[MUSIC] The model of a continuous-time system using a differential equation would have this function f and this function h. These functions will need to be defined in the right state, and input spaces similar to this discrete-time model. And we will define that as specifically when we specialize this model to the physical model for our CPS. For the time being, I'd like to give you an example, and actually, come back to the vehicle model. That we illustrate it for this discrete-time model. Similarly, we can have a positional state x1, x2 and if this is the position of the vehicle Or the vehicle may be pointing in some direction We can add now to that orientation. This will be the orientation of the vehicle, and we're going to define the angle to be relative to the axis corresponding to x2. The questions of motion that will lead to a differential equation corresponds to, Assignments of the variation of these quantities. Now, variations over time, remember that these have variations with respect to ordinary time. And if we have an input that corresponds to the forward velocity, I will label that as v1. Then we will have the following variation of the position in this direction given by this sine of the angle. In the direction of x2 we will have also the following variation. And if we assume that there's another input that controls the angular velocity of the vehicle, we will have the following variation for the angle itself. These equations that define the differential equation will be uniquely defining the function f, where now, X is three dimensional state. And then v is, Second order state. Let make that right, so where v is equal to v1, v2 and x is equal to x1, x2, x3. One reason we want to look at this example is because, in practice, we don't have the capability of applying any forward velocity or any desired angular velocity. So v1 is the forward velocity, Might be restricted or constrained, To the range, -v1 max, v1 max, where this parameter is positive. Similarly, v2 which defines the rate of angular change, Might have a constraint. Would be omega, might be positive quantity. So in practical problems, we not only have the differential equation that will cover the change of this variables continuously according to the input of the state, but also will have constraints on the inputs. These constraints will basically define restrictions on the ranges that we can apply the input. And our model for reasonable physical assistant in a CPS should also incorporate this type of constraints. We will add this in the next video, but to recap with the model of our continuous time model, continuous time system given, An initial state, x0, which now will correspond to initial position in the two directions, an initial angle. And an input, Which now the input will be, Correspondent to the forward velocity and the angular velocity. The trajectory of the system, Is also a function of time that describes position x1, x2 and angle over time. And satisfying, This differential equation here How do we determine that a state trajectory or, again, we call it solution because we're talking about systems. Depends on how good we are at finding a function that satisfy these conditions for a given input and for a given initial estate. That can be done in mainly two ways, one way is to have some intuition about the system. Guess the type of function that will satisfy such dynamical equation, differential equation. And then check whether that's the case for the given input of a given initial state. Typically in that guessing, we have freedom. We have some parameters and some functions that are probably undefined. And then through the process of checking, we can solve for them. We are going to do that for a slightly simpler model of a temperature system later on. The other approach, is to actually use a tool that would allow us to propagate forward in time numerically. The evolution of the state according to the differential equation. That is typically called simulation or numerical analysis of the system. And that requires software or a code that will actually implement such computation. There are basically integration schemes that will provide the trajectory to the system. [MUSIC]
