An example of discrete-time model, corresponds to a ground vehicle, where we have three variables, one of them is the horizontal position, the other one is the vertical position, and the orientation is a third variable. So this is the orientation of vehicle and the angle between these and the horizontal will correspond to the survival. So you can probably think of something more sophisticated here but, think about here, there is a vehicle that, you see there moving in this direction at this particular point. And we can have control along this direction but, we can also control the orientation. So, we can move in a straight line but also, we can make rotations. So, this is position, this is also position. Think about this on the plane, call it horizontal, vertical, or however you want to call it. And this is the angle, defining the orientation. So, these three variables will define our vector X. So this vector will correspond to your vector in dimension three, in this case n is equal to three, from the previous notation. And the inputs to the system, we could have both, as we described in the previous video. We could have the force or thrust, and we can have the angular change. So, we will call this one, V1, and we will call this one, V2, so therefore, we will have input vector that has dimension two. So, now we are saying that, n is equal to two. So this is the state, these are the inputs, and let's say that the output is only the angle. So this will be vector equal to X3, which is the angle, therefore our function h of X,V is no more than the start component of the state. And this is a value that belongs to R, therefore, p will be equal to one. How do we relate these inputs, state an output is, what we need to come out for with G and h. But again, from the previous video, we had these signals, when the inputs, these output being in this case the angle and we will have a model that corresponds to the following equations. So, what are G and h? We can use first principles to derive this model. The idea is that, since the model is discreet, first principle will give us a continuous time model, which will arrive in a future video. And the discretization of that model will give us the model that I'm going to write, right here. The intuition is very simple, if you think about the angle, what's going to happen is that the model is going to give us the current angle, plus some variation of the angle which will be captured by V2, and then multiplied by sum constant. I'll tell you later more about the constant. But, basically what this is saying, is that the variation is zero of the angle, and then when the speed is zero, then the angle keeps the same at every k. Similarly, we can do a model for X1, and for X2 dynamics. And for those models, we will have also some other constant, call it d, this could be different. And the change will be according to the sinusoidal function of the angle, for X1 and the cosine for X3. So, what this is basically saying is that if I have a particular rate of change, the sinusoidal will project the variation over the right plane, and the cosinusoidal will project the variation over the right plane according to the angle of the vehicle. And necessarily, we can tune these with the velocity or the force thrust that will correspond to moving on that particular line. So, if now we look at this dynamical model, which again we will look at we will come up with the continuous time version, from here you can read out the function G. So the function G is essentially this expression that you see right here, d and c are constants that depend on what is called sampling time. As we said before given initial state, in this case initial position and initial angle, given this constant that corresponds to the model itself, and given that inputs V1 and V2 as a function of k, we can now generate the change of the state according to these difference equation. And generate the output according to this equation. And, in this case, generate how the angle will change over time. As you probably realize, this model is not very sophisticated. It's a good discrete time model of a vehicle. But, the benefit of now wrapping these around cyber component, which will in particular implement a control algorithm, will be that we will assign V2 in particular, let's say go from a desire point into space no matter where we start. So, they in particular a control problem will be to drive the vehicle to this particular point, with this particular orientation from any region of the space. And that will require the second piece, the cyber component that we have right here, the interfaces and a way to design the entire system correspond to a cyber physical system.
