Now, we want to look at definiteness of functions. Definiteness is basically a measure of something being positive. If it's scalers, it's easy. If it's three, then 3 times x squared, some squared measure, always gives you a positive. It's 0 when x is 0. These definite functions should always be 0 at your reference. And if it's posit definite, positive everywhere else. So x squared is one we like a lot. You will see today also with energy, it's all of these quadratic measures. So it's just kind of a bowl. That's what we're looking at, right? What if it's semi-definite? What does that mean? Is it Tosh? What does positive 70 definite mean? Okay, back row? >> That, actually, I'm not completely sure. >> Kevin? Right, what's semi-definite mean? >> B through 0 7 times. >> Yeah, so Ian, we're going to be using these terms today so please if you haven't watched the other lectures, catch up. because otherwise none of this is going to make sense. You need to re-review these definitions every time you get to this classes, every time before you watch these lectures. I'm installing it, it won't sink in otherwise. We'll have all this arguments. I'll keep asking questions, as this stable log runs why? Once the basic concepts come in, as we're doing it, that's all you need. You'll get all the insights you want. So, semi-definite means you touch zero, but you never go negative. The opposite of positive definite is negative definite and in fact, sometimes we need that. That could be a function that's kind of like this. It's 0 when you add 0, and negative everywhere else. Negative semi-definite means you touch it occasionally, so let me throw in a different function. Let me throw in x cubed. Is x cubed positive definite, positive semi-definite, negative definite, negative semi-definite? Trevor, you're shaking your hands. >> It's one of those. >> It's one of those, I agree. It's definitely one of those. That definitely narrows it down. [INAUDIBLE] those want. Is it none of those? >> None of those. >> So I think it was one of those. I was like, why are you saying that. Okay, that makes much more sense. Okay, it's none of those, and there's a word for that. It's called indefinite, right? Sometimes you have functions like that that just are going to be this. Or you look at in the homework there's multi-dimensional stuff. And some students always argue, well, this is positive definite if I only consider positive xs. But if we [INAUDIBLE] the stability, we can't guarantee that people will only bump it to one side, not the other, right? Remember, you always have to be able to draw a ball, a finite neighborhood around the thing that you're studying. And you can't say, well, along this trajectory, we would always be positive here, we'd be negative, we will just never to perturb to the negative side. Life doesn't work that way, that's not a stability argument. So this one would be indefinite, right? And the theorems we have wouldn't apply. It doesn't mean it's unstable, it just means we cannot say if it is stable or not. We'll see, okay, Jordan. >> So, for the last lecture I was watching it, and you basically drew a problem below zero and above zero and you said that for the above you could shift your coordinates system. Can you also do that for if it's below? You can do it both ways? >> Well, to some point. This one, you could never shift up or down to make it always positive or negative, right? if your V function has the bowl shape you're looking for, but it's not zero where it needs to be, typically what you do is a coordinate change. And then you can talk about the stability, not about zero meters, but the stability about one meters. And that's where this function actually happens. So really without loss and generality, we're always going to be talking about driving either delta xs and the tracking problem or just xs and a regulation problem to 0. because we've assumed we've got a coordinate shift such that if there's an equilibrium right at the origin but somewhere away, you've made a new coordinate system there and you're driving things back. Yep, so that can happen, so no good. When you're looking at these functions and homeworks and trying to figure out what's happening here, is it local, is it global, plot them out. Go to math lab, go to mathematical something. Plot these functions and you will quickly see visually, too, wait a minute. Just think about it, is definite about this point, so where is going to go to zero. And what's happening there and is there a finite region where it's guaranteed always positive. That's something you have to consider, so plots are actually very, very helpful in this stuff. Especially when you only have one or two coordinates. The F15, well, that's not a problem. You have to look at the mathematics of it. Okay, so these are definiteness, definition that we have, basically making things positive. There was also things about if you have a matrix, if you got a 3 by 3 matrix if it's positive definite then this is the matrix version of it, this will always be 0 unless x is all zeros right? That's the origin. If x is all zeroes, this is [INAUDIBLE] zero. But anything else, this scalar answer will give you something positive. And this works. We've seen such functions. One half, let's see. That's omega transposed I omega. So a condition for a matrix to be, much as a function, but a matrix to be positive definite. You can go look up on Wiki, there's different definitions. But a very popular one is the eigenvalues have to be all positive real numbers. And for inertia tensor your principal axis are always positive and real, so the inertia tensors are actually symmetric positive definite matrix that we have. And kinetic energy makes sense, kinetic energy is either zero if you have no rates or it's positive, we never have kinetic energy all of a sudden. So that would be a positive definite function, and an example of that. What does it mean to be a function now? That's kind of where we ended up with the lectures. >> The derivative was negative definite? >> Okay, there was three conditions, you're jumping to number three. >> Well, the function itself is positive. >> You need V of X to be positive definite. V dot had to be what? >> Negative semi-definite. >> Negative semi-definite which we write like this typically. If I say greater than zero, it is always understood at x equal to zero it is zero. It has to be, it's away from that. What was the other one that we needed? >> [INAUDIBLE] >> At least one's differentiable, right? So, there's a smoothness condition in there. >> And continuous derivative, right? >> Yes, exactly, continuous derivatives is the exact wording that we have in there. So continuous, dts, derivatives, shorthand for that, right? That was the condition, and the beauty was, now, all those stability definitions that we have with balls and neighborhoods and cones and torusi, whatever you have, we actually have to solve differential equations and prove these properties. Which is very hard to do with non-linear systems. The theory basically boils down to if you can come up with one of these functions. That proves these three properties around your state of interest. Which could be an equilibrium or a tracking problem, a reference motion. Then this system is stable, which means the output on stable, all right? Doesn't guarantee convergence, but it is the output on stable.
