‫But what will happen if I make my imaginary parts a zero like this and like this, let's see what's
‫going to happen.
‫And so you can see that even though the tracking is very slow, but there is no oscillation.
‫So this is the Z dimension.
‫You see, you don't have any oscillation, but you track the Z dimension very slowly y well because
‫you don't have any imaginary parts.
‫So you cannot have oscillation, but your real part is very small.
‫So it's minus zero point one.
‫And now let's go back to our complex polls and now let's make the real poll not minus zero point one,
‫but plus zero point zero one.
‫So a positive number, but 10 times smaller, zero point zero one.
‫And let's see what's going to happen.
‫So the drone oscillates in the Z dimension and this oscillation is growing.
‫You see this constellation is growing in the Z dimension.
‫And the reason for that is because you have complex polls here, imaginary parts give you a solution
‫and then you have a small, positive, real part that will increase your error.
‫And now I want to show you something else.
‫How about if I put zero here and zero here?
‫So now you can imagine that you think you would oscillate and your amplitude wouldn't change and, well,
‫that's the result.
‫However, as you can see, the amplitude does change.
‫So this is the Z dimension and you can see that the amplitude grows even though your real part is zero.
‫And why is it like that?
‫Well, the explanation that I can give you is the following, that theoretically it shouldn't grow,
‫because what happens is that your lump, the one equals zero plus I times zero point three and then
‫your lump, the two equals zero minus eight times zero point three.
‫That's for your Z dimension, right?
‫For your use of Z.
‫And so your K one value here would be minus zero point zero nine.
‫I know that because I looked it up and then K two would be zero.
‫So in your Z dimension differential equation, you OK?
‫One would have this value and your K two would have this value.
‫And so if you calculate your error as a function of time, then you will have an expression that only
‫consists of cosigns and signs and you will not have something like this where you have the oil and no
‫to the power of their real part of the polls times time.
‫You will not have that because this one will be zero.
‫So this entire thing will be won.
‫So the amplitude of the oscillation shouldn't grow.
‫This shouldn't grow, but it does.
‫Well, the reason for that is that these key values here, they will give you the right e double dot
‫in the Z dimension so that your error would simply oscillate, but so that its amplitude wouldn't change.
‫But then this error, double dot needs to be converted to Phi R, Theta R and you won.
‫And not only this, of course, in the end, all your three erodable dots will need to be converted
‫into PHY are, Seeta are and you won and then you one will go to plant and then fire and Sittar, they
‫will go to the NPC controller and one happens in the NPC controller.
‫These values are your reference values, right.
‫But then the true PHY and Seeta values.
‫Right phy true and feet are true.
‫These are the values that belong to the drone.
‫These are the values that need to catch these phy are and Setara values.
‫They do track your fly are and Setara values, but they might not track them perfectly.
‫And because of this imperfection, your error here still grows.
‫So at least that's how I see it, because at the beginning I was surprised by it.
‫But then I thought about it.
‫And I think that that's the reason.
‫If I put here a very small negative, real number minus zero point zero one and then minus zero point
‫zero one, then you can see that your amplitude is not changing, at least it's not visible.
‫Maybe it's changing a little bit, but it's not visible.
‫You see, this is your Z dimension.
‫It's not very visible that your Z dimension is changing, maybe only a little bit.
‫Well, in fact, yes, it is changing a little bit here.
‫The distance is a little bit bigger than, for example, here.
‫But my point is that if you put a small, negative, real number here, then it might cancel out this
‫imperfection, this one here that we talked about.
‫And now let's make our Z dimension poles just real numbers, minus one and minus two, and let's make
‫our X dimension poles complex and let's see what's going to happen then.
‫You can see that now you have oscillation and then damping out, but in the X dimension.
‫You see now this is your X dimension.
‫You don't have that in the Y dimension and you don't have that in the Z dimension because now you have
‫strong, negative, real pulse.
‫And these are the angles here, the control inputs and the omega's.
‫And now let's do the same thing, but let's give complex pause to Y dimension.
‫So we're going to comment at the X dimension and we're going to give complex pause to the Y dimension.
‫So let's see what's going to happen.
