1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:18,450 at ocw.mit.edu. 8 00:00:28,320 --> 00:00:31,020 DENNIS FREEMAN: Hello and welcome. 9 00:00:31,020 --> 00:00:33,330 As before, as you might be expecting, 10 00:00:33,330 --> 00:00:36,030 the big new news is, of course-- well it's not new news. 11 00:00:36,030 --> 00:00:38,640 The big news is exam, tomorrow night. 12 00:00:38,640 --> 00:00:40,980 This should be very familiar stuff to you. 13 00:00:40,980 --> 00:00:43,470 I apologize, a couple people asked me right before 14 00:00:43,470 --> 00:00:45,770 lecture about Fourier because there was Fourier 15 00:00:45,770 --> 00:00:47,670 on one of the previous exams. 16 00:00:47,670 --> 00:00:50,310 There is no Fourier. 17 00:00:50,310 --> 00:00:53,620 I should have probably axed that problem. 18 00:00:53,620 --> 00:00:56,850 So in my head I was trying to be complete disclosure. 19 00:00:56,850 --> 00:00:59,400 This is what the exam looked like. 20 00:00:59,400 --> 00:01:02,200 I didn't mean to send the false message that there is Fourier. 21 00:01:02,200 --> 00:01:05,099 There is no Fourier. 22 00:01:05,099 --> 00:01:07,650 Next time we'll have lots of opportunities 23 00:01:07,650 --> 00:01:10,930 to ask Fourier questions, so no Fourier this time. 24 00:01:10,930 --> 00:01:13,800 On the other hand, yes, Bode, Root Locus, that kind of stuff, 25 00:01:13,800 --> 00:01:14,880 yes. 26 00:01:14,880 --> 00:01:18,960 So, that kind of stuff you should get on top of. 27 00:01:18,960 --> 00:01:21,750 Questions or comments about the exam? 28 00:01:21,750 --> 00:01:24,360 I guess the one thing to remember is it's in Walker. 29 00:01:24,360 --> 00:01:26,460 It's not on the third floor of 26. 30 00:01:26,460 --> 00:01:29,460 Other than that, you should be kind of in a good position 31 00:01:29,460 --> 00:01:30,912 to know roughly what to expect. 32 00:01:30,912 --> 00:01:31,412 Yes? 33 00:01:31,412 --> 00:01:33,450 AUDIENCE: When you say two pages of notes, 34 00:01:33,450 --> 00:01:36,270 do you mean two separate 8 and a half by 11 cards? 35 00:01:36,270 --> 00:01:37,680 DENNIS FREEMAN: Yes. 36 00:01:37,680 --> 00:01:41,010 So the idea is if you made one 8 and a half by 11 front 37 00:01:41,010 --> 00:01:44,280 and back for the last exam, please just bring that. 38 00:01:44,280 --> 00:01:46,387 Don't bother making a new one. 39 00:01:46,387 --> 00:01:48,720 But you might want to make a new one that contains stuff 40 00:01:48,720 --> 00:01:50,190 that we've done since then. 41 00:01:50,190 --> 00:01:51,330 That's the idea. 42 00:01:51,330 --> 00:01:53,730 But we of course don't care what you do. 43 00:01:53,730 --> 00:01:58,420 Micro, you know, if you want to bring-- so, no electronics. 44 00:01:58,420 --> 00:02:00,854 But optical magnification is fine. 45 00:02:00,854 --> 00:02:03,020 If you want to bring a little microscope, it's fine. 46 00:02:03,020 --> 00:02:04,480 We don't mind. 47 00:02:04,480 --> 00:02:07,420 If you can have an all optics, no electronic, microfiche 48 00:02:07,420 --> 00:02:10,120 reader, fine. 49 00:02:10,120 --> 00:02:13,960 So the idea is just to send the message 50 00:02:13,960 --> 00:02:16,540 that you shouldn't be spending a lot of time 51 00:02:16,540 --> 00:02:18,100 leafing through 300 pages. 52 00:02:18,100 --> 00:02:20,364 That's what we don't want you to do. 53 00:02:20,364 --> 00:02:21,340 AUDIENCE: Question. 54 00:02:21,340 --> 00:02:21,550 DENNIS FREEMAN: Yes 55 00:02:21,550 --> 00:02:23,530 AUDIENCE: Can we bring our abacus, or is that not allowed? 56 00:02:23,530 --> 00:02:24,400 DENNIS FREEMAN: Can you bring your? 57 00:02:24,400 --> 00:02:25,349 AUDIENCE: Abacus. 