1 00:00:00,620 --> 00:00:01,460 ‫Welcome back. 2 00:00:02,150 --> 00:00:09,380 ‫And now let's continue with our inner loop now we're here in MPAC simplification and we're going to 3 00:00:09,380 --> 00:00:12,410 ‫enter this NPC simplification function. 4 00:00:13,080 --> 00:00:16,760 ‫So this function is here, support that NPC simplification. 5 00:00:17,150 --> 00:00:24,800 ‫It takes in your discrete ABC and the matrices and your horizon period, and it gives you these huge 6 00:00:24,800 --> 00:00:33,170 ‫matrices each double bar, then double bar transposed, then C double bar and then A with two hats. 7 00:00:34,010 --> 00:00:41,900 ‫So this NPC simplification function, I literally copied it from my previous course. 8 00:00:42,710 --> 00:00:50,420 ‫I did not change anything in this function compared to the previous course, because in the previous 9 00:00:50,420 --> 00:00:53,570 ‫course, I built it in a scalable way. 10 00:00:53,600 --> 00:01:00,410 ‫So if I have bigger matrices, then this function will take that into account automatically. 11 00:01:01,100 --> 00:01:09,320 ‫Remember that the point of the NPC simplification was to go from your discrete A, B, C and D matrices. 12 00:01:10,550 --> 00:01:16,100 ‫To first of all, these matrices, que double bar te double bar are double bar. 13 00:01:17,120 --> 00:01:18,450 ‫So you can see them here. 14 00:01:18,470 --> 00:01:21,200 ‫Cue double bar, T double bar, double bar. 15 00:01:21,650 --> 00:01:23,480 ‫And that's the process for that. 16 00:01:23,510 --> 00:01:30,500 ‫So first of all, you take your disk with a b, c d matrices, then you augment them. 17 00:01:31,520 --> 00:01:34,880 ‫And remember that this is the augmentation process. 18 00:01:35,180 --> 00:01:42,890 ‫So you see, I create these eight will then be till the C deal, then detailed the like this and I'm 19 00:01:42,890 --> 00:01:44,960 ‫doing the same thing here in the code. 20 00:01:45,770 --> 00:01:47,260 ‫Then these ones, here they are. 21 00:01:47,280 --> 00:01:50,990 ‫My MBC weight matrices, QSR. 22 00:01:52,070 --> 00:01:54,590 ‫And then I also need these terms here. 23 00:01:54,890 --> 00:02:02,360 ‫C tilde transpose times, queue times, C2 and then also see till there, transpose times s time still 24 00:02:02,360 --> 00:02:12,500 ‫there and then also queue times Ctrl there and then also s time see tilde and all that is created here 25 00:02:13,310 --> 00:02:14,240 ‫in this block. 26 00:02:15,200 --> 00:02:18,380 ‫Then queue, double bar, double bar and double bar. 27 00:02:19,220 --> 00:02:20,960 ‫That's this one here. 28 00:02:21,260 --> 00:02:24,950 ‫Well, actually here I simply create the big zero matrices. 29 00:02:25,760 --> 00:02:31,730 ‫And then here in this loop, I will start filling in these zero matrices. 30 00:02:32,660 --> 00:02:40,490 ‫So all these elements in these big matrices, these diagonal elements, they will enter into these Q 31 00:02:40,490 --> 00:02:53,330 ‫double bar and T double bar matrices here in this loop in your RW bar matrix is then created here, 32 00:02:53,330 --> 00:02:55,460 ‫which is inside this loop as well. 33 00:02:56,330 --> 00:03:01,670 ‫And then I also need my C double bar matrix and then a with two hats. 34 00:03:01,820 --> 00:03:06,470 ‫So these are these matrices here, and they are created here. 35 00:03:07,190 --> 00:03:14,360 ‫So for example, if you take a look at this C double bar matrix, then it becomes pretty long. 36 00:03:15,260 --> 00:03:19,010 ‫But these are just positions, so it's c double bar matrix. 37 00:03:19,700 --> 00:03:23,150 ‫And this would be the role for this matrix. 38 00:03:23,360 --> 00:03:27,350 ‫And then this would be the column for this matrix. 39 00:03:27,650 --> 00:03:36,080 ‫And then that element in the matrix equals this, which is some kind of element like, for example, 40 00:03:36,080 --> 00:03:40,040 ‫b tilde or maybe a deal, the squared times B tilde. 41 00:03:40,820 --> 00:03:48,500 ‫And then once they have those matrices, then I also need to get my h double bar and f double bar transposed. 