1 00:00:00,460 --> 00:00:01,310 ‫Welcome back. 2 00:00:02,110 --> 00:00:09,130 ‫And so, as you have seen, it's not really apparent how your errors behave when you choose your key 3 00:00:09,160 --> 00:00:15,190 ‫values like one and then K two in your differential equation. 4 00:00:15,890 --> 00:00:25,270 ‫However, you can very well predict the behavior of your error using the power constants, lambda one 5 00:00:25,420 --> 00:00:26,920 ‫and lambda two. 6 00:00:27,670 --> 00:00:38,200 ‫If you want your error to go to zero, then choose lambda one to be less than zero and lambda two to 7 00:00:38,200 --> 00:00:39,230 ‫be less than zero. 8 00:00:39,250 --> 00:00:45,880 ‫Remember it's an and it must be and and you cannot choose. 9 00:00:45,880 --> 00:00:50,350 ‫Both of them must be less than zero if you want oscillation. 10 00:00:51,250 --> 00:01:00,760 ‫Well then make them complex, make your lambdas complex numbers and then if you want your error to go 11 00:01:00,760 --> 00:01:10,930 ‫to infinity will then either make your lambda one greater than zero or make your lambda to greater than 12 00:01:10,930 --> 00:01:17,170 ‫zero or make them both greater than zero. 13 00:01:17,410 --> 00:01:19,940 ‫And then your error will go to infinity. 14 00:01:20,650 --> 00:01:21,430 ‫Of course. 15 00:01:21,700 --> 00:01:22,990 ‫Why would you want that? 16 00:01:23,290 --> 00:01:26,830 ‫The entire point of control engineering is to get errors to zero. 17 00:01:27,190 --> 00:01:35,630 ‫Countless books have been written just to achieve that one thing, to get your errors to zero. 18 00:01:36,490 --> 00:01:43,180 ‫But anyway, you choose your lambdas based on how you want your system to behave. 19 00:01:43,780 --> 00:01:52,970 ‫And then from those Londis, you compute your key values in your differential equation. 20 00:01:53,800 --> 00:02:01,690 ‫Essentially, you design and build your differential equation from your lambda values. 21 00:02:02,050 --> 00:02:07,080 ‫So you construct your differential equation by choosing your lambdas. 22 00:02:07,870 --> 00:02:14,050 ‫And since your lambdas can be a real and complex numbers, we can create a map for them. 23 00:02:14,770 --> 00:02:16,870 ‫A two dimensional map. 24 00:02:17,410 --> 00:02:25,840 ‫One dimension is for real numbers and the other dimension is for imaginary numbers. 25 00:02:26,710 --> 00:02:31,270 ‫And on that map, I can place my lambda values. 26 00:02:32,050 --> 00:02:37,450 ‫I can place one lambda value here and another lambda value here. 27 00:02:38,170 --> 00:02:45,280 ‫So let's say that I chose my lambda one to be minus three. 28 00:02:45,740 --> 00:02:56,020 ‫It's a real number and I chose my lambda two to be minus four and in both instances the imaginary part 29 00:02:56,230 --> 00:02:57,010 ‫is zero. 30 00:02:57,790 --> 00:03:03,790 ‫And so I have marked my lambda one with a cross and lambda two with a cross. 31 00:03:04,750 --> 00:03:07,420 ‫And these lambdas actually have a name. 32 00:03:07,960 --> 00:03:09,160 ‫They are called. 33 00:03:10,090 --> 00:03:20,890 ‫Polls, these lambdas, they are called polls, so polls are nothing else but the power Constance lambdas 34 00:03:21,340 --> 00:03:28,390 ‫and in control engineering, there is a method called poll placement to stabilize the system. 35 00:03:28,960 --> 00:03:33,730 ‫And poll placement simply means choosing the line does. 36 00:03:34,450 --> 00:03:35,020 ‫That's it. 37 00:03:35,830 --> 00:03:43,060 ‫When you place the polls on this map, you choose your lambdas and this map itself. 38 00:03:43,420 --> 00:03:52,030 ‫This two dimensional map that we have here, this map is called and it's plain. 39 00:03:52,360 --> 00:03:53,020 ‫All right. 40 00:03:53,330 --> 00:03:55,060 ‫So it's called and explain. 41 00:03:55,810 --> 00:03:59,500 ‫And it's also called Le Plus Domain. 42 00:04:00,320 --> 00:04:05,710 ‫OK, so you can also call this two dimensional map Le plus domain. 