58 00:02:25,349 --> 00:02:26,890 DENNIS FREEMAN: Abacuses are welcome. 59 00:02:26,890 --> 00:02:29,620 In fact, I would love to race somebody with an abacus. 60 00:02:29,620 --> 00:02:31,900 I will take you on with a slide rule. 61 00:02:31,900 --> 00:02:32,950 [LAUGHTER] 62 00:02:32,950 --> 00:02:36,587 Okay, so I learned with a slide rule and the question was-- 63 00:02:36,587 --> 00:02:38,170 Well, no, you don't want to know that. 64 00:02:38,170 --> 00:02:40,810 I don't want to tell you that, actually. 65 00:02:40,810 --> 00:02:42,860 Okay, other questions, comments? 66 00:02:42,860 --> 00:02:46,090 Abacuses, fine. 67 00:02:46,090 --> 00:02:47,755 Non electronic devices. 68 00:02:53,150 --> 00:02:58,060 So today is the final lecture on feedback. 69 00:02:58,060 --> 00:03:00,910 What we've done over the course of the last week 70 00:03:00,910 --> 00:03:03,730 and a half or so is looked at a variety of applications. 71 00:03:03,730 --> 00:03:06,100 Last time we looked at how to perform-- 72 00:03:06,100 --> 00:03:07,900 how to improve performance with feedback. 73 00:03:07,900 --> 00:03:10,316 We looked at how you could increase the bandwidth of an op 74 00:03:10,316 --> 00:03:13,390 amp, increase the speed of an op amp, 75 00:03:13,390 --> 00:03:16,780 change a motor so that its natural output, which 76 00:03:16,780 --> 00:03:18,790 is a voltage controlled velocity, 77 00:03:18,790 --> 00:03:21,580 could be perturbed into a voltage controlled position. 78 00:03:21,580 --> 00:03:23,740 Today I want to just do a few more examples. 79 00:03:23,740 --> 00:03:26,590 Because what I want you to do is leave this place thinking 80 00:03:26,590 --> 00:03:28,480 feedback's really useful. 81 00:03:28,480 --> 00:03:29,680 It's not just torture. 82 00:03:29,680 --> 00:03:30,736 I mean it is torture. 83 00:03:30,736 --> 00:03:32,860 We do torture you and you're supposed to figure out 84 00:03:32,860 --> 00:03:34,568 how it works quantitatively and all that. 85 00:03:34,568 --> 00:03:36,010 But it's not just torture. 86 00:03:36,010 --> 00:03:37,870 It's actually very useful. 87 00:03:37,870 --> 00:03:41,800 And that's what I want you to take away. 88 00:03:41,800 --> 00:03:44,650 The first application today that I want to think about 89 00:03:44,650 --> 00:03:47,390 is reducing unwanted parameter variation. 90 00:03:47,390 --> 00:03:50,140 So the idea that I want you to think about 91 00:03:50,140 --> 00:03:53,590 is what do you have in a power amplifier of the type that you 92 00:03:53,590 --> 00:03:55,900 would drive speakers-- 93 00:03:55,900 --> 00:03:58,270 of the type that you would use to drive speakers. 94 00:03:58,270 --> 00:04:01,090 That's a kind of a complicated electronic device. 95 00:04:01,090 --> 00:04:03,250 It's kind of hard to make high power. 96 00:04:03,250 --> 00:04:05,500 You end up having to generate high power in order 97 00:04:05,500 --> 00:04:10,517 to make a loud speaker that fills this hall for example. 98 00:04:10,517 --> 00:04:12,100 And if you were to try to do something 99 00:04:12,100 --> 00:04:14,935 bigger like a concert, it's even harder. 100 00:04:14,935 --> 00:04:17,050 And the problem with those kinds of devices 101 00:04:17,050 --> 00:04:19,779 is that they are active. 102 00:04:19,779 --> 00:04:21,370 They generate signals that are bigger 103 00:04:21,370 --> 00:04:23,560 that generate more power at their output 104 00:04:23,560 --> 00:04:25,870 than there was available at the input. 105 00:04:25,870 --> 00:04:28,280 And we're very good at such things. 106 00:04:28,280 --> 00:04:32,260 Unfortunately, such things are intrinsically non-linear. 