1 00:00:00,480 --> 00:00:01,240 ‫Welcome back. 2 00:00:01,800 --> 00:00:05,340 ‫So we had our non-linear states based equations. 3 00:00:06,400 --> 00:00:12,560 ‫These ones here, and we managed to put them in this form. 4 00:00:13,450 --> 00:00:19,370 ‫Now this state's basic wagin form here was created for linear systems. 5 00:00:20,020 --> 00:00:21,290 ‫It makes sense, right? 6 00:00:21,880 --> 00:00:32,640 ‫You have matrices here and then you also have vectors here like this and vectors and matrices. 7 00:00:32,680 --> 00:00:34,890 ‫They are all part of linear algebra. 8 00:00:35,650 --> 00:00:45,070 ‫So normally you use this form for linear systems, either for LTI systems where your ABCDE matrices 9 00:00:45,070 --> 00:00:53,050 ‫are completely constant or for LTV systems where these matrices directly depend on time. 10 00:00:53,920 --> 00:00:59,530 ‫So this format here is a linear states based equation format. 11 00:01:00,400 --> 00:01:08,860 ‫However, what we did in the previous video, we took our nonlinear equations, the ones that we have 12 00:01:08,860 --> 00:01:18,700 ‫here, these are nonlinear equations and we managed to reformulate them in such a way that now the state's 13 00:01:18,700 --> 00:01:22,140 ‫basic equations appear to be linear. 14 00:01:22,390 --> 00:01:30,040 ‫They are not linear, but they appear to be linear because you have the A, B, C and D matrices and 15 00:01:30,040 --> 00:01:32,820 ‫separate state and input vectors. 16 00:01:33,550 --> 00:01:39,330 ‫So because of that, the system appears to be linear, but in reality they are not. 17 00:01:40,000 --> 00:01:45,770 ‫You have simply packed all the nonlinearities into the A matrix. 18 00:01:46,600 --> 00:01:50,200 ‫In other words, the equations are exactly the same. 19 00:01:50,990 --> 00:01:55,860 ‫You haven't made any approximations, you haven't made any linear ization. 20 00:01:56,530 --> 00:01:58,570 ‫The system is exactly the same. 21 00:01:59,290 --> 00:02:07,200 ‫You simply rearranged the equations by moving all the nonlinearities inside the Matrix. 22 00:02:08,050 --> 00:02:17,860 ‫You're using a standard linear state based equation format on a nonlinear system by keeping all these 23 00:02:17,860 --> 00:02:21,460 ‫nonlinearities hidden inside the Matrix. 24 00:02:22,500 --> 00:02:31,290 ‫And there is a reason why you do it, the nonlinearities that you have inside The Matrix or FYE, that 25 00:02:31,590 --> 00:02:38,350 ‫seat on that and Omega, these variables are measured in real time. 26 00:02:39,300 --> 00:02:47,230 ‫That means that they have to get updated every zero point one second. 27 00:02:48,210 --> 00:02:59,400 ‫That means that in every inner cycle you have a different five dot, Seeta, Dot and Omega and therefore 28 00:02:59,430 --> 00:03:04,820 ‫you have a different a matrix every zero point one second you have a different a matrix. 29 00:03:05,610 --> 00:03:13,050 ‫But then again, the NPC algorithm that you learned in the previous course doesn't care about it. 30 00:03:13,770 --> 00:03:21,120 ‫For the NPC, every inner loop iteration, every zero point one second is a bright new day. 31 00:03:22,130 --> 00:03:32,840 ‫Let's say that at sample time, K, your non constant variables in the Matrix are five, that equals 32 00:03:33,260 --> 00:03:43,300 ‫Phi Dot at K Theta, that equals Theta Dot at K and Omega equals Omega at K. 33 00:03:44,120 --> 00:03:48,800 ‫And so all these variables will go into the A matrix. 34 00:03:49,610 --> 00:03:57,050 ‫And for this particular a zero point, one second inter-group interval, the a matrix is constant. 35 00:03:58,010 --> 00:04:09,170 ‫So in this particular loop at K, the NPC treats the system as an LTI system because all the matrices 36 00:04:09,170 --> 00:04:21,260 ‫are constant, because Phi K, Seeta, Sub K and Omega Sub K are real time value set K. 37 00:04:22,200 --> 00:04:29,670 ‫You get them from a plant or you measure them with sensors, but they are real time values that go into 38 00:04:29,670 --> 00:04:30,720 ‫this, a matrix. 39 00:04:31,800 --> 00:04:34,050 ‫Then zero point one seconds later. 40 00:04:35,060 --> 00:04:42,690 ‫You get your new fi dot theater dot and Omega at Cape plus one. 41 00:04:43,640 --> 00:04:52,100 ‫So now this fire, that variable in this matrix will equal to find that at K plus one, which is a different 42 00:04:52,100 --> 00:04:54,560 ‫value from five dot at K. 43 00:04:55,310 --> 00:05:05,900 ‫The same thing here, theater that becomes Seeta Dot and K plus one and Omega will become Omega at Cape 44 00:05:06,080 --> 00:05:07,640 ‫plus one as well. 45 00:05:08,490 --> 00:05:15,740 ‫Again, these are concrete numeric values that will enter the Matrix. 46 00:05:16,460 --> 00:05:24,260 ‫You're a matrix will now be different compared to the previous one from zero point one seconds before. 