‫And now let's leave the polls in the X dimension alone and let's take the Y dimension and let's see
‫what we're going to have here.
‫Let's change this minus one to plus zero point zero one and let's see what's going to happen.
‫So the drone starts following the trajectory.
‫It seems doing pretty well, so from here, it seems that it was able to track the trajectory quite
‫well.
‫But let's look at the dimensions separately.
‫So the X dimension went very well and well, if you look at the Y dimension, then you see that in the
‫end there is a small difference that is growing.
‫Here is just you haven't had enough time to let this difference grow.
‫But you see it is starting to grow here.
‫And the reason is because you have a very weak, positive poll here, but then a very strong negative
‫poll here.
‫So how about if I make this minus zero point five?
‫Let's see what happens then so you can see that you already have some problems here.
‫The tracking is not nearly as good as it was before.
‫The extermination is fine.
‫However, the Y dimension is not fine.
‫You see now the distance is getting bigger here and it's already visible.
‫And now let's go back to our minus one and minus two polls and let's run another trajectory, which
‫is my favorite one.
‫I call it the crown, the crown trajectory.
‫It's not actually that hard to do if you look at your X and Y dimension, then they're just your cosine
‫and sine functions.
‫The only difference is that now the Z dimension is also oscillating.
‫So as your drone goes in the circle, it also goes up and down.
‫You see, that's how it looks like.
‫That's the X dimension.
‫That's the Y dimension.
‫And that's the Z dimension you see now, Z dimension is also sinusoidal.
‫Before our drone was following a Z trajectory, that was a straight line, which means that your drone
‫would go up steadily, but now it just goes up and down in a sinusoidal way.
‫And these are the angles here and the control inputs and then the omega's.
‫What happens if for the Z dimension, Paul, I'm going to put here plus zero point zero one so you can
‫see that something is not going well at all, it is able to follow the X dimension and then the Y dimension.
‫But because we changed the poles in the Z dimension and we made one pole positive, then now the error
‫seems to be growing.
‫So it has gone past the ground already actually.
‫So it's under the ground.
‫So as you can see, didn't go very well.
‫That's your X dimension.
‫That's your y dimension and then the Z dimension again, you see the positive pole is messing things
‫up.
‫So remember, to get the errors to grow, both poles need to be negative.
‫And if you have some kind of system where you have 10 poles, so you have some kind of tenth order differential
‫equation, then all those 10 poles need to be negative.
‫But now I want to make this poll zero and then this poll zero and then also the same thing for the wider
‫mention this one and this one.
‫Now, my polls still are not complex.
‫They are still real polls.
‫I don't have an imaginary part here, but I've made everything zero.
‫And now I've gone back to our spiral trajectory so you can see that something is not going quite right.
‫So the drone is failing in the end.
‫This is how the drone goes.
‫This is what it was supposed to do in blue, and this is what it actually did.
‫You see, when polls are zero and I mean completely zero, both the real and imaginary part are zero,
‫then it's kind of like a special situation.
‫You really need to start looking at the initial conditions then, because, for example, here in this
‫case, you have this initial condition, two meters, the air, the blue line, minus the red line.
‫At the beginning, when time equals zero, your error is two meters and your error dot, which is this
‫blue X dot reference line minus this red true X dot line at time equals zero seconds.
‫Their difference is zero meters per second.
‫So the error dot is zero meters per second.
‫Now if you choose your lump, the one to be zero and lump the two to be zero, then if you put them
‫inside the equations that you had previously derived.
‫Then you will find that your Q1 and Q2 will be zero as well, both of them will be zero.
‫So if your Q1 and Q2 are zeros, then what will be your use, OpEx and use of WI if all these constants
‫here are zeros.
‫Of course.
‫Use of ax and use of Y will also be zeroes, which means that your error double dot X will be zero and
‫error double dot Y will be zero.
‫And so earlier we talked that our error double dot influences the error dot and then the error that
‫influences the error rate.
‫So it's like a chain.
‫So if this is always zero, then that means that this guy here never changes.
‫So in this X dimension case, if your error that is zero at the beginning.
‫Then theoretically, it should stay zero, and that means that if your error at the beginning is two
‫metres, then it will stay two metres, because since you're Eridani equals zero, that means that your
‫error will stay the same.
