WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.720 continue to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.320 To make a donation, or view additional materials 00:00:13.320 --> 00:00:17.280 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.280 --> 00:00:18.450 at ocw.mit.edu. 00:00:30.707 --> 00:00:32.290 PROFESSOR: I wanted to give a lecture, 00:00:32.290 --> 00:00:36.819 because as I told you in presentations, I love feedback. 00:00:36.819 --> 00:00:38.235 In fact, I love it so much, that I 00:00:38.235 --> 00:00:40.890 think the examples we're going to do, 00:00:40.890 --> 00:00:45.900 we can do analytically are not maybe 00:00:45.900 --> 00:00:50.860 sufficiently compelling for you to believe how exciting control 00:00:50.860 --> 00:00:51.360 can be. 00:00:51.360 --> 00:00:54.090 So let me just start by giving a few minutes of a research 00:00:54.090 --> 00:00:57.181 result we had a few years ago. 00:00:57.181 --> 00:00:59.430 That's why you've got color images-- everybody noticed 00:00:59.430 --> 00:01:02.670 the color images on the slides. 00:01:02.670 --> 00:01:06.180 I brought this robot in to the first recitation 00:01:06.180 --> 00:01:07.610 of [INAUDIBLE]. 00:01:07.610 --> 00:01:11.350 It's a big 2 meter wing span ornithopter, and in my case, 00:01:11.350 --> 00:01:13.950 we've been trying to design control systems to make 00:01:13.950 --> 00:01:16.710 airplanes move more like birds. 00:01:16.710 --> 00:01:18.880 Here's one example of that. 00:01:18.880 --> 00:01:21.080 One thing birds do that planes don't do very well 00:01:21.080 --> 00:01:24.340 is land on a perch. 00:01:24.340 --> 00:01:26.880 So we asked the question, can we take a simple airplane 00:01:26.880 --> 00:01:29.280 with fixed wings, no flapping allowed, 00:01:29.280 --> 00:01:31.260 and make it land on a perch-- 00:01:31.260 --> 00:01:33.880 UAV stands for unmanned aerial vehicle-- 00:01:33.880 --> 00:01:36.730 land on a perch like a bird. 00:01:36.730 --> 00:01:39.730 The story I want to tell you is that with feedback design, 00:01:39.730 --> 00:01:41.670 you can. 00:01:41.670 --> 00:01:43.890 Here's why it's interesting and hard. 00:01:43.890 --> 00:01:47.600 The reason airplanes don't land on a perch 00:01:47.600 --> 00:01:51.136 is because, when your wing is at a low angle, which 00:01:51.136 --> 00:01:53.120 a plane normally is when it's flying, 00:01:53.120 --> 00:01:56.350 the air flow around the wing is relatively easy to model. 00:01:56.350 --> 00:01:59.730 It's easy to write a block diagram description of. 00:01:59.730 --> 00:02:02.590 But if you go to moderate angles, that's still true. 00:02:02.590 --> 00:02:05.210 The air stays attached to the wings and everything-- 00:02:05.210 --> 00:02:12.050 [INAUDIBLE] 00:02:12.050 --> 00:02:16.660 At a low angle, the air is still attached to the wing, 00:02:16.660 --> 00:02:19.610 and so we have very good models of the airflow around here, 00:02:19.610 --> 00:02:22.100 and the lift and drag forces that you get when 00:02:22.100 --> 00:02:23.900 you're flying in these regimes. 00:02:23.900 --> 00:02:27.020 But if you go up too far, like this, then everything changes. 00:02:27.020 --> 00:02:30.380 So you think, the air can't quite stay around the wing. 00:02:30.380 --> 00:02:31.490 You get separation. 00:02:31.490 --> 00:02:33.860 You get big vortices pulling off the back. 00:02:33.860 --> 00:02:36.920 It's very nonlinear, very unsteady. 00:02:36.920 --> 00:02:38.810 Very complicated flight regime. 00:02:38.810 --> 00:02:43.040 And as such, our best aircrafts to date 00:02:43.040 --> 00:02:45.710 mostly stay down in this regime where we have good models 00:02:45.710 --> 00:02:49.740 and we can design conventional control systems for. 00:02:49.740 --> 00:02:51.780 But birds don't do that at all. 00:02:51.780 --> 00:02:54.440 So birds are, often, when they're landing on a perch, 00:02:54.440 --> 00:02:56.570 they're up even far beyond this. 00:02:56.570 --> 00:02:57.980 Maybe even 90 degrees. 00:02:57.980 --> 00:02:59.840 "Angle of attack", it's called. 00:02:59.840 --> 00:03:02.390 They're way into this deep stall regime-- 00:03:02.390 --> 00:03:06.100 that's called stalling your wings when that happens. 00:03:06.100 --> 00:03:08.900 And it seems like they're doing a lot better landing on a perch 00:03:08.900 --> 00:03:11.420 because we don't see our airplanes do it. 00:03:11.420 --> 00:03:13.310 But you can actually try to quantify that. 00:03:13.310 --> 00:03:16.190 So if you want to compare the performance of a bird 00:03:16.190 --> 00:03:18.807 landing versus a plane landing, the first thing you have to do 00:03:18.807 --> 00:03:21.140 is you have to take out the differences in mass and wing 00:03:21.140 --> 00:03:22.820 area and all these things. 00:03:22.820 --> 00:03:24.130 But you can do that. 00:03:24.130 --> 00:03:27.320 And a fair comparison is the distance 00:03:27.320 --> 00:03:30.850 averaged drag coefficient, which is just a way 00:03:30.850 --> 00:03:33.650 to scale out the effects that you'd expect from having 00:03:33.650 --> 00:03:36.950 a bigger wing or a bigger mass. 00:03:36.950 --> 00:03:38.960 You can plot this drag coefficient 00:03:38.960 --> 00:03:41.090 for a few different vehicles. 00:03:41.090 --> 00:03:44.330 This is a standard runway landing of a 747, 00:03:44.330 --> 00:03:45.830 would get the drag coefficient, just 00:03:45.830 --> 00:03:50.480 so you have some calibration, of about a 0.16. 00:03:50.480 --> 00:03:54.920 The X-31 was a super maneuverable research vehicle. 00:03:54.920 --> 00:03:58.256 It was designed to do super short runway landings, 00:03:58.256 --> 00:04:00.380 and it got a drag coefficient during those landings 00:04:00.380 --> 00:04:02.967 of about 0.3. 