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‫Welcome back.

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‫But now, if you remember, we could rewrite this vector here like this.

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‫But now everything have till date because they are all augmented.

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‫So this matrix here was called C Double Bar.

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‫And then this one here was called like this a double hat.

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‫So be till there, then a till the times b till there, then a till the squared times B till there and

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‫then a till the cubed times B till there.

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‫They all have dimensions nine by three.

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‫That means that C double bar is thirty six by 12 and then a till there and then a till the squared,

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‫a tilde cubed and then a tilde to the power of four.

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‫They were all nine by nine.

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‫So that means that this entire global matrix, a double had it's going to be 36 rows and nine columns.

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‫So essentially, you have rewritten this vector here x still the global in terms of augmented presence,

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‫state vector, which is this one and the control input increments like these ones here.

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‫So you rewrite this vector like this and then this vector transposed, well, it's this one here.

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‫And then again, this vector here, you rewrite it like this and remember that this c double bar and

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‫a double hat, they use easier assumption.

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‫The A Tilde Matrix does not get updated here as we predict future states from K plus one up until K

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‫plus four.

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‫And by the way, if in the future you have to use this non simplified LP and PC version, then here

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‫again, when you augment your system, meaning that you go from U2, you three and you for two Delta,

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‫U2, Delta you three and Delta you four.

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‫Then of course, in here you will have deltas everywhere like this.

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‫So Delta, you zero, delta, you want Delta, you two and Delta you three, and then all your mattresses

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‫will be augmented and also all your air mattresses will be augmented, just like we had discussed it

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‫here.

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‫You see, this is your eighth.

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‫Will there be till there?

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‫See Tilda and Tilda, you're eight.

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‫Till that consists of these discrete A and B matrices and also your zero matrix and then identity matrix.

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‫Your B tilde has this discrete B matrix and an identity matrix and then your C tilde.

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‫It contains your discrete C matrix and then a zero matrix and then your D tilde.

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‫It's just a zero matrix that you can neglect.

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‫OK, but now, since your control inputs are delta use and not use, then you might ask how you update

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‫your global omega in the matrix.

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‫You got your omega zero global from the plant in the previous loop.

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‫But remember, you also had to obtain Omega one global omega two global in omega three global that you

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‫would put inside your A1 matrices, a two matrices and a three matrices.

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‫Just like this one went into your A0 matrix.

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‫Well, remember you need it, you one, you two three, and you fought for that.

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‫And so you won.

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‫That was always the same.

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‫It only changed after each outer loop, but during the inner loop iterations you won, that came from

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‫your state feedback minimization controller.

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‫During the inner loops, it remained unchanged.

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‫But in order to get omega one global, you will use to Q3 and Q4 values from this predicted augment

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‫the state vector X till the sub one.

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‫As you can see, I've also put Tilda's here and also here, where your present state vector is in order

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‫to indicate that they are augmented.

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‫So remember X still that one?

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‫It's this one here.

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‫And so your omega one global, it will depend on the unchanged u one and these values here and now in

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‫order to get actual.

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‫Until then, you will also use this Delta Use Zero, which is the first left over from the previous

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‫loop.

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‫So in the previous loop, you received this solution from your AMPK controller.

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‫You only used this one here.

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‫You did not use this one here.

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‫And so this was your first left over in the previous loop.

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‫But now in this loop, it became Delta U zero, just like this one became Delta U minus one.

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‫And so you use this one here for this Delta use zero in order to obtain your ex tilde one.

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‫And then from that, you will get these utu you three and you for values.

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‫And together with the unchanged u one value, you will get your omega one global.

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‫Similarly, in order to get your Omega two global, you will use the unchanged U one value and then

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‫you will have ex the two your predicted.

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‫AUGMENT the state vector two time steps into the future, and it looks like this.

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‫And so you will also use these values together with your unchanged U.

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‫One value in order to obtain your omega two global and in order to get your ex still the two.

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‫You will also use this variable here, which is your second left over from the previous loop.

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‫And so in this loop, it would be Delta U.

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‫One.

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‫And so this variable, it goes here, and that's how you get your ex tilde to finally, in order to

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‫get your omega three global for that, you are going to need it.

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‫Still, the three and it looks like this.

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‫And so in order to get your omega three global again, you use your unchanged you one value and these

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‫values here and also in order to compute your ex.

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‫Still, the three for that, you will take your final left over from the previous loop, which was Delta

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‫you three there.

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‫But now in this loop, it's Delta, you two.

