So, last time, we saw that it was possible to destabilize a hybrid system
by switching between different modes even if the different subsystems or modes were
stable or asymptotically stable themselves.
in this lecture I want to harp on this theme a little more and, in fact, I've
called it Danger, Beware! because this is actually something that
we need to be aware of and it is not always so easy to ensure that the overall
hybrid system is stable even if the subsystems are unstable.
In fact, like I saw, we looked at a counter example that allowed us to
actually destabilize, which means drive the state off to infinity, by being
unlucky or unclever in how we transitioned in and out of the different
modes. So, if we ignore the resets, meaning
there are no reset maps, the state doesn't jump when you're making
transitions, we get something called a switch system where you have x dot if is
f(x,u) but now I have this a little sigma index here and sigma is what's known as
the switch signal. So, sigma is going to tell me which mode
the system is in. So, if sigma is 1 I am in mode 1.
If sigma is 10, it's in mode 10 or if it's p it's in mode p.
So, all I'm doing is I'm switching between different modes and this switch
signal can be, be rather different. It can be just random.
It can be driven by time. It can be driven by things happening in
the state. But it's a very general way of describing
a switch system. Now, if you have that, we can actually
talk about different kinds of stability. And I just wanted to point out that these
things actually exist and we should be aware of them.
The first is what's known as universal asymptotic stability. And universal means
that there is nothing I can do to destabilize the system.
No matter what, this upside-down A means for all sigma, so x will go to 0 no
matter how I switch. That's called universal stability.
The other notion is existential stability, which means there exists the
flipped E it's called an existential quantifier.
It means, there exists a switch signal that makes the system stable.
So, not all of them, but at least there is one that makes it go down to zero.
And these are the two main ways in which people want to deal with switch systems.
In our case, we don't have, there exists a switch signal typically, or for all, we
have what's called hybrid stability and that we actually have a hybrid system
that is itself generating the, the switch signal and this is known as hybrid
stability. So, x goes to zero, not for any all sigma
or for all sigma, but for the one that happens to be the one that we have in our
hybrid system. So, here are some results.
Let's say that I have a hybrid system where all the individual modes are
asymptotically stable. Well, then, can we guarantee that it is
at least existentially stable? Well, clearly, we can, right?
Because what we do is we don't switch. If I design a switch signal that just
picks one mode and then stays with that forever and ever and ever, well, we're
never switching but all the individual modes are asymptotically stable so if all
the modes are asymptotically stable, we never switch and voila,
we do have an existentially stable system.
But, as we will have, as we have seen, they're not always universally
asymptotically stable. And the reason for this is well, this
counter example. The reason is that we can actually
destabilize the system by an unfortunate switching between the modes.
So, what do we do about this? Well, there is something in nonlinear control known
as a common Lyapunov function. I'm not going to talk about this in this
course, mainly because the common Lyapunov
function is an elusive beast that you can almost never find.
But theoretically, that's what you're going to have to hunt for.
Practically speaking though, what you need to do is the following.
First, never design unstable controllers because then you're going to be in
trouble probably, right? So, design stabilizing controllers for
the subsystems. And then, we design the switching logic,
meaning how are we going to switch between the different modes and if we're
lucky or we have a lot amount of free time on our hands, we can go find, or try
to find these common Lyapunov functions. Now, like I said, finding that is really
more art than science in the sense that it's very hard in general to find it.
So, the most important thing here is really, we need to be aware of the fact
that stable subsystems do not ensure asymptotic stability of the hybrid
systems. So, we need to be aware of it and test,
test, test, test, test. In the sense that, run it, see what
happens, do we get instabilities induced? And if we do, we need to start messing
with our switching logic. But this is really key and I cannot
underemphasize the testing aspect of the hybrid system.
In fact, when people design avionics software, for instance, the majority of
the time is spent not on test, on, on the development of the controllers, but on
testing that the switching logic in combination with the controllers does not
induce instability. So, that's what we need to keep an eye
out for.