00:04:02.967 --> 00:04:04.550 There's a few other projects out there 00:04:04.550 --> 00:04:06.870 about trying to make perching planes that 00:04:06.870 --> 00:04:08.621 were getting similar numbers. 00:04:08.621 --> 00:04:10.370 But then we went out and looked at nature. 00:04:10.370 --> 00:04:12.230 I have some collaborators at Harvard 00:04:12.230 --> 00:04:14.667 that work with real birds. 00:04:14.667 --> 00:04:16.250 It turns out-- I wanted him to tell me 00:04:16.250 --> 00:04:18.200 the numbers from some really elite bird, 00:04:18.200 --> 00:04:21.179 like at least a hawk or something like this. 00:04:21.179 --> 00:04:22.220 But they work on pigeons. 00:04:22.220 --> 00:04:24.440 So that was the only number we could get, was a pigeon. 00:04:24.440 --> 00:04:25.550 They actually convinced me by the end 00:04:25.550 --> 00:04:26.650 that pigeons are really good. 00:04:26.650 --> 00:04:28.608 The way they can sort of dive through the fence 00:04:28.608 --> 00:04:30.200 and get your lunch by the food trucks? 00:04:30.200 --> 00:04:32.100 I mean, that's actually-- 00:04:32.100 --> 00:04:35.210 they're seriously skilled birds. 00:04:35.210 --> 00:04:36.680 We even had-- one of the fun things 00:04:36.680 --> 00:04:38.240 about doing this kind of research is you get visitors. 00:04:38.240 --> 00:04:41.181 We had will.i.am from the Black Eyed Peas come to the lab. 00:04:41.181 --> 00:04:42.930 And he said-- after we told him the story, 00:04:42.930 --> 00:04:45.410 he said pigeons are ghetto birds. 00:04:45.410 --> 00:04:47.030 They got mad stop. 00:04:47.030 --> 00:04:50.390 That sort of summarizes it. 00:04:53.390 --> 00:04:54.620 You already know the answer. 00:04:54.620 --> 00:04:57.800 But if you look at the drag coefficient 00:04:57.800 --> 00:05:01.430 that a bird gets when it's landing, they get a 10. 00:05:01.430 --> 00:05:03.620 So they're doing orders of magnitude 00:05:03.620 --> 00:05:05.730 better than our best planes in terms of stopping. 00:05:05.730 --> 00:05:07.460 If you just want to be fun about it, 00:05:07.460 --> 00:05:09.290 you could say, what would it take for a 747 00:05:09.290 --> 00:05:12.090 to get a 10 to impress will.i.am. 00:05:12.090 --> 00:05:14.240 And it turns out that 747 would have 00:05:14.240 --> 00:05:17.600 to go-- they fly at about 450 miles an hour cruise speed. 00:05:17.600 --> 00:05:19.700 It would have to stop in 40 meters 00:05:19.700 --> 00:05:21.145 to do what a bird's doing. 00:05:21.145 --> 00:05:22.520 The wings would probably pop off. 00:05:22.520 --> 00:05:23.645 There's problems with that. 00:05:23.645 --> 00:05:26.782 But the dynamics are impressive, of the birds. 00:05:26.782 --> 00:05:28.490 And what's more, when they're doing that, 00:05:28.490 --> 00:05:30.890 they're getting incredible accuracy. 00:05:30.890 --> 00:05:35.474 This is one of my favorite videos of a perching owl. 00:05:35.474 --> 00:05:37.640 If you watch really closely, you can see the airflow 00:05:37.640 --> 00:05:38.870 get complicated on his wings. 00:05:38.870 --> 00:05:41.390 You can see his leading edge feathers start to flip over. 00:05:49.940 --> 00:05:50.677 Boom. 00:05:50.677 --> 00:05:52.760 The way you do that is you put food by the camera. 00:05:55.910 --> 00:05:57.740 We tried to build these simple planes 00:05:57.740 --> 00:05:59.855 and ask if could they do what the owls are doing, 00:05:59.855 --> 00:06:00.980 what the pigeons are doing. 00:06:00.980 --> 00:06:02.780 Very simple planes. 00:06:02.780 --> 00:06:05.660 We did them in an indoor environment. 00:06:05.660 --> 00:06:09.800 We made it so it was basically a boring plane, 00:06:09.800 --> 00:06:12.545 but it can do very interesting things in pitch, up and down. 00:06:15.590 --> 00:06:18.530 Then we spent some time trying to build a dynamical systems 00:06:18.530 --> 00:06:20.060 representation of the plane. 00:06:20.060 --> 00:06:21.341 And how do you do that? 00:06:21.341 --> 00:06:23.090 Well, you shoot the plane a bunch of times 00:06:23.090 --> 00:06:25.340 into a fishing net. 00:06:25.340 --> 00:06:26.840 And you collect a bunch of data. 00:06:26.840 --> 00:06:28.464 And like we talked about in recitation, 00:06:28.464 --> 00:06:30.230 you can potentially get from that data 00:06:30.230 --> 00:06:31.700 a model of the dynamics. 00:06:31.700 --> 00:06:33.920 And turns out, we had to do a nonlinear dynamical 00:06:33.920 --> 00:06:35.810 model for the vehicle, but we could 00:06:35.810 --> 00:06:39.830 do a linear actuator model plus delays and everything. 00:06:39.830 --> 00:06:43.520 We built a pretty good model of the plane, which is sort of-- 00:06:43.520 --> 00:06:45.560 for an aerodynamics crowd, this would 00:06:45.560 --> 00:06:48.170 tell you that this is a surprisingly good fit 00:06:48.170 --> 00:06:53.960 to the lift and drag forces over a very large range of angles. 00:06:53.960 --> 00:06:57.250 Then we had a 6003 problem, basically. 00:06:57.250 --> 00:06:59.785 The nonlinear terms make it a little different. 00:06:59.785 --> 00:07:01.160 And what we did as a group was we 00:07:01.160 --> 00:07:04.820 innovated a way to design a feedback controller that 00:07:04.820 --> 00:07:07.370 could make this thing very accurate during very 00:07:07.370 --> 00:07:09.680 high performance maneuvers. 00:07:09.680 --> 00:07:12.260 And a long story short, we can now shoot an airplane 00:07:12.260 --> 00:07:14.090 into a motion capture arena. 00:07:14.090 --> 00:07:16.390 This is slowed down about 11 times. 00:07:16.390 --> 00:07:18.210 It's this airplane right here. 00:07:18.210 --> 00:07:20.869 And it goes into a very deep stall. 00:07:20.869 --> 00:07:22.910 And we can land with an accuracy, enough accuracy 00:07:22.910 --> 00:07:25.205 to land on a perch. 