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‫And so you take this variable and you put it here where your delta you two is, and that's how you obtained

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‫X2.

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‫Still the three.

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‫And that's where we could stop, because with all that, we could obtain our A0, A1, A2 and A3 matrices

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‫for the non simplified NPV method.

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‫But the thing is that you don't need to put till this year in order to know whether your mattresses

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‫are augmented or not.

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‫And that's because when you see in the equations that you have Delta use zero delta, you want Delta,

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‫you two and Delta you three.

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‫In other words, when you see that you're working with these control inputs and not with these control

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‫inputs, then you automatically know that you're a bee.

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‫And then also see matrices are augmented.

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‫You don't have to put till this year.

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‫It's like we did it with this discrete versus continuous version.

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‫This was your discrete format.

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‫This was your continuous format.

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‫When your equations were in this form where you had x dot equals and then everything else.

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‫Then you knew that your A and B matrices that they were continuous.

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‫But when your equations were in this form, so you had XSET K plus one equals and then here you had

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‫exit K and you had K, then you knew that your A and B matrices that they were discrete, they were

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‫discrete ties and the relationships between the discrete ties and continues a, b and C matrices.

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‫They were derived in the first car course and you can see their relationships here.

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‫These could equals all this and this is your continuous.

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‫A discrete B equals all this.

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‫This is your continuous B and then discrete C equals continuous C.

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‫So you did not need to write a D or B D or c d for your discrete matrices because you saw it from the

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‫equations you were working with.

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‫And the same thing with system augmentation.

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‫If your system is in this form, then it's not augment it.

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‫But if it's in this form, then it is augmented.

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‫Why?

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‫Because here you have use up K and here you have delta use of K.

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‫So you know that these two a matrices, they are not the same.

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‫And you know that these two B matrices, they are not the same.

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‫Just like you knew that these two matrices are not the same.

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‫And these to be matrices are not the same in the first one.

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‫This a is a discrete a matrix.

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‫And this b that these could be matrix because your control input is use of K.

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‫But in the second equation, your control input is delta.

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‫Use of K the change of your control input.

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‫That's why you know that this a matrix.

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‫It's actually your augmented a matrix.

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‫And this matrix, it's your augmented B matrix.

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‫But you don't need to put these till this here, because the fact that you have delta use of K here,

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‫it already tells you that you're A and B matrices and also your C matrix that they are all augmented.

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‫So you don't actually need to put till this year.

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‫You know that you are working with an augmented system because you can see deltas here.

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‫You know that this guy, this guy and this guy, they are both discrete A, B and C matrices because

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‫here you have K plus one K and K, and they are also augmented A, B and C matrices because you have

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‫delta here.

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‫And next, we will again open up the parenthesis, just like in the previous course.

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‫And so if you now have your prime cost function and you open up the parentheses here, then this is

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‫what you will get.

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‫It's exactly like in the previous course only instead of deltas you have use.

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‫But the logic and the procedure is exactly the same.

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‫And again, you will have some constant terms that you don't need for optimization purposes.

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‫So you can just say that they are zero.

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‫You can just lose them.

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‫And if you do that, if you lose those constant terms, then you will have a new cost function, a cost

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‫function without these constant terms.

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‫And we're going to name that J double prime.

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‫And then exactly like you did in the previous course, you take this J double prime that you have here,

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‫which is this one here and you put it in a form where your control input increments are like that.

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‫So you have a control input increment here.

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‫And then here you have the same thing, but transposed and then in the middle you have a collection

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‫of matrices.

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‫If you put them together like this, you will get your h double bar.

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‫And then you take the remaining terms and you make sure that your control input increment is here.

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‫Your present state vector, which is augmented, is here, then your reference value vector is here.

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‫And then in the middle, you will have this collection of matrices that we called f double bar transposed.

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‫And the same thing happens for us is just instead of Delta Delta us, you're going to have Delta use.

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‫And so that's how your final G double prime cost function looks like.

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‫That's its final form.

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‫This vector here is 12 by one.

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‫This one here is one by 12.

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‫That means that you're h double bar is 12 by 12.

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‫Then this one is one by nine.

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‫This one is one by 12.

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‫That means that this entire thing is one by twenty one.

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‫This one is twelve by one.

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‫That means that this one is twenty one by twelve.

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‫That means that this entire thing here will be one by one because you would cancel out 20 1s and 12s

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‫like this.

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‫And of course, here you would also have one by one because you would cancel out these buttons and these

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‫12s like this.