1 00:00:00,720 --> 00:00:01,530 ‫Welcome back. 2 00:00:02,130 --> 00:00:10,350 ‫So in the previous videos, we talked about the fact that there is a more advanced method to integrate 3 00:00:10,350 --> 00:00:13,440 ‫states and then method is called. 4 00:00:14,610 --> 00:00:17,580 ‫The fourth order, wrong Akutan method. 5 00:00:18,780 --> 00:00:28,800 ‫So so far, we have been using this formula here in order to compute our new state at K one, using 6 00:00:28,980 --> 00:00:36,930 ‫the present state at K, plus the time derivative of the state at K times, the sample time interval 7 00:00:36,930 --> 00:00:38,880 ‫divided by N. 8 00:00:39,880 --> 00:00:48,970 ‫And again, this end chops this sample time interval into smaller pieces, and so this method is called 9 00:00:49,390 --> 00:00:50,620 ‫the oilor method. 10 00:00:51,610 --> 00:00:58,960 ‫And this method is not wrong, however, people tend to use wrong Akutan method now because the other 11 00:00:58,960 --> 00:01:00,730 ‫method has certain issues. 12 00:01:01,990 --> 00:01:09,640 ‫Here you can see a graph, you have time on the horizontal axis and you have a random state on the vertical 13 00:01:09,640 --> 00:01:10,150 ‫axis. 14 00:01:11,210 --> 00:01:14,690 ‫Let's imagine that you're at time t. 15 00:01:15,810 --> 00:01:20,310 ‫And this point here is T plus T. 16 00:01:20,310 --> 00:01:26,220 ‫S, so this interval here, that's your sample time interval. 17 00:01:27,390 --> 00:01:38,700 ‫So on the vertical axis, this point here is your state at and this point here is your state at K plus 18 00:01:38,700 --> 00:01:39,150 ‫one. 19 00:01:40,370 --> 00:01:48,200 ‫And so you are using the other method to be able to predict that new state at K plus one. 20 00:01:49,180 --> 00:01:57,370 ‫And you do it like this, you compute the time derivative here in this point, which is the slope, 21 00:01:58,210 --> 00:02:02,290 ‫and you projected to T plus Tietz. 22 00:02:03,360 --> 00:02:13,110 ‫And so you're going to end up being here, so this WidePoint here, that's your true new state. 23 00:02:13,990 --> 00:02:26,370 ‫So it's this point here and this Redpoint here, that's your predicted new state by oilor method. 24 00:02:27,030 --> 00:02:29,100 ‫So it's this one here. 25 00:02:30,280 --> 00:02:38,920 ‫As you can see, this method can give you quite an imprecise answer, because this would be your error 26 00:02:38,920 --> 00:02:39,370 ‫here. 27 00:02:40,530 --> 00:02:44,380 ‫This difference in purple, that would be your error. 28 00:02:45,450 --> 00:02:57,180 ‫So when you project the derivative that you get at time T til time T plus two seconds, you might have 29 00:02:57,180 --> 00:03:04,890 ‫an error between the true state and your predicted state that is unacceptably large. 30 00:03:05,960 --> 00:03:14,030 ‫Then in the previous course, we solved this problem like this, we took this time sample interval and 31 00:03:14,030 --> 00:03:23,110 ‫we chopped it into several pieces and we used the exact same method is just now. 32 00:03:23,900 --> 00:03:26,480 ‫We did that on a smaller interval. 33 00:03:27,860 --> 00:03:30,500 ‫So we computed a derivative here. 34 00:03:31,520 --> 00:03:43,130 ‫And we projected it to here to this point, then here we computed another derivative and we projected 35 00:03:43,130 --> 00:03:47,170 ‫it till this second point here like this. 36 00:03:48,500 --> 00:03:57,650 ‫Then again, we computed a new derivative in this point here, and we projected it till this point here. 37 00:03:58,910 --> 00:04:04,540 ‫And then a new derivative here that was projected till this point here. 38 00:04:05,480 --> 00:04:12,110 ‫And finally, a derivative here that was projected till this point here. 39 00:04:13,050 --> 00:04:17,110 ‫That approach did increase the precision and reduce the error. 