1 00:00:00,540 --> 00:00:01,330 ‫Welcome back. 2 00:00:01,860 --> 00:00:06,410 ‫So let's fill in the system now and let's start with the B matrix. 3 00:00:07,320 --> 00:00:15,630 ‫If you look at these equations here, then in each equation you have your last term and these last terms 4 00:00:15,630 --> 00:00:16,230 ‫here. 5 00:00:16,230 --> 00:00:18,960 ‫They are relevant to our B matrix. 6 00:00:20,010 --> 00:00:28,110 ‫So you have your control inputs here and your B matrix, and you have to be able to use them. 7 00:00:29,190 --> 00:00:36,710 ‫To construct these terms here with you to Q3 and Q4 and to do that, it's quite easy. 8 00:00:37,320 --> 00:00:42,640 ‫You can see that the control inputs, they are divided by the mass moments of inertia. 9 00:00:43,350 --> 00:00:53,260 ‫And so if you take this last term in the first equation, for example, then you can put one over Isobar, 10 00:00:53,280 --> 00:00:58,260 ‫X, X here and then you can have zero here and here. 11 00:00:59,130 --> 00:01:03,090 ‫Then you would multiply one over, I suspect six. 12 00:01:03,180 --> 00:01:11,370 ‫You would multiply that by you two and then you will have some other terms from here and then you will 13 00:01:11,370 --> 00:01:14,340 ‫get your five double dots and there you go. 14 00:01:14,340 --> 00:01:23,490 ‫You have your five double dot equals something plus you two over ice up X X four seater double knot. 15 00:01:24,120 --> 00:01:25,050 ‫It's like this. 16 00:01:25,050 --> 00:01:33,390 ‫You have zero here, one over I suppose Y y and then zero here. 17 00:01:34,080 --> 00:01:42,060 ‫Now you would multiply this one over mass moment of inertia, but the body frame y axis, you would 18 00:01:42,060 --> 00:01:44,520 ‫multiply by you three. 19 00:01:45,390 --> 00:01:55,200 ‫And that's why you have this last term here like this and finally for Citable Dot, you have it like 20 00:01:55,200 --> 00:02:05,880 ‫this one over I, sub Z and Z and that thing, you multiplied by four. 21 00:02:06,690 --> 00:02:10,200 ‫And here is the last term of the last equation. 22 00:02:10,200 --> 00:02:16,460 ‫EUFOR divided by the mass movement of inertia, but the body from Z axis. 23 00:02:17,340 --> 00:02:19,340 ‫So the B matrix was easy. 24 00:02:19,350 --> 00:02:21,130 ‫And how about the A matrix. 25 00:02:21,150 --> 00:02:21,480 ‫Now. 26 00:02:22,450 --> 00:02:27,910 ‫So the Matrix looks like this and let's analyze it a bit. 27 00:02:28,880 --> 00:02:38,570 ‫You have your fi double dot equals and then you have these terms here and then the first two terms, 28 00:02:39,360 --> 00:02:41,550 ‫they are relevant to the A matrix. 29 00:02:42,740 --> 00:02:52,160 ‫So if you take this element here and then you multiply it by theta dot. 30 00:02:53,370 --> 00:02:57,360 ‫Then you will get the second term here, right, this one. 31 00:02:58,330 --> 00:03:10,690 ‫And then if you take this element here and then you multiply it by side that, well, then you get this 32 00:03:10,690 --> 00:03:12,630 ‫first term here, right? 33 00:03:13,710 --> 00:03:21,930 ‫Because that would be inside your state vector, but then everything else will be in the Matrix. 34 00:03:22,890 --> 00:03:29,500 ‫And you can already see that it's a nonlinear system because, look, you have one state inside the 35 00:03:29,610 --> 00:03:30,690 ‫Matrix as well. 36 00:03:31,530 --> 00:03:33,930 ‫You had this setar dot here. 37 00:03:34,900 --> 00:03:43,300 ‫And now if you take to double that equals and then you take this element here and you multiply it by 38 00:03:43,300 --> 00:03:52,150 ‫five, that will then you get this term here right in this second equation, second term. 39 00:03:52,990 --> 00:03:55,180 ‫And you see you have a minus sign here. 40 00:03:55,600 --> 00:03:59,570 ‫And that's why you have a minus sign here as well. 41 00:04:00,460 --> 00:04:12,370 ‫And then if you take this element here and you multiply it by side dot, then you will get this first 42 00:04:12,370 --> 00:04:18,820 ‫term here in the second equation side that again, would be in the state vector and then everything 43 00:04:18,820 --> 00:04:20,530 ‫else would be in the A matrix. 44 00:04:21,280 --> 00:04:26,000 ‫So you see, you have a state inside the Matrix as well, this five dot. 45 00:04:26,860 --> 00:04:30,220 ‫And finally, what are you going to do about the third equation here? 46 00:04:31,240 --> 00:04:35,430 ‫Well, you can do it in several ways, but this is how I did it. 47 00:04:36,130 --> 00:04:47,950 ‫I take this element here and then I multiply it by this one here, and then I take this element here, 48 00:04:48,490 --> 00:04:52,570 ‫and then I multiply it by the state here. 49 00:04:53,320 --> 00:04:58,990 ‫And you see you have your theta that state here in your file, that state here. 50 00:04:59,860 --> 00:05:01,510 ‫And you can perfectly do it. 51 00:05:01,900 --> 00:05:03,990 ‫If you write it out, then you will see why. 52 00:05:04,960 --> 00:05:07,570 ‫So you're upside double dot equals. 