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Now, we want to look at definiteness of functions.

Â Definiteness is basically a measure of something being positive.

Â If it's scalers, it's easy.

Â If it's three, then 3 times x squared,

Â some squared measure, always gives you a positive.

Â It's 0 when x is 0.

Â These definite functions should always be 0 at your reference.

Â And if it's posit definite, positive everywhere else.

Â So x squared is one we like a lot.

Â You will see today also with energy, it's all of these quadratic measures.

Â So it's just kind of a bowl.

Â That's what we're looking at, right?

Â 2:20

Is it none of those?

Â >> None of those.

Â >> So I think it was one of those.

Â I was like, why are you saying that.

Â Okay, that makes much more sense.

Â Okay, it's none of those, and there's a word for that.

Â It's called indefinite, right?

Â Sometimes you have functions like that that just are going to be this.

Â Or you look at in the homework there's multi-dimensional stuff.

Â And some students always argue, well,

Â this is positive definite if I only consider positive xs.

Â But if we [INAUDIBLE] the stability,

Â we can't guarantee that people will only bump it to one side, not the other, right?

Â Remember, you always have to be able to draw a ball,

Â a finite neighborhood around the thing that you're studying.

Â And you can't say, well, along this trajectory, we would always be positive

Â here, we'd be negative, we will just never to perturb to the negative side.

Â Life doesn't work that way, that's not a stability argument.

Â So this one would be indefinite, right?

Â And the theorems we have wouldn't apply.

Â It doesn't mean it's unstable,

Â it just means we cannot say if it is stable or not.

Â We'll see, okay, Jordan.

Â >> So, for the last lecture I was watching it,

Â and you basically drew a problem below zero and above zero and you said that for

Â the above you could shift your coordinates system.

Â Can you also do that for if it's below?

Â You can do it both ways?

Â >> Well, to some point. This one, you could never shift up or

Â down to make it always positive or negative, right?

Â if your V function has the bowl shape you're looking for, but it's

Â not zero where it needs to be, typically what you do is a coordinate change.

Â And then you can talk about the stability, not about zero meters, but

Â the stability about one meters.

Â And that's where this function actually happens.

Â So really without loss and generality,

Â we're always going to be talking about driving either delta xs and

Â the tracking problem or just xs and a regulation problem to 0.

Â because we've assumed we've got a coordinate shift such that if

Â there's an equilibrium right at the origin but somewhere away,

Â you've made a new coordinate system there and you're driving things back.

Â Yep, so that can happen, so no good.

Â When you're looking at these functions and homeworks and trying to figure out what's

Â happening here, is it local, is it global, plot them out.

Â Go to math lab, go to mathematical something.

Â Plot these functions and you will quickly see visually, too, wait a minute.

Â Just think about it,

Â is definite about this point, so where is going to go to zero.

Â And what's happening there and

Â is there a finite region where it's guaranteed always positive.

Â That's something you have to consider, so

Â plots are actually very, very helpful in this stuff.

Â Especially when you only have one or two coordinates.

Â The F15, well, that's not a problem.

Â You have to look at the mathematics of it.

Â Okay, so these are definiteness, definition that we have,

Â basically making things positive.

Â There was also things about if you have a matrix, if you got a 3 by 3 matrix

Â if it's positive definite then this is the matrix version of it,

Â this will always be 0 unless x is all zeros right?

Â That's the origin.

Â If x is all zeroes, this is [INAUDIBLE] zero.

Â But anything else, this scalar answer will give you something positive.

Â And this works.

Â We've seen such functions.

Â 5:22

One half, let's see.

Â That's omega transposed I omega.

Â So a condition for a matrix to be, much as a function, but

Â a matrix to be positive definite.

Â You can go look up on Wiki, there's different definitions.

Â But a very popular one is the eigenvalues have to be all positive real numbers.

Â And for inertia tensor your principal axis are always positive and real, so

Â the inertia tensors are actually symmetric positive definite matrix that we have.

Â And kinetic energy makes sense, kinetic energy is either zero if you have no

Â rates or it's positive, we never have kinetic energy all of a sudden.

Â So that would be a positive definite function, and an example of that.

Â 6:22

V dot had to be what?

Â >> Negative semi-definite.

Â >> Negative semi-definite which we write like this typically.

Â If I say greater than zero, it is always understood at x equal to zero it is zero.

Â It has to be, it's away from that.

Â What was the other one that we needed?

Â >> [INAUDIBLE] >> At least one's differentiable, right?

Â So, there's a smoothness condition in there.

Â >> And continuous derivative, right?

Â >> Yes, exactly, continuous derivatives is the exact wording that we have in there.

Â So continuous, dts, derivatives, shorthand for that, right?

Â That was the condition, and the beauty was, now, all those stability definitions

Â that we have with balls and neighborhoods and cones and torusi, whatever you have,

Â we actually have to solve differential equations and prove these properties.

Â Which is very hard to do with non-linear systems.

Â The theory basically boils down to if you can come up with one of these functions.

Â That proves these three properties around your state of interest.

Â Which could be an equilibrium or a tracking problem, a reference motion.

Â Then this system is stable, which means the output on stable, all right?

Â Doesn't guarantee convergence, but it is the output on stable.

Â