00:07:25.205 --> 00:07:27.830 It turns out if you can build a good enough control system that 00:07:27.830 --> 00:07:30.440 can handle the complexity of the dynamics, 00:07:30.440 --> 00:07:33.631 then you can make these things happen. 00:07:33.631 --> 00:07:35.630 This is just-- to convince you that the dynamics 00:07:35.630 --> 00:07:38.240 are complicated, this is the flow visualization. 00:07:38.240 --> 00:07:41.570 We built a wind tunnel releasing smoke from the leading edge 00:07:41.570 --> 00:07:43.850 and took a picture of it to show you how 00:07:43.850 --> 00:07:45.230 complicated the dynamics were. 00:07:45.230 --> 00:07:49.580 So control is potentially an incredibly powerful idea. 00:07:49.580 --> 00:07:52.940 You can make-- we try to make robots that run like ostriches. 00:07:52.940 --> 00:07:55.490 We try to make all kinds of things happen. 00:07:55.490 --> 00:07:57.999 You could imagine improving wind energy. 00:07:57.999 --> 00:07:59.540 You could imagine all kinds of things 00:07:59.540 --> 00:08:01.206 where control is an integral part of it. 00:08:05.410 --> 00:08:08.330 The feedback in there was absolutely essential. 00:08:08.330 --> 00:08:12.590 If I took the same plane and thought about the model a lot 00:08:12.590 --> 00:08:16.340 and then designed a controller, just a set of commands 00:08:16.340 --> 00:08:20.240 to the elevator, to try to make it land on the perch, 00:08:20.240 --> 00:08:23.210 it misses the perch every single time. 00:08:23.210 --> 00:08:24.830 And with feedback, we can hit-- it's 00:08:24.830 --> 00:08:26.246 only with feedback that we can hit 00:08:26.246 --> 00:08:27.600 the perch every single time. 00:08:27.600 --> 00:08:28.850 And there's a reason for that. 00:08:28.850 --> 00:08:34.010 You guys probably heard the idea that fighter jets 00:08:34.010 --> 00:08:36.200 are unstable without the control system, right? 00:08:36.200 --> 00:08:39.020 Fighter jets are-- a lot of times, the systems that 00:08:39.020 --> 00:08:41.330 have peak maneuverability, that's 00:08:41.330 --> 00:08:43.919 often at odds with stability. 00:08:43.919 --> 00:08:46.460 In fact, when you try to make a very high performance fighter 00:08:46.460 --> 00:08:49.730 jet, and you want to be able to turn, you actually-- 00:08:49.730 --> 00:08:53.150 if the control system is off, this thing should be unstable. 00:08:53.150 --> 00:08:57.200 It's not a complete pathology, but that's pretty true, 00:08:57.200 --> 00:09:00.200 because what you want, you can elicit a very fast turn 00:09:00.200 --> 00:09:02.540 if you basically let the system go unstable. 00:09:02.540 --> 00:09:05.330 Then you can let the unstable dynamics of the system 00:09:05.330 --> 00:09:07.880 make a very rapid turn. 00:09:07.880 --> 00:09:12.370 In fact, it's common to build a system that is unstable, 00:09:12.370 --> 00:09:14.640 and it's only stable when there's feedback in the loop 00:09:14.640 --> 00:09:17.071 so that you can have very high maneuverability when 00:09:17.071 --> 00:09:17.570 you need it. 00:09:21.710 --> 00:09:25.520 You've already done feedback if you took 601, right? 00:09:25.520 --> 00:09:28.190 So what I want to do today is-- 00:09:28.190 --> 00:09:30.620 I taught 601 when some of you were in there. 00:09:30.620 --> 00:09:33.920 I want to go through the example that you already worked through 00:09:33.920 --> 00:09:36.050 in the 601 lab for those of you that took it. 00:09:36.050 --> 00:09:39.110 I think it's sufficiently complete here 00:09:39.110 --> 00:09:40.430 if you didn't take it. 00:09:40.430 --> 00:09:41.480 But we want to use that-- 00:09:41.480 --> 00:09:42.950 I'm going to use that as an example to build 00:09:42.950 --> 00:09:45.366 on what you already know and to show us with the new tools 00:09:45.366 --> 00:09:48.050 how far you can get in thinking about what that robot did 00:09:48.050 --> 00:09:49.240 for you last year. 00:09:49.240 --> 00:09:52.490 Do you remember this example of the little pioneer robot that 00:09:52.490 --> 00:09:56.240 had to go to the wall, the foam wall, 00:09:56.240 --> 00:09:58.130 using the noisy sonar sensor? 00:09:58.130 --> 00:10:00.500 And we wanted to get a desired distance from the wall? 00:10:00.500 --> 00:10:02.450 And by the end, you guys were moving the board around 00:10:02.450 --> 00:10:03.950 and it was trying to track the wall? 00:10:06.860 --> 00:10:09.590 In that exercise, in the 601 lab, 00:10:09.590 --> 00:10:11.930 you guys tried a bunch of different gains. 00:10:11.930 --> 00:10:14.810 For some gains, we saw responses that sort of looked like this. 00:10:14.810 --> 00:10:16.630 If you gave a desired distance to a wall, 00:10:16.630 --> 00:10:18.491 it would go up to that desired distance. 00:10:18.491 --> 00:10:20.240 For other gains, you saw some oscillation. 00:10:20.240 --> 00:10:22.760 You saw some big oscillations if you tried 00:10:22.760 --> 00:10:25.490 all the gains we recommended. 00:10:25.490 --> 00:10:27.114 Now you guys have a deep understanding 00:10:27.114 --> 00:10:29.405 of what kind of things can cause that type of response. 00:10:32.960 --> 00:10:35.876 The way we told you to think about it in 601 00:10:35.876 --> 00:10:37.250 is actually the way that we often 00:10:37.250 --> 00:10:39.590 think about control systems. 00:10:39.590 --> 00:10:42.470 Typically, we have something called the plant, which 00:10:42.470 --> 00:10:44.480 describes the dynamics of the robot or the thing 00:10:44.480 --> 00:10:46.160 we're trying to care about. 00:10:46.160 --> 00:10:49.690 You guys know why it's called the plant often? 00:10:49.690 --> 00:10:52.580 What would be the history of calling it "plant"? 00:10:52.580 --> 00:10:55.634 It's a weird name for it, right? 00:10:55.634 --> 00:10:57.800 Actually, some of this stuff grew up in the chemical 00:10:57.800 --> 00:10:59.424 industry, in chemical plants. 