107 00:04:32,260 --> 00:04:36,430 So we actually take advantage of non-linearities in materials 108 00:04:36,430 --> 00:04:38,030 in order to do that. 109 00:04:38,030 --> 00:04:40,000 The result though is that at a high level 110 00:04:40,000 --> 00:04:41,350 the thing's still non-linear. 111 00:04:41,350 --> 00:04:43,720 And we can see some of that. 112 00:04:43,720 --> 00:04:46,120 And not only non-linear, but other things. 113 00:04:46,120 --> 00:04:48,480 They can be temperature dependent. 114 00:04:48,480 --> 00:04:50,560 So if you were to make a high power amplifier 115 00:04:50,560 --> 00:04:52,540 it's very likely that the performance 116 00:04:52,540 --> 00:04:55,270 would depend on temperature. 117 00:04:55,270 --> 00:04:57,090 Who cares, it's always room temperature. 118 00:04:57,090 --> 00:05:00,310 No, not inside the amplifier. 119 00:05:00,310 --> 00:05:02,050 It may start out at room temperature 120 00:05:02,050 --> 00:05:04,300 every time you turn it on, but within a few minutes 121 00:05:04,300 --> 00:05:06,200 it's going to be much higher than room temperature. 122 00:05:06,200 --> 00:05:07,866 And if you didn't do something about it, 123 00:05:07,866 --> 00:05:10,730 the gain would change. 124 00:05:10,730 --> 00:05:13,280 If you didn't do something about it in your car, 125 00:05:13,280 --> 00:05:15,500 it would track the weather, which may be good. 126 00:05:15,500 --> 00:05:17,530 Maybe it would provide a distraction 127 00:05:17,530 --> 00:05:18,980 and provide utility to winter. 128 00:05:18,980 --> 00:05:21,740 But short of that normally we don't 129 00:05:21,740 --> 00:05:25,940 want the sound level that comes out of a stereo system 130 00:05:25,940 --> 00:05:28,380 to depend on temperature. 131 00:05:28,380 --> 00:05:30,770 So how do you take a device that intrinsically 132 00:05:30,770 --> 00:05:31,980 depends on temperature? 133 00:05:31,980 --> 00:05:34,580 Here I am imagining that I have a power amplifier whose 134 00:05:34,580 --> 00:05:38,600 gain can go between 8 and 12, say, depending 135 00:05:38,600 --> 00:05:40,340 on what the temperature is. 136 00:05:40,340 --> 00:05:43,040 How could you fix that so that its behavior 137 00:05:43,040 --> 00:05:47,570 was more consistent from day to day, from hour to hour. 138 00:05:47,570 --> 00:05:50,480 And of course, not too surprisingly, the answer 139 00:05:50,480 --> 00:05:53,272 is use feedback. 140 00:05:53,272 --> 00:05:54,920 So how do you do that? 141 00:05:54,920 --> 00:05:57,800 So you want to connect the amplifier to the speaker. 142 00:05:57,800 --> 00:05:59,847 After all, it's inside the amplifier 143 00:05:59,847 --> 00:06:01,430 that you have all the transistors that 144 00:06:01,430 --> 00:06:03,020 are able to drive the speaker. 145 00:06:03,020 --> 00:06:06,790 So you don't want to perturb that connection. 146 00:06:06,790 --> 00:06:08,280 But what you can do with feedback 147 00:06:08,280 --> 00:06:12,180 is monitor what was the voltage that actually got produced 148 00:06:12,180 --> 00:06:15,020 and try to rig the feedback system to coerce it 149 00:06:15,020 --> 00:06:18,410 to the voltage you wish it were. 150 00:06:18,410 --> 00:06:21,180 So the way you can do that is to generate a feedback loop. 151 00:06:21,180 --> 00:06:26,130 Precede the power amplifier with a voltage amplifier 152 00:06:26,130 --> 00:06:28,890 that does not necessarily significantly increase 153 00:06:28,890 --> 00:06:29,400 the power. 