[MUSIC] We then understand enough of this script time model from the previous video. Now we can move on to a more advanced model. One that will allow us to characterize the state, the input and the output of the system continuously, where the variable now is ordinary time for the parameterization of time. As we did before, we will have inputs and outputs, and these type of models are the models that we will use to model the dynamics of the physical component of a cyberphysical system. Denoting the input as v and the output as eta, we will now have a state as well for the physical component that we will label as x. And the difference between the discrete time model and the continuous time model is that now these signals will be having values at every time t. Where t x values on the positive real line which is denoted as the symbol R with a subscript greater or equal then zero. If the input that we are applying is of continuous time form, then it will be given by a curve potentially, with points of non differentiability and with values for every t in this range of time. The output will also have values that, Are defined for every t. The key question is, how do we relate the input, To the output, And the state? Remember that to define such a relationship, we need to come up with a model, and moreover, we need to determine what the initial state of the system would be. The model that we will use will be, Based on continuous time, And governed by differential equations. The way we model this is by defining the state derivative with respect to time. We're going to write this down as x dot, but this is short for the derivative of x with respect to time. That change of the state, which is no more than the velocity, will be governed by some function labeled as F that is evaluated at the state, and at the input to the physical system, which will be called v. The output will similarly be defined as we did for the discrete time model as a function, let's label it as h as well, that depends also on the state and on the input. So with this symbol, where, again, x dot will represent variation with respect to time, F is a function and h is a function. Given, An initial state, Denoted as x sub 0 and an input, Which will be an input of time, The trajectory, Or solution, To the system, Is a function, Of time, Which we will denote also as x(t), satisfying, The differential equation, That we have right here. This is the so called initial value problem with an input. Therefore, it's called the forced initial value problem. IVP for short of initial value problem. And as you can imagine, if these functions are well defined and initial condition is given and the input is defined for all the time, now we can come back here to this system. And come up with a plot of the state trajectory, or, again, the solution to the system from that particular initial value over time. [MUSIC]
