1 00:00:00,730 --> 00:00:08,620 ‫And now you have all the information you need in order to calculate your one, your predicted state 2 00:00:08,620 --> 00:00:11,830 ‫vector one time step into the future. 3 00:00:12,490 --> 00:00:16,480 ‫This predicted XB one, it will look like this. 4 00:00:17,110 --> 00:00:26,860 ‫You have your new Phi Phi Don Theta on that and side that that will be here inside your X1 vector. 5 00:00:27,400 --> 00:00:33,760 ‫And now you want to predict your future state vector two time steps into the future. 6 00:00:34,360 --> 00:00:42,430 ‫And so this is the equation a one times X1 plus b times you won. 7 00:00:43,060 --> 00:00:49,120 ‫Now this you want here, that's your second left over from the previous loop. 8 00:00:49,810 --> 00:00:53,530 ‫That's this one here or in this loop. 9 00:00:54,130 --> 00:00:55,510 ‫We named it like this. 10 00:00:56,140 --> 00:01:02,020 ‫So your second left over you two from the previous loop in this loop, it's you one. 11 00:01:02,620 --> 00:01:06,790 ‫And so this one here, then of course, the beam matrix. 12 00:01:07,450 --> 00:01:09,040 ‫It's constant everywhere. 13 00:01:09,670 --> 00:01:14,110 ‫Then this X1 vector here, it's this one here. 14 00:01:14,470 --> 00:01:18,100 ‫Your future predicted state one time step into the future. 15 00:01:18,760 --> 00:01:22,420 ‫And with that, you can now update your a matrix. 16 00:01:23,110 --> 00:01:33,610 ‫Now, from here, you will take 5.1 and 3.1 from this X1 vector and you put it inside your A1 matrix, 17 00:01:34,270 --> 00:01:39,670 ‫but you also need to update your omega inside your A1 matrix. 18 00:01:40,360 --> 00:01:43,810 ‫Now this omega zero, here it was calculated. 19 00:01:44,410 --> 00:01:53,830 ‫Using this you one control input from feedback notation box and using these you two use three and you 20 00:01:53,830 --> 00:01:58,000 ‫for values that you apply to the plan in the previous loop. 21 00:01:58,570 --> 00:01:59,860 ‫You see you one. 22 00:01:59,890 --> 00:02:05,980 ‫It came from your feedback linear zation box, but then you to use three and you four. 23 00:02:06,670 --> 00:02:12,160 ‫These are the values that you had applied to your plan box in the previous loop. 24 00:02:12,790 --> 00:02:15,670 ‫And out of all that, you got your Omega. 25 00:02:16,240 --> 00:02:18,280 ‫And that's how you got your omega zero. 26 00:02:18,880 --> 00:02:27,850 ‫But now, in order to update our Omega and have some kind of omega one that would go inside your A1 27 00:02:27,850 --> 00:02:28,420 ‫matrix. 28 00:02:28,930 --> 00:02:39,610 ‫For that, we will use you to Q3 and Q4 values from here, from use of one in the previous loop or in 29 00:02:39,610 --> 00:02:41,860 ‫other words, you sub zero in this loop. 30 00:02:42,490 --> 00:02:46,990 ‫So Omega Zero was calculated using this information. 31 00:02:47,650 --> 00:02:51,970 ‫Omega one will be calculated using this information. 32 00:02:52,570 --> 00:02:58,420 ‫And don't forget we're talking about our global omega, the omega that we are talking about here. 33 00:02:59,050 --> 00:03:05,770 ‫It's this one here, the omega that is composed of all this omega super water. 34 00:03:06,490 --> 00:03:13,870 ‫And so if you look at this global control architecture, then you can see that your you one, it will 35 00:03:13,870 --> 00:03:22,180 ‫be the same during all the inner loop iterations, you will get a new you one value only after you start 36 00:03:22,180 --> 00:03:29,350 ‫with a new outer loop, but during your inner loop iterations, your U1 will stay the same. 37 00:03:29,950 --> 00:03:36,280 ‫And so in order to calculate your omega one global, your you one will remain unchanged. 