0:00

So, in the last lecture, we investigated this innocent-looking model or system,

Â which was a ball bouncing on a surface. And we saw something rather strange

Â there, which was that, that the, the ball ended up bouncing an infinite number of

Â times in finite time. And this is part of another potential

Â complication that comes from hyberdizing your model, namely, that you have these

Â kinds of infinitely many swicthes. And this is known as the Zeno Phenomenon.

Â And in today's lecture, we're going to dig a little deeper into the, the Zeno

Â Phenomenon and see what we can do about it and if you can understand it.

Â But fundamentally, what I would like to point out is that Zeno is bad, because if

Â you're actually running something that's asked to do an infinite amount of things

Â in finite time, it crashes. If you're running this on the computer,

Â the simulations crash. another thing is that we know that there

Â is something inaccurate or wrong with our model because the ball, if I drop a ball,

Â it doesn't bounce an infinite number of times,

Â it bounces 17 times and then it stops bouncing.

Â So, there's something wrong with our model.

Â That's another warning flag. And the third warning flag is that we

Â don't actually know what the system does beyond the, the Zeno point, meaning the

Â time up to which we have an infinite number of switches.

Â So, since we can't really define what the system is doing beyond that point, things

Â like asymptotic stability is meaningless because time is not allowed to really

Â progress off to infinity. So, first of all, why is it called the

Â Zeno phenomenon? Well, there was a Greek philosopher, Zeno, Zeno of Elea who spent

Â a lot of time thinking about movement and the dynamic world and basically his point

Â was that our perception of the world is wrong because clearly there are all these

Â problems out there. For instance, here's one of his famous paradoxes.

Â We have a hare racing a tortoise. And the tortoise is a little slower so the

Â tortoise gets a head start. In fact, the tortoise starts there and

Â then, the race is on. And at some point, the hare reaches the

Â point were the tortoise started from but at that point, right, the tortoise has

Â moved, not much but it has moved a little bit.

Â This is how far the tortoise has moved. Okay. The race goes on.

Â And at some point, the hare catches up to where the tortoise was last time but now,

Â the tortoise has moved a little bit more, not much, and then this repeats.

Â In fact, here is the, the paradox. The paradox is that the hare never catches up

Â with the tortoise because every time it reaches the step that the tortoise was

Â last time, the tortoise would have moved a tiny bit.

Â Now mathematically, this is nothing.

Â We know now about convergent series. We know that even though there are

Â infinitely many of these small intervals the sum of them will converge and there

Â is indeed a point where the hare will catch the tortoise.

Â but the problem for us is that if I model this as a hybrid system, I have, again,

Â infinitely many switches in finite time. So, this is why this kind of infinite

Â amount of switches is called the Zeno Phenomenon because it can be traced back

Â to Zeno's many paradoxes about motion. Now, let's look at another example, one

Â that's not a hare and a tortoise but one that's rather innocent-looking.

Â 3:16

Let's say that x-dot is negative, for +x - 1 and it's positive for -x.

Â Okay, so we have this. If I write this as a this is a hybrid

Â automoton, so let's see what's going there.

Â An if I draw this little plot here, right, here is here is time and here's x.

Â Let's say that x starts there. Well, it starts positive so x-dot is

Â going to be -1, so it's going to decay down with a slope of -1 and then it

Â becomes tiny bit negative, and then oh, it's going to switch back up to plus, and

Â then it goes up and in a second, it becomes just a tiny bit positive again,

Â it switches down and, in fact, really what's happening is that once it hits 0,

Â it starts switching like crazy here. In practice, it would chatter but in

Â theory, it starts switching like crazy here and this is actually not good at

Â all. So, this is really a Super-Zeno

Â Phenomenon because not only do we have infinitely many switches in finite time,

Â we have it at the single time instance, which is when the system actually hits

Â x=0. So, for that reason, we typically talk

Â about two different kinds of Zeno types to, so type 1 Zeno, which is what I now

Â call the Super-Zeno. It says that you get infinitely many switches in a single time

Â instant. In this case, again, I want to reiterate

Â this, the ball came down here and then, not the ball, this system came down here

Â and then it started switching infinitely many times right there.

Â Now, type 2 is Zeno by not type 1, meaning, you have infinitely many

Â switches but you have that over a time interval and the bouncing ball is really

Â an example of that. So.

Â there are some good news and bad news in all these rather messy switching

Â situation. The bad news is that Zeno is a problem as

Â we've seen. However, type 1 which is arguably the

Â more common type is not only detectable, meaning, it is easy to see if you're

Â going to end up in a situation where you are going to switch infinitely many times

Â at a single time instant. But the other good news about type 1 is

Â that we can actually deal with it because you know what,

Â what should this system do? It should go down to zero and then it should stay at

Â 0. It's clear that that's what we want the

Â system to do and, in fact, you can do that as you will see in the next lecture.

Â Now, the last piece of bad news though is that type 2,

Â the bouncing ball type, that is hard, it's hard to deal with a bouncing ball.

Â It's hard to detect it, it hard, it's hard to remedy it.

Â and this is again, a situation where you really need to test your system and see

Â do I get something like this where you start seeing an accumulation of switch

Â times. And if you do, you need to go back and

Â revisit your model. But in the next lecture, we will see how

Â to indeed overcome the, the problems that a type 1 Zeno system will will cause us.

Â