154 00:06:29,400 --> 00:06:31,566 That makes that kind of an amplifier somewhat easier 155 00:06:31,566 --> 00:06:34,400 to make. 156 00:06:34,400 --> 00:06:36,860 So imagine that this power amplifier 157 00:06:36,860 --> 00:06:38,720 has big chunky transistors because it 158 00:06:38,720 --> 00:06:43,160 has to generate amps in order to make your speakers work. 159 00:06:43,160 --> 00:06:46,460 This, by comparison, could be a much less power hungry kind 160 00:06:46,460 --> 00:06:47,060 of amplifier. 161 00:06:47,060 --> 00:06:48,850 And it's easier to make, and in particular 162 00:06:48,850 --> 00:06:51,260 would be easy to make it with a gain of 10 or 100 163 00:06:51,260 --> 00:06:52,730 or 1,000 or 10,000. 164 00:06:52,730 --> 00:06:56,270 We're very good at doing that kind of stuff. 165 00:06:56,270 --> 00:07:00,440 Then the idea of feedback is to compare the signal 166 00:07:00,440 --> 00:07:04,490 that you generated to the signal that you might have desired. 167 00:07:04,490 --> 00:07:06,080 So here's the signal you generated. 168 00:07:06,080 --> 00:07:07,430 Here is the signal you desired. 169 00:07:07,430 --> 00:07:09,650 I'm comparing it through a monitor 170 00:07:09,650 --> 00:07:14,710 that's feeding back beta times the output voltage. 171 00:07:14,710 --> 00:07:16,750 The reason I do that can be seen if you just 172 00:07:16,750 --> 00:07:17,666 work through the math. 173 00:07:17,666 --> 00:07:21,700 So if I just write the closed loop transfer function 174 00:07:21,700 --> 00:07:27,890 for x goes to y, the easy way to think about that's 175 00:07:27,890 --> 00:07:29,630 Black's formula, of course, right? 176 00:07:29,630 --> 00:07:33,500 So think about the forward path, k times f naught divided 177 00:07:33,500 --> 00:07:35,580 by 1 plus the loop gain. 178 00:07:35,580 --> 00:07:42,342 So 1 plus the gain around the loop is beta k f naught. 179 00:07:42,342 --> 00:07:43,800 And now just think about what would 180 00:07:43,800 --> 00:07:47,280 happen if I coerced it so that I made k the thing that's 181 00:07:47,280 --> 00:07:51,230 under my control big. 182 00:07:51,230 --> 00:07:53,750 If I make k big enough, then I can 183 00:07:53,750 --> 00:07:57,660 arrange so that beta k f naught, independent of beta and k, 184 00:07:57,660 --> 00:08:00,110 I can always choose k so that the product is big 185 00:08:00,110 --> 00:08:03,600 compared to the one. 186 00:08:03,600 --> 00:08:07,460 If I do that, I can ignore the one. 187 00:08:07,460 --> 00:08:09,650 If I ignore the one, the k's cancel, 188 00:08:09,650 --> 00:08:13,100 the f naughts cancel, and I'm left with whatever beta. 189 00:08:13,100 --> 00:08:15,170 Beta didn't depend on f naught. 190 00:08:15,170 --> 00:08:18,230 That's what I wanted. 191 00:08:18,230 --> 00:08:22,390 It didn't even depend on k, other than k 192 00:08:22,390 --> 00:08:25,960 had to be big enough so that the division worked. 193 00:08:25,960 --> 00:08:32,110 So the result is that you can make a stable gain 194 00:08:32,110 --> 00:08:35,799 by incorporating an amplifier with an unstable gain. 195 00:08:35,799 --> 00:08:38,080 Unstable meaning that it varies in time, not that it's 196 00:08:38,080 --> 00:08:40,496 unstable in the sense that we often use the word unstable. 197 00:08:40,496 --> 00:08:41,350 I should avoid that. 198 00:08:41,350 --> 00:08:43,630 The parameter of the gain changes. 199 00:08:43,630 --> 00:08:46,300 It changes between 8 and 12. 200 00:08:46,300 --> 00:08:48,740 And I can illustrate that with this kind of a picture. 201 00:08:48,740 --> 00:08:52,840 If I plot the variation at f naught. 202 00:08:52,840 --> 00:08:54,850 I've showed a range here from 0 to 20. 203 00:08:54,850 --> 00:08:57,220 I'm imagining that in a practical situation 204 00:08:57,220 --> 00:09:00,850 it could vary between 8 and 12. 