38 00:03:36,910 --> 00:03:46,120 ‫But for you to use in Q4, you use these values here and then you just put them inside these formulas 39 00:03:46,120 --> 00:03:46,600 ‫here. 40 00:03:47,230 --> 00:03:54,100 ‫You get your omega one, two, three and four for smaller one, two, three and four, and then you 41 00:03:54,100 --> 00:04:00,640 ‫add them together using this formula here and this one here that will be your omega one global. 42 00:04:01,210 --> 00:04:03,970 ‫And so you're a matrix is now updated. 43 00:04:03,970 --> 00:04:05,110 ‫We call it A1. 44 00:04:05,740 --> 00:04:12,010 ‫It now contains 5.1 theta that one and omega one global. 45 00:04:12,610 --> 00:04:15,760 ‫And now you can calculate your X2. 46 00:04:16,450 --> 00:04:23,560 ‫This is your predicted state vector two time steps into the future, but now you want to predict your 47 00:04:24,220 --> 00:04:27,340 ‫state vector three time steps into the future. 48 00:04:27,910 --> 00:04:36,820 ‫And so for that, your equations will be a two times x two plus b times you two. 49 00:04:38,200 --> 00:04:40,780 ‫And so you simply followed the same drill. 50 00:04:41,350 --> 00:04:42,850 ‫You have this equation here. 51 00:04:43,600 --> 00:04:45,310 ‫This you two here. 52 00:04:45,940 --> 00:04:50,680 ‫That's your third and final leftover from the previous loop. 53 00:04:51,310 --> 00:04:54,460 ‫It was you three in the previous loop. 54 00:04:55,120 --> 00:04:58,900 ‫And now in this loop, it has become you two. 55 00:04:59,530 --> 00:05:04,510 ‫And that's why here I call you to your Vector X2. 56 00:05:05,170 --> 00:05:06,250 ‫It looks like this. 57 00:05:06,910 --> 00:05:08,100 ‫And just like you took. 58 00:05:08,650 --> 00:05:15,160 ‫Find out one and see to that one in order to update your image x here. 59 00:05:15,790 --> 00:05:18,970 ‫Inform your A1 matrix the same thing. 60 00:05:18,970 --> 00:05:30,550 ‫Here you're going to take your 5.2 and feet under two that comes from this x two state vector and you're 61 00:05:30,550 --> 00:05:33,550 ‫going to update your a matrix. 62 00:05:34,030 --> 00:05:38,020 ‫With these two variables, 5.2 and thethird are two. 63 00:05:38,650 --> 00:05:46,540 ‫But of course, you also have to update your global omega inside your matrix in order to form your A2 64 00:05:46,540 --> 00:05:47,050 ‫matrix. 65 00:05:47,680 --> 00:05:54,910 ‫You need your omega to global, and in order to do that, you go back to these equations. 66 00:05:55,570 --> 00:06:03,040 ‫Your you one will be the same because your new one will change only after you go through this feedback 67 00:06:03,100 --> 00:06:08,230 ‫unionisation box again, but you two using your four, they will change. 68 00:06:08,860 --> 00:06:13,600 ‫And so now you will take this sub vector here. 69 00:06:13,930 --> 00:06:22,450 ‫Your second left over and you will use that together with your unchanged you one in order to calculate 70 00:06:22,450 --> 00:06:24,520 ‫your omega to global. 71 00:06:25,120 --> 00:06:28,390 ‫So this second left over, it's this one here. 72 00:06:29,020 --> 00:06:33,610 ‫Or if you think in terms of a next loop, then it's this one here. 73 00:06:34,210 --> 00:06:42,250 ‫And so you put all these used here inside your formula, you calculate your omega one, two, three 74 00:06:42,250 --> 00:06:50,470 ‫and four for your model one, two, three and four, and then you form your global omega, which will 75 00:06:50,470 --> 00:06:55,930 ‫be your omega to global that will also go inside this AA matrix. 76 00:06:56,560 --> 00:07:00,600 ‫And so your A2 matrix is now complete and then you'll be. 77 00:07:00,610 --> 00:07:02,650 ‫Matrix remains unchanged. 