43 00:04:06,640 --> 00:04:16,750 ‫So Le plus Domain is a two dimensional map on which you can place polls or your Londis or your power 44 00:04:16,750 --> 00:04:17,880 ‫constants here. 45 00:04:18,610 --> 00:04:21,480 ‫So let's have another Le plus domain here. 46 00:04:21,490 --> 00:04:27,970 ‫This is the real number dimension and this is the imaginary number dimension. 47 00:04:28,000 --> 00:04:28,530 ‫All right. 48 00:04:29,290 --> 00:04:37,560 ‫And let's say that we want our system to oscillate and damp out at the same time. 49 00:04:37,960 --> 00:04:46,690 ‫So that means that we need complex poles, complex Londis and those complex lambdas need to have a negative 50 00:04:46,690 --> 00:04:47,510 ‫real number. 51 00:04:48,190 --> 00:04:58,870 ‫So, for example, we need LUNDELL one to be minus four plus eight times three, and then we want Lunda 52 00:04:59,020 --> 00:05:04,730 ‫two to be minus four, minus eight times three. 53 00:05:05,350 --> 00:05:13,750 ‫And remember, when you're dealing with complex poles, then in both cases you're both Poles have the 54 00:05:13,750 --> 00:05:16,330 ‫same real number, which is minus four. 55 00:05:17,300 --> 00:05:25,550 ‫And so in the real number line, let's say that minus four is here and then on the imaginary number 56 00:05:25,550 --> 00:05:33,850 ‫line, you will have plus three here and then minus three here. 57 00:05:34,580 --> 00:05:38,470 ‫And so one of your poles will be here. 58 00:05:39,260 --> 00:05:42,270 ‫That would be your loved one. 59 00:05:42,920 --> 00:05:44,150 ‫It's this one here. 60 00:05:44,840 --> 00:05:47,980 ‫And the second lambda will be here. 61 00:05:48,320 --> 00:05:51,230 ‫That would be your lambda, too. 62 00:05:52,070 --> 00:05:59,020 ‫And so if you place your poles here, then your system response will be something like this. 63 00:05:59,840 --> 00:06:03,060 ‫It will oscillate and it will dump itself out. 64 00:06:04,040 --> 00:06:11,320 ‫And now let's have one more Laplanche domain here or one more splain. 65 00:06:11,570 --> 00:06:16,940 ‫So this is the real number axis and this is the imaginary no axis. 66 00:06:17,690 --> 00:06:24,320 ‫And now let's say that you have a fifth order, homogeneous LTI differential equation. 67 00:06:24,620 --> 00:06:29,240 ‫That means that you have to have five lambdas. 68 00:06:29,990 --> 00:06:32,360 ‫In other words, you need to have five poles. 69 00:06:32,840 --> 00:06:41,090 ‫And let's assume that they are on the Laplanche domain like this one pole is here, the other pole is 70 00:06:41,090 --> 00:06:41,620 ‫here. 71 00:06:42,410 --> 00:06:46,880 ‫Then you have two poles that are conjugates of each other. 72 00:06:46,890 --> 00:06:48,200 ‫They're complex poles. 73 00:06:48,200 --> 00:06:53,750 ‫They have the same real number and then the opposite imaginary numbers. 74 00:06:54,500 --> 00:06:57,370 ‫And then one of the poles is here. 75 00:06:58,250 --> 00:07:03,890 ‫So you see that there is one pole on the right side of the imaginary axis. 76 00:07:04,490 --> 00:07:14,450 ‫That means that you have one lambda that is greater than zero and that means that your entire system 77 00:07:14,600 --> 00:07:16,040 ‫becomes unstable. 78 00:07:16,700 --> 00:07:24,710 ‫The error will go to infinity because one lambda this one here is greater than zero. 79 00:07:25,190 --> 00:07:29,700 ‫One pole is greater than zero in the real number axis. 80 00:07:30,440 --> 00:07:38,420 ‫So if you want stability, then you have to keep all poles on the left side of the imaginary axis. 81 00:07:38,750 --> 00:07:40,550 ‫All poles need to be here. 82 00:07:41,470 --> 00:07:50,380 ‫It means that all your polls would then be less than zero, and in the next video, we will derive formulas 83 00:07:50,560 --> 00:07:56,580 ‫to connect the lambdas or the polls with the key values. 84 00:07:57,580 --> 00:08:05,620 ‫So you choose your polls to achieve the desired error of behavior, and then you will have formulas 85 00:08:06,010 --> 00:08:12,770 ‫to be able to compute the key values which are here in the differential equation. 86 00:08:13,450 --> 00:08:15,790 ‫So that's what we're going to do in the next video. 87 00:08:16,060 --> 00:08:16,990 ‫Thank you very much.