Last time we saw that not only we have to worry about Zeno. This zeno problem or the zeno phenomenon comes in two different flavors. And one thing that I stated was that type 1 zeno which is infinitely many switches in zero time or in one timing sense here represented by this system where. Again, if I have x here, and time here, and I start positive, then x is going to be -1 until x becomes, just, just negative. And then, we're going to get this infinitely many switches going on right there. that's type 1. Well, it turns out, that we can actually remedy this. And the reason for it is, you know what is very, very clear what should happen. The system shouldn't grind to a halt. It should just nicely keep continuing on like this Like x is equal to 0. So how do I take this intuitive notion of that x should just keep saying and be equal to 0 and make that mathematically sound. What, what's the topic of today's lecture, and this construction by which we can continue beyond the zeno point in the type 1 zeno system, using something that's known as. Sliding mode control. So, let's be a little bit general here. Let's say that I have one system, x dot is f1(x) and then I have a switching surface, g(x). And when g is negative, I switch to f two, and when it becomes positive I switch back to f1. So here it is, here's my switching surface, g(x) = 0, that's where the action is. Now on this side I'm going to be using f1, and da, da, da, da, da, da, da, when I hit this point, well let's say that f1 is pointing inwards. Well on this part of the world I am going to be using f2. Well, let's say that f2 here points outwards. This means that when I hit this point again, I grind to a halt. so this is really what's going on is that both of the vector fields both f1 and f2 point in the wrong direction. So f1 points over in to the f2 territory. f2 points over in the f1 territory, but again it's clear what should really happen. We should somehow slide along the switching surface here. That's clear because f1 and f2 are pulling in different directions and this is why it's known as sliding mode control because what we do is we slide along the switching surface. So let's see how to actually make this sliding happen. Again, I have g positive on this side, g negative on this side, and the switching surface is g = 0. f2 wants to drive me in this direction, and f1 wants to drive me in this direction. I want to slide along the surface. That should be the right solution. Well first of all what are the conditions under which I'm going to slide. Well f2 needs to point in the positive direction of g because on this side g is positive. So what I'm going to do is I'm going to find this thing, the vector that's normal to the switching surface and it turns out that luckily for us this is the gradient. The partial derivative of g with respect to x, transpose. And now I take what's called the inner product, so, with this thing and this thing. So the inner product is just a multiplication. and if this inner product is positive, it means that this one and this one are pointing. in the same direction. I also take the inter-product with this and that, and if the inter-product is negative, it means that they're pointing in, in different directions. So what this means is I actually have a condition for sliding. I need dg the x, where this is actually dg, dx1, blah, blah, blah, blah, blah, dg, the xn a roll vector like this times f1, well f11 to f1n, 'cause these are all vectors. If this is negative, that is code for having this arrow and this arrow pointing in different directions. So, f1 points into negative g territory. f2 points into positive g territory. if this happens at the switching surface, then we have sliding. And, one way we can think about this object here, it's the derivative of g, in the direction f1. And this is the derivative of g in the direction f2, and there's actually a fancy term for this. it's called the Lie derivative. So the derivative of G in the direction f, we're going to write this Lfg which is simply code for dg/dx * f. So when I write Lfg, this is what I mean. It's the lead derivative of g in the direction of f. So, using our slightly fancier notation we know that sliding occurs. Meaning we should slide if the derivative, g, in the f1 direction is negative, which, again, means this and this have different directions, and the derivative of g in the f2 direction is positive, which means this and this have the same signs meaning they point in the same, in the same directions. So this actually tells us whether or not we have sliding and wallah, we actually have a test for type 1 zeno that says that sliding occurs. If Lf1g is negative and Lf2g is positive, and this is at g(x) = 0 so this is along the switching surface when we're inside the different mode regime. We don't have to worry about this. But on the switching surface, this our zeno type 1 probe that we have to use to see whether or not we slide. And this nice because this is something that's easily implemented. We still don't know what actually happens beyond the zeno point, meaning, how do we slide. And that is going to be the topic of next lecture.