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‫Welcome back.

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‫So let's solve this exercise now, the task was to put these equations into this vector matrix form.

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‫And so let's first take this first role.

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‫So you have your X plus one, which is this one here.

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‫Your X up key would be here.

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‫Use of K minus one would be here and then Delta use up K, it would be here.

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‫So that means that this a here it will go here.

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‫This B here it will go here and the same B which is here, it will also go here.

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‫And now we take the second equation, USB key which is here use K minus one, that's this one here.

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‫And Delta use up K, that's this one here.

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‫Note that use of K is a three by one vector, three rows, one column and the same thing is this one

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‫use of K minus one three by one vector.

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‫The state vector is of course six by one because we had six states, this one as well.

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‫That means that this a matrix, it's six by six and this B matrix is six by three because you would

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‫multiply this B by this three by one vector.

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‫That means what you have here is an identity matrix, which in the scalar world would be one, and since

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‫this is three by one and this is three by one, that means that your identity matrix would be a three

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‫by three identity matrix because you would multiply it.

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‫This one here, that's your identity matrix.

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‫You would multiply by this vector, which is this one here and the same thing here.

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‫You also have an identity matrix here.

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‫That means that you have a three by three identity matrix as the second element in this global matrix.

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‫It's like a sub matrix within a big global matrix.

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‫And well, since you have three roles in this vector, in this use of K vector, but this a matrix has

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‫six columns, then that means that you will have a zero matrix here with three rows and six columns.

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‫And now what about the output?

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‫Remember that you have six states and then out of those six states you would select three outputs and

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‫you would do that with a C matrix.

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‫So you would have to multiply the C matrix by your state vector, which is here.

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‫And since you have three outputs and six states, then the C matrix will be three by six.

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‫So this three by six C matrix will come here.

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‫Now you have three outputs and then three control inputs.

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‫That means that you will have a three by three zero matrix.

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‫And when it comes to your output, then there is no contribution from this vector here.

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‫So three outputs and then three delta use.

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‫That means that you will have a three by three.

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‫Zero matrix, in other words, you can just ignore this term, we would call this Matrix D still there,

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‫but you can ignore it, but you cannot ignore other matrices.

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‫This global matrix here, we call it a tilde.

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‫It's an augmented a matrix.

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‫This is beating the Yog, meant to be matrix.

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‫And this is C Tilde, the augmented C matrix.

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‫This here would be your augmented presence state vector.

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‫This would be your augmented state vector.

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‫But the zero point one seconds in the future, the new augmented state vector.

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‫Again, this is the augmented presence state vector.

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‫And so this form here, it's equivalent to this form here.

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‫We have simply put it in the vector matrix form.

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‫So, you know, the dimensions of the sub matrices, A, B, C, the identity matrix and also sub vectors.

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‫So you had a six by one state vector and then a three by one control input vector and then also a three

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‫by one, the increment of the control input vector.

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‫But the dimensions of the augmented vectors and matrices are the following.

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‫This one here now has nine rows and one column.

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‫So it's this one, the same thing for this thing.

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‫The augmented state vector has nine rows in one column.

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‫Now, the output, of course, was three by one.

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‫Now, since you have nine rows in one column here and nine rows in one column here, then this a there,

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‫it will be a nine by nine matrix.

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‫And you can also see from the sub matrices if you add up their dimensions.

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‫Now, since you have three rows in your delta, you vector and then nine rows in your augment a state

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‫vector, then that means that your be toolbar is nine by three and then you have three rows in your

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‫output vector and then nine rows in your augment a state vector.

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‫That means that your Kielder is three by nine, three rows, nine columns.

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‫So remember when you see a still there, it means augmented matrices and augmented state vectors.

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‫In addition, remember one thing since the Horizon period of our NPC is for your NPC will find a sequence

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‫of control inputs that looks like this.

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‫That's what your NPC will find.

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‫So if we take a random element here just to see how it looks like, then it looks like this delta you

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‫to add K plus to Delta U three at K plus two and Delta U for at K plus two.

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‫So that's how each element looks like and therefore, since this one is three by one, well, then you

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‫know that this entire vector will be 12 by one, 12 rows and one column.

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‫However, remember that in each NPC duration, only the first sub vector is considered and the rest

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‫is neglected.

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‫And also remember that the plant has to receive the real control inputs, not its increments.

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‫So it has to receive use up key and not Delta use up.

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‫That's not what the plant wants.

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‫The plant wants this, but you can achieve that very easily with this operation.

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‫Use up K equals use up K minus one plus Delta.

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‫You sup K or in the vector matrix form it would be like this use of K equals use up K minus one plus.

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‫Then you have a matrix here and this is your global control input sequence.

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‫It's this one here.

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‫So if you want to extract this element here, then you will have an identity matrix here, which is

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‫three by three.

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‫And then in order to cancel out the other elements, you just need to have zero matrices here.

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‫A three by three zero matrix for this one, a three by three zero matrix for this one here, and a three

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‫by three zero matrix for this one here.

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‫So this term here is this term here and then this entire term here, it would be this term here like

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‫this.

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‫And then you get this vector and this vector will go into your plant in the vehicle case, your control

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‫input was a scalar.

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‫So you could do this extraction with a simple roll vector with ones and zeros.

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‫So that's how we did it.

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‫In the previous course.

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‫You didn't have an identity matrix, you just had one and then you had zeros here because these elements

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‫here, they were scalars, they were not vectors, they were scalars.

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‫All right.

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‫So in the next video, we're going to go back to our cost function and we're going to continue working

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‫with that.

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‫Thank you very much.