1 00:00:00,730 --> 00:00:09,760 ‫Welcome back in this video, I would like to expand on the LPV concept, we ended up with this formulation 2 00:00:10,390 --> 00:00:19,900 ‫that allowed us to predict our four future states x one x two x three and X four in one go. 3 00:00:20,500 --> 00:00:25,360 ‫However, when it comes to LP V, then this is a simplified approach. 4 00:00:26,020 --> 00:00:28,480 ‫This is your global control schematics. 5 00:00:29,140 --> 00:00:36,940 ‫You have your plant, the planner feedback organization, and in this box you have LP and PC. 6 00:00:37,570 --> 00:00:45,550 ‫You already know that you have four inner loops through one outer loop, so the Feedback Loop unionization 7 00:00:45,550 --> 00:00:53,740 ‫controller gives you a new five R and Theta R every 0.4 seconds that you have to track. 8 00:00:54,340 --> 00:01:04,990 ‫And then NPC using the LP formulation tries to track fire and thetr together with CI R that comes out 9 00:01:04,990 --> 00:01:13,510 ‫of the planner and it tries to finish tracking it during four inner loops and each inner loop lasts 10 00:01:13,510 --> 00:01:15,130 ‫for 0.1 second. 11 00:01:15,700 --> 00:01:21,550 ‫So 0.4 seconds when outer loop 0.1 second one inner loop. 12 00:01:22,090 --> 00:01:31,150 ‫Therefore, you have four inner loops in one outer loop, so every 0.1 second, the LP v NPC controller 13 00:01:31,780 --> 00:01:41,170 ‫gives you a new U2, Q3 and Q4 control inputs that go into the plant together with you, one that comes 14 00:01:41,170 --> 00:01:43,330 ‫out of this feedback linear Asian box. 15 00:01:43,990 --> 00:01:52,870 ‫And then the plant gives you new states every 0.1 seconds, every in the Loop LP, the NPC gives you 16 00:01:53,530 --> 00:01:58,570 ‫new U2, Q3 and Q4 control inputs and every inner loop. 17 00:01:59,020 --> 00:02:00,670 ‫The plan gives you new states. 18 00:02:01,360 --> 00:02:10,840 ‫Of course, you one stays the same during all the four inner loops you want changes every outlook. 19 00:02:11,470 --> 00:02:21,390 ‫And now remember that our continuous a matrix in our LP formulation depends on the variables fi that 20 00:02:21,810 --> 00:02:25,810 ‫standard and also the omega. 21 00:02:26,380 --> 00:02:34,930 ‫And so because of that, the continuous a matrix changes every 0.1 second, every inner loop, not the 22 00:02:34,930 --> 00:02:43,620 ‫B matrix, though the B matrix remains unchanged because it does not have any changing variables there. 23 00:02:44,170 --> 00:02:48,340 ‫Only I double x, I double Y and I double Z. 24 00:02:49,000 --> 00:02:55,450 ‫These are your mass moments of inertia about the body from X, Y and z axis. 25 00:02:56,050 --> 00:03:02,110 ‫And because of the fact that they are measured with respect to your body frame, they stay constant. 26 00:03:02,680 --> 00:03:07,380 ‫But the matrix changes every 0.1 second, every inner loop. 27 00:03:08,020 --> 00:03:15,640 ‫But that also means that in the prediction formula, both of these gigantic matrices. 28 00:03:16,210 --> 00:03:25,960 ‫This one here and this one here, they would also change every 0.1 second because the matrix is inside 29 00:03:25,960 --> 00:03:29,170 ‫them would change every 0.1 second. 30 00:03:29,740 --> 00:03:36,880 ‫Now, if your horizon period is very short, like in our case, Horizon period equals four, then it's 31 00:03:36,880 --> 00:03:37,360 ‫OK. 32 00:03:37,960 --> 00:03:42,640 ‫You can use this formulation, which is what we will do in this course. 