00:10:59.424 --> 00:11:00.340 Is that what you said? 00:11:00.340 --> 00:11:01.730 No? 00:11:01.730 --> 00:11:04.560 It's actually-- it has a history in chemical plants. 00:11:04.560 --> 00:11:09.020 But now everybody calls their robot plants. 00:11:09.020 --> 00:11:12.140 The plant is the dynamics of our robot, for instance. 00:11:12.140 --> 00:11:13.490 You've got some output of that. 00:11:13.490 --> 00:11:15.830 In this case, it's the position to the wall. 00:11:15.830 --> 00:11:17.330 A sensor that reads it. 00:11:17.330 --> 00:11:19.820 It may have its own dynamics. 00:11:19.820 --> 00:11:23.090 And your goal is to take some reference command, a position 00:11:23.090 --> 00:11:24.784 you want to go to, let's say, compare it 00:11:24.784 --> 00:11:27.200 with what the sensor is saying and build a controller that 00:11:27.200 --> 00:11:31.371 takes that error and makes the robot or control system do what 00:11:31.371 --> 00:11:32.120 you want it to do. 00:11:35.560 --> 00:11:39.370 In this example, the one you studied in 601, 00:11:39.370 --> 00:11:41.470 the plant was a very simple model 00:11:41.470 --> 00:11:43.690 of the dynamics of the robot. 00:11:43.690 --> 00:11:45.700 It was just a first-order model. 00:11:45.700 --> 00:11:48.010 We said that the output, the distance 00:11:48.010 --> 00:11:51.970 from the wall, the distance in front of the wall, at time n 00:11:51.970 --> 00:11:56.530 was just the distance in front of the wall time minus-- 00:11:56.530 --> 00:11:58.210 because, you remember the frustration 00:11:58.210 --> 00:11:59.590 of the flipped sign, too? 00:11:59.590 --> 00:12:01.730 But I kept it consistent here. 00:12:01.730 --> 00:12:03.399 So it's T times the velocity, just 00:12:03.399 --> 00:12:05.440 happens that since the velocity is going this way 00:12:05.440 --> 00:12:06.970 and the distance is getting smaller, 00:12:06.970 --> 00:12:08.890 you get a minus T. This looks almost 00:12:08.890 --> 00:12:13.590 like the first-order approximation of a CT system, 00:12:13.590 --> 00:12:15.130 right? 00:12:15.130 --> 00:12:17.131 So that's the dynamics of the vehicle. 00:12:17.131 --> 00:12:18.880 The sensor, we're going to assume for now, 00:12:18.880 --> 00:12:21.070 is just perfect, that it just immediately 00:12:21.070 --> 00:12:23.650 tells me the distance to the wall 00:12:23.650 --> 00:12:25.570 and gives the perfect feedback. 00:12:25.570 --> 00:12:27.917 And then we designed a controller, which 00:12:27.917 --> 00:12:30.250 just takes the error, the difference between the desired 00:12:30.250 --> 00:12:32.560 position and the actual position, 00:12:32.560 --> 00:12:35.800 multiplies it by a constant K and comes 00:12:35.800 --> 00:12:38.170 up with a velocity command that goes to the motors. 00:12:41.420 --> 00:12:45.810 You can visualize that as a block diagram, of course. 00:12:45.810 --> 00:12:49.880 You get that this plant model here-- 00:12:49.880 --> 00:12:51.890 it's got a minus one here and a minus one here, 00:12:51.890 --> 00:12:54.740 so that's equivalent having a delay in the forward path. 00:12:54.740 --> 00:12:57.210 And then it's got feedback, because the previous signal 00:12:57.210 --> 00:12:58.880 was directly put around to the feedback. 00:12:58.880 --> 00:13:02.570 So this part here is the model of the plant. 00:13:05.190 --> 00:13:07.130 Then we put the T in, with a gain 00:13:07.130 --> 00:13:10.100 here, so I guess this part here is the plant. 00:13:10.100 --> 00:13:14.334 Here's our simple controller and our sensor is perfect. 00:13:14.334 --> 00:13:15.500 So that's our block diagram. 00:13:15.500 --> 00:13:17.541 We've just turned our robot into a block diagram. 00:13:17.541 --> 00:13:21.740 And we know everything about how to analyze those things. 00:13:21.740 --> 00:13:23.390 You can use the operator notation. 00:13:23.390 --> 00:13:28.070 You could think of it as a system function, too. 00:13:28.070 --> 00:13:29.900 This guy here you know is-- 00:13:33.032 --> 00:13:34.990 you guys can do this in your sleep now, almost? 00:13:34.990 --> 00:13:38.126 But if I just take that by itself, 00:13:38.126 --> 00:13:39.250 what does that come out to? 00:13:41.800 --> 00:13:42.850 It's a plus here. 00:13:42.850 --> 00:13:44.920 This is x and y. 00:13:44.920 --> 00:13:52.630 So I get y is R x plus y. 00:13:52.630 --> 00:14:01.720 Or y over x is R over 1 minus R. Then 00:14:01.720 --> 00:14:06.144 I multiply that by K and a negative t. 00:14:06.144 --> 00:14:07.810 And then I do the exact same computation 00:14:07.810 --> 00:14:09.550 again to get this loop around. 00:14:09.550 --> 00:14:14.060 And this is our input output system function. 00:14:14.060 --> 00:14:16.510 Simplify it a little bit, I get this. 00:14:16.510 --> 00:14:20.080 Simplify it a little bit more to identify the pole, 00:14:20.080 --> 00:14:22.540 and you can see that the pole now looks like 1 plus KT. 00:14:25.100 --> 00:14:29.200 T represents the time step between updates, 00:14:29.200 --> 00:14:30.300 K is our feedback gain. 00:14:33.679 --> 00:14:35.220 Now, we can start thinking about what 00:14:35.220 --> 00:14:37.050 happens if we choose different-- let's just 00:14:37.050 --> 00:14:39.360 lump K and T together, because the K you choose 00:14:39.360 --> 00:14:41.664 is going to be intimately connected to time. 00:14:41.664 --> 00:14:43.830 In this system, there's no point in separating them. 00:14:43.830 --> 00:14:46.980 So we just talk about KT as a system. 00:14:46.980 --> 00:14:48.150 If we chose KT-- 00:14:48.150 --> 00:14:50.100 KT is almost always going to be negative. 00:14:50.100 --> 00:14:52.410 You want negative feedback, in general, 00:14:52.410 --> 00:14:54.960 for making things stable. 