205 00:09:00,850 --> 00:09:03,880 And then in red, I'm showing the gain 206 00:09:03,880 --> 00:09:07,850 that would result if there were no feedback. 207 00:09:07,850 --> 00:09:11,770 And as you can see that over the range of 8 to 12, 208 00:09:11,770 --> 00:09:15,700 the output gain, the total gain of the system, 209 00:09:15,700 --> 00:09:17,710 varies between 8 and 12, not surprisingly. 210 00:09:17,710 --> 00:09:19,720 That's what it would mean to have no feedback. 211 00:09:19,720 --> 00:09:25,270 But if I crank in feedback, if I set the k to 100 and feedback 212 00:09:25,270 --> 00:09:30,820 ratio to 1/10, the 100 is sufficient to make the one 213 00:09:30,820 --> 00:09:34,660 negligible which means that my overall gain is 1 over 10 214 00:09:34,660 --> 00:09:37,780 determined only by the feedback ratio, beta. 215 00:09:37,780 --> 00:09:41,050 And so the result is this line shows 216 00:09:41,050 --> 00:09:44,830 the solution to that equation. 217 00:09:44,830 --> 00:09:47,850 And you can see that it's ruler straight 218 00:09:47,850 --> 00:09:50,370 through the interesting region. 219 00:09:50,370 --> 00:09:51,540 Does that makes sense? 220 00:09:51,540 --> 00:09:54,090 So what I've been able to do is use feedback 221 00:09:54,090 --> 00:09:57,240 in order to control the value of a parameter that 222 00:09:57,240 --> 00:10:00,480 was otherwise variable, or sensitive to something, 223 00:10:00,480 --> 00:10:01,590 often temperature. 224 00:10:01,590 --> 00:10:04,020 Is that all clear? 225 00:10:04,020 --> 00:10:07,110 That kind of a thing-- 226 00:10:07,110 --> 00:10:09,150 Question? 227 00:10:09,150 --> 00:10:12,891 I got rid of all the variations, except beta. 228 00:10:12,891 --> 00:10:14,140 What about variations in beta? 229 00:10:16,970 --> 00:10:19,247 Right, the premise was I couldn't control this, 230 00:10:19,247 --> 00:10:20,830 it was going to vary between 8 and 12. 231 00:10:20,830 --> 00:10:23,120 But what if beta changes. 232 00:10:23,120 --> 00:10:25,550 What if beta went from 1/8th to 1/12th, 233 00:10:25,550 --> 00:10:28,960 would that affect things? 234 00:10:28,960 --> 00:10:30,675 AUDIENCE: So yeah, so we could put 235 00:10:30,675 --> 00:10:31,900 a feedback loop [INAUDIBLE]. 236 00:10:31,900 --> 00:10:34,191 DENNIS FREEMAN: You could put in another feedback loop. 237 00:10:34,191 --> 00:10:37,510 That doesn't sound like it's going to converge quickly. 238 00:10:37,510 --> 00:10:38,824 Yes? 239 00:10:38,824 --> 00:10:42,185 AUDIENCE: Typically we want to be able to control beta anyway. 240 00:10:42,185 --> 00:10:43,060 DENNIS FREEMAN: Right 241 00:10:43,060 --> 00:10:44,240 AUDIENCE: There's nothing users control. 242 00:10:44,240 --> 00:10:45,790 DENNIS FREEMAN: That's right, we'd want to control it. 243 00:10:45,790 --> 00:10:46,950 Is there any reason we-- 244 00:10:46,950 --> 00:10:48,520 but that's what we wanted to do with f naught too. 245 00:10:48,520 --> 00:10:49,603 We wanted to control that. 246 00:10:49,603 --> 00:10:51,730 We'd like that to be 10. 247 00:10:51,730 --> 00:10:52,700 Yes. 248 00:10:52,700 --> 00:10:56,095 AUDIENCE: It will depend on how that beta would change. 249 00:10:56,095 --> 00:10:58,520 Changes, say, from 1/8th to 1/12th, 250 00:10:58,520 --> 00:10:59,975 probably nothing to worry about. 251 00:10:59,975 --> 00:11:04,957 If it changes from 1/8th to 1 over 8,000, 252 00:11:04,957 --> 00:11:07,739 then you may want to think about what k and f are. 253 00:11:07,739 --> 00:11:09,780 DENNIS FREEMAN: So what do you think will happen? 254 00:11:09,780 --> 00:11:11,571 Do you think it will vary a lot or a little 255 00:11:11,571 --> 00:11:13,590 compared to 8 to 12? 