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Last time we saw that not only we have to worry about Zeno.

Â This zeno problem or the zeno phenomenon comes in two different flavors.

Â And one thing that I stated was that type 1 zeno which is infinitely many switches

Â in zero time or in one timing sense here represented by this system where.

Â Again, if I have x here, and time here, and I start positive, then x is going to

Â be -1 until x becomes, just, just negative.

Â And then, we're going to get this infinitely many switches going on right

Â there. that's type 1.

Â Well, it turns out, that we can actually remedy this.

Â And the reason for it is, you know what is very, very clear what should happen.

Â The system shouldn't grind to a halt. It should just nicely keep continuing on

Â like this Like x is equal to 0.

Â So how do I take this intuitive notion of that x should just keep saying and be

Â equal to 0 and make that mathematically sound.

Â What, what's the topic of today's lecture, and this construction by which

Â we can continue beyond the zeno point in the type 1 zeno system, using something

Â that's known as. Sliding mode control.

Â So, let's be a little bit general here. Let's say that I have one system, x dot

Â is f1(x) and then I have a switching surface, g(x).

Â And when g is negative, I switch to f two, and when it becomes positive I

Â switch back to f1. So here it is, here's my switching

Â surface, g(x) = 0, that's where the action is.

Â Now on this side I'm going to be using f1, and da, da, da, da, da, da, da, when

Â I hit this point, well let's say that f1 is pointing inwards.

Â Well on this part of the world I am going to be using f2.

Â Well, let's say that f2 here points outwards.

Â This means that when I hit this point again, I grind to a halt.

Â so this is really what's going on is that both of the vector fields both f1 and f2

Â point in the wrong direction. So f1 points over in to the f2 territory.

Â f2 points over in the f1 territory, but again it's clear what should really

Â happen. We should somehow slide along the

Â switching surface here. That's clear because f1 and f2 are

Â pulling in different directions and this is why it's known as sliding mode control

Â because what we do is we slide along the switching surface.

Â So let's see how to actually make this sliding happen.

Â Again, I have g positive on this side, g negative on this side, and the switching

Â surface is g = 0. f2 wants to drive me in this direction,

Â and f1 wants to drive me in this direction.

Â I want to slide along the surface. That should be the right solution.

Â Well first of all what are the conditions under which I'm going to slide.

Â Well f2 needs to point in the positive direction of g because on this side g is

Â positive. So what I'm going to do is I'm going to

Â find this thing, the vector that's normal to the switching surface and it turns out

Â that luckily for us this is the gradient. The partial derivative of g with respect

Â to x, transpose. And now I take what's called the inner

Â product, so, with this thing and this thing.

Â So the inner product is just a multiplication.

Â and if this inner product is positive, it means that this one and this one are

Â pointing. in the same direction.

Â I also take the inter-product with this and that, and if the inter-product is

Â negative, it means that they're pointing in, in different directions. So what this

Â means is I actually have a condition for sliding.

Â I need dg the x, where this is actually dg, dx1, blah, blah, blah, blah, blah,

Â dg, the xn a roll vector like this times f1, well f11 to f1n, 'cause these are all

Â vectors. If this is negative, that is code for

Â having this arrow and this arrow pointing in different directions.

Â So, f1 points into negative g territory. f2 points into positive g territory.

Â if this happens at the switching surface, then we have sliding.

Â And, one way we can think about this object here, it's the derivative of g, in

Â the direction f1. And this is the derivative of g in the

Â direction f2, and there's actually a fancy term for this.

Â it's called the Lie derivative. So the derivative of G in the direction

Â f, we're going to write this Lfg which is simply code for dg/dx * f.

Â So when I write Lfg, this is what I mean. It's the lead derivative of g in the

Â direction of f. So, using our slightly fancier notation

Â we know that sliding occurs. Meaning we should slide if the

Â derivative, g, in the f1 direction is negative, which, again, means this and

Â this have different directions, and the derivative of g in the f2 direction is

Â positive, which means this and this have the same signs meaning they point in the

Â same, in the same directions. So this actually tells us whether or not

Â we have sliding and wallah, we actually have a test for type 1 zeno that says

Â that sliding occurs. If Lf1g is negative and Lf2g is positive,

Â and this is at g(x) = 0 so this is along the switching surface when we're inside

Â the different mode regime. We don't have to worry about this.

Â But on the switching surface, this our zeno type 1 probe that we have

Â to use to see whether or not we slide. And this nice because this is something

Â that's easily implemented. We still don't know what actually happens

Â beyond the zeno point, meaning, how do we slide.

Â And that is going to be the topic of next lecture.

Â