00:14:54.960 --> 00:14:59.280 If we choose KT equals 0.5, then with that system function, 00:14:59.280 --> 00:15:00.780 there's a delay in the forward path, 00:15:00.780 --> 00:15:02.154 so it's going to be offset by one 00:15:02.154 --> 00:15:06.000 and then it's just got the simple single pole response. 00:15:06.000 --> 00:15:09.222 The unit step response similarly looks like this. 00:15:09.222 --> 00:15:10.680 So the question is, what determines 00:15:10.680 --> 00:15:14.030 the speed of that response? 00:15:14.030 --> 00:15:15.780 Here you go. 00:15:15.780 --> 00:15:17.530 I have to get you to talk to your neighbor 00:15:17.530 --> 00:15:18.904 the way Denny always seems to get 00:15:18.904 --> 00:15:20.587 you to talk to your neighbor. 00:15:20.587 --> 00:15:21.170 Take a minute. 00:15:21.170 --> 00:15:23.570 Figure out which of these or none of them do you think 00:15:23.570 --> 00:15:26.420 would give you the best response for this simple block diagram 00:15:26.420 --> 00:15:26.920 system. 00:15:31.889 --> 00:15:33.180 Talk out loud to your neighbor. 00:16:48.501 --> 00:16:49.000 OK. 00:16:49.000 --> 00:16:50.770 Show of fingers, what do you think it is? 00:16:54.760 --> 00:16:55.260 OK. 00:16:55.260 --> 00:16:56.520 A lot of right answers. 00:16:56.520 --> 00:16:57.885 Let's just do it real quick. 00:17:01.100 --> 00:17:03.830 The fastest possible convergence is 00:17:03.830 --> 00:17:06.470 going to be at the pole zero. 00:17:06.470 --> 00:17:08.730 Pole zero, what's it going to look like? 00:17:08.730 --> 00:17:11.240 It's going to give you-- 00:17:11.240 --> 00:17:13.280 the best you can do if you zero this out, 00:17:13.280 --> 00:17:15.391 you get the impulse response. 00:17:15.391 --> 00:17:16.849 The unit sample response would just 00:17:16.849 --> 00:17:21.780 be R. You're going to have a delay of 1, that's inevitable. 00:17:21.780 --> 00:17:27.340 You get one non-zero entry and then zeroes everywhere else. 00:17:27.340 --> 00:17:29.090 In discrete time systems, you can actually 00:17:29.090 --> 00:17:30.560 kill things in a single step. 00:17:30.560 --> 00:17:35.560 You can set a pole at 0 that's the fastest possible response. 00:17:35.560 --> 00:17:37.550 But if you think about the different responses 00:17:37.550 --> 00:17:40.700 for different values of K, you can use the pole zero diagrams 00:17:40.700 --> 00:17:42.230 to pretty much understand everything 00:17:42.230 --> 00:17:45.020 there is to know about it. 00:17:45.020 --> 00:17:49.220 For KT less than 0 to negative 1-- like I said, 00:17:49.220 --> 00:17:51.140 we want to think about negative feedback-- 00:17:51.140 --> 00:17:54.860 that's going to take us from out here on the unit circle back 00:17:54.860 --> 00:17:55.760 towards the origin. 00:17:55.760 --> 00:17:58.400 If I keep making KT more negative, 00:17:58.400 --> 00:17:59.620 it's going to go out here. 00:17:59.620 --> 00:18:01.697 If I make KT too negative, bad things 00:18:01.697 --> 00:18:02.780 are going to happen again. 00:18:02.780 --> 00:18:04.820 The poles are going to go outside the unit circle 00:18:04.820 --> 00:18:06.236 and will actually have alternating 00:18:06.236 --> 00:18:07.340 diverging responses. 00:18:12.650 --> 00:18:14.570 The answer is negative 1. 00:18:14.570 --> 00:18:16.880 Negative 1 puts you at a pole of 0. 00:18:21.810 --> 00:18:24.540 Here's to think about it physically. 00:18:24.540 --> 00:18:30.690 If I choose K correctly, since KT was negative 1, 00:18:30.690 --> 00:18:33.130 that means K's going to be negative 10. 00:18:33.130 --> 00:18:36.660 That's going to be commanding exactly the right speed so 00:18:36.660 --> 00:18:41.580 that the robot, after the one tenth of a second, let's say, 00:18:41.580 --> 00:18:43.610 it gets you exactly to the right position. 00:18:43.610 --> 00:18:48.090 Unit sample response in this case would be saying, 00:18:48.090 --> 00:18:50.257 the robot's at zero, I want it to be at 1, 00:18:50.257 --> 00:18:51.840 and then I want it to be back at zero. 00:18:51.840 --> 00:18:54.660 The command is going 0, 1, 0. 00:18:54.660 --> 00:18:57.130 And the robot is going to be almost doing exactly that. 00:18:57.130 --> 00:19:00.300 It's going to go from 0 to 1 in a single step. 00:19:00.300 --> 00:19:03.180 And then back to 0 in a single step. 00:19:03.180 --> 00:19:04.830 It's just going to be delayed by one. 00:19:04.830 --> 00:19:07.814 So you give it a command saying, go here, go here, go back. 00:19:07.814 --> 00:19:10.230 And it's going to do exactly the right thing-- boom, boom, 00:19:10.230 --> 00:19:12.480 except for one step delayed, because you can't get rid 00:19:12.480 --> 00:19:14.490 of that R. 00:19:14.490 --> 00:19:18.002 So if I plot it-- let me draw a stem diagram, 00:19:18.002 --> 00:19:20.460 but coming down in time so that I can line up with the axes 00:19:20.460 --> 00:19:21.440 up here. 00:19:21.440 --> 00:19:24.570 If I start with an initial position here, 00:19:24.570 --> 00:19:28.020 and I command it to go to the desired front position, 00:19:28.020 --> 00:19:30.720 it's going to go boom right to that front position 00:19:30.720 --> 00:19:32.460 with one unit of delay. 00:19:32.460 --> 00:19:37.880 And then it's going to stay there for the rest of the time. 00:19:37.880 --> 00:19:40.260 It would be a unit step response. 00:19:44.220 --> 00:19:47.890 But that's not what we got to see on the robots in 601, 00:19:47.890 --> 00:19:48.390 right? 00:19:48.390 --> 00:19:51.670 The real robot didn't work like that. 00:19:51.670 --> 00:19:55.420 And the way we made the robot model more realistic was we 00:19:55.420 --> 00:19:58.240 said, OK, but your sensor's got some delay. 00:19:58.240 --> 00:20:00.310 And actually, if you knew what was going on 00:20:00.310 --> 00:20:02.850 behind those 601 robots, it's actually had a lot of delay. 