1 00:00:00,730 --> 00:00:08,950 ‫But remember, x tilde global equals C double bar times, Delta, you global. 2 00:00:09,610 --> 00:00:13,780 ‫Plus a double circle Flex Times X to the zero. 3 00:00:14,500 --> 00:00:20,920 ‫That means that y global star, it equals C till the star. 4 00:00:20,950 --> 00:00:22,900 ‫Global Times. 5 00:00:22,900 --> 00:00:29,050 ‫X still the global or it equals C to the star. 6 00:00:29,060 --> 00:00:32,830 ‫Global Times C Double Bar. 7 00:00:33,490 --> 00:00:33,940 ‫Times. 8 00:00:33,940 --> 00:00:34,990 ‫Delta you global. 9 00:00:35,110 --> 00:00:37,590 ‫Plus C till Death Star. 10 00:00:37,600 --> 00:00:39,490 ‫Global Times. 11 00:00:39,490 --> 00:00:42,100 ‫A Double Circle Flex Times. 12 00:00:42,100 --> 00:00:43,150 ‫It's still the zero. 13 00:00:43,780 --> 00:00:48,430 ‫Remember that your C double bar and a double circle flex. 14 00:00:48,790 --> 00:00:54,190 ‫These are these matrices here, c double bar and a double circle flex. 15 00:00:54,790 --> 00:00:57,610 ‫And so what we have done here? 16 00:00:58,240 --> 00:01:02,140 ‫We did the exact same thing in the previous car course. 17 00:01:02,800 --> 00:01:07,210 ‫You see why global star equals and then all this. 18 00:01:07,840 --> 00:01:11,410 ‫We took this c till the global star. 19 00:01:11,950 --> 00:01:14,950 ‫We multiplied it by this vector here. 20 00:01:15,640 --> 00:01:22,930 ‫That means that this red matrix it got distributed among these two terms here. 21 00:01:23,530 --> 00:01:25,630 ‫This is your C double bar matrix. 22 00:01:26,290 --> 00:01:30,430 ‫This is your a double circle flex from the previous car course. 23 00:01:31,090 --> 00:01:38,590 ‫And then when you performed this multiplication, then you can see how these C2, the star matrices, 24 00:01:39,160 --> 00:01:43,120 ‫how they were distributed inside these matrices here. 25 00:01:43,690 --> 00:01:50,290 ‫And as you can see in the previous car course, we used the non simplified LP approach. 26 00:01:50,530 --> 00:01:56,740 ‫That's why your C double bar and a double serve conflicts matrices are a bit different there. 27 00:01:57,280 --> 00:02:05,950 ‫We had to do it that way because our horizon period in that course was considerably longer than in this 28 00:02:05,950 --> 00:02:06,460 ‫course. 29 00:02:07,090 --> 00:02:20,080 ‫And now, since our y star global vector contains the predicted U2 Q3 and you values one, two, three 30 00:02:20,080 --> 00:02:29,040 ‫and four time steps into the future, what I can do now I can reformulate my constraints like that y 31 00:02:29,050 --> 00:02:34,870 ‫global star, minimum y global star and y global star maximum. 32 00:02:35,470 --> 00:02:43,810 ‫So this vector here, it simply contains all the minimal values that I can have here in this vector. 33 00:02:44,410 --> 00:02:53,230 ‫And then this vector here, it contains all the maximum values that I can have here and both for minimum 34 00:02:53,230 --> 00:02:54,640 ‫and maximum values. 35 00:02:55,210 --> 00:03:00,100 ‫U2 zero, U2 one, U2 two and U2 three. 36 00:03:00,790 --> 00:03:01,930 ‫They would be the same. 37 00:03:02,530 --> 00:03:04,420 ‫Then you three zero U2. 38 00:03:04,420 --> 00:03:07,450 ‫You want U3 two and you three three. 39 00:03:08,140 --> 00:03:11,380 ‫They would have the same maximum and minimum value as well. 40 00:03:12,010 --> 00:03:17,200 ‫And finally, you for zero, you for one, you for two and you for three. 41 00:03:17,830 --> 00:03:22,350 ‫They will also have the same maximum and minimum of values. 