42 00:03:49,400 --> 00:03:51,800 ‫And so I get them here. 43 00:03:52,040 --> 00:03:59,180 ‫Once I have my other matrices, I can construct my h double bar and then f double bar transposed, then 44 00:03:59,180 --> 00:04:00,410 ‫happens all here. 45 00:04:01,310 --> 00:04:09,050 ‫So, for example, h double bar equals this c double bar transpose times Q Double Bar Times C Double 46 00:04:09,050 --> 00:04:10,850 ‫Bar, plus R double bar. 47 00:04:11,750 --> 00:04:14,330 ‫And with these two operations, I will get that. 48 00:04:15,170 --> 00:04:23,570 ‫So first of all, I multiply the CW bar transpose with Q Double Bar and then whatever I get here, I 49 00:04:23,570 --> 00:04:28,460 ‫multiply by C Double Bar and then I add this double bar. 50 00:04:29,300 --> 00:04:37,040 ‫And well, these four lines here they are here to create this f double bar transposed matrix. 51 00:04:37,880 --> 00:04:45,110 ‫And then you give all these matrices to the main file and your main file receives each double bar, 52 00:04:45,320 --> 00:04:49,340 ‫double bar transpose C double bar and A with two hats. 53 00:04:50,240 --> 00:04:55,910 ‫And then if you take this row vector and you multiply it by f double bar transpose. 54 00:04:56,910 --> 00:04:59,970 ‫Then you will get your small f transposed. 55 00:05:00,360 --> 00:05:02,670 ‫So it's this one here. 56 00:05:03,450 --> 00:05:08,910 ‫So this role, that's this vector times this matrix. 57 00:05:09,810 --> 00:05:17,610 ‫And then to get your delta use, you had to take the gradient of your cost function and then you had 58 00:05:17,610 --> 00:05:21,360 ‫to take the inverse of your h double bar matrix. 59 00:05:22,140 --> 00:05:24,270 ‫And then you also had the minus sign here. 60 00:05:25,370 --> 00:05:35,920 ‫And then this f double bar times this vector that will be the transpose of this small F.T. 61 00:05:36,800 --> 00:05:41,300 ‫And so this formula here that you see in the code. 62 00:05:41,900 --> 00:05:43,400 ‫The formula is this one. 63 00:05:44,150 --> 00:05:47,780 ‫So you see I take the inverse of the double bar matrix. 64 00:05:48,440 --> 00:05:51,080 ‫I also take the transpose of my F.T. 65 00:05:51,530 --> 00:05:56,750 ‫And then I multiply them together and I put the minus sign here as well. 66 00:05:57,350 --> 00:05:58,970 ‫And then I get my dues. 67 00:05:59,870 --> 00:06:01,400 ‫Remember that here. 68 00:06:01,670 --> 00:06:03,900 ‫My due is not a scalar. 69 00:06:03,920 --> 00:06:05,060 ‫It's a vector. 70 00:06:05,930 --> 00:06:15,140 ‫And not only I have four time instances here, not only I have their K and then keep one and then K 71 00:06:15,140 --> 00:06:18,290 ‫plus two and then K plus three. 72 00:06:19,040 --> 00:06:24,840 ‫But for each time period, I have three different control input increments. 73 00:06:24,860 --> 00:06:34,460 ‫So for example, four K plus two I have Delta You two, Delta U three and Delta U four. 74 00:06:35,150 --> 00:06:38,590 ‫And that's true for K K plus one, K plus two and K plus three. 75 00:06:38,600 --> 00:06:43,280 ‫That means that this vector will have twelve elements. 76 00:06:44,240 --> 00:06:48,710 ‫But of course, for the plant, I need the original control inputs. 77 00:06:49,820 --> 00:06:53,180 ‫So I take my previous you to Q3 and Q4. 78 00:06:54,230 --> 00:06:59,600 ‫And then out of this deal vector, I take the first element for each control input. 79 00:07:00,770 --> 00:07:03,290 ‫So essentially, I'm doing this thing here. 80 00:07:03,920 --> 00:07:07,400 ‫You see, I only want to take this vector. 81 00:07:07,400 --> 00:07:11,570 ‫I have this big vector and then I have the small sub vectors. 82 00:07:12,230 --> 00:07:15,440 ‫And here you have an example four K plus two. 83 00:07:15,470 --> 00:07:17,390 ‫That's how this vector looks like. 84 00:07:18,050 --> 00:07:20,630 ‫And I only want to take this first vector. 85 00:07:21,720 --> 00:07:31,440 ‫And in the cold, I do it here and then here in this Utah little matrix, I keep track of all my control 86 00:07:31,440 --> 00:07:34,200 ‫inputs in order to be able to plot them later. 87 00:07:35,040 --> 00:07:39,640 ‫So you see, whenever I have my new, you want you to use three and you four. 88 00:07:39,840 --> 00:07:43,710 ‫They get concatenated to this you total matrix.