1 00:00:00,560 --> 00:00:01,410 ‫Welcome back. 2 00:00:01,940 --> 00:00:08,450 ‫I now want to caution you about something, suppose that you have a differential equation like this. 3 00:00:09,670 --> 00:00:14,530 ‫This one here, and let's put it in the state space form. 4 00:00:15,600 --> 00:00:29,370 ‫So your X dot equals X dot and then your X double dot equals minus two times X minus three times three 5 00:00:29,370 --> 00:00:37,370 ‫times X dot and then plus sine T like this. 6 00:00:38,190 --> 00:00:49,710 ‫And so we will have our X don't here X double dot here you will have your A matrix here and then you 7 00:00:49,710 --> 00:01:01,800 ‫will have your state vector X and then X dot and then plus you will have your B matrix here times sine 8 00:01:02,400 --> 00:01:02,770 ‫T.. 9 00:01:03,330 --> 00:01:12,840 ‫So here you will have zero and one because X equals X dot and then here you will have minus two and 10 00:01:12,840 --> 00:01:20,880 ‫then here you will have minus three times T and then in the B matrix you will have zero here and one 11 00:01:20,880 --> 00:01:21,330 ‫here. 12 00:01:22,320 --> 00:01:32,220 ‫So you can see that this differential equation here, it's linear, but it's linear time variant, so 13 00:01:32,220 --> 00:01:33,530 ‫it's LTV. 14 00:01:34,140 --> 00:01:37,160 ‫That's because you're a matrix. 15 00:01:37,170 --> 00:01:39,360 ‫Here is a function of time. 16 00:01:39,900 --> 00:01:42,780 ‫It's an LTV system. 17 00:01:43,660 --> 00:01:48,470 ‫Let's now find the systems, poles or eigenvalues. 18 00:01:48,660 --> 00:01:52,960 ‫And remember, the eigenvalues are found like this. 19 00:01:53,790 --> 00:02:05,070 ‫You take this a matrix, and then from that you will subtract a matrix where you have these eigenvalues 20 00:02:05,310 --> 00:02:13,240 ‫like this, and then you take the determinant of it and then you equate it to zero. 21 00:02:14,100 --> 00:02:17,160 ‫So the difference of these matrices will be this one. 22 00:02:17,460 --> 00:02:19,650 ‫And then you take the determinant of it. 23 00:02:20,280 --> 00:02:22,320 ‫And again, you equate it to zero. 24 00:02:22,980 --> 00:02:38,100 ‫And so now you will have minus lambda times, minus three times T minus lambda and then minus one times 25 00:02:38,220 --> 00:02:39,660 ‫minus two. 26 00:02:39,990 --> 00:02:42,540 ‫So that's how you compute the determinant. 27 00:02:42,870 --> 00:02:50,920 ‫This element times, this element minus this element times this element and that equals to zero. 28 00:02:51,930 --> 00:03:04,290 ‫So if you write it out then you will have three times lambda times T plus lambda squared and plus two 29 00:03:04,620 --> 00:03:15,600 ‫equals zero or in other way you can write it like this lambda squared plus three times T and then that 30 00:03:15,600 --> 00:03:21,210 ‫is multiplied by lambda plus two equals zero. 31 00:03:22,470 --> 00:03:25,870 ‫And so your lamda and you will have two of them. 32 00:03:26,370 --> 00:03:36,540 ‫It will be minus three times D, which is this one here, divided by two plus minus, and then you will 33 00:03:36,540 --> 00:03:46,710 ‫have square root where you will have three times tea over to you, square it and then you subtract two 34 00:03:46,710 --> 00:03:47,450 ‫from it. 35 00:03:48,390 --> 00:03:51,120 ‫So that will be your Lunda one. 36 00:03:52,180 --> 00:03:54,820 ‫And that will be your lamda to hear. 37 00:03:55,950 --> 00:04:00,570 ‫As you can see, your polls now are not close to numbers. 38 00:04:00,990 --> 00:04:02,820 ‫They are a function of time. 39 00:04:02,830 --> 00:04:05,520 ‫You have the time variable here and here. 40 00:04:06,210 --> 00:04:13,500 ‫So your loved one is a function of time and then your number two is a function of time as well. 41 00:04:14,220 --> 00:04:15,300 ‫So what does it mean? 42 00:04:16,450 --> 00:04:23,710 ‫Well, it means that the polls are not fixed in the Laplanche domain. 