1 00:00:00,480 --> 00:00:09,990 ‫And now let's leave the polls in the X dimension alone and let's take the Y dimension and let's see 2 00:00:09,990 --> 00:00:11,400 ‫what we're going to have here. 3 00:00:11,730 --> 00:00:20,370 ‫Let's change this minus one to plus zero point zero one and let's see what's going to happen. 4 00:00:21,440 --> 00:00:24,470 ‫So the drone starts following the trajectory. 5 00:00:25,570 --> 00:00:33,130 ‫It seems doing pretty well, so from here, it seems that it was able to track the trajectory quite 6 00:00:33,130 --> 00:00:33,660 ‫well. 7 00:00:34,300 --> 00:00:37,420 ‫But let's look at the dimensions separately. 8 00:00:38,170 --> 00:00:45,850 ‫So the X dimension went very well and well, if you look at the Y dimension, then you see that in the 9 00:00:45,850 --> 00:00:49,840 ‫end there is a small difference that is growing. 10 00:00:49,840 --> 00:00:56,400 ‫Here is just you haven't had enough time to let this difference grow. 11 00:00:56,650 --> 00:00:58,830 ‫But you see it is starting to grow here. 12 00:00:59,590 --> 00:01:08,020 ‫And the reason is because you have a very weak, positive poll here, but then a very strong negative 13 00:01:08,020 --> 00:01:08,860 ‫poll here. 14 00:01:09,470 --> 00:01:13,780 ‫So how about if I make this minus zero point five? 15 00:01:13,930 --> 00:01:21,490 ‫Let's see what happens then so you can see that you already have some problems here. 16 00:01:22,450 --> 00:01:26,110 ‫The tracking is not nearly as good as it was before. 17 00:01:26,920 --> 00:01:28,830 ‫The extermination is fine. 18 00:01:29,920 --> 00:01:33,340 ‫However, the Y dimension is not fine. 19 00:01:33,340 --> 00:01:39,400 ‫You see now the distance is getting bigger here and it's already visible. 20 00:01:40,430 --> 00:01:49,220 ‫And now let's go back to our minus one and minus two polls and let's run another trajectory, which 21 00:01:49,220 --> 00:01:50,360 ‫is my favorite one. 22 00:01:51,430 --> 00:01:55,210 ‫I call it the crown, the crown trajectory. 23 00:01:56,230 --> 00:02:03,880 ‫It's not actually that hard to do if you look at your X and Y dimension, then they're just your cosine 24 00:02:03,880 --> 00:02:05,140 ‫and sine functions. 25 00:02:05,770 --> 00:02:11,320 ‫The only difference is that now the Z dimension is also oscillating. 26 00:02:12,340 --> 00:02:18,070 ‫So as your drone goes in the circle, it also goes up and down. 27 00:02:19,060 --> 00:02:20,920 ‫You see, that's how it looks like. 28 00:02:21,960 --> 00:02:23,790 ‫That's the X dimension. 29 00:02:24,970 --> 00:02:26,620 ‫That's the Y dimension. 30 00:02:27,760 --> 00:02:33,430 ‫And that's the Z dimension you see now, Z dimension is also sinusoidal. 31 00:02:34,460 --> 00:02:44,060 ‫Before our drone was following a Z trajectory, that was a straight line, which means that your drone 32 00:02:44,060 --> 00:02:50,230 ‫would go up steadily, but now it just goes up and down in a sinusoidal way. 33 00:02:51,170 --> 00:02:56,870 ‫And these are the angles here and the control inputs and then the omega's. 34 00:02:58,000 --> 00:03:08,560 ‫What happens if for the Z dimension, Paul, I'm going to put here plus zero point zero one so you can 35 00:03:08,560 --> 00:03:17,290 ‫see that something is not going well at all, it is able to follow the X dimension and then the Y dimension. 36 00:03:17,300 --> 00:03:28,570 ‫But because we changed the poles in the Z dimension and we made one pole positive, then now the error 37 00:03:28,570 --> 00:03:29,760 ‫seems to be growing. 38 00:03:30,760 --> 00:03:34,530 ‫So it has gone past the ground already actually. 39 00:03:34,540 --> 00:03:35,640 ‫So it's under the ground. 40 00:03:36,580 --> 00:03:39,040 ‫So as you can see, didn't go very well. 41 00:03:39,850 --> 00:03:41,350 ‫That's your X dimension. 