Now we are ready to define our general model for the physical components of a Cyber-Physical System. We will model the physics mathematically. Now the physical, which will have inputs and outputs, will relate those inputs and outputs on the state using a differential equation model. The inputs are going to be given by a vector that we're going to label as U and takes values in dimension M_sub_P, P for physical. The output, we're going to be labeling it as Y and it takes values in dimension R_sub_P. And the state of this physical component will also be a vector. We're going to label it as Z and both take value in dimension N_sub_P. The model that we will put inside will be a differential equation model, where Z changes according to the differential equation, where the function that defines the differential equation is going to be F_sub_P and depends on the state and on the input. The output to this which is Y will be relating the state and the input through a function H_sub_P. The initial state will be denoted as Z0. So this is the model of the physical component of a Cyber-Physical System. We will have inputs, initial state that will generate the trajectory Z of the system and that will also lead to an output of the system. As we defined before, we will have this function of P defined on the right space. This theta space on the input space, mapping back to the state space, as it assigns the velocity of the state variable. HP will have definitions also on the same. The outputs, we say that will bring dimension RP. And, these are the key functions of my system, so this is the output function and this is defining the change of Z over time. Now, the one thing that we motivated with a vehicle model for a particular physical system, was that potentially, the state and the input of a system might have some restrictions. In order to model those restrictions, were going to include a constraint, which will involve the state and the input. That constraint can be written in many ways, a convenient way to write it is by adding a set inclusion that restricts Z and U. So this is subject to this particular constraint. So this set that defines a constraint which we call C_sub_P, is a subset of the state space cross the input space, and defines the constraints on Z and the input U. To give you an example, back to the vehicle model, we have that V1 was restricted to a range minus V1 max plus V1 max and that V2 was restricted to a range minus W plus W. These two constraints can be actually be captured via set CP that is not restricted in any way the state, by making sure that the two components of the input in this case will be U, will be in the range minus V1 max, V2 max. And then, these set cross further with W range for V2. We could further add constraints on the state that will further restrict the space for the Z component of the set CP. Certainly, if a set that defines the constraints is equal to the entire state and input space on the set has no constraints and we are certainly using a traditional differential equation model without constraints. These as you see, for the input case and you could probably visualize when you have obstacles or you have objects in the space that you don't want to reach. You can actually enlarge that type of constraints that you can model with the set CP. At the end of the day, for us will be whether we have a solution to an initial value problem with an input and with constraints.