43 00:04:25,290 --> 00:04:31,640 ‫These polls, they move with time, they are not in the same location all the time. 44 00:04:33,020 --> 00:04:37,310 ‫So with time, the polls can become complex. 45 00:04:38,540 --> 00:04:44,570 ‫Or one of them can become positive or both of them can become positive. 46 00:04:45,550 --> 00:04:47,070 ‫And that's not good. 47 00:04:48,160 --> 00:04:53,500 ‫And things get even worse when you have non-linear differential equations. 48 00:04:54,650 --> 00:05:01,910 ‫Suppose that you have this differential equation here and you see now here, you will have X squared. 49 00:05:03,060 --> 00:05:12,690 ‫Let's put it in the state space for, again, your X dot equals X dot and then your X double dot equals 50 00:05:13,530 --> 00:05:24,450 ‫minus two times X squared, minus three times T times X dot and then plus D squared. 51 00:05:25,710 --> 00:05:29,940 ‫So here you will have X dot and then X double dot. 52 00:05:31,160 --> 00:05:34,280 ‫Your state vector is X and X dot. 53 00:05:35,570 --> 00:05:46,100 ‫Your input function is T squared, and so here you will have zero and one, and now here, since you 54 00:05:46,100 --> 00:05:58,370 ‫have this X squared here, then you will have minus two times X here and then minus three times T here. 55 00:05:58,910 --> 00:06:05,540 ‫So if you write it out, then you will see that minus two times X, you would multiply it by X and then 56 00:06:05,540 --> 00:06:07,190 ‫you will have X squared. 57 00:06:08,080 --> 00:06:11,290 ‫And then here you will have zero and one. 58 00:06:12,830 --> 00:06:18,980 ‫So you can see that this is a nonlinear system and let's find the Poles now. 59 00:06:20,420 --> 00:06:29,630 ‫So this difference here, this matrix minus this lamda matrix, and then you take the determinant of 60 00:06:29,630 --> 00:06:33,740 ‫it and then you equate it to zero, so you will have. 61 00:06:34,970 --> 00:06:39,950 ‫This matrix, and then you take the determinate of it and you equate it to zero. 62 00:06:40,990 --> 00:06:44,380 ‫Taking the out of it will give you this expression. 63 00:06:45,370 --> 00:06:49,630 ‫After some mathematical manipulation, you will have this expression here. 64 00:06:50,790 --> 00:06:59,520 ‫And then your polls are in this form, one with a plus sign and one with a minus sign, so you can see 65 00:06:59,760 --> 00:07:07,680 ‫that your polls now depend on time and also on the state itself. 66 00:07:08,680 --> 00:07:11,360 ‫Which in turn depends on time as well. 67 00:07:11,980 --> 00:07:22,810 ‫So I can write it down like this, No one is a function of time and also a function of the state, which 68 00:07:22,990 --> 00:07:25,000 ‫itself is a function of time. 69 00:07:25,950 --> 00:07:32,980 ‫And the same thing for London, too, so again, it's not fixed. 70 00:07:33,450 --> 00:07:42,270 ‫Again, the polls can become complex or positive or both with time like these polls here, they can 71 00:07:42,270 --> 00:07:50,700 ‫move around, they can go like this closer to each other, and then they can become complex like this. 72 00:07:51,720 --> 00:07:58,980 ‫And then both of them can even move here and then they can be complex and unstable. 73 00:08:00,420 --> 00:08:09,560 ‫So your system might start behaving very unpredictably, so it might start oscillating, become unstable, 74 00:08:09,570 --> 00:08:12,300 ‫and a lot of trouble can come from that. 75 00:08:13,200 --> 00:08:22,410 ‫So my point is that this Paul Placemen method should only be used when you have an LTI differential 76 00:08:22,410 --> 00:08:23,210 ‫equation. 77 00:08:23,490 --> 00:08:34,680 ‫In other words, if you put it in the state's space form, X dot vector equals eight times X vector 78 00:08:35,040 --> 00:08:39,060 ‫plus B times and then the input vector. 