42 00:03:42,350 --> 00:03:51,800 ‫That's your y dimension and then the Z dimension again, you see the positive pole is messing things 43 00:03:51,800 --> 00:03:52,230 ‫up. 44 00:03:53,210 --> 00:04:01,130 ‫So remember, to get the errors to grow, both poles need to be negative. 45 00:04:01,580 --> 00:04:08,990 ‫And if you have some kind of system where you have 10 poles, so you have some kind of tenth order differential 46 00:04:08,990 --> 00:04:12,860 ‫equation, then all those 10 poles need to be negative. 47 00:04:13,930 --> 00:04:24,760 ‫But now I want to make this poll zero and then this poll zero and then also the same thing for the wider 48 00:04:24,760 --> 00:04:28,010 ‫mention this one and this one. 49 00:04:28,960 --> 00:04:31,710 ‫Now, my polls still are not complex. 50 00:04:31,720 --> 00:04:33,880 ‫They are still real polls. 51 00:04:34,180 --> 00:04:39,190 ‫I don't have an imaginary part here, but I've made everything zero. 52 00:04:40,240 --> 00:04:49,950 ‫And now I've gone back to our spiral trajectory so you can see that something is not going quite right. 53 00:04:50,320 --> 00:04:54,010 ‫So the drone is failing in the end. 54 00:04:54,010 --> 00:04:55,830 ‫This is how the drone goes. 55 00:04:55,840 --> 00:05:00,610 ‫This is what it was supposed to do in blue, and this is what it actually did. 56 00:05:01,610 --> 00:05:11,140 ‫You see, when polls are zero and I mean completely zero, both the real and imaginary part are zero, 57 00:05:11,420 --> 00:05:15,500 ‫then it's kind of like a special situation. 58 00:05:16,400 --> 00:05:23,300 ‫You really need to start looking at the initial conditions then, because, for example, here in this 59 00:05:23,660 --> 00:05:31,280 ‫case, you have this initial condition, two meters, the air, the blue line, minus the red line. 60 00:05:31,850 --> 00:05:41,450 ‫At the beginning, when time equals zero, your error is two meters and your error dot, which is this 61 00:05:41,570 --> 00:05:51,860 ‫blue X dot reference line minus this red true X dot line at time equals zero seconds. 62 00:05:52,070 --> 00:05:55,720 ‫Their difference is zero meters per second. 63 00:05:56,210 --> 00:05:59,450 ‫So the error dot is zero meters per second. 64 00:06:00,320 --> 00:06:09,590 ‫Now if you choose your lump, the one to be zero and lump the two to be zero, then if you put them 65 00:06:09,890 --> 00:06:13,940 ‫inside the equations that you had previously derived. 66 00:06:14,930 --> 00:06:24,090 ‫Then you will find that your Q1 and Q2 will be zero as well, both of them will be zero. 67 00:06:24,950 --> 00:06:36,650 ‫So if your Q1 and Q2 are zeros, then what will be your use, OpEx and use of WI if all these constants 68 00:06:36,650 --> 00:06:38,190 ‫here are zeros. 69 00:06:39,170 --> 00:06:39,890 ‫Of course. 70 00:06:40,940 --> 00:06:50,300 ‫Use of ax and use of Y will also be zeroes, which means that your error double dot X will be zero and 71 00:06:50,300 --> 00:06:53,530 ‫error double dot Y will be zero. 72 00:06:54,440 --> 00:07:04,790 ‫And so earlier we talked that our error double dot influences the error dot and then the error that 73 00:07:04,790 --> 00:07:07,100 ‫influences the error rate. 74 00:07:07,400 --> 00:07:08,540 ‫So it's like a chain. 75 00:07:09,320 --> 00:07:15,990 ‫So if this is always zero, then that means that this guy here never changes. 76 00:07:16,580 --> 00:07:23,060 ‫So in this X dimension case, if your error that is zero at the beginning. 77 00:07:24,310 --> 00:07:36,490 ‫Then theoretically, it should stay zero, and that means that if your error at the beginning is two 78 00:07:36,490 --> 00:07:45,190 ‫metres, then it will stay two metres, because since you're Eridani equals zero, that means that your 79 00:07:45,640 --> 00:07:47,610 ‫error will stay the same. 80 00:07:47,620 --> 00:07:52,110 ‫It will not change because error that is the change of error with respect to time. 81 00:07:52,870 --> 00:07:58,540 ‫And that's why you see that essentially your error here is zero. 82 00:07:59,080 --> 00:08:06,250 ‫Well, theoretically it should be zero, but then in practice you have numerical errors. 