is the function of that state space.

Â 9:58

So if you have a multidimensional, x1 and x2, there'll be a plane.

Â And yes, there might be rotations and stuff, but

Â then still you'd have to think of the balls.

Â And balls for 1D become just symmetric line segments,

Â 2D it's circles, 3D it is a ball, 4D it's hyperballs.

Â That's where it goes.

Â And so now, good, good question.

Â So work with this, this will, again,

Â with this class I'm kind of showing you the core things needed.

Â 10:31

Following on that, positive semi definite functions is almost the same definition.

Â The only thing added is instead of being greater than or equal to zero,

Â or greater than zero, it says greater than or equal to zero.

Â Again, we are ignoring this point in that argument.

Â So if we looked at these plots that we've already done,

Â like Matt was talking about, this blue one here,

Â it was everything up to where it crossed zero then, but only included that one.

Â If you're talking semi-definite,

Â you could include those points and it would still be good.

Â 11:19

This would be something that is now, for example, globally positive semi-definite.

Â I'm always positive or I'm zero.

Â And semi-definite is away from the zero part.

Â Yes, Matt?

Â >> Does the ball have to be [INAUDIBLE] so like the x cubed, for example?

Â Instead of the ball or it has to be definite?

Â >> No, it has to be finite, yeah,

Â because otherwise you're talking about the equilibrium.

Â You've just discovered that page.

Â Hey this is good if I have zero disturbances.

Â That is what the ball being zero would mean.

Â And for stability we have to look at finite epsilon disturbances.

Â Yep, it can be really, really small but it has to be finite.

Â Very good question.

Â Okay, good.

Â So you could see, there's things we could play, but

Â there's nice visual representations.

Â That's always positive, this could be positive or zero, right?

Â And they could be local or it could be global.

Â Then simple examples could be things like these where often have these kinds of

Â functions.

Â Where if this were mass, this would be like potential energy of a spring, k

Â times x squared over 2 would be potential energy, this could be kinetic energy.

Â So potential and kinetic energy, what happens there, that's an example?

Â So with more than just one, you can create these functions.

Â This function v is only 0 if x is 0 and x square is 0.

Â If either of them is perturbed, this function is non-zero, all right?

Â But it is positive.

Â So this would actually be a globally positive definite function.

Â This function also never goes to zero away from the origin.

Â It doesn't do that bend down and then go up again and stuff.

Â So it's not semi definite, it's actually globally positive definite in a sense.

Â So you can do it on functions.

Â Yes, Marta?

Â >> In all of these examples, xr is origin, like the equilibrium is origin.

Â What if it's not 0,0?

Â What if it's not- >> Then you do a coordinate change.

Â And xr would be, for example, assume we had the linearization we said.

Â We took x minus xr and then the new coordinates were delta u or

Â delta x actually in that case.

Â That is the departure motion about the reference.

Â If you have a tracking problem, that's why I always have these xrs,

Â that's my reference.

Â Reference could be an equilibrium and then you could define your frame to be

Â at that equilibrium so everything drives to zero.

Â So you can always do that.

Â But for the tracking problem, your reference keeps moving,

Â I want to be here then I have to be here.

Â So now, you define your state relative to this time varying things.

Â So, that is a delta x.

Â So, if you did tracking problem, for example, down here,

Â all you would have to do is replace x with delta x.

Â And now you have a positive definite function in terms of your tracking errors,

Â my tracking position errors and tracking rate errors.

Â If I had delta x squared and delta x dot squared.

Â 14:09

Now, great.

Â You guys are thinking ahead.

Â Now, there's a common way we might matrix stuff as well.

Â And you hear, you can go to open a Wiki and look and figure out what does it mean

Â for matrix k to be positive definite, but it basically boils down to this math here.

Â If you could prove that for any states x, times this matrix of numbers.

Â So you x transpose this times x.

Â It's like 3 times x squared.

Â For scalars we know, well, 3, there's a positive number times x squared, that

Â is actually positive definite function in the sense because it's 0 at 0 and

Â positive everywhere else times 3 doesn't change that result.

Â If I had minus 3 times x squared,

Â it would not be positive definite because it would flip the whole sign, right?

Â You can do that now, just for scalars, you can do it for matrices.

Â You can come up with conditions on these matrices such that x transposes matrix

Â times x.

Â It is going to be positive.

Â And we've seen one such matrix already, your inertia tenser.

Â One of the requirements for

Â this to be positive definite is your eigenvalues have to positive.

Â And for any real spacecraft like this, rigid spacecraft, you do your inertial

Â tensers, we know the principle, inertia is all going to be three real eigenvalues.

Â So inertial tenser will use it a bunch, is a positive definite matrix.

Â 15:29

If you can go to zero sometimes, that means you had a zero eigenvalue somewhere.

Â But it's still positive.

Â Semi-definite and if somewhere it goes negative or

Â negative definites, different combinations of the stuff, all right?

Â So, I'll let you look at that.

Â But what we'll use this form a lot.

Â Instead of general functions, we will create this, this way.

Â And we'll have our definitions.

Â