40 00:04:17,450 --> 00:04:26,270 ‫As you can see, your final error is now much smaller because you did this integration over a smaller 41 00:04:26,270 --> 00:04:27,770 ‫time, sample period. 42 00:04:28,900 --> 00:04:37,750 ‫So maybe now your final error would be this one here, this distance, so a much better improvement, 43 00:04:37,750 --> 00:04:38,080 ‫right? 44 00:04:39,010 --> 00:04:46,010 ‫However, the truth is that in real life, this approach can be impractical. 45 00:04:47,080 --> 00:04:55,660 ‫One of the reasons is computational cost when your system is very complex and requires a lot of computations. 46 00:04:56,050 --> 00:05:04,820 ‫Then you have to understand that calculations, even if made by computers, they need time. 47 00:05:05,470 --> 00:05:13,870 ‫So when time is an issue, then you really want to minimize the amount of mathematical operations and 48 00:05:13,870 --> 00:05:21,130 ‫the amount of iterations in the program, especially when you're doing something real time and calculations 49 00:05:21,130 --> 00:05:23,020 ‫need to happen real time. 50 00:05:23,950 --> 00:05:34,930 ‫Meaning that you want your for or while loops in the program run less cycles, and so luckily, instead 51 00:05:34,930 --> 00:05:45,700 ‫of breaking down teams or the sample time interval into multiple pieces, we can avoid doing that by 52 00:05:45,700 --> 00:05:50,350 ‫using the fourth order wrong kouta method. 53 00:05:51,220 --> 00:05:55,330 ‫It is a very popular and widely used method in engineering. 54 00:05:56,230 --> 00:06:02,860 ‫And like I mentioned before, Matlab, for example, has an integrative function which is called. 55 00:06:03,940 --> 00:06:14,710 ‫Oh, the forty five and Oddisee stands for ordinary differential equations and four words for fourth 56 00:06:14,710 --> 00:06:15,870 ‫order wrong kouta. 57 00:06:16,960 --> 00:06:23,050 ‫And so that shows that Runga Ceuta is a pretty popular thing and something that is good to know. 58 00:06:23,990 --> 00:06:31,640 ‫And by the way, when you work in Matlab and you want to use this function, then you have to have your 59 00:06:31,640 --> 00:06:36,320 ‫differential equations in the form of states based equations. 60 00:06:37,290 --> 00:06:43,950 ‫Meaning that they all have to be first order differential equations, which we have in discourse, in 61 00:06:43,950 --> 00:06:52,800 ‫other words, you cannot have something like this when your state is X and then you have something else, 62 00:06:53,220 --> 00:06:56,220 ‫and then all that equals X double dot. 63 00:06:56,700 --> 00:06:59,820 ‫That's not a first order differential equation. 64 00:07:00,450 --> 00:07:08,550 ‫A first order differential equation would be if your state plus something else would equal X dot. 65 00:07:09,680 --> 00:07:17,930 ‫Or if you want to use X double dot, then that means that your state has to be X dot and only then you 66 00:07:17,930 --> 00:07:20,600 ‫can use this Matlab function. 67 00:07:21,020 --> 00:07:23,750 ‫At least that's based on my experience. 68 00:07:24,860 --> 00:07:29,900 ‫And you can find information on math works about how this function works. 69 00:07:31,110 --> 00:07:38,940 ‫But again, my point is that it uses the fourth order wrong Akutan method, so I'm going to teach you 70 00:07:38,940 --> 00:07:43,530 ‫this method now so that it would not be a mystery to you. 71 00:07:44,400 --> 00:07:50,160 ‫And just to reiterate, in this course, I use wrong Akutan method in my code. 72 00:07:51,230 --> 00:08:00,500 ‫And when I use my wrong Akutan method in my code, then I did not break this sample time interval into 73 00:08:00,530 --> 00:08:01,790 ‫several pieces. 74 00:08:02,670 --> 00:08:07,570 ‫And the code worked very well and the simulation also ran very well. 75 00:08:08,310 --> 00:08:14,620 ‫So it's a good method to learn and it can save you some computational time. 76 00:08:15,090 --> 00:08:16,530 ‫So let's get started.