‫It will not change because error that is the change of error with respect to time.
‫And that's why you see that essentially your error here is zero.
‫Well, theoretically it should be zero, but then in practice you have numerical errors.
‫And then in real life, of course, you never have pure theoretical results.
‫But you see these two lines are pretty much aligned with each other.
‫And since they're almost zero, OK, there is a small difference here, but the error is almost zero
‫and it doesn't change.
‫And because of that, since your error at the beginning is two metres, it will stay being two metres.
‫You see, it doesn't change.
‫How about the Y dimension, though?
‫You see with the Y dimension is different because here you also have an initial error, but now there
‫is also a difference in error.
‫Dot again, since your constants here are zeros because your poles are zeros, then that means that
‫your error double dot y will be zero.
‫All this was about the X dimension.
‫All right.
‫And now you're erodable.
‫That Y is zero, which means that your error dot will not change.
‫But now your error, that is not zero.
‫Now there is a difference here and the difference here is something like zero point three one four meters
‫per second.
‫In other words, it's pie over ten.
‫That's what it is.
‫It's pie over ten.
‫So it's three point fourteen divided by ten.
‫Now, because of this thing, this thing will not change, it will keep being over 10, but since the
‫error itself depends on error dot, that means that this error that will keep changing, this error
‫and this error will change all the time like this at the beginning.
‫The error is one meter.
‫All right.
‫So the distance here is one meters because this blue line starts at zero meters.
‫And then the real Y position of the drone at time equals zero seconds is minus one meters in the Y dimension.
‫So the error is one meter.
‫So that's at the beginning.
‫But then you have your error dot, and then since it's not zero, you have to multiply it by time.
‫So remember your error, that is your d e divided by d t so your error here is one plus dí over deti
‫times time.
‫In other words, your error in the Y dimension.
‫And remember now we're talking about the Y dimension.
‫You're error here now is one plus so pi over 10 times time.
‫And so this difference here between the blue line and the red line is growing according to this function
‫here.
‫So I have graphed the error function here.
‫Here in green, you have this error function.
‫It's this one here.
‫Error as a function of time equals one plus pi over ten times T.
‫So that's the one in green.
‫And so when you make your polls zero, then you have to be extra careful and pay attention to these
‫kind of things.
‫But of course, in control engineering, your objective is to minimize the errors so it doesn't make
‫sense for you to choose polls that are positive or Xeros.
‫You always want negative polls because you want your errors to become zero.
‫That's your objective as a control engineer.
‫And now I want to play a little bit with complex polls.
‫I want to see what's going to happen then.
‫So we're going to comment about the real polls and we're going to uncommented the complex polls, so
‫as you can see here, when we talk about complex polls, then the real part is the same for all of them,
‫minus zero point one here and minus zero point one here, minus zero point zero one here and minus zero
‫point zero one here.
‫And then the imaginary parts are the conjugates plus zero point three times and minus zero point three
‫times, Jay, if I put here plus one point three times, Jay, I have to put here minus one point three
‫times, Jay.
‫Let's for a moment have real X and Y dimension polls and then for the Z dimension, let's have complex
‫polls.
‫And now I'm going to make my real polls here like this minus zero point one and also here minus zero
‫point one.
‫And so I've taken out another trajectory and we're going to see how it follows this straight line trajectory.
‫So, as you can see, it's not being very successful.
‫If you look at this, the Z dimension, it seems to be oscillating.
‫So that's how it followed the trajectory.
‫You can see that since the Z dimension is up and since we had complex polls, but still a small negative
‫real number because of that, the drone oscillated in the Z dimension, but it damped itself out.
‫That's because our Lumb the one was minus zero point one plus eight times zero point three and then
‫LAMDA two was minus zero point one minus eight times zero point three.
‫So these ones give you oscillation and this one here damps this oscillation out.
‫You see, it's like this, you don't really have that in the X dimension because you had real polls
‫there, no imaginary parts, the same thing for the Y dimension, but your Z dimension did oscillate
‫before it became stable.
‫And you can see this.
‫You want control input, which is mainly responsible for tracking the altitude because the propellers
‫give you the thrust upwards and you want is your thrust control input.
‫And you see there is also some oscillation in this.
‫You want control input.