256 00:11:13,590 --> 00:11:14,970 AUDIENCE: Probably not that much. 257 00:11:14,970 --> 00:11:15,930 DENNIS FREEMAN: Why? 258 00:11:15,930 --> 00:11:18,602 AUDIENCE: Because the k is already-- 259 00:11:18,602 --> 00:11:21,506 1/8th and 1/12th is not really much of a difference 260 00:11:21,506 --> 00:11:22,357 in terms of-- 261 00:11:22,357 --> 00:11:24,690 DENNIS FREEMAN: So why do I think about beta differently 262 00:11:24,690 --> 00:11:27,650 from the way I think about f naught? 263 00:11:27,650 --> 00:11:30,950 Is there a good reason for that or is that just 264 00:11:30,950 --> 00:11:32,810 wishful thinking, right? 265 00:11:32,810 --> 00:11:34,230 Engineering by wishful thinking. 266 00:11:34,230 --> 00:11:36,320 I wish I could get a beta that was exactly right. 267 00:11:36,320 --> 00:11:38,135 Yes? 268 00:11:38,135 --> 00:11:41,105 AUDIENCE: For amplification, beta needs to be less than 1 269 00:11:41,105 --> 00:11:44,060 [INAUDIBLE] so [INAUDIBLE] the resistor, it's not 270 00:11:44,060 --> 00:11:46,060 going to have a very big temperature [INAUDIBLE] 271 00:11:46,060 --> 00:11:47,390 DENNIS FREEMAN: Precisely. 272 00:11:47,390 --> 00:11:52,150 So the problem is getting gain in a stable and reliable 273 00:11:52,150 --> 00:11:56,360 fashion, not in getting attenuation. 274 00:11:56,360 --> 00:12:02,370 We can control the value of a resistor with great precision. 275 00:12:02,370 --> 00:12:05,490 And so it's easy, by comparison, it's 276 00:12:05,490 --> 00:12:12,310 relatively easy to control the gain of a resistor network, 277 00:12:12,310 --> 00:12:17,140 and much harder to control the gain of a transistor network. 278 00:12:17,140 --> 00:12:19,150 Transistors have gain. 279 00:12:19,150 --> 00:12:22,870 Gain is hard to make in a non temperature dependent fashion. 280 00:12:22,870 --> 00:12:25,990 Division by a resistor, it's not hard to make a resistor. 281 00:12:25,990 --> 00:12:28,810 I mean it's not trivial, but it's not hard. 282 00:12:28,810 --> 00:12:32,020 Even in the dark ages-- 283 00:12:32,020 --> 00:12:35,980 even in the dark ages, resistors were made by winding wires, 284 00:12:35,980 --> 00:12:37,750 and they were very stable with temperature 285 00:12:37,750 --> 00:12:41,080 or could be made to be very stable with temperatures. 286 00:12:41,080 --> 00:12:45,460 So one of the tricks in feedback is that we usually-- 287 00:12:45,460 --> 00:12:47,500 so we move the-- 288 00:12:47,500 --> 00:12:52,120 we move the parameter dependence out of the forward path 289 00:12:52,120 --> 00:12:55,330 and into the feedback path with the idea 290 00:12:55,330 --> 00:12:57,310 that we're moving it out of the part that 291 00:12:57,310 --> 00:13:00,550 has active parts like transistors and into the part 292 00:13:00,550 --> 00:13:02,920 that we can build with passive parts. 293 00:13:02,920 --> 00:13:05,040 And we almost always do feedback that way. 294 00:13:05,040 --> 00:13:06,290 Not always, but almost always. 295 00:13:09,200 --> 00:13:11,930 Another important point is that this could 296 00:13:11,930 --> 00:13:15,320 work at a very high speed. 297 00:13:15,320 --> 00:13:17,270 As an illustration of that I want 298 00:13:17,270 --> 00:13:22,160 to think about what would be inside of that f naught box. 299 00:13:22,160 --> 00:13:23,900 Here's a very simple idea of what 300 00:13:23,900 --> 00:13:27,770 you could put in there that's based on two power transistors. 301 00:13:27,770 --> 00:13:30,040 This is not a course in power transistors. 302 00:13:30,040 --> 00:13:32,060 I am not going to give you a very good model 303 00:13:32,060 --> 00:13:32,810 for a transistor. 