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‫Welcome back.

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‫And so let's go back to our cost function.

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‫This was the initial version for the vehicle.

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‫This was the cost function that we had in the previous course.

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‫The initial version and of course, now in our case, instead of these deltas here, we're going to

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‫have the you control input vectors like this, right?

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‫And obviously because of that and because of the different dimensions of the error vectors, we will

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‫have different dimensions in the weight matrix here.

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‫So the S Q and our weight matrices, these three, you will have different dimensions.

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‫However, since we started treating the input increments as our control inputs, we replace the use

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‫of Cape Plus I vectors here with.

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‫Delta use up, Kate, plus I vectors like this next.

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‫Just like in the previous course, we start manipulating the cost function.

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‫We can replace the error variables like this.

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‫So this is our error at K Plus End, which is this vector here.

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‫And that would be the reference value vector at K plus and minus C till they're times X tilde vector

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‫at K plus end.

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‫And so if we also take this vector here, then we can rewrite it like this.

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‫The reference value vector at K plus ie minus C till that times x till the vector at K plus i.

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‫The reference vector simply contain your reference values.

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‫So the reference factor at Cape Plus RN equals Phi are at Cape Plus N Seta are at Cape Plus NW and PSI

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‫are at Cape Plus and.

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‫And then right, Kate, plus eye equals fire at Capas Eye Theatre are at Kate plus I as well and I are

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‫at Kate.

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‫Plus I now remember since your input increments the deltas, since they are your control inputs, now

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‫you need to have your augmented matrices and vectors, so you need to have your seat still there and

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‫then your ex there.

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‫Not C.

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‫And X.

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‫So now this would be wrong, because now you have delta use in your cost function.

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‫And so now you take the cost function that you have here and then you will very simply replace the error

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‫vectors with these expressions here.

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‫And if you do that, then this is how your cost function will look like this one here in the parentheses.

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‫That's your e sub K plus n, and then this one here is the same thing, but transposed.

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‫And then this expression here, that's this vector.

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‫And then here, it's just the same thing, but transposed as well.

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‫And here you have your control input increments.

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‫Here you have the column vector.

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‫And then here you have the roll vector because you have this transposed here and then you have to open

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‫up the parentheses in the exact same way, like we did it in the previous course.

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‫In fact, now I would like to challenge you.

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‫I would like to give you a big exercise, and if you do then exercise, then you will for sure, solidify

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‫your knowledge about the controller.

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‫In the previous course, I showed you how you can go from this original form, this original cost function

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‫that you had until this final form of the cost function.

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‫In this final cost function form, you had huge matrices like H Double Bar and then F Double Bar transposed.

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‫That was the final cost function form here.

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‫And you can see that you have a constant value here, which is your present state.

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‫It's not a variable because you know this value already, it's a specific number, your present state

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‫now and then also your reference vectors.

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‫They are also known, so they are constant values.

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‫They are not variables.

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‫So in this final cost function form, the only independent variables that you had was the delta deltas.

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‫And then you took the gradient of it.

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‫And then once you had your gradient, you were able to use this linear algebra technique where you just

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‫take the inverse of your h double bar matrix and then you compute your control inputs sequence like

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‫this.

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‫So then was the stuff from the previous course.

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‫So my exercise for you is this I've already done the first step.

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‫I have replaced the deltas with the two vectors, and then I've also made them increment you vectors.

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‫So I've put deltas here and I've also rewritten the cost function like this where I have replaced the

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‫error variables.

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‫So from this point on trying to achieve this form only for the UAV.

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‫So instead of Delta Delta equals something you would have Delta, you equals something.

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‫But don't just write this expression, go through.

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‫The entire derivation that we had in the previous course used the material from the previous course

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‫as much as needed and go through the entire derivation process until you reach this form only for the

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‫US case now.

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‫So you would start by opening up the parentheses and you do it exactly like you did it in the previous

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‫course.

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‫And again, you don't have to have it all memorized if you don't remember it.

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‫Then just look it up from the previous course and go through the exact same steps like you went through

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‫in the previous course and then try to get this final form for the UAV.

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‫And then in the next two videos, I will show you how it should be done.

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‫It really will be like a recap of the previous course when it comes to the embassy controller derivation.

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‫So thank you very much and see you in the next video.