42 00:03:23,050 --> 00:03:30,760 ‫But now what I can do, I can still have my y global star minimum here, and then I can still have my 43 00:03:30,910 --> 00:03:32,920 ‫y global star maximum here. 44 00:03:33,550 --> 00:03:40,390 ‫But now, instead of this one here, I'm going to replace it with this one here. 45 00:03:40,990 --> 00:03:43,540 ‫So I'm going to have C to the star. 46 00:03:43,540 --> 00:03:54,490 ‫Global Times C Double Bar Times, Delta U Global Plus C to the Star Global Times A Double Circle Flex 47 00:03:55,150 --> 00:03:57,970 ‫Times X2 still the zero next. 48 00:03:57,970 --> 00:04:04,330 ‫I can take this second term here and I can put it on this side and on that side. 49 00:04:04,900 --> 00:04:19,330 ‫And so I will have y global star minimum minus C till the Global Star Times, a double circle flex times 50 00:04:19,630 --> 00:04:29,350 ‫till the zero, then all that is less than or equal to then I have here C till the star Global Times 51 00:04:29,530 --> 00:04:35,530 ‫C Double Bar Times, Delta U Global, which is this one here. 52 00:04:36,160 --> 00:04:46,080 ‫And then that is less than or equal to Y Global Max star and then minus C to the Star Global. 53 00:04:46,720 --> 00:04:50,110 ‫A double circle flex and X tilde zero. 54 00:04:50,830 --> 00:04:52,480 ‫And now it becomes interesting. 55 00:04:53,080 --> 00:04:57,970 ‫Now I only have a term here that has Delta Global in it. 56 00:04:58,540 --> 00:04:59,920 ‫What I can do now. 57 00:05:00,580 --> 00:05:02,410 ‫I can separate them like this. 58 00:05:02,920 --> 00:05:04,840 ‫So first of all, we take this one. 59 00:05:05,530 --> 00:05:07,540 ‫Then we take this one here. 60 00:05:08,260 --> 00:05:09,880 ‫This will be our second part. 61 00:05:10,390 --> 00:05:12,960 ‫So I will have see to the star. 62 00:05:12,970 --> 00:05:14,590 ‫Global Times. 63 00:05:14,590 --> 00:05:15,750 ‫See Double Bar. 64 00:05:16,420 --> 00:05:16,840 ‫Times. 65 00:05:16,840 --> 00:05:17,870 ‫Delta, you global. 66 00:05:18,520 --> 00:05:23,800 ‫And that's less than or equal to why Star Global Max minus. 67 00:05:24,400 --> 00:05:25,900 ‫And then this term here. 68 00:05:26,530 --> 00:05:30,910 ‫And then the second part here, it will be like this again. 69 00:05:31,240 --> 00:05:32,490 ‫See till this star. 70 00:05:32,500 --> 00:05:34,180 ‫Global Times. 71 00:05:34,180 --> 00:05:35,140 ‫See Double Bar. 72 00:05:35,740 --> 00:05:36,770 ‫Times Delta. 73 00:05:36,770 --> 00:05:38,020 ‫You Global. 74 00:05:38,680 --> 00:05:40,600 ‫It's greater than or equal to. 75 00:05:41,260 --> 00:05:44,620 ‫And then you have all this that you have over here. 76 00:05:45,250 --> 00:05:48,580 ‫But now remember all our inequality signs. 77 00:05:48,970 --> 00:05:50,620 ‫They had to point to the left. 78 00:05:51,010 --> 00:05:55,510 ‫So that means that I have to multiply this side by minus one. 79 00:05:56,140 --> 00:05:57,760 ‫And this side by minus one. 80 00:05:58,390 --> 00:06:02,920 ‫And if I do that, then this sign will become like this. 81 00:06:03,550 --> 00:06:09,040 ‫In other words, I can simply write that I have a minus sign here. 82 00:06:09,700 --> 00:06:11,740 ‫Then the inequality sign is like this. 83 00:06:12,340 --> 00:06:15,730 ‫Then I'm going to put a minus sign here and here. 84 00:06:15,760 --> 00:06:17,020 ‫I will have a plus sign. 85 00:06:17,620 --> 00:06:20,890 ‫And now you can just formulate them like this. 86 00:06:21,400 --> 00:06:22,930 ‫I have this matrix here. 87 00:06:23,590 --> 00:06:29,860 ‫Times Delta, you global less than or equal to why global star max. 88 00:06:30,490 --> 00:06:34,870 ‫And then all this and then minus y global star men. 89 00:06:35,500 --> 00:06:36,940 ‫And then plus all this. 90 00:06:37,570 --> 00:06:38,560 ‫And there you have it. 91 00:06:39,160 --> 00:06:43,240 ‫You have your matrix and you have your h vector. 92 00:06:44,020 --> 00:06:50,560 ‫And now all we have to do is to give this g matrix and each vector to the solver. 