1 00:00:00,420 --> 00:00:01,210 ‫Welcome back. 2 00:00:01,680 --> 00:00:13,530 ‫Since our horizon period equals four in our case here, then the global reference vector, the global 3 00:00:13,530 --> 00:00:23,730 ‫augmented state vector and the global control input vector, which is the control moment increment, 4 00:00:24,180 --> 00:00:26,160 ‫would be like this, respectively. 5 00:00:26,910 --> 00:00:29,070 ‫That's your global reference vector. 6 00:00:29,910 --> 00:00:37,370 ‫That's your global augmented state vector, and that's your global control moment increment vector. 7 00:00:38,220 --> 00:00:43,980 ‫So let's see how an element in each of these vectors looks like. 8 00:00:44,940 --> 00:00:48,180 ‫Let's, for example, pick this element. 9 00:00:48,190 --> 00:00:52,110 ‫Here are some key plus two equals. 10 00:00:52,620 --> 00:01:02,640 ‫Five are at case plus two seats are at K plus two and Passi are at K plus two. 11 00:01:03,390 --> 00:01:06,180 ‫So that's how this vector looks like. 12 00:01:07,050 --> 00:01:15,450 ‫If you remember, then our Feed the calendarization controller gave these reference values to the NPC 13 00:01:15,450 --> 00:01:21,050 ‫controller Phi Are, Sittar are and then Passi are. 14 00:01:21,810 --> 00:01:27,180 ‫So it gave an array of five values to the NPC controller. 15 00:01:27,990 --> 00:01:34,130 ‫Now this global reference vector, it only has the future reference values. 16 00:01:34,980 --> 00:01:45,620 ‫So for FY R it would be this for a seat R it would be this and for PCR it would be this. 17 00:01:46,410 --> 00:01:51,170 ‫So these four elements here, these are these four points here. 18 00:01:52,050 --> 00:02:02,250 ‫And so if we take this element here then this Phi are at K plus two, it would be this one here, theta 19 00:02:02,250 --> 00:02:12,180 ‫art K plus two would be this one here and then are at K plus two would be this one here because here 20 00:02:12,180 --> 00:02:19,470 ‫you have K then this point here that would be your K plus one, then this point here would be K plus 21 00:02:19,470 --> 00:02:23,790 ‫two, then this point here would be K plus three. 22 00:02:24,420 --> 00:02:29,760 ‫And finally the final point would be K plus four like this. 23 00:02:30,630 --> 00:02:40,120 ‫And you can also see that these reference values for Phi R are all equal and the same thing for Theta, 24 00:02:40,140 --> 00:02:41,700 ‫R and R. 25 00:02:42,480 --> 00:02:55,530 ‫That means that Phi are at K plus one which is inside this vector here equals Phi are at K plus two, 26 00:02:55,530 --> 00:03:06,240 ‫which is inside this vector here equals Phi are at K plus three, which is inside this vector here and 27 00:03:06,240 --> 00:03:14,670 ‫that equals Phi are at K plus four which is inside this sub vector here. 28 00:03:15,270 --> 00:03:26,400 ‫And then of course equals Phi R at K which is inside your let's say, present reference value. 29 00:03:26,400 --> 00:03:34,590 ‫But this element here is not present in this global reference vector here because it doesn't make sense, 30 00:03:34,590 --> 00:03:37,590 ‫because you need reference values for the future. 31 00:03:38,250 --> 00:03:47,430 ‫You already have your present state value x2 that at K you need your future reference values in order 32 00:03:47,430 --> 00:03:50,580 ‫to compute your new augmented state values. 33 00:03:51,330 --> 00:03:54,630 ‫And you can see that they're all equal because you have a straight line here. 34 00:03:54,990 --> 00:04:00,870 ‫And the same thing is true for theta hours and hours. 35 00:04:01,410 --> 00:04:07,770 ‫You have a straight line here and here and that's why you would have the same kind of equality here. 36 00:04:08,610 --> 00:04:17,220 ‫Now, since our sub vector here is three by one, that means that this global reference value vector 37 00:04:17,220 --> 00:04:23,120 ‫is twelve by one twelve rows and one column. 38 00:04:23,970 --> 00:04:31,290 ‫Let's now take an augmented state vector, this one here at K plus three. 39 00:04:31,500 --> 00:04:33,720 ‫Let's see how it looks like. 40 00:04:34,410 --> 00:04:36,240 ‫That's how it looks like. 41 00:04:36,420 --> 00:04:51,330 ‫You see you have your six states here that your X at K plus three and this is your U vector at K plus 42 00:04:51,330 --> 00:04:59,580 ‫to remember in order to get your you add K you have you at. 43 00:05:00,000 --> 00:05:07,500 ‫K minus one, plus Delta, you at K like this? 44 00:05:08,550 --> 00:05:26,820 ‫Or you at now equals you at now, minus one, plus Delta you at now, so your previous control input 45 00:05:27,360 --> 00:05:37,920 ‫was here inside this augmented state vector and since the subsector is nine rows in one column, then 46 00:05:37,920 --> 00:05:47,790 ‫this entire global augmented state vector will be thirty six rows and one column because you have nine 47 00:05:47,790 --> 00:05:52,200 ‫rows in one sub vector but you have four vectors here. 48 00:05:52,920 --> 00:05:57,380 ‫And finally, let's take a look at this element here. 49 00:05:58,080 --> 00:06:07,470 ‫Delta U at K and here it is, Delta you two at K, Delta U, three at K and Delta you four at K. 50 00:06:08,220 --> 00:06:14,970 ‫Remember that they are control moment increments because your Q2, Q3 and Q4, there are moments right 51 00:06:14,970 --> 00:06:16,740 ‫there, control moments there. 52 00:06:16,740 --> 00:06:18,960 ‫Knewton times meaders. 53 00:06:19,410 --> 00:06:23,820 ‫So since you have Delta here than their control moment increments.