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‫Welcome back.

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‫So let's start with the derivation then, or to be more precise, a recap of the derivation.

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‫So this is the screenshot from the previous course.

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‫And then what I'm going to do here, I'm going to replace this delta with a U.

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‫And also this delta here with the U.

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‫So it's going to be Delta U at Karplus I vector and then here Delta U at K plus I vector transposed.

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‫Now in our case, in the other case, the horizon period and therefore end equals four.

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‫Then here in green you replace the error variables that are here inside the parentheses.

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‫So here again, I'm going to replace this delta with a U.

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‫And here as well, this is going to be Delta U at K plus I and here Delta you add K plus I transposed.

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‫And so when you took this thing here, this parenthesis which was transposed, then you could write

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‫it out like this, like we did it in the previous course.

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‫So this is a screenshot from the previous course is the same thing.

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‫The only difference now is that inside your reference vectors and augment state vectors, you have different

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‫variables there and also the dimensions of the matrices, like Kielder, for example, they are different.

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‫And also instead of Delta deltas, you have Delta use here.

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‫And so when you have this expression transposed, then what you can do, you can rewrite it like this,

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‫where you distribute this transpose sine among both terms.

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‫And then here, when you open up this parenthesis with this transpose sine, then according to linear

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‫algebra, this augment is state vector.

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‫That was the second variable here.

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‫It will be the first variable here and then the C tilde, it will be the second.

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‫And also both of them will be transposed as well.

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‫Right.

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‫And then you took this entire first term here in purple and then you rewrote it like this.

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‫You simply open up the parentheses and that's it.

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‫Since your s matrix here was diagonal, you could rewrite this thing like this, which was exactly the

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‫same term, like this one here.

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‫And then you could put them together.

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‫And then that means that this one half here and one half here would disappear.

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‫And now, since you have rewritten this term here that I have encircled with this purple line, since

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‫you've been able to rewrite it like this, then you could rewrite the entire cost function in this form

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‫where I'm just going to make a small correction instead of Delp.

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‫I'm going to put you here and also here instead of Delta.

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‫I'm going to put you here.

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‫So Delta you at K plus I vector and here the same thing only transposed.