00:20:02.850 --> 00:20:05.200 There's Python running serial interfaces 00:20:05.200 --> 00:20:10.480 to over the serial link to the fairly old controller 00:20:10.480 --> 00:20:11.370 in the pioneer robot. 00:20:11.370 --> 00:20:13.440 So the delay is real. 00:20:13.440 --> 00:20:18.670 There's a delay of about 1/10 of a second in the sensor. 00:20:18.670 --> 00:20:22.180 If I take that exact same controller, exact same gain 00:20:22.180 --> 00:20:24.850 that I already did, now put it in this new system that 00:20:24.850 --> 00:20:27.320 has an extra delay, then what happens? 00:20:27.320 --> 00:20:30.640 I get a velocity of 10 after one step, looks good. 00:20:30.640 --> 00:20:32.260 But then, uh-oh, there was some delay 00:20:32.260 --> 00:20:34.245 in the sensor I didn't realize was here, 00:20:34.245 --> 00:20:36.370 so I'm still going to take corrective action trying 00:20:36.370 --> 00:20:37.060 to get me there. 00:20:37.060 --> 00:20:38.976 It's going to move me all the way to the wall, 00:20:38.976 --> 00:20:40.660 smash into the wall, and then it's 00:20:40.660 --> 00:20:41.920 going to realize it was zero. 00:20:41.920 --> 00:20:44.003 It's going to-- there's some delay in seeing that. 00:20:44.003 --> 00:20:45.460 It's going to move me back. 00:20:45.460 --> 00:20:49.620 And you're going to get oscillations. 00:20:49.620 --> 00:20:52.650 You can see that now by just adding the model to our block 00:20:52.650 --> 00:20:54.790 diagram. 00:20:54.790 --> 00:20:59.550 If we put the delay now in the feedback path, 00:20:59.550 --> 00:21:03.630 otherwise, keep the block diagram exactly the same. 00:21:03.630 --> 00:21:07.020 Now, you can write the system function. 00:21:07.020 --> 00:21:08.524 What's the resulting system function 00:21:08.524 --> 00:21:10.440 given that this thing is in the feedback path? 00:22:48.317 --> 00:22:50.900 Go ahead and put your fingers up when you think you've got it. 00:22:54.600 --> 00:22:55.170 All right. 00:22:55.170 --> 00:22:55.770 Fantastic. 00:23:02.060 --> 00:23:03.810 Just like I did here, you can just quickly 00:23:03.810 --> 00:23:07.350 replace the accumulator there with the equivalent block 00:23:07.350 --> 00:23:08.790 diagram. 00:23:08.790 --> 00:23:09.750 Do the loop again. 00:23:09.750 --> 00:23:11.610 It's going to be exactly what we did before, 00:23:11.610 --> 00:23:15.910 but it's got a new R in the feedback path. 00:23:15.910 --> 00:23:19.720 Giving us the R-squared here, just on the feedback path. 00:23:19.720 --> 00:23:21.920 It's exactly the same otherwise except for that R, 00:23:21.920 --> 00:23:24.750 and that simplifies out to this. 00:23:24.750 --> 00:23:27.130 So the answer was 4. 00:23:27.130 --> 00:23:29.640 That's just operator notation, polynomial algebra. 00:23:34.720 --> 00:23:37.750 If we want to find the poles of that system, 00:23:37.750 --> 00:23:43.270 we can just go ahead and factor the quadratic form 00:23:43.270 --> 00:23:45.289 in the denominator. 00:23:45.289 --> 00:23:47.080 The roots of the denominator look something 00:23:47.080 --> 00:23:50.290 like this, which is a little ugly to think about. 00:23:50.290 --> 00:23:53.980 But we can think it through. 00:23:53.980 --> 00:24:00.190 For general KT-- when KT's small, KT's about zero, 00:24:00.190 --> 00:24:03.300 then I'll go ahead and simplify that a little bit to say, 00:24:03.300 --> 00:24:04.990 KT could sneak inside the-- 00:24:04.990 --> 00:24:07.420 KT and KT-squared aren't so different. 00:24:07.420 --> 00:24:09.340 We're going to sneak it inside here. 00:24:09.340 --> 00:24:12.090 And then you can see that the poles end up at-- around K 00:24:12.090 --> 00:24:17.130 equal to zero you get a pole at the origin. 00:24:17.130 --> 00:24:20.860 But you also get a pole up by the unit circle. 00:24:20.860 --> 00:24:24.620 Around 1 and around 0. 00:24:24.620 --> 00:24:30.560 Remember, when you've got multiple poles in the system, 00:24:30.560 --> 00:24:34.220 the total response is going to be dominated 00:24:34.220 --> 00:24:38.204 by whatever is the slower pole. 00:24:38.204 --> 00:24:39.620 The total system response is going 00:24:39.620 --> 00:24:40.760 to be dominated by the slower pole. 00:24:40.760 --> 00:24:43.250 In this case, the slower pole's the one closer to the unit 00:24:43.250 --> 00:24:43.749 circle. 00:24:48.540 --> 00:24:52.410 What about if KT equals negative 0.25? 00:24:52.410 --> 00:24:54.390 We can pop that in and solve it. 00:24:54.390 --> 00:24:57.510 Exactly-- math works out nicely. 00:24:57.510 --> 00:25:02.820 We get poles at a half, two poles at a half. 00:25:02.820 --> 00:25:05.940 And in fact, there's a smooth transition between-- if you 00:25:05.940 --> 00:25:10.080 look at the numbers between KT equals zero and KT equals 0.25, 00:25:10.080 --> 00:25:12.630 you'll see as you vary that gain, 00:25:12.630 --> 00:25:17.040 the poles move together along the real axis 00:25:17.040 --> 00:25:18.990 until they come together at a half. 00:25:21.860 --> 00:25:29.270 So here, you've got a purely real response dominated 00:25:29.270 --> 00:25:31.130 by these poles at a half. 00:25:31.130 --> 00:25:33.080 System's stable. 00:25:33.080 --> 00:25:36.170 If you keep changing K though, the poles came together 00:25:36.170 --> 00:25:40.310 and then they split off and start going this way. 00:25:40.310 --> 00:25:42.260 And in fact, if you look at negative 1, which 00:25:42.260 --> 00:25:44.960 is the one that was the best response, 00:25:44.960 --> 00:25:50.300 it put a pole exactly at 0, for the system with no delay. 00:25:50.300 --> 00:25:52.130 If you put it at the system with delay, 00:25:52.130 --> 00:25:55.280 they land exactly on the unit circle. 00:25:55.280 --> 00:25:57.260 Right there. 00:25:57.260 --> 00:26:00.230 Complex poles on the unit circle. 00:26:00.230 --> 00:26:02.269 You're going to get a stable oscillation. 