‫Welcome back.
‫Since our horizon period equals four in our case here, then the global reference vector, the global
‫augmented state vector and the global control input vector, which is the control moment increment,
‫would be like this, respectively.
‫That's your global reference vector.
‫That's your global augmented state vector, and that's your global control moment increment vector.
‫So let's see how an element in each of these vectors looks like.
‫Let's, for example, pick this element.
‫Here are some key plus two equals.
‫Five are at case plus two seats are at K plus two and Passi are at K plus two.
‫So that's how this vector looks like.
‫If you remember, then our Feed the calendarization controller gave these reference values to the NPC
‫controller Phi Are, Sittar are and then Passi are.
‫So it gave an array of five values to the NPC controller.
‫Now this global reference vector, it only has the future reference values.
‫So for FY R it would be this for a seat R it would be this and for PCR it would be this.
‫So these four elements here, these are these four points here.
‫And so if we take this element here then this Phi are at K plus two, it would be this one here, theta
‫art K plus two would be this one here and then are at K plus two would be this one here because here
‫you have K then this point here that would be your K plus one, then this point here would be K plus
‫two, then this point here would be K plus three.
‫And finally the final point would be K plus four like this.
‫And you can also see that these reference values for Phi R are all equal and the same thing for Theta,
‫R and R.
‫That means that Phi are at K plus one which is inside this vector here equals Phi are at K plus two,
‫which is inside this vector here equals Phi are at K plus three, which is inside this vector here and
‫that equals Phi are at K plus four which is inside this sub vector here.
‫And then of course equals Phi R at K which is inside your let's say, present reference value.
‫But this element here is not present in this global reference vector here because it doesn't make sense,
‫because you need reference values for the future.
‫You already have your present state value x2 that at K you need your future reference values in order
‫to compute your new augmented state values.
‫And you can see that they're all equal because you have a straight line here.
‫And the same thing is true for theta hours and hours.
‫You have a straight line here and here and that's why you would have the same kind of equality here.
‫Now, since our sub vector here is three by one, that means that this global reference value vector
‫is twelve by one twelve rows and one column.
‫Let's now take an augmented state vector, this one here at K plus three.
‫Let's see how it looks like.
‫That's how it looks like.
‫You see you have your six states here that your X at K plus three and this is your U vector at K plus
‫to remember in order to get your you add K you have you at.
‫K minus one, plus Delta, you at K like this?
‫Or you at now equals you at now, minus one, plus Delta you at now, so your previous control input
‫was here inside this augmented state vector and since the subsector is nine rows in one column, then
‫this entire global augmented state vector will be thirty six rows and one column because you have nine
‫rows in one sub vector but you have four vectors here.
‫And finally, let's take a look at this element here.
‫Delta U at K and here it is, Delta you two at K, Delta U, three at K and Delta you four at K.
‫Remember that they are control moment increments because your Q2, Q3 and Q4, there are moments right
‫there, control moments there.
‫Knewton times meaders.
‫So since you have Delta here than their control moment increments.