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So, last time, we saw that it was possible to destabilize a hybrid system

Â by switching between different modes even if the different subsystems or modes were

Â stable or asymptotically stable themselves.

Â in this lecture I want to harp on this theme a little more and, in fact, I've

Â called it Danger, Beware! because this is actually something that

Â we need to be aware of and it is not always so easy to ensure that the overall

Â hybrid system is stable even if the subsystems are unstable.

Â In fact, like I saw, we looked at a counter example that allowed us to

Â actually destabilize, which means drive the state off to infinity, by being

Â unlucky or unclever in how we transitioned in and out of the different

Â modes. So, if we ignore the resets, meaning

Â there are no reset maps, the state doesn't jump when you're making

Â transitions, we get something called a switch system where you have x dot if is

Â f(x,u) but now I have this a little sigma index here and sigma is what's known as

Â the switch signal. So, sigma is going to tell me which mode

Â the system is in. So, if sigma is 1 I am in mode 1.

Â If sigma is 10, it's in mode 10 or if it's p it's in mode p.

Â So, all I'm doing is I'm switching between different modes and this switch

Â signal can be, be rather different. It can be just random.

Â It can be driven by time. It can be driven by things happening in

Â the state. But it's a very general way of describing

Â a switch system. Now, if you have that, we can actually

Â talk about different kinds of stability. And I just wanted to point out that these

Â things actually exist and we should be aware of them.

Â The first is what's known as universal asymptotic stability. And universal means

Â that there is nothing I can do to destabilize the system.

Â No matter what, this upside-down A means for all sigma, so x will go to 0 no

Â matter how I switch. That's called universal stability.

Â The other notion is existential stability, which means there exists the

Â flipped E it's called an existential quantifier.

Â It means, there exists a switch signal that makes the system stable.

Â So, not all of them, but at least there is one that makes it go down to zero.

Â And these are the two main ways in which people want to deal with switch systems.

Â In our case, we don't have, there exists a switch signal typically, or for all, we

Â have what's called hybrid stability and that we actually have a hybrid system

Â that is itself generating the, the switch signal and this is known as hybrid

Â stability. So, x goes to zero, not for any all sigma

Â or for all sigma, but for the one that happens to be the one that we have in our

Â hybrid system. So, here are some results.

Â Let's say that I have a hybrid system where all the individual modes are

Â asymptotically stable. Well, then, can we guarantee that it is

Â at least existentially stable? Well, clearly, we can, right?

Â Because what we do is we don't switch. If I design a switch signal that just

Â picks one mode and then stays with that forever and ever and ever, well, we're

Â never switching but all the individual modes are asymptotically stable so if all

Â the modes are asymptotically stable, we never switch and voila,

Â we do have an existentially stable system.

Â But, as we will have, as we have seen, they're not always universally

Â asymptotically stable. And the reason for this is well, this

Â counter example. The reason is that we can actually

Â destabilize the system by an unfortunate switching between the modes.

Â So, what do we do about this? Well, there is something in nonlinear control known

Â as a common Lyapunov function. I'm not going to talk about this in this

Â course, mainly because the common Lyapunov

Â function is an elusive beast that you can almost never find.

Â But theoretically, that's what you're going to have to hunt for.

Â Practically speaking though, what you need to do is the following.

Â First, never design unstable controllers because then you're going to be in

Â trouble probably, right? So, design stabilizing controllers for

Â the subsystems. And then, we design the switching logic,

Â meaning how are we going to switch between the different modes and if we're

Â lucky or we have a lot amount of free time on our hands, we can go find, or try

Â to find these common Lyapunov functions. Now, like I said, finding that is really

Â more art than science in the sense that it's very hard in general to find it.

Â So, the most important thing here is really, we need to be aware of the fact

Â that stable subsystems do not ensure asymptotic stability of the hybrid

Â systems. So, we need to be aware of it and test,

Â test, test, test, test. In the sense that, run it, see what

Â happens, do we get instabilities induced? And if we do, we need to start messing

Â with our switching logic. But this is really key and I cannot

Â underemphasize the testing aspect of the hybrid system.

Â In fact, when people design avionics software, for instance, the majority of

Â the time is spent not on test, on, on the development of the controllers, but on

Â testing that the switching logic in combination with the controllers does not

Â induce instability. So, that's what we need to keep an eye

Â out for.

Â