33 00:03:43,240 --> 00:03:50,800 ‫However, if the horizon period is longer, which was the case in the previous course in this control 34 00:03:50,800 --> 00:03:59,140 ‫system series called Applied Control Systems, two autonomous cars 360 tracking where the horizon appeared 35 00:03:59,140 --> 00:04:03,640 ‫equal 10, then you have to do it in a more advanced way. 36 00:04:04,210 --> 00:04:09,130 ‫You also have to update your A matrices in your predictions. 37 00:04:10,340 --> 00:04:14,780 ‫So I'm going to draw here, my LPV and BusyBox, my plants. 38 00:04:15,380 --> 00:04:20,030 ‫For now, let's still consider that our Horizon period equals four. 39 00:04:20,630 --> 00:04:30,650 ‫So let's say that your MPAC controller finds this vector of control inputs you global that looks like 40 00:04:30,650 --> 00:04:30,950 ‫this. 41 00:04:31,640 --> 00:04:38,210 ‫Note that this U0 it contains your first you to use three and you four control inputs. 42 00:04:38,900 --> 00:04:40,160 ‫Then you one. 43 00:04:40,850 --> 00:04:46,130 ‫It contains your second to Q3 and Q4 inputs and et cetera. 44 00:04:46,160 --> 00:04:49,160 ‫You two, it would contain your third. 45 00:04:49,160 --> 00:04:52,760 ‫You two use three and you four inputs and then you three. 46 00:04:52,760 --> 00:04:57,290 ‫It would continue to force you to Q3 and Q4 inputs. 47 00:04:57,950 --> 00:05:08,210 ‫And so you already know that NPC only takes the first sub vector and gives it to the plant and it discards 48 00:05:08,390 --> 00:05:09,140 ‫the rest. 49 00:05:09,920 --> 00:05:13,430 ‫These are your leftovers that you're not going to use for your plant. 50 00:05:13,970 --> 00:05:20,940 ‫Then your plant will compute a new state vector that's called X1, and then you're going to start with 51 00:05:20,940 --> 00:05:21,680 ‫the new loop. 52 00:05:22,280 --> 00:05:28,580 ‫So what I will do now, I'm going to put here a red line, and I'm going to say that everything that 53 00:05:28,580 --> 00:05:35,390 ‫crosses this red line, it means that for that variable, a new loop has started. 54 00:05:35,990 --> 00:05:39,770 ‫So for example, this is your new state vector X1. 55 00:05:40,430 --> 00:05:46,610 ‫But once it crosses this red line, then you are in your new loop already. 56 00:05:47,180 --> 00:05:53,330 ‫So this x1 the same vector here it will be X0. 57 00:05:53,450 --> 00:06:01,250 ‫So your present state, it's exactly the same vector is just because it has crossed this red line. 58 00:06:01,730 --> 00:06:10,880 ‫Now I consider this vector to be a present state vector because now I'm in my new loop and I'm concerned 59 00:06:10,880 --> 00:06:17,150 ‫with finding new control inputs in order to calculate a new state vector. 60 00:06:17,720 --> 00:06:23,240 ‫And by the way, this present state vector, it will also enter into your plant. 61 00:06:23,870 --> 00:06:31,610 ‫In other words, once you cross this red line, your new state becomes your present state because you're 62 00:06:31,610 --> 00:06:32,750 ‫in a new loop now. 63 00:06:33,290 --> 00:06:41,000 ‫And then if we look at these control input leftovers, you won you two and you three the ones that you 64 00:06:41,000 --> 00:06:45,940 ‫did not use for your plants, the ones that you discarded. 65 00:06:46,520 --> 00:06:50,570 ‫Well, at the beginning, they are you one, you two and three. 