0:00

So last time we saw that we could actually test for type1 zeno and the way

Â we could test this was to see whether or not the gradient dg, dx transpose points

Â in the same direction as f2, in the opposite direction of f1.

Â So if I have that, then I do have type 1 zeno.

Â And we came up with a little probe here to check whether or not, this is indeed

Â happening. So, the first thing we do is we check

Â LF1G negative, and LF. 2 G positive.

Â Where LFG was this thing we called the lead derivative.

Â Which was simply just fancy speak for DG, DX times, in this case, F1 of X.

Â That's all it means. So this is the probe that checks whether

Â or not sliding occurs. Today, I want to talk about what's called

Â regularizations, which is. Code for okay, let's say we have that, we

Â have sliding, how do we compute how we slide? And it turns out that this is

Â going to be very important when we move on to robotics because this occurs quite

Â a bit when you actually start running robotic navigation systems.

Â So today's lecture is all about How to be actually slide along this surface.Well,

Â lets actually try to be a little careful in [UNKNOWN].

Â So if I'm sliding here, what is characterizing that motion well, i know

Â that g(x)=0 which means that if i take the time, the relative of g, we respect

Â to t Then it should be 0 because g is not going to change when I'm sliding because

Â I'm staying at g=0. So one thing we need to do is say that

Â dg/dt=0. Well, let's see where that takes us,

Â dg/dt It is equal to dgdx*xdot. So now what is x dot along here.

Â Well, what I am going to postulate is that it's a combination of f2 and f1.

Â In fact let's say that is equal to sigma1*f1+sigma2+f2.

Â So it's a convex combination of, of these two Vector fields.

Â So if I do that, I get instead of x dots, I get this thing here sigma 1 f1 plus

Â sigma 2 f2. Well, since these are scalars, I can

Â actually pull them out here, which means that I can write this expression that's

Â is a little bit messy looking like sigma 1 times the lead derivative of g along f1

Â plus sigma 2 times the lead derivative of g along f2.

Â So that's what the time derivative of g actually is.

Â But I want that to be equal to 0. This needs to be equal to 0 because that

Â was our condition. So if I put this thing equal 0 then I can

Â solve for instance for sigma 2 and I get. This expression right here.

Â So I know how Ïƒ2 is going to depend on Ïƒ1.

Â Well that's a good start. We also know that both Ïƒ need to be

Â positive because I'm not allowed to always start flowing backwards and they

Â also should sum up to 1 because otherwise they can go super fast along this

Â direction which I typically don't want. I want it to respect the dynamics.

Â So. I have additional constraint too.

Â So the sigma needs to be positive and they need to sum up to 1.

Â Well. let's see how we can actually solve it.

Â So, let's go back to our old friend, the example.

Â And again this example. Looks like this, where, you know what?

Â I'm sliding down with slope -1. Here's x and here's time.

Â And when I hit the switching surface, I stay with x=0 for the duration.

Â Okay, what is the switching surface, first of all? Well, it's x=0, so g(x) is

Â simply x=0. Okay,

Â let's compute some of these Lee derivatives.

Â So, lf1g. It's, it's the derivative of g, with

Â respect to x. So, the derivative of that with respect

Â to x, is simply 1. F1 is simply negative 1.

Â So, the Lee derivative is negative 1. Which is 1 of the things we needed for

Â zeno. We needed, or for sliding.

Â We needed this to be negative for, type 1 zeno to occur.

Â That's the first. Well let's do the same for f 2.

Â Sorry this is an f 2 right here, apologize about that.

Â l f 2 g well it's a dgdx which equals the derivative of that with respect to x

Â which is 1, and f 2 which is plus 1. Right, so I get 1*1, which is equal to 1,

Â which is positive, which was the other condition for having Type 1 Zeno.

Â So we know that we actually have Type 1 Zeno, or sliding, here.

Â okay. So now, let's try figuring out what what

Â the The sigmas should actually be. Well, we had a formula for sigma 2..

Â It is equal to -sigma1 and these two Lee derivatives divided by eachother.

Â Well, this was -1 and this was +1 so sigma 2 is simple equal to sigma 1.

Â Rather simple. [INAUDIBLE].

