So the first question we want to start answering is how do we model cyber-physical system? So there are many ways the model cyber-physical systems, and we will cover many of these. The most elementary yet powerful type of model is what is called a discrete-time model. In discrete-time model, as the model suggests, that time is discrete, the values of the variables might take in discrete sets, but also potentially on dense sets, and the models essentially uses allotted time, and you go from time equals zero to time equal one, to time equal two, and so on. And I will tell you more of the details in a second. A more structured type of discrete model is finite-state machines. And as you realize from the name, the model has a state that take value from a finite set, and the model itself will allow only transition between the different states using some sort of a law, and we will define that very soon. The next type of model that adds now the capability of having continuous or infinite number of time instances is what is called continuous-time models, and this typically boils down to objects that are related to differential equations, and that's the type of model we will use mainly for the physics component of a cyber-physical system. Those type of models can be further extended if we now allow the variables of a continuous-time model to have jumps, and those are impulsive continuous-time models. Those jumps could be time-triggered or could be event-based-on-a-state-triggered. So event-triggered continuous-time models is another class. 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.