00:26:07.720 --> 00:26:09.580 Which is exactly what we saw there. 00:26:13.400 --> 00:26:16.490 Just a quick-- you know this like the back of your hand now, 00:26:16.490 --> 00:26:19.955 but what's the period of that oscillation? 00:26:26.260 --> 00:26:31.660 You've got two poles on the unit circle right there. 00:26:31.660 --> 00:26:33.855 What's the period of oscillation? 00:26:39.300 --> 00:26:41.370 Put your fingers up when you think you know. 00:26:52.380 --> 00:26:53.120 Yep. 00:26:53.120 --> 00:26:54.290 Good. 00:26:54.290 --> 00:26:56.650 Most people got it. 00:26:56.650 --> 00:27:00.820 This thing was a half, so it's 1/2 square root of 3 over 2. 00:27:00.820 --> 00:27:03.497 So that thing had to be at pi over 3. 00:27:03.497 --> 00:27:05.080 It's actually the same pole that I was 00:27:05.080 --> 00:27:09.782 using in recitation yesterday. 00:27:09.782 --> 00:27:11.240 So if you have a pole at pi over 3, 00:27:11.240 --> 00:27:15.140 it makes it around in six steps. 00:27:15.140 --> 00:27:17.340 The period of the oscillation is six. 00:27:26.140 --> 00:27:28.840 This is generally true, that if you 00:27:28.840 --> 00:27:32.110 put a controlled gain, a feedback gain, 00:27:32.110 --> 00:27:36.700 into the closed loop dynamics, then even a simple gain 00:27:36.700 --> 00:27:40.240 can allow you to really shift around the poles of the system. 00:27:40.240 --> 00:27:42.850 And since we know the response of the system, the zeros 00:27:42.850 --> 00:27:46.570 matter, but for convergence, for the rate of convergence, 00:27:46.570 --> 00:27:49.410 it's the biggest pole that dominates. 00:27:49.410 --> 00:27:53.436 It's very nice to understand how those poles move with K. 00:27:53.436 --> 00:27:54.810 If you change the system, the way 00:27:54.810 --> 00:27:56.880 they move with K is different. 00:27:56.880 --> 00:28:00.000 And you can just change K to tune the response 00:28:00.000 --> 00:28:03.330 to be what you want, from KT equals 0 to infinity, 00:28:03.330 --> 00:28:06.810 the poles go towards here and then off in both 00:28:06.810 --> 00:28:08.608 directions, actually, to infinity. 00:28:14.450 --> 00:28:19.740 So, KT equals negative 1 was the fastest possible response 00:28:19.740 --> 00:28:22.020 for the system without any delay in the sensor. 00:28:22.020 --> 00:28:24.700 What's the fastest possible response for this one? 00:28:40.200 --> 00:28:41.250 Oscillations are allowed. 00:28:41.250 --> 00:28:43.050 I just want the fastest possible response. 00:29:14.840 --> 00:29:15.340 Yeah. 00:29:15.340 --> 00:29:15.880 Looks good. 00:29:15.880 --> 00:29:19.431 So where do I want the poles to be for the fastest possible 00:29:19.431 --> 00:29:19.930 response? 00:29:26.470 --> 00:29:31.180 These poles are still stable and oscillations are 00:29:31.180 --> 00:29:33.370 fine, but the absolute-- 00:29:33.370 --> 00:29:35.920 the magnitude of those poles is larger 00:29:35.920 --> 00:29:38.260 when they're out here in the complex. 00:29:38.260 --> 00:29:39.910 When I'm down here, I have got a pole 00:29:39.910 --> 00:29:41.960 over by one, that's going to dominate. 00:29:41.960 --> 00:29:45.400 So the best I can do is if I put the poles at a half. 00:29:45.400 --> 00:29:48.130 That gives us the largest pole, the smallest 00:29:48.130 --> 00:29:49.950 possible magnitude. 00:29:49.950 --> 00:29:54.640 And that happened if the poles-- the double poles at a half 00:29:54.640 --> 00:29:57.940 happened when KT was negative 0.25. 00:29:57.940 --> 00:30:00.289 Most of you said, the answer is 2. 00:30:08.560 --> 00:30:11.920 In general, delay is a bad thing. 00:30:11.920 --> 00:30:14.530 In DT systems, we have good representations of delay. 00:30:14.530 --> 00:30:17.290 It's even worse in continuous time. 00:30:17.290 --> 00:30:18.250 Delay is a bad thing. 00:30:18.250 --> 00:30:23.770 It tends to make control systems not work as well. 00:30:23.770 --> 00:30:27.460 If you just took the ideal sensor, we had a K equals 1, 00:30:27.460 --> 00:30:30.580 we had a response that started here, we could put it anywhere 00:30:30.580 --> 00:30:33.534 we wanted along the real axis. 00:30:33.534 --> 00:30:35.200 As far negative as we wanted, of course, 00:30:35.200 --> 00:30:38.980 what we chose was to put it right at the origin. 00:30:38.980 --> 00:30:41.380 But as soon as we just added that one piece of delay, 00:30:41.380 --> 00:30:44.140 the things we could do with proportional feedback 00:30:44.140 --> 00:30:46.630 changed completely, and ultimately, 00:30:46.630 --> 00:30:51.320 got worse because I have two poles now, first of all, 00:30:51.320 --> 00:30:54.130 and I can't simultaneously get both of those poles 00:30:54.130 --> 00:30:55.450 to go to zero. 00:30:55.450 --> 00:30:58.949 In fact, as I change them, they go off into complex. 00:30:58.949 --> 00:31:00.490 The fact that it's complex isn't bad. 00:31:00.490 --> 00:31:03.156 But the fact that there's two of them, and the best I can do now 00:31:03.156 --> 00:31:05.620 is get to a half, which is a much slower response, the best 00:31:05.620 --> 00:31:06.480 possible response. 00:31:10.030 --> 00:31:13.600 If I added even more delay, things get even worse. 00:31:13.600 --> 00:31:16.004 If I had two units of delay and I 00:31:16.004 --> 00:31:17.920 went through the same exercise, what you'd see 00:31:17.920 --> 00:31:21.780 is that the poles would start in the same place. 00:31:21.780 --> 00:31:23.735 They'd come together and go there, 00:31:23.735 --> 00:31:25.360 and there's another pole that goes off. 00:31:25.360 --> 00:31:27.870 Just because there are three poles in that system. 