1 00:00:00,560 --> 00:00:01,370 ‫Welcome back. 2 00:00:02,000 --> 00:00:10,880 ‫In the previous course, we also derived this formula here for being able to predict future states at 3 00:00:10,880 --> 00:00:18,350 ‫some kind of future time sample, we made these kind of substitutions where, for example, we wrote 4 00:00:18,650 --> 00:00:24,700 ‫X1 in terms of X zero and Delta zero. 5 00:00:25,520 --> 00:00:34,190 ‫Now, in the other case, we can do it exactly like we did it in the previous course by employing this 6 00:00:34,190 --> 00:00:35,400 ‫formula here. 7 00:00:36,200 --> 00:00:44,270 ‫The only difference now, of course, is that instead of your Delta being as your control inputs, you 8 00:00:44,270 --> 00:00:47,110 ‫have this vector as your control inputs. 9 00:00:47,900 --> 00:00:56,480 ‫So for example, you're you zero would be like this, you two zero three zero and EUFOR zero. 10 00:00:57,230 --> 00:01:02,170 ‫Now note that here the letter K has a different meaning. 11 00:01:02,540 --> 00:01:14,480 ‫Usually we say that if you have X, sub K, we say that it's a present state and X at K plus one. 12 00:01:15,290 --> 00:01:19,640 ‫It is a state one time sample in the future. 13 00:01:20,060 --> 00:01:24,610 ‫So zero point one seconds in the future in our case. 14 00:01:25,460 --> 00:01:36,410 ‫But in this formula, if you say that your K equals four, then let's say that you have X four instead 15 00:01:36,410 --> 00:01:46,880 ‫of X, OK, you have except for then here in this formula it means that this X four is the predicted 16 00:01:47,120 --> 00:01:51,000 ‫state vector four times samples in the future. 17 00:01:51,800 --> 00:02:00,350 ‫So this thing here, it would be now plus in our case, four times sample time intervals. 18 00:02:00,530 --> 00:02:06,350 ‫So that would be now plus zero point four seconds later. 19 00:02:07,250 --> 00:02:14,810 ‫So in the convention that you are used to, it would be X, sub K plus four. 20 00:02:15,130 --> 00:02:15,640 ‫Right. 21 00:02:16,460 --> 00:02:25,910 ‫But in this formula here, this key means how many times samples into the future you have predicted 22 00:02:25,910 --> 00:02:27,170 ‫your states. 23 00:02:27,950 --> 00:02:30,030 ‫You can also think of it like that. 24 00:02:30,650 --> 00:02:37,100 ‫So this is your state vector and that zero plus four or zero plus. 25 00:02:37,300 --> 00:02:40,940 ‫OK, so that's the convention with this specific formula. 26 00:02:41,660 --> 00:02:46,960 ‫So let's say that we want to predict the states four times samples in the future. 27 00:02:47,660 --> 00:02:50,130 ‫So this would be your now here. 28 00:02:50,900 --> 00:02:54,110 ‫This would be your X up zero. 29 00:02:54,950 --> 00:02:57,800 ‫So this is your present state except zero. 30 00:02:58,490 --> 00:03:04,070 ‫And that's why I said that you can think of it like this when you have this K, you can think of it 31 00:03:04,520 --> 00:03:11,800 ‫in this way, X up zero plus K because a zero means your present state in this formula. 32 00:03:12,410 --> 00:03:23,840 ‫And then the zero point one seconds later you will have your predicted state X one X up to accept three 33 00:03:24,290 --> 00:03:26,330 ‫and X for. 34 00:03:27,110 --> 00:03:33,770 ‫And also remember that your NPC horizon period is also four. 35 00:03:34,560 --> 00:03:45,620 ‫That means that the NPC finds control inputs, you zero, you one, you two and you three. 36 00:03:46,460 --> 00:03:53,150 ‫But the plant only receives this one, the first one and then it discards the rest. 37 00:03:54,050 --> 00:04:01,370 ‫However, if you want to predict your future states, right, because remember, these are the predicted 38 00:04:01,730 --> 00:04:02,840 ‫future states. 39 00:04:03,820 --> 00:04:10,400 ‫This is what you know, this is your present state and these are the predicted states and you're using 40 00:04:10,550 --> 00:04:16,370 ‫the mathematical model of your system in order to predict these future states into the future. 41 00:04:16,790 --> 00:04:24,590 ‫In order to do that, you do need to have these control inputs as well, because you have to plug them 42 00:04:24,590 --> 00:04:34,640 ‫in into this vector in order to be able to find your predicted future states like this one here, for 43 00:04:34,640 --> 00:04:42,410 ‫example, if you want to find except for then in this vector here, your control inputs must go from 44 00:04:42,410 --> 00:04:44,090 ‫zero up until three.