0:16

where we have three variables,

Â one of them is the horizontal position,

Â the other one is the vertical position,

Â and the orientation is a third variable.

Â So this is the orientation of

Â vehicle and the angle between

Â these and the horizontal will correspond to the survival.

Â So you can probably think of something more sophisticated here but, think about here,

Â there is a vehicle that,

Â you see there moving in this direction at this particular point.

Â And we can have control along this direction but,

Â we can also control the orientation.

Â So, we can move in a straight line but also,

Â we can make rotations.

Â So, this is position,

Â this is also position.

Â Think about this on the plane, call it horizontal,

Â vertical, or however you want to call it.

Â And this is the angle, defining the orientation.

Â So, these three variables will define our vector X.

Â So this vector will correspond to your vector in dimension three,

Â in this case n is equal to three,

Â from the previous notation.

Â And the inputs to the system,

Â we could have both, as we described in the previous video.

Â We could have the force or thrust,

Â and we can have the angular change.

Â 2:33

So, we will call this one,

Â V1, and we will call this one, V2, so therefore,

Â we will have input vector that has dimension two.

Â So, now we are saying that,

Â n is equal to two.

Â So this is the state,

Â these are the inputs,

Â and let's say that the output is only the angle.

Â So this will be vector equal to X3, which is the angle,

Â therefore our function h of X,V is no more than the start component of the state.

Â And this is a value that belongs to R,

Â therefore, p will be equal to one.

Â How do we relate these inputs,

Â state an output is,

Â 3:43

what we need to come out for with G and h. But again,

Â from the previous video,

Â we had these signals, when the inputs,

Â these output being in this case the angle and we will have a model that

Â corresponds to

Â the following equations.

Â So, what are G and h?

Â We can use first principles to derive this model.

Â The idea is that,

Â since the model is discreet,

Â first principle will give us a continuous time model,

Â which will arrive in a future video.

Â And the discretization of

Â that model will give us the model that I'm going to write, right here.

Â The intuition is very simple,

Â if you think about the angle,

Â what's going to happen is that the model is going to give us the current angle,

Â plus some variation of the angle which will be captured by V2,

Â and then multiplied by sum constant.

Â I'll tell you later more about the constant.

Â But, basically what this is saying,

Â is that the variation is zero of the angle,

Â and then when the speed is zero,

Â then the angle keeps the same at every k. Similarly,

Â we can do a model for X1,

Â and for X2 dynamics.

Â And for those models, we will have also some other constant,

Â call it d, this could be different.

Â And the change will be according to the sinusoidal function of the angle,

Â for X1 and the cosine for X3.

Â So, what this is basically saying is that if I have a particular rate of change,

Â the sinusoidal will project the variation over the right plane,

Â and the cosinusoidal will project the variation over

Â the right plane according to the angle of the vehicle.

Â And necessarily, we can tune these with the velocity

Â or the force thrust that will correspond to moving on that particular line.

Â So, if now we look at this dynamical model,

Â which again we will look at we will come up with the continuous time version,

Â from here you can read out the function G. So

Â the function G is essentially this expression that you see right here,

Â d and c are constants that depend on

Â what is called sampling time.

Â As we said before given initial state,

Â in this case initial position and initial angle,

Â given this constant that corresponds to the model itself,

Â and given that inputs V1 and V2 as a function of k,

Â we can now generate the change of the state according to these difference equation.

Â And generate the output according to this equation.

Â And, in this case,

Â generate how the angle will change over time.

Â As you probably realize,

Â this model is not very sophisticated.

Â It's a good discrete time model of a vehicle.

Â But, the benefit of now wrapping these around cyber component,

Â which will in particular implement a control algorithm,

Â will be that we will assign V2 in particular,

Â let's say go from a desire point into space no matter where we start.

Â So, they in particular a control problem will

Â be to drive the vehicle to this particular point,

Â with this particular orientation from any region of the space.

Â And that will require the second piece,

Â the cyber component that we have right here,

Â the interfaces and a way to design

Â the entire system correspond to a cyber physical system.

Â