Â But what, what are they? Well, recall that the need to sum up to 1.

Â Which means that they both have to be a half.

Â What this actually means, then, is that we can compute what the induced mode,

Â sliding mode, actually is. Well, I take sigma 1 * f1, which is half

Â of f1. And then i take sigma2 times F2 which is

Â half of F2, well F1 is negative one and F2 is plus one so i get -0.5 + 0.5 and

Â that's equal to 0. So I know that my induced mode, in this

Â case, is x dot = 0, which is exactly what we wanted.

Â because remember this picture that we've drawn over and over again.

Â We want to start here, get down there, and then keep staying at 0.

Â And the math turns out to work out, in this case.

Â Now, let's find the induced mode in general.

Â I know that the general formula for sigma2 and I, I want to point out that

Â this is not a formula any one in their right mind should memorize, but we should

Â know where it comes from and be able to use it when we need to.

Â But, here is the general formula for the, for the induced mode.

Â Well, we also have that sigma 1 plus sigma 2, is equals to 1, which means that

Â I can take sigma 2 here, and plug it in, because this is what sigma 2 is.

Â So now I have an expression on sigma 1 so if I solve that, I acutally know what

Â sigma 1 is. I encourage you to go through the math

Â yourself, it's a little bit of a mouth full, but what its, what it tells me is

Â that I compute sigma 1, and I can compute sigma 2, just as well.

Â And, if I now sum all of this up together, I get this expression inside

Â the pink box there. And I recognize again, this is a little

Â bit of a mouthful, but what this tells me is Exactly what the induced mode is.

Â So I am going to put a sweet heart around this thing, not because it's particularly

Â pretty but because it's systematic and it tell us how to actually find the sliding

Â mode controller or the induced mode which means that we know Exactly, not only when

Â type 1 zeno occurs. But how to progress beyond it.

Â So, let's figure it out. If I have, this is my hybrid system.

Â And, if this is a type 1 zeno hybrid automaton.

Â How do I regularize it? What do I do to add this extra sliding mode? Well, this

Â is what I do. Now, let's parse this, even though it

Â looks a little bit like a mouthful, let's figure out what it means.

Â So let's say I'm here, g is positive. G is positive and all is, all is well.

Â Then, if g becomes 0 and the sliding condition is satisfied.

Â Then I move in to what we just computed here which is the induce mode.

Â And again it's a little bit of a mouth full but we will see later on that in

Â robotics we have no choice but to actually use this.

Â So then I'm going to use this mode until if g becomes positive I jump back to f1.

Â If g becomes negative I jump to f2. And of course we don't always have Type 1

Â Zeno some switches are nice if I simply end up g negative, I jump directly from

Â mode 1 to mode 2 and vice versa. So, this is how you take a hybrid system

Â and make it immune to the Nauseous and bad effects of Type 1 Zeno.

Â And the nice thing is that this is completely general, and we don't ever

Â again have to worry about Zeno. Type 1 Zeno, Type 2 Zeno we already said

Â them about. Having said that this actually brings me

Â to the summary of this entire model and hybrid systems.

Â So, what do we have, we have models, we have very rich models which are the

Â hybrid [UNKNOWN] models. We have something what we call Stability

Â Awareness or just We're aware of the fact that, just because the submodules, or the

Â submodes are stable themselves doesn't mean that the hybrid system is stable.

Â And we need to be aware of it, test for it.

Â We also have seen zeno as another awkward hybridization that occurs.

Â Or an awkward phenomenon that can occur when you go hybrid.

Â we have 2 classes of zeno. 1, which is type 2, which is the bouncing

Â ball. Infinitely many switch, many switches in

Â finite, but not, not zero time. That's bad and scary, and we can't do

Â anything about it except, look out for it.

Â But then, we have type 1. And we now know, not only how to check

Â for it. But how to get around it using these

Â things that I call regularizations or the induced sliding mode and with that if I

Â had, you know a confetti, I would toss it up into the air because this ends the

Â massive part and the pre-robotics part really of this course.

Â So what we're going to do in the next module is go back to robotics.

Â Apply, all of our new and awesome tools, and see, how we can unleash them to

Â actually make mobile ro, robots do cool things in the real world.

Â Well done.

Â