304 00:13:32,810 --> 00:13:34,851 There's other courses that do much better justice 305 00:13:34,851 --> 00:13:36,300 to that topic. 306 00:13:36,300 --> 00:13:38,270 But what I do want to say is that the idea 307 00:13:38,270 --> 00:13:41,030 would be that turning those transistors on and off 308 00:13:41,030 --> 00:13:43,370 represents a significant source of distortion, 309 00:13:43,370 --> 00:13:45,420 or potentially a source of distortion. 310 00:13:45,420 --> 00:13:48,140 So for example, the simplest possible model 311 00:13:48,140 --> 00:13:51,860 I could think about would say that in order 312 00:13:51,860 --> 00:13:54,740 to generate, in order to push this voltage to a higher 313 00:13:54,740 --> 00:13:59,600 number, what I want to do is turn this transistor on 314 00:13:59,600 --> 00:14:01,760 because this will pull this voltage up 315 00:14:01,760 --> 00:14:04,470 toward the positive rail. 316 00:14:04,470 --> 00:14:07,110 So if the input is such that I'd like this to go up 317 00:14:07,110 --> 00:14:09,570 what I would like to have happen is that this transistor 318 00:14:09,570 --> 00:14:12,900 would come on and conduct current this way, which would 319 00:14:12,900 --> 00:14:15,930 suck this voltage up to there. 320 00:14:15,930 --> 00:14:20,130 To do that, the voltage here has to exceed the voltage there. 321 00:14:20,130 --> 00:14:23,590 For a BJT, that's not the only kind of transistor 322 00:14:23,590 --> 00:14:26,950 I'm just illustrating one particular kind of example. 323 00:14:26,950 --> 00:14:31,780 For a bipolar transistor of this type, 324 00:14:31,780 --> 00:14:33,660 the way you would turn that transistor on 325 00:14:33,660 --> 00:14:35,880 would be to have this voltage exceed 326 00:14:35,880 --> 00:14:39,150 this voltage by at least some threshold voltage, nominally 327 00:14:39,150 --> 00:14:43,530 0.6 volts, which means that the kind of relationship 328 00:14:43,530 --> 00:14:48,360 between v out and v in that you would get is illustrated here. 329 00:14:48,360 --> 00:14:51,450 If the input voltage differed from the output voltage 330 00:14:51,450 --> 00:14:54,480 by less than plus or minus the threshold voltage, then 331 00:14:54,480 --> 00:14:57,070 nothing would happen because neither transistor would be on. 332 00:14:59,760 --> 00:15:03,270 If the input exceeded what was desired for the output 333 00:15:03,270 --> 00:15:06,150 by more than 0.6, the transistor on the top 334 00:15:06,150 --> 00:15:10,960 would start to come on, and you would get roughly linear growth 335 00:15:10,960 --> 00:15:14,830 in the output voltage in relation to the input voltage. 336 00:15:14,830 --> 00:15:16,090 Similarly with negative. 337 00:15:16,090 --> 00:15:20,500 If you make the voltage at the input sufficiently negative, 338 00:15:20,500 --> 00:15:22,010 you can cause the bottom transistor 339 00:15:22,010 --> 00:15:26,636 to come on and suck the voltage down toward minus 50. 340 00:15:26,636 --> 00:15:28,510 But again, you have to exceed the difference. 341 00:15:28,510 --> 00:15:30,885 The difference has to exceed something like the threshold 342 00:15:30,885 --> 00:15:34,960 voltage, normally about 0.6 volts, which means then 343 00:15:34,960 --> 00:15:39,280 that you get this piecewise linear approximation 344 00:15:39,280 --> 00:15:43,400 to the relation between the input and the output voltage. 345 00:15:43,400 --> 00:15:46,070 The problem with that is that it distorts the waveform. 346 00:15:46,070 --> 00:15:49,040 So if I imagine a purely sinusoidal input, 347 00:15:49,040 --> 00:15:53,720 what I get at the output then is not a purely sinusoidal output. 