66 00:06:51,200 --> 00:06:59,950 ‫But then once you start with your new loop, once you cross this red line, they become you zero, you 67 00:07:00,020 --> 00:07:02,000 ‫one and you two. 68 00:07:02,630 --> 00:07:05,990 ‫So this one here is this one here. 69 00:07:06,650 --> 00:07:12,530 ‫But in the next loop, then this one here, it's this one here, but in the next loop. 70 00:07:13,130 --> 00:07:17,870 ‫And then this one here, that's this one here in the next loop. 71 00:07:18,500 --> 00:07:25,370 ‫And actually, the same thing happens with this you zero the sum vector that you did apply to your plant. 72 00:07:26,030 --> 00:07:27,950 ‫At the beginning, it was you zero. 73 00:07:28,280 --> 00:07:38,300 ‫But then once you go and enter a new loop, this you zero in the new loop, it will be you minus one. 74 00:07:38,900 --> 00:07:43,430 ‫Since you're in a new loop, everything shifts one step back. 75 00:07:44,900 --> 00:07:52,760 ‫And so as you go through your new loop, you enter into your PC box and inside your LPV, the NPC box, 76 00:07:53,390 --> 00:08:05,330 ‫you have to predict your Future State Vector X1 with this formula, A0 times your present state vector, 77 00:08:05,960 --> 00:08:08,780 ‫plus B times use zero. 78 00:08:09,380 --> 00:08:15,380 ‫So this X1, this is your predicted state vector one time step into the future. 79 00:08:16,010 --> 00:08:20,150 ‫Now, keep in mind that this use zero here. 80 00:08:20,840 --> 00:08:24,110 ‫That's not the one that you had applied to your plans. 81 00:08:24,740 --> 00:08:25,940 ‫One loop before. 82 00:08:26,540 --> 00:08:28,400 ‫Remember, you're in a new loop now. 83 00:08:28,940 --> 00:08:31,730 ‫So this you zero here four. 84 00:08:31,730 --> 00:08:35,840 ‫Which is better if I use green, it's this one here. 85 00:08:36,440 --> 00:08:42,020 ‫Or in other words, in the previous loop, it was your first leftover. 86 00:08:42,590 --> 00:08:43,700 ‫Is this one here? 87 00:08:44,300 --> 00:08:44,960 ‫You won? 88 00:08:45,560 --> 00:08:50,330 ‫But now, since you're in your new loop, this new one became you zero. 89 00:08:50,930 --> 00:08:56,980 ‫And since you're predicting your future state vectors in your new loop, this use zero here. 90 00:08:57,620 --> 00:09:02,420 ‫It's this one here or your first left over from the previous loop. 91 00:09:02,960 --> 00:09:04,680 ‫Then this vector X0. 92 00:09:04,680 --> 00:09:07,490 ‫And that's your present state vector. 93 00:09:07,490 --> 00:09:10,370 ‫In the previous loop, it was your new state vector. 94 00:09:10,670 --> 00:09:18,990 ‫But now in this loop, it's your present state vector, and it looks like this transposed because actually 95 00:09:18,990 --> 00:09:20,120 ‫it's your column vector. 96 00:09:20,690 --> 00:09:31,070 ‫And so from this present state vector, you take 5.0 and citadels zero and you put them inside your 97 00:09:31,070 --> 00:09:32,300 ‫A0 matrix. 98 00:09:32,960 --> 00:09:37,880 ‫You see, this was your continuous a matrix in the LPT format. 99 00:09:38,450 --> 00:09:40,700 ‫But remember that this is zero here. 100 00:09:41,390 --> 00:09:43,880 ‫This is your discrete A0. 101 00:09:44,450 --> 00:09:48,260 ‫So actually, it's like this this aid zero. 102 00:09:48,500 --> 00:09:53,360 ‫It's this one here, and it equals the identity matrix. 103 00:09:53,360 --> 00:10:02,360 ‫Plus the continuous a matrix times the sample time interval and this one here. 104 00:10:03,020 --> 00:10:04,490 ‫That's this one here. 105 00:10:05,090 --> 00:10:09,020 ‫And quite frankly, you don't need these notations here. 106 00:10:09,560 --> 00:10:13,550 ‫You can automatically see that this is a discrete a matrix. 