93 00:06:51,220 --> 00:06:59,890 ‫Together with your cost function information, so don't you global that considers the constraints it 94 00:06:59,890 --> 00:07:00,550 ‫equals? 95 00:07:01,180 --> 00:07:12,310 ‫Solve underscore kewpie h double bar small if transposed g and then H transposed and then in addition, 96 00:07:12,970 --> 00:07:17,350 ‫you will have your solver equals c accept. 97 00:07:18,010 --> 00:07:18,700 ‫And that's it. 98 00:07:19,330 --> 00:07:26,050 ‫That's how you find the solution that minimizes your cost function and respects your constraints. 99 00:07:26,050 --> 00:07:32,020 ‫At the same time, from the solvers point of view, you constrained your delta use. 100 00:07:32,680 --> 00:07:38,860 ‫But in reality, you know that our journey started with Omegas. 101 00:07:39,520 --> 00:07:42,610 ‫You first had a minimum and maximum omega. 102 00:07:43,270 --> 00:07:47,590 ‫That's what we started with, with minimum and maximum Omegas. 103 00:07:48,190 --> 00:07:53,270 ‫Then from there, we went to minimum and maximum use. 104 00:07:53,890 --> 00:07:59,950 ‫And once we had that, we were able to go from here till this form here. 105 00:08:00,550 --> 00:08:01,840 ‫So that was the journey. 106 00:08:02,440 --> 00:08:07,360 ‫And now I would like to talk about a problem that you face for sure. 107 00:08:07,960 --> 00:08:17,230 ‫It can happen many times that when you work with constraints, then there simply are no solutions that 108 00:08:17,230 --> 00:08:24,400 ‫minimize the cost function and respect all the constraints at the same time to give you good intuition 109 00:08:24,400 --> 00:08:25,570 ‫about this problem. 110 00:08:26,110 --> 00:08:30,340 ‫In the previous car course, I had a very good example for that. 111 00:08:30,940 --> 00:08:32,920 ‫I will show it to you here as well. 112 00:08:33,490 --> 00:08:40,480 ‫So if you have already covered the previous car course and I mean the second concourse, then you can 113 00:08:40,480 --> 00:08:42,220 ‫skip the next two videos. 114 00:08:42,670 --> 00:08:45,910 ‫But if not, then feel free to watch them here. 115 00:08:46,450 --> 00:08:52,630 ‫So let's forget about the drone during two videos, and let's focus on this car example. 116 00:08:53,200 --> 00:08:57,310 ‫Even though this problem can occur with any system. 117 00:08:58,300 --> 00:08:59,470 ‫You can do it by hand. 118 00:08:59,950 --> 00:09:03,310 ‫I'm going to give you a Matrix H double bar. 119 00:09:03,940 --> 00:09:04,930 ‫It looks like this. 120 00:09:05,590 --> 00:09:10,960 ‫Then I'm going to give you a small f transposed that looks like this. 121 00:09:11,650 --> 00:09:16,510 ‫Then I'm going to give you a g matrix that looks like this. 122 00:09:17,140 --> 00:09:24,400 ‫And then I'm going to give you the H Vector transposed that looks like this. 123 00:09:25,030 --> 00:09:26,530 ‫So you have two questions. 124 00:09:27,160 --> 00:09:30,250 ‫Question number one, do you have a solution? 125 00:09:30,850 --> 00:09:41,260 ‫Delta you global that minimizes your cost function, j, and respects all your constraints. 126 00:09:41,830 --> 00:09:43,360 ‫So that's the first question. 127 00:09:43,990 --> 00:09:46,770 ‫And Question two is why. 128 00:09:47,320 --> 00:09:50,020 ‫Whatever answer you have for your question one. 129 00:09:50,650 --> 00:09:52,060 ‫Why is it like that? 130 00:09:52,630 --> 00:09:53,920 ‫So give it some thought. 131 00:09:54,460 --> 00:09:56,560 ‫Try doing it by yourself first. 132 00:09:57,130 --> 00:09:59,350 ‫Don't watch the next video right away. 133 00:09:59,950 --> 00:10:03,760 ‫And then when you have thought about it, then I'll see you in the next video. 134 00:10:03,910 --> 00:10:04,810 ‫Thank you very much.