78 00:07:03,220 --> 00:07:11,920 ‫And with all that, you can compute your EXP three, which is your predicted state vector three time 79 00:07:11,930 --> 00:07:13,300 ‫steps into the future. 80 00:07:13,900 --> 00:07:22,750 ‫And in order to predict for time steps into the future, you will have this equation a three times Exp 81 00:07:22,750 --> 00:07:26,920 ‫three plus B times use of three. 82 00:07:27,550 --> 00:07:33,310 ‫Now, the thing is that there is no left over left for your U3. 83 00:07:33,880 --> 00:07:36,070 ‫We have used up all the use here. 84 00:07:36,700 --> 00:07:39,790 ‫The first one went into the plant in the previous loop. 85 00:07:40,510 --> 00:07:45,100 ‫Then in the next loop, your you one became your use zero and went here. 86 00:07:45,700 --> 00:07:52,240 ‫Then you two became u one and went here and then your U3. 87 00:07:52,900 --> 00:07:55,240 ‫It became you two and it went here. 88 00:07:55,840 --> 00:07:57,400 ‫So you have no more use left. 89 00:07:57,910 --> 00:08:06,190 ‫But that's OK because your main goal is not to calculate, except for your main goal is to update this 90 00:08:06,580 --> 00:08:18,730 ‫matrix and get your A3 matrix and you get this A3 matrix updated by using information from Excerpt three. 91 00:08:19,270 --> 00:08:29,200 ‫So my ace up is here and then from X up three, you will simply take five, three and three to the three 92 00:08:29,530 --> 00:08:34,060 ‫because this is how your x three looks like transposed. 93 00:08:34,510 --> 00:08:38,470 ‫And so you will take this one and this one. 94 00:08:39,100 --> 00:08:43,690 ‫And now you also need to calculate your omega three global. 95 00:08:44,320 --> 00:08:52,930 ‫And in order to do that, you take this final left over here that you had discarded in the previous 96 00:08:52,930 --> 00:08:53,350 ‫loop. 97 00:08:53,980 --> 00:09:01,210 ‫And you're going to use that to calculate your omega three global together with your unchanged you one 98 00:09:01,210 --> 00:09:08,650 ‫value because again, you one only changes when your next outlook starts. 99 00:09:09,310 --> 00:09:16,900 ‫And so you put all that inside your omega formulas, you calculate your global omega, which will be 100 00:09:16,900 --> 00:09:18,340 ‫your omega three global. 101 00:09:18,970 --> 00:09:25,510 ‫And with that, you update your a matrix and you will have your A3 matrix. 102 00:09:26,170 --> 00:09:30,700 ‫And since our horizon period equals four, then this is where we stop. 103 00:09:32,050 --> 00:09:43,060 ‫And so in the end, you have an eight zero matrix that depends on a 5.0 seat, the zero and omega zero 104 00:09:43,420 --> 00:09:54,280 ‫global, then you have a one that is a function of find out one seat and at one and omega one, global 105 00:09:54,880 --> 00:10:02,020 ‫A2 is a function of fine, a two seater that two and omega two global. 106 00:10:02,560 --> 00:10:05,020 ‫And finally, a three. 107 00:10:05,680 --> 00:10:13,360 ‫It's a function of fine, a three seater that three and omega three global. 108 00:10:13,960 --> 00:10:22,720 ‫So even if you are inside your NPC box predicting your state vectors into the future internally, even 109 00:10:22,720 --> 00:10:25,540 ‫then we are updating our matrices. 110 00:10:26,170 --> 00:10:35,950 ‫So let's now put them in a global matrix vector form where you only need your use and your present state 111 00:10:35,950 --> 00:10:46,360 ‫vector in order to compute your X1, X2, X3 and X4 in one go. 112 00:10:46,960 --> 00:10:48,670 ‫Let's do that in the next video. 113 00:10:48,700 --> 00:10:49,570 ‫Thank you very much.