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‫So to be honest with you, the difference between the oilor and wrong Akutan method is not that big.

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‫This is your oilor method, and this X is just a random state, and note that now I didn't put an end

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‫here, I don't have an end here.

4
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‫So that means that I'm not chopping this sample time interval into multiple pieces.

5
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‫And so this ex dot sub, kay, that's your state time derivative, that's your Delta X over Tietz.

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‫And you know what, this slope here.

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‫It's the only difference between the oilor and the cuter method.

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‫The only difference.

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‫This is the only place where the difference comes in.

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‫In the Oilor method, you only have one slope, which is the time derivative of your state when your

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‫state is X, OK, so your present state, but in a fourth order, wrong Akutan method, you have four

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‫different slopes for different time, derivatives of your states.

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‫And the final slope that you will put here is the weighted average of those four slopes.

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‫So let's just say that you're for slope's are X dots up, K sub one, then X dot stuff K sub to X.com

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‫case sub three and X dots up, case up for these are your four slope's.

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‫And don't worry, you will see how to get them.

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‫But assuming that, you know, you're for Slope's, you will compute their weighted average like this.

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‫So your weighted average slope, let's call it just X dots up K and W A. which stands for a weighted

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‫average, and then the formula is like this.

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‫One sixth time's the first slope plus.

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‫Two times the second slope plus.

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‫Two times the third slope and plus.

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‫The fourth slope like this.

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‫And of course, if you open up your parenthesis, then you will have this equation like this.

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‫And so these numbers in front of the slopes, they are the weights.

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‫And you can see that the second slope and the third slope, they have bigger weights and that means

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‫that they have stronger influence over the final answer that you will get when you compare your weighted

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‫average slope and apply it in this formula here.

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‫And so the first and the fourth slope, because of their smaller weights, they influence the final

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‫result less.

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‫And also it makes sense that you have six in the denominator here, because when you add up all the

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‫weights, then you have to get one.

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‫So it's like one sixth plus two sixth plus two sixth plus one sixth and all that equals one.

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‫So your weights, they have to be normalized.

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‫They have to be like a percentage.

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‫Now, if you're curious about why the weights are exactly like that, I assume that they were deduced

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‫through experimentation that two German mathematicians.

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‫Carl Runga and Martin Kutta, when they develop these methods, perhaps they tested it with different

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‫weights and decided that the weights which you see here, are very good for the four different slopes.

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‫However, that's just my opinion.

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‫I haven't researched that.

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‫Exactly.

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‫But what I can tell you is this, again, we have our state time, Grauwe.

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‫And our first slope is equivalent to the one that we had when we used the Oiler method at time equals

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‫30 seconds.

46
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‫So right here.

47
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‫So, in fact, the first slope that you have here is the same slope that you had in the Oiler method

48
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‫is just wrong.

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‫Akutan method, you will have three more slopes.

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‫So this first slope is calculated in this point at K. at time equals T. Then the second and third slope

51
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‫they are computed at time T plus T.

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‫S over to.

53
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‫So this point here would be t plus tips, and so both the second and the third slope, both of them,

54
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‫they are computed in the middle of your sample time interval here.

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‫And then the fourth slope is computed at time T plus two seconds.

56
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‫So if you look at these four weights, then I guess in order to accurately predict the new true state.

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‫It's more important to consider what's happening in the middle of your sample time interval than on

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‫the edges.

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‫So that's why I guess the second and then the third slope have bigger weights in order to assign more

60
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‫importance to them.

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‫And so once you have your weighted average slope, this one here, you compute the new state in the

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‫same way like in the Euler case, meaning that this thing here, it will go into this thing here.

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‫So that's the difference.

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‫Now you just use a weighted average slope that consists of four different slopes.

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‫And so if you want to, you can also rewrite this method like this and note this red part, that's your

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‫weighted average slope.

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00:07:06,090 --> 00:07:17,040
‫And that's the difference between Guta and the Oiler method, not that I did not divide the time interval

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‫into different sub intervals, that means that I can get useful results with less iterations.

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00:07:25,620 --> 00:07:40,080
‫Also, note that when I write X up K and X up K plus one in our context, the K and K plus one, they

70
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‫do not represent time here.

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00:07:42,240 --> 00:07:48,020
‫They just represent the state now and the next state.

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00:07:49,020 --> 00:08:00,060
‫So in the Euler case, the state now was up K and then the next state except K plus one.

73
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‫Was at time T plus T as over nd that's when we chopped the T into several pieces.

74
00:08:12,030 --> 00:08:21,630
‫So let's say if you're T equals zero point one seconds and then you're N equals five, then that means

75
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‫that your true interval after which you're going to compute a new state, you're going to compute a

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‫new state after zero point zero two seconds.

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00:08:33,500 --> 00:08:45,260
‫So when you use this approach, then your subindex K means your state now and your subindex K plus one

78
00:08:45,710 --> 00:08:51,020
‫means your state zero point zero two seconds later.

79
00:08:52,220 --> 00:09:00,260
‫But now in our Runga Kutta scenario, since we are not shopping our teams into multiple pieces.

80
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‫The now ex up K still means the state now, so the present state, but in our wrong Akutan scenario,

81
00:09:11,080 --> 00:09:19,210
‫except K plus one now means a new state at a time plus T.

82
00:09:19,330 --> 00:09:28,720
‫S or zero point one seconds later, not zero point zero two seconds later, zero point one seconds later.

83
00:09:30,050 --> 00:09:38,480
‫So the subindices, Kate and Kate, plus one, they are not for time, they are just to tell you what's

84
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‫now, what's present and what's the next state, no matter what time interval you choose.

85
00:09:46,580 --> 00:09:54,020
‫And so in the wrong Kakuta scenario, we chose our time interval to be zero point one seconds.

86
00:09:54,890 --> 00:10:04,850
‫But since I use the weighted average slope, I still get precise results even by not cutting the ties

87
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‫into several pieces.

88
00:10:07,620 --> 00:10:15,750
‫And now we are going to use our state's basic equations to show you how exactly this slope calculations

89
00:10:15,750 --> 00:10:19,200
‫happen, because this is what the game will be now.

90
00:10:19,590 --> 00:10:24,130
‫The game will be about finding these four slopes.

91
00:10:24,540 --> 00:10:30,960
‫Well, actually, you already know how to get the first slope is the same one that you had in the earlier

92
00:10:30,960 --> 00:10:31,800
‫method case.

93
00:10:32,220 --> 00:10:36,920
‫But now the game will be about finding the rest of the four slopes.

94
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‫We will go through the entire process from the states at K up until we get to the States at K plus one.

95
00:10:49,140 --> 00:10:55,650
‫And by the way, again, I use the same logic in my code, so.

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‫Let's get started on that in the next video.

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‫Thank you very much.