107 00:10:13,940 --> 00:10:14,510 ‫Why? 108 00:10:15,140 --> 00:10:25,670 ‫Because if you look at this form, then it's X at K plus one equals eight times x at the K plus b times 109 00:10:26,330 --> 00:10:27,350 ‫use up K. 110 00:10:27,950 --> 00:10:29,750 ‫This is a discrete form. 111 00:10:30,380 --> 00:10:39,380 ‫A continuous form would be like this x dot equals eight times x plus b times you. 112 00:10:39,980 --> 00:10:47,300 ‫If you have x dot here, then that means that you're A and B matrices are continuous. 113 00:10:47,840 --> 00:10:57,560 ‫If you have X at Kate plus one here and exit K here and you add K here, then your A and B matrices 114 00:10:57,560 --> 00:10:58,430 ‫are discrete. 115 00:10:59,060 --> 00:11:06,320 ‫So I don't have to specify whether my A matrix is discrete or continuous. 116 00:11:06,980 --> 00:11:11,390 ‫You can see from this form this one, it's like your Capa's one. 117 00:11:11,810 --> 00:11:15,200 ‫And this zero, it's like your K and also the zero. 118 00:11:15,200 --> 00:11:16,520 ‫It's like your K here. 119 00:11:17,120 --> 00:11:23,660 ‫But if you look at this form here, then you can see that here you have your ex dot, you have your 120 00:11:23,840 --> 00:11:29,960 ‫five dot fi double down theater dot to double dot signed on side double dot. 121 00:11:30,500 --> 00:11:33,890 ‫So this is clearly in the continuous format. 122 00:11:34,370 --> 00:11:38,480 ‫Therefore, these matrices here, they're continuous. 123 00:11:39,080 --> 00:11:47,570 ‫And so obviously this B matrix here, it's also your discrete P matrix, which is your continuous B 124 00:11:47,570 --> 00:11:49,790 ‫matrix times t. 125 00:11:50,690 --> 00:11:51,840 ‫So that's the formula. 126 00:11:52,370 --> 00:11:57,650 ‫We covered the derivations in the first part in this control system series. 127 00:11:58,280 --> 00:12:01,040 ‫You see, that's how you obtain your disk with a matrix. 128 00:12:01,400 --> 00:12:04,400 ‫And this could be matrix and then you're discrete. 129 00:12:04,550 --> 00:12:08,510 ‫See, Matrix was equal to your continuous see matrix. 130 00:12:09,140 --> 00:12:19,140 ‫And now, once you have put your 5.0 and 10.0 inside your discrete A0 matrix, you take your latest 131 00:12:19,140 --> 00:12:23,870 ‫omega that you have calculated inside your plant. 132 00:12:24,500 --> 00:12:25,640 ‫It's right here. 133 00:12:26,270 --> 00:12:32,180 ‫First, you calculated your omega one omega to omega three and omega four, and then you calculated 134 00:12:32,210 --> 00:12:33,650 ‫your global omega. 135 00:12:34,280 --> 00:12:36,680 ‫It happened here inside the plant. 136 00:12:37,250 --> 00:12:43,940 ‫You see you had your you want you to use three and you four, and you used them to compute your global 137 00:12:43,940 --> 00:12:44,090 ‫or. 138 00:12:45,130 --> 00:12:50,560 ‫And so you take the global Omega that you had calculated in your plant in the previous loop. 139 00:12:51,280 --> 00:13:00,280 ‫And you also put it inside your A0 matrix here because as you can see that in the continuous LP, a 140 00:13:00,310 --> 00:13:03,040 ‫matrix, you also have omega here. 141 00:13:03,610 --> 00:13:09,790 ‫And that's why you also have to put it inside your discrete a matrix in the previous loop this omega 142 00:13:09,790 --> 00:13:10,300 ‫zero. 143 00:13:10,930 --> 00:13:16,540 ‫It was actually omega one because it was your new omega that you had calculated inside your plant. 144 00:13:16,840 --> 00:13:21,460 ‫But now, since you started with a new loop, it became your present omega.