‫Well, it seems to track it pretty well.
‫You do have oscillation in the Y dimension, but it damps out pretty quickly.
‫And now let's give all palls complex pause.
‫OK, let's make sure that the X, Y and Z dimension, all of them have a negative real value and some
‫kind of imaginary value as well.
‫Let's see what's going to happen.
‫So the drone is starting to follow the trajectory and you can see that in the Z dimension, it has some
‫kind of oscillation.
‫But it's approaching the trajectory, so in the end, you have this kind of tracking, you have oscillation
‫in the X dimension, but it damps itself out thanks to negative real numbers.
‫You have the same thing in the Y Dimension Oscillation, but it disappears and you have the same thing
‫with the Z dimension where you have a little bit of oscillation and then it damps itself out thanks
‫to.
‫The negative real numbers here that are minus one and minus one, values are here quite big, so they
‫cancel out this oscillation here pretty quickly.
‫But now what happens if I put here minus 10 and also here minus 10?
‫What's going to happen?
‫The real number in the Z dimension, now it's minus 10 and now I'm going to run the code and look,
‫the code stops running.
‫So what is it?
‫Say you can't take a square root of a negative number.
‫The problem might be that the trajectory is too chaotic or it might have large discontinues jumps.
‫Try to make a smoother trajectory without discontinuous jumps.
‫Are the possible causes might be values for variables such as the sample time interval or horizon period
‫or other variables here, including poles.
‫So they might be problematic.
‫So in our case, the problem is Poles.
‫So what has happened?
‫Well, this has happened.
‫You cannot take a square root of a negative number if you remember.
‫Then in the end of the third chapter, you had to compute your omega one, Omega to Omega three and
‫Omega four.
‫And so you ended up with equations that had square root of something, a one for Megahed, two for Omega
‫three and four Megafaun, so that was the end of Chapter three.
‫So the program stops working when you have negative numbers under the square root here.
‫So what does it mean physically?
‫It means that the controllers tell you that in order to perform a maneuver at the given time, you need
‫to have control inputs.
‫You one, you to use three and you four.
‫You need to have certain values for them calculated by the controller.
‫But then the model that we have derived tells you that there is no combination of Omega one, omega
‫two, omega three and Omega four that the drone can produce to achieve the required.
‫You won you to use three and you four.
‫That's why you have negative numbers under the square root.
‫It means that there is no real solution for the Omega's that usually happens when your trajectory changes
‫very abruptly or if you give the controller very strong pulls, forcing the ovei to do two sharp maneuvers
‫that it cannot handle.
‫So your trajectory must be smooth enough and the polls need to stay within a certain area in the Laplanche
‫domain.
‫So this is your plus domain.
‫And you need to make sure that your polls don't go very far away from this origin, because even if
‫they are negative but they are very far away from this origin point, then you might force your drone
‫to perform very aggressive maneuvers and then your controller will tell you that, OK, I need this.
‫You want this, you to do this, you three and this you four.
‫But then the model will tell you that, sorry, man, I cannot give you the omega one and two and three
‫and four that will allow you to achieve your required.
‫You want you to use three and you four.
‫So the conclusion is, have your trajectories be smooth, don't have big poles, be in the negative
‫region if you want your errors to approach zero.
‫And if you don't want oscillation, then be on the negative side and on this real number line, don't
‫go into the imaginary area.
‫Right.
‫OK, so you want your error and error not to go to zero for that, you chose the appropriate lump the
‫one and lump the two, you computed the right key one and key to values.
‫And that gave you the error double dot that will then make error and error dot approach zero.
‫But this error doubled.
‫That is just a required value.
‫In order to achieve this error double dot, you need to have the right you one you to use three and
‫you four control inputs.
‫And so that means that you need to find a connection between error dot and then you one by R and FITA
‫are, which are the control inputs of this feedback lionization controller.
‫And then you one will go straight to the plant and then fire and theatergoer along with your cirE will
‫go into your NPC controller.
‫That will give you the remaining you two, three and four control inputs.
‫And so in the next video, we're going to learn how to make a connection between your required E double
‫dot and then you one fire.
‫And Sittar, thank you very much.