83 00:08:06,250 --> 00:08:13,720 ‫And then in real life, of course, you never have pure theoretical results. 84 00:08:14,680 --> 00:08:18,250 ‫But you see these two lines are pretty much aligned with each other. 85 00:08:19,330 --> 00:08:28,090 ‫And since they're almost zero, OK, there is a small difference here, but the error is almost zero 86 00:08:28,120 --> 00:08:29,470 ‫and it doesn't change. 87 00:08:30,010 --> 00:08:38,310 ‫And because of that, since your error at the beginning is two metres, it will stay being two metres. 88 00:08:38,710 --> 00:08:40,950 ‫You see, it doesn't change. 89 00:08:41,800 --> 00:08:43,630 ‫How about the Y dimension, though? 90 00:08:44,500 --> 00:08:52,810 ‫You see with the Y dimension is different because here you also have an initial error, but now there 91 00:08:52,810 --> 00:08:54,790 ‫is also a difference in error. 92 00:08:54,790 --> 00:09:05,440 ‫Dot again, since your constants here are zeros because your poles are zeros, then that means that 93 00:09:05,440 --> 00:09:10,850 ‫your error double dot y will be zero. 94 00:09:11,830 --> 00:09:14,440 ‫All this was about the X dimension. 95 00:09:14,440 --> 00:09:14,950 ‫All right. 96 00:09:15,460 --> 00:09:17,680 ‫And now you're erodable. 97 00:09:17,680 --> 00:09:23,740 ‫That Y is zero, which means that your error dot will not change. 98 00:09:24,640 --> 00:09:27,390 ‫But now your error, that is not zero. 99 00:09:27,550 --> 00:09:37,450 ‫Now there is a difference here and the difference here is something like zero point three one four meters 100 00:09:37,450 --> 00:09:38,310 ‫per second. 101 00:09:38,620 --> 00:09:42,070 ‫In other words, it's pie over ten. 102 00:09:42,580 --> 00:09:43,420 ‫That's what it is. 103 00:09:43,420 --> 00:09:44,650 ‫It's pie over ten. 104 00:09:45,520 --> 00:09:49,120 ‫So it's three point fourteen divided by ten. 105 00:09:50,090 --> 00:09:59,150 ‫Now, because of this thing, this thing will not change, it will keep being over 10, but since the 106 00:09:59,150 --> 00:10:08,210 ‫error itself depends on error dot, that means that this error that will keep changing, this error 107 00:10:08,570 --> 00:10:12,690 ‫and this error will change all the time like this at the beginning. 108 00:10:12,860 --> 00:10:14,870 ‫The error is one meter. 109 00:10:15,170 --> 00:10:15,740 ‫All right. 110 00:10:16,490 --> 00:10:23,480 ‫So the distance here is one meters because this blue line starts at zero meters. 111 00:10:23,480 --> 00:10:33,780 ‫And then the real Y position of the drone at time equals zero seconds is minus one meters in the Y dimension. 112 00:10:33,800 --> 00:10:36,530 ‫So the error is one meter. 113 00:10:37,340 --> 00:10:38,780 ‫So that's at the beginning. 114 00:10:39,080 --> 00:10:46,670 ‫But then you have your error dot, and then since it's not zero, you have to multiply it by time. 115 00:10:47,360 --> 00:11:01,070 ‫So remember your error, that is your d e divided by d t so your error here is one plus dí over deti 116 00:11:01,610 --> 00:11:03,420 ‫times time. 117 00:11:04,280 --> 00:11:07,810 ‫In other words, your error in the Y dimension. 118 00:11:08,120 --> 00:11:11,780 ‫And remember now we're talking about the Y dimension. 119 00:11:12,290 --> 00:11:19,760 ‫You're error here now is one plus so pi over 10 times time. 120 00:11:20,610 --> 00:11:30,990 ‫And so this difference here between the blue line and the red line is growing according to this function 121 00:11:30,990 --> 00:11:31,460 ‫here. 122 00:11:32,220 --> 00:11:35,280 ‫So I have graphed the error function here. 123 00:11:35,490 --> 00:11:39,030 ‫Here in green, you have this error function. 