79 00:08:40,320 --> 00:08:48,210 ‫Then you're A and B matrixes, they should be constant, they shouldn't vary with time, and they shouldn't 80 00:08:48,210 --> 00:08:51,660 ‫depend on other states either. 81 00:08:52,560 --> 00:08:54,360 ‫They should be constant matrixes. 82 00:08:55,930 --> 00:09:01,900 ‫Then your polls will be fixed in place and you can rely on this method. 83 00:09:03,000 --> 00:09:11,130 ‫And now, if you remember then in the previous section, when we dealt with more predictive control, 84 00:09:11,460 --> 00:09:19,290 ‫then we in fact had this situation, our matrix was not constant. 85 00:09:19,740 --> 00:09:21,990 ‫In fact, it had states in it. 86 00:09:22,680 --> 00:09:30,210 ‫And we called that method linear parameter, varying method or LTV. 87 00:09:31,200 --> 00:09:38,430 ‫And so what I want to tell you now is that LTV is a robust control method. 88 00:09:39,690 --> 00:09:45,090 ‫There are books written about it and essentially what l.P is doing. 89 00:09:46,430 --> 00:09:55,310 ‫In the A and B matrix, it is treating those variables that are not fixed. 90 00:09:55,730 --> 00:09:59,000 ‫It is treating them as parameters. 91 00:10:00,680 --> 00:10:09,740 ‫Let's say parameter one or parameter two, and then also here, parameter one and parameter two, and 92 00:10:09,740 --> 00:10:17,090 ‫those parameters, they consist of these variables that are not fixed in the A and B matrices. 93 00:10:17,240 --> 00:10:19,130 ‫In this case, only a matrix. 94 00:10:20,220 --> 00:10:28,560 ‫And essentially, the linear parameter varying theory as part of robust control is trying to determine. 95 00:10:29,840 --> 00:10:35,460 ‫How much these parameters can vary in order to have acceptable results? 96 00:10:37,070 --> 00:10:45,170 ‫So just as a quick example, let's say that you have two axes here and then here you will have X and 97 00:10:45,170 --> 00:10:50,660 ‫then here you will have T, the two variables that vary in this matrix. 98 00:10:51,890 --> 00:10:59,390 ‫And then this linear parameter varying theory tries to determine the regions. 99 00:11:00,540 --> 00:11:02,700 ‫For these varying parameters. 100 00:11:03,830 --> 00:11:13,720 ‫So, for example, if you're somewhere here, so your ex is here and then your time is here and so you 101 00:11:13,730 --> 00:11:15,790 ‫are here, then that's OK. 102 00:11:16,760 --> 00:11:26,420 ‫But then if you're somewhere here where your ex is kind of OK, but then you are out of this box because 103 00:11:26,420 --> 00:11:29,000 ‫your time has varied too much. 104 00:11:29,450 --> 00:11:37,870 ‫So in the end, you are in this region here, then that's not OK because you are out of this region. 105 00:11:39,220 --> 00:11:47,080 ‫So there is an entire theory about it, but this is a very initial superficial introduction to it. 106 00:11:48,150 --> 00:11:56,130 ‫In our case, in our course, for the trajectories that we have, this LPT method works very well. 107 00:11:57,070 --> 00:12:03,130 ‫And the variables that are inside our NPC, a matrix. 108 00:12:04,070 --> 00:12:07,790 ‫That, if you remember, was a six by six matrix. 109 00:12:09,010 --> 00:12:15,460 ‫It contained five dot and theater dot and then also Omega. 110 00:12:16,420 --> 00:12:22,420 ‫In our case, these varying variables, they didn't vary enough. 111 00:12:23,480 --> 00:12:33,170 ‫To ruin the performance of our drone, but this OPV method, this is definitely one of the methods that 112 00:12:33,800 --> 00:12:41,030 ‫you should study if you want to become more advanced in control and perhaps focus on robust control. 113 00:12:42,250 --> 00:12:49,010 ‫And so in the next video, we're going to focus more on our specific feedback lionization controller. 114 00:12:49,540 --> 00:12:51,680 ‫We're going to start wrapping things up. 115 00:12:52,180 --> 00:12:53,110 ‫Thank you very much.