47 00:05:25,100 --> 00:05:25,940 ‫But so what? 48 00:05:26,690 --> 00:05:29,170 ‫It has concrete values inside it. 49 00:05:29,180 --> 00:05:37,310 ‫And so the NPC controller that you're familiar with will take these A.B.C.. 50 00:05:37,310 --> 00:05:38,750 ‫The Matrix is. 51 00:05:39,780 --> 00:05:49,410 ‫This critize them and then compute your use three and you four and then zero point one seconds later 52 00:05:49,650 --> 00:05:59,790 ‫at K plus two, you will have a new Firedog Theta Dot and Omega, either your plant or your sensors 53 00:05:59,790 --> 00:06:03,090 ‫in your real UELI will update these values. 54 00:06:03,900 --> 00:06:12,240 ‫And so your a matrix will get updated automatically and then the NPC controller will use that updated 55 00:06:12,240 --> 00:06:17,820 ‫one for computing the new control input values, UTU, U3 and EUFOR. 56 00:06:18,660 --> 00:06:23,820 ‫And that will go on every inner loop till the end of the trajectory. 57 00:06:24,630 --> 00:06:27,420 ‫Remember, the NPC is just an algorithm. 58 00:06:27,420 --> 00:06:38,010 ‫It can I think it just needs constant A, B, C and D matrices matrices with numeric values as their 59 00:06:38,010 --> 00:06:44,340 ‫elements, not functions nor independent variables, but concrete numeric values. 60 00:06:45,120 --> 00:06:49,740 ‫And it will give you some kind of you to use three and EUFOR in return. 61 00:06:50,640 --> 00:06:57,960 ‫And you know what, it turns out that this real time based control where you update the variables in 62 00:06:57,960 --> 00:07:06,930 ‫your a matrix as you go in real time, it turns out that it works very well in many cases. 63 00:07:07,230 --> 00:07:10,680 ‫And in fact, it is a very popular methodology. 64 00:07:11,370 --> 00:07:18,410 ‫My professor at the university once told me that one area where they use it is the aerospace industry, 65 00:07:18,900 --> 00:07:26,130 ‫this kind of methodologies being used in controlling airplanes, for example, due to the fact that 66 00:07:26,130 --> 00:07:31,110 ‫many systems in the aerospace engineering industry are highly nonlinear. 67 00:07:31,830 --> 00:07:37,020 ‫One option would be to use non-linear MPAC techniques for that. 68 00:07:37,890 --> 00:07:46,740 ‫However, as I've heard, these techniques require a lot of computational time because non-linear NPC 69 00:07:46,890 --> 00:07:51,330 ‫simply requires more time for optimizing the cost function. 70 00:07:52,140 --> 00:07:58,380 ‫And that can be problematic when you need your controller to run fast and control the system in real 71 00:07:58,380 --> 00:08:03,160 ‫time, especially if the frequency of a controlled loop is high. 72 00:08:03,930 --> 00:08:09,810 ‫So you need to compute the new you to use three and you four at a very high rate. 73 00:08:10,710 --> 00:08:19,080 ‫So you can either get a faster computer that does computations faster or you can bypass this and use 74 00:08:19,080 --> 00:08:22,680 ‫the easiest NPC that you already know. 75 00:08:23,550 --> 00:08:31,350 ‫By using the methodology that you have just learned here, you put the nonlinear states based equations 76 00:08:31,890 --> 00:08:40,680 ‫in the linear format by keeping the nonlinearities inside the A Matrix elements, meaning that you will 77 00:08:40,680 --> 00:08:44,070 ‫have independent variables in the Matrix. 78 00:08:45,000 --> 00:08:54,480 ‫But then you give those variables a specific value every zero point one second, and that value comes 79 00:08:54,480 --> 00:09:03,780 ‫either from the plant if you're simulating the OV flight or from the sensors of a real USV, if you 80 00:09:03,780 --> 00:09:05,790 ‫are doing real flight. 81 00:09:06,790 --> 00:09:15,490 ‫And so you update your a matrix every zero point one second based on the data that you get in real time, 82 00:09:16,450 --> 00:09:24,310 ‫then you democratize the system and you feed it into your familiar NPC to get your you to use three 83 00:09:24,310 --> 00:09:25,210 ‫and you four. 84 00:09:26,190 --> 00:09:34,950 ‫To get your three control inputs, this strategy might not give you the best optimum solution, a nonlinear 85 00:09:34,950 --> 00:09:39,230 ‫NPC technique might give you more optimal solutions. 86 00:09:39,690 --> 00:09:43,770 ‫However, very often this strategy is good enough. 87 00:09:44,550 --> 00:09:52,490 ‫But the advantage of this strategy is that it runs faster than non-linear NPC techniques. 88 00:09:53,400 --> 00:10:02,760 ‫It arrives to the solutions your you to use three and you for it arrives there faster, making it more 89 00:10:02,760 --> 00:10:04,370 ‫practical in real life. 90 00:10:05,070 --> 00:10:06,720 ‫And this strategy has a name. 91 00:10:06,930 --> 00:10:11,580 ‫It's called a linear parameter. 92 00:10:12,580 --> 00:10:17,440 ‫Varying methodology or LPT for short.