And combining essentially all of these type of modes,

one can reach to a more general type of modeling framework called hybrid automata.

Hybrid automata, in simple words,

combines elements of finite-state machines and

elements of continuous-time models with the impulses.

And in order to not claim that this is an extensive list,

I should say that there are other models and we will touch on some of them.

At some point, we're going to naturally arrive to a model that is powerful,

combines many of these aspects and elements in a holistic way,

and allow us to model continuous,

cyber, and the interfaces.

I would like to clarify some of the aspects on this model and this model,

so we have it in written.

So let me just write down here that time is

discrete, time is continuous.

And what we are going to be using in order to represent time being discrete will be

a variable that we'll call k. And that variable k will be either 0,

1, 2, 3 and so on.

This particular set of elements which are the natural numbers, including zero,

we will denote it as this symbol N with a double bar in the beginning,

while when we look at continuous-time models,

when time is continuous,

we will use the typical variable t to denote time,

and this will take values in the dense set from zero to infinity,

which is the positive and zero real numbers for which we will use

the following symbol which corresponds to the R with double bar of the symbol,

and then this greater or equal than zero,

meaning that we are looking at the positive line.

So for us, the model of a cyber-physical system,

the entire system will involve discrete-time to keep track of

the cyber steps or the computations and

continuous-time to keep track of the physics evolution,

and it will essentially combine the two of them

into a notion of time which we will call hybrid time.