124 00:11:39,750 --> 00:11:41,070 ‫It's this one here. 125 00:11:41,370 --> 00:11:47,190 ‫Error as a function of time equals one plus pi over ten times T. 126 00:11:48,250 --> 00:11:50,620 ‫So that's the one in green. 127 00:11:51,710 --> 00:12:00,410 ‫And so when you make your polls zero, then you have to be extra careful and pay attention to these 128 00:12:00,410 --> 00:12:01,360 ‫kind of things. 129 00:12:02,300 --> 00:12:10,400 ‫But of course, in control engineering, your objective is to minimize the errors so it doesn't make 130 00:12:10,400 --> 00:12:14,870 ‫sense for you to choose polls that are positive or Xeros. 131 00:12:15,620 --> 00:12:21,300 ‫You always want negative polls because you want your errors to become zero. 132 00:12:21,950 --> 00:12:24,610 ‫That's your objective as a control engineer. 133 00:12:25,340 --> 00:12:30,880 ‫And now I want to play a little bit with complex polls. 134 00:12:31,220 --> 00:12:33,230 ‫I want to see what's going to happen then. 135 00:12:34,200 --> 00:12:43,890 ‫So we're going to comment about the real polls and we're going to uncommented the complex polls, so 136 00:12:43,890 --> 00:12:51,750 ‫as you can see here, when we talk about complex polls, then the real part is the same for all of them, 137 00:12:51,990 --> 00:13:01,200 ‫minus zero point one here and minus zero point one here, minus zero point zero one here and minus zero 138 00:13:01,200 --> 00:13:02,700 ‫point zero one here. 139 00:13:03,640 --> 00:13:10,300 ‫And then the imaginary parts are the conjugates plus zero point three times and minus zero point three 140 00:13:10,300 --> 00:13:20,410 ‫times, Jay, if I put here plus one point three times, Jay, I have to put here minus one point three 141 00:13:20,410 --> 00:13:21,310 ‫times, Jay. 142 00:13:32,970 --> 00:13:42,120 ‫Let's for a moment have real X and Y dimension polls and then for the Z dimension, let's have complex 143 00:13:42,120 --> 00:13:42,690 ‫polls. 144 00:13:43,740 --> 00:13:52,530 ‫And now I'm going to make my real polls here like this minus zero point one and also here minus zero 145 00:13:52,530 --> 00:13:53,670 ‫point one. 146 00:13:54,540 --> 00:14:05,580 ‫And so I've taken out another trajectory and we're going to see how it follows this straight line trajectory. 147 00:14:06,180 --> 00:14:09,170 ‫So, as you can see, it's not being very successful. 148 00:14:09,540 --> 00:14:14,220 ‫If you look at this, the Z dimension, it seems to be oscillating. 149 00:14:15,120 --> 00:14:17,640 ‫So that's how it followed the trajectory. 150 00:14:18,360 --> 00:14:28,890 ‫You can see that since the Z dimension is up and since we had complex polls, but still a small negative 151 00:14:28,890 --> 00:14:37,680 ‫real number because of that, the drone oscillated in the Z dimension, but it damped itself out. 152 00:14:38,550 --> 00:14:47,730 ‫That's because our Lumb the one was minus zero point one plus eight times zero point three and then 153 00:14:47,880 --> 00:14:55,120 ‫LAMDA two was minus zero point one minus eight times zero point three. 154 00:14:55,920 --> 00:15:06,150 ‫So these ones give you oscillation and this one here damps this oscillation out. 155 00:15:07,220 --> 00:15:13,580 ‫You see, it's like this, you don't really have that in the X dimension because you had real polls 156 00:15:13,580 --> 00:15:22,280 ‫there, no imaginary parts, the same thing for the Y dimension, but your Z dimension did oscillate 157 00:15:22,940 --> 00:15:24,650 ‫before it became stable. 158 00:15:25,400 --> 00:15:26,420 ‫And you can see this. 159 00:15:26,420 --> 00:15:34,010 ‫You want control input, which is mainly responsible for tracking the altitude because the propellers 160 00:15:34,010 --> 00:15:39,830 ‫give you the thrust upwards and you want is your thrust control input. 161 00:15:40,580 --> 00:15:44,840 ‫And you see there is also some oscillation in this. 162 00:15:44,840 --> 00:15:46,370 ‫You want control input.