53 00:05:08,600 --> 00:05:16,550 ‫So this one is this element here, and I'm going to multiply by five dot, which is this one here. 54 00:05:17,530 --> 00:05:27,730 ‫Plus, this element here, Times seat, that which is this one here, and then, of course, you take 55 00:05:27,730 --> 00:05:38,400 ‫this element here, one over I up, and then you multiply it by your EUFOR, which would be here. 56 00:05:39,400 --> 00:05:46,890 ‫Now you can see that the first and then the second term, they actually have a common denominator. 57 00:05:47,020 --> 00:05:47,530 ‫Right. 58 00:05:47,890 --> 00:05:50,650 ‫And then they also have the same states here. 59 00:05:50,650 --> 00:05:58,210 ‫You have theatergoer and find that here and then fita that and find that here just in another order, 60 00:05:58,210 --> 00:05:59,130 ‫which doesn't matter. 61 00:05:59,830 --> 00:06:06,570 ‫And also in both terms you have X, X minus I y, y, so you have it here and here. 62 00:06:07,510 --> 00:06:13,000 ‫So your numerator is also the same and that means that you can add them up. 63 00:06:13,970 --> 00:06:21,860 ‫And so if you add them up the first and then the second term, you will have it like this, plus you 64 00:06:21,870 --> 00:06:26,860 ‫for over, I sup Z and Z. 65 00:06:27,560 --> 00:06:33,920 ‫And if you look at this form now, then it's exactly like your third equation here. 66 00:06:34,400 --> 00:06:36,760 ‫BPCI double dot equals everything else. 67 00:06:37,580 --> 00:06:39,370 ‫So that's how it looks like. 68 00:06:40,010 --> 00:06:43,670 ‫But now again, is it an LTI form? 69 00:06:44,240 --> 00:06:56,960 ‫Well, no, it's not because well the B matrix and also your C and D matrices, while they're all constant 70 00:06:57,770 --> 00:07:04,430 ‫and you can see that your C matrix was constant, you only had ones and zeros there and inside the Matrix, 71 00:07:04,430 --> 00:07:05,750 ‫you only had zeros there. 72 00:07:05,930 --> 00:07:07,270 ‫So they are also constant. 73 00:07:08,060 --> 00:07:14,840 ‫So while you're A, B, C and D matrices are constant, you're a matrix is not. 74 00:07:15,890 --> 00:07:25,580 ‫You have variables, dot theta that and Omega inside the A Matrix and those things, they change. 75 00:07:26,490 --> 00:07:36,270 ‫The Matrix is not even an LTV, linear time, varying system, you're a matrix does not directly depend 76 00:07:36,270 --> 00:07:37,020 ‫on time. 77 00:07:38,010 --> 00:07:46,380 ‫Your matrix has other states in it and then even a variable that we didn't classify as a state and all 78 00:07:46,380 --> 00:07:51,630 ‫of them depend on time, so it is obviously a nonlinear system. 79 00:07:52,550 --> 00:08:01,910 ‫And just to be clear, we don't have to specify all variables that change, like, for example, this 80 00:08:01,910 --> 00:08:02,490 ‫Omega. 81 00:08:03,050 --> 00:08:06,290 ‫We don't have to specify them all as states. 82 00:08:07,250 --> 00:08:17,120 ‫Remember, in the previous course, I said that the states are the variables that change that you absolutely 83 00:08:17,120 --> 00:08:20,720 ‫need to have in order to control the system. 84 00:08:21,590 --> 00:08:31,310 ‫In other words, you can have many variables that change, but you don't have to classify them all as 85 00:08:31,310 --> 00:08:32,000 ‫states. 86 00:08:32,240 --> 00:08:37,640 ‫If you can, for example, classify some of them as states and you're able to control your system, 87 00:08:38,540 --> 00:08:39,470 ‫then that's OK. 88 00:08:39,470 --> 00:08:44,920 ‫And you can leave other variables B without classifying them as states. 89 00:08:45,440 --> 00:08:47,720 ‫So that's what we have done here with an Omega. 90 00:08:48,530 --> 00:08:55,970 ‫This variable changes, but we don't need to classify it as a state in order to be able to control our 91 00:08:55,970 --> 00:08:56,450 ‫system. 92 00:08:57,260 --> 00:09:05,270 ‫And that makes our lives easier because thanks to that, our state vector is smaller and our A and B 93 00:09:05,270 --> 00:09:06,710 ‫matrices are smaller. 94 00:09:07,610 --> 00:09:16,040 ‫But OK, so we have now established that we have a nonlinear system and we cannot have a small ÀNGEL 95 00:09:16,050 --> 00:09:25,160 ‫approximation with Phi Dot or Setar that they are not going to be small values when the drone follows 96 00:09:25,160 --> 00:09:26,090 ‫the trajectory. 97 00:09:27,200 --> 00:09:34,940 ‫And there is no reason to believe that the omega value will be small as well, because especially if 98 00:09:34,940 --> 00:09:42,200 ‫you have a huge difference between counterclockwise and clockwise, rather rotations, than this omega 99 00:09:42,200 --> 00:09:44,430 ‫value will be quite large. 100 00:09:45,050 --> 00:09:49,940 ‫So you cannot approximate it to be small or zero. 101 00:09:50,840 --> 00:09:57,350 ‫So does that mean that we cannot use the NPC strategy that you learned in the previous course? 102 00:09:58,220 --> 00:10:02,530 ‫Well, based on the information you have so far, no. 103 00:10:03,080 --> 00:10:11,450 ‫However, in this course, you will learn that you, in fact, can use that easiest NPC strategy even 104 00:10:11,450 --> 00:10:11,940 ‫here. 105 00:10:12,770 --> 00:10:14,800 ‫Let's talk about it in the next video.