00:31:27.870 --> 00:31:32.230 Two poles come together and split off this way. 00:31:32.230 --> 00:31:35.290 And this one goes off this way. 00:31:35.290 --> 00:31:37.104 And the place you probably want, depending 00:31:37.104 --> 00:31:39.020 on how fast this one splits off, but the place 00:31:39.020 --> 00:31:41.630 you probably want to put it is when the two poles are together 00:31:41.630 --> 00:31:42.270 right there. 00:31:42.270 --> 00:31:44.370 That's going to give you the fastest response. 00:31:44.370 --> 00:31:48.570 But that fastest response is still slower than what we had-- 00:31:48.570 --> 00:31:50.780 it's a bigger number than a half. 00:31:50.780 --> 00:31:54.340 It's 0.682. 00:31:54.340 --> 00:31:57.890 It's going to be a worse response than when 00:31:57.890 --> 00:32:00.421 I had a delay of one, which is intuitive. 00:32:05.420 --> 00:32:08.460 That's a quick reminder of something you already 00:32:08.460 --> 00:32:13.080 did in 601, of using feedback and the tools we've already 00:32:13.080 --> 00:32:16.072 got, which is poles and zeros and everything, to understand 00:32:16.072 --> 00:32:17.280 how to design feedback gains. 00:32:17.280 --> 00:32:20.020 We're going to get more into it in the next couple of lectures. 00:32:20.020 --> 00:32:23.490 But let me just convince you that this stuff is real. 00:32:23.490 --> 00:32:25.620 And I showed this diagram once in 601, 00:32:25.620 --> 00:32:27.940 but it probably means even more to you now. 00:32:33.170 --> 00:32:36.810 This is an F-14. 00:32:36.810 --> 00:32:41.900 It's one of the best modern engineering control 00:32:41.900 --> 00:32:44.750 systems ever built. It was built for this F-14. 00:32:44.750 --> 00:32:49.220 They got more research money and modeling this vehicle, 00:32:49.220 --> 00:32:53.480 designing ultimate gains for it. 00:32:53.480 --> 00:32:56.420 It's such a success story that you can actually, 00:32:56.420 --> 00:32:59.900 if you're in MATLAB, you can open up the F-14 demo 00:32:59.900 --> 00:33:04.940 and see what a flight control system for an F-14 looks like. 00:33:04.940 --> 00:33:07.190 MATLAB-- I don't know if you've played with Simulink-- 00:33:07.190 --> 00:33:11.000 MATLAB has a language called Simulink, a graphical language 00:33:11.000 --> 00:33:13.460 that allows you to draw the block diagrams 00:33:13.460 --> 00:33:17.430 and simulate them and even design controllers for them. 00:33:17.430 --> 00:33:19.310 And it turns out you can make a block diagram 00:33:19.310 --> 00:33:27.860 description of an F-14 using only tools from 601 and 6003. 00:33:27.860 --> 00:33:31.520 You can see all the same adders and gains, 00:33:31.520 --> 00:33:36.480 transfer functions, system functions like this. 00:33:36.480 --> 00:33:39.710 The only essential difference is that in some of the diagrams, 00:33:39.710 --> 00:33:41.810 you see multiple inputs coming in. 00:33:41.810 --> 00:33:43.610 In this class, we've restricted ourselves 00:33:43.610 --> 00:33:46.850 so far to thinking about single input, single output systems, 00:33:46.850 --> 00:33:48.190 which keeps everything clean. 00:33:48.190 --> 00:33:51.410 All the intuition scales to multiple inputs 00:33:51.410 --> 00:33:53.120 and multiple outputs. 00:33:53.120 --> 00:33:58.570 But that's sort of the only big addition of complexity 00:33:58.570 --> 00:34:02.200 when you go to this model from what we've done before. 00:34:02.200 --> 00:34:06.759 If you zoom in onto the controller here, you can-- 00:34:06.759 --> 00:34:08.300 these block diagrams in this language 00:34:08.300 --> 00:34:10.777 allow you to abstract away a bunch of hidden things 00:34:10.777 --> 00:34:12.110 with a single transfer function. 00:34:12.110 --> 00:34:16.670 If I zoom in, then you can see things you recognize. 00:34:16.670 --> 00:34:18.770 It's got a low-pass filter, which 00:34:18.770 --> 00:34:21.139 is just a system with a pole at negative 1, 00:34:21.139 --> 00:34:25.150 exactly what we did in recitation the other day. 00:34:25.150 --> 00:34:27.590 This is what people-- 00:34:27.590 --> 00:34:29.510 this is what MATLAB uses on an F-14. 00:34:29.510 --> 00:34:31.468 I guess it might not be what the military uses. 00:34:31.468 --> 00:34:35.739 But it's modeled after what the unclassified documents say 00:34:35.739 --> 00:34:36.739 is happening on an F-14. 00:34:39.949 --> 00:34:41.659 Everything here, you understand, right? 00:34:41.659 --> 00:34:44.330 A proportional controller in the end. 00:34:44.330 --> 00:34:47.300 One difference in F-14s is that-- 00:34:47.300 --> 00:34:50.060 and in general for these more complicated systems-- 00:34:50.060 --> 00:34:52.520 they'll design slightly different controllers 00:34:52.520 --> 00:34:53.929 given the situation. 00:34:53.929 --> 00:34:59.746 So if you have an altimeter on there and pressure sensors, 00:34:59.746 --> 00:35:03.720 an inclinometer will tell you the angle of the plane-- 00:35:03.720 --> 00:35:07.490 given those sensors, they'll pick a different K 00:35:07.490 --> 00:35:09.230 for a proportional controller out 00:35:09.230 --> 00:35:12.500 of a pot of a library of controllers 00:35:12.500 --> 00:35:14.300 they've already designed. 00:35:14.300 --> 00:35:16.610 So it's called gain scheduled control. 00:35:16.610 --> 00:35:19.340 But the analysis and design of each K 00:35:19.340 --> 00:35:23.310 is a linear systems design and analysis. 00:35:23.310 --> 00:35:24.900 This is super powerful stuff. 00:35:24.900 --> 00:35:28.310 I think-- signal processing is good, too, 00:35:28.310 --> 00:35:30.170 but control is where it's at. 00:35:32.900 --> 00:35:35.810 I guess I went a little fast, but that's 00:35:35.810 --> 00:35:39.260 your introduction to feedback. 00:35:39.260 --> 00:35:42.410 We'll do DT and CT feedback in the next couple of lectures. 00:35:42.410 --> 00:35:44.450 And if anybody hasn't picked up their exams, 00:35:44.450 --> 00:35:45.930 we have them over here. 00:35:45.930 --> 00:35:48.670 And we'll see you in recitation tomorrow.