348 00:15:53,720 --> 00:15:56,170 And, in fact, that's quite objectionable 349 00:15:56,170 --> 00:15:57,760 for many purposes. 350 00:15:57,760 --> 00:15:59,560 In particular, in the example where 351 00:15:59,560 --> 00:16:02,560 we're listening to a sound, if we were to listen to this sound 352 00:16:02,560 --> 00:16:05,500 it would sound very different from the way 353 00:16:05,500 --> 00:16:09,340 a sine wave would sound. 354 00:16:09,340 --> 00:16:11,350 So the way I can do that is, not surprisingly, 355 00:16:11,350 --> 00:16:14,000 fix it with feedback. 356 00:16:14,000 --> 00:16:16,670 Again imagine that this is the complicated part. 357 00:16:16,670 --> 00:16:19,230 v f naught, followed by the speaker, 358 00:16:19,230 --> 00:16:22,030 put some gain in front of it, gain's cheap. 359 00:16:22,030 --> 00:16:24,250 We can do that run relatively well. 360 00:16:24,250 --> 00:16:26,320 And put in feedback so that I can directly 361 00:16:26,320 --> 00:16:29,650 compare how is the speaker voltage compare to the voltage 362 00:16:29,650 --> 00:16:32,440 that I'd like to listen to. 363 00:16:32,440 --> 00:16:35,380 The result then is that, depending 364 00:16:35,380 --> 00:16:38,710 on how big is the gain, I can change 365 00:16:38,710 --> 00:16:40,630 how big is this dead zone. 366 00:16:40,630 --> 00:16:47,060 The zone of input voltages that produce no change in output. 367 00:16:47,060 --> 00:16:50,380 So if I turn up the gain, here I am illustrating k equals one. 368 00:16:50,380 --> 00:16:53,680 If I turn it up to k equals two the dead zone gets smaller-- 369 00:16:53,680 --> 00:16:57,520 k equals four, eight, 10, whatever. 370 00:16:57,520 --> 00:16:59,950 So the idea is that by adjusting the gain, 371 00:16:59,950 --> 00:17:02,710 I control how much feedback there is. 372 00:17:02,710 --> 00:17:07,410 And I can control how distorted the wave form looks. 373 00:17:07,410 --> 00:17:11,910 And I can illustrate how bad it is to not do that 374 00:17:11,910 --> 00:17:15,540 by comparing an original clip of music 375 00:17:15,540 --> 00:17:17,550 to different values of the feedback. 376 00:17:17,550 --> 00:17:20,670 So what I'm going to do is play this many clips, 377 00:17:20,670 --> 00:17:22,920 some original-- 378 00:17:22,920 --> 00:17:27,310 the original that came off the mp3. 379 00:17:27,310 --> 00:17:29,310 Put it through the system with no feedback, that 380 00:17:29,310 --> 00:17:30,660 is to say there isn't a k here. 381 00:17:30,660 --> 00:17:31,701 There's no feedback loop. 382 00:17:31,701 --> 00:17:35,220 You just put v in a v out. 383 00:17:35,220 --> 00:17:38,415 Then k of two, four, eight, 16, and then back to the original. 384 00:17:38,415 --> 00:17:40,290 And if you listen, you should be able to hear 385 00:17:40,290 --> 00:17:42,220 the effects of the distortion. 386 00:17:42,220 --> 00:17:42,720 OK. 387 00:17:42,720 --> 00:17:44,494 Ready? 388 00:17:44,494 --> 00:17:46,486 [MUSIC PLAYING] 389 00:18:00,960 --> 00:18:01,590 You hear? 390 00:18:01,590 --> 00:18:03,586 [MUSIC PLAYING] 391 00:19:05,471 --> 00:19:05,970 OK. 392 00:19:05,970 --> 00:19:07,840 Everybody get to hear it? 393 00:19:07,840 --> 00:19:10,510 So the point-- although the acoustics in this room are 394 00:19:10,510 --> 00:19:12,130 terrible-- 395 00:19:12,130 --> 00:19:16,480 but even somebody with as old ears as mine, I 396 00:19:16,480 --> 00:19:19,610 can hear a very big difference even between k 397 00:19:19,610 --> 00:19:23,230 equals 16 and the original if I listen to it over headphones, 398 00:19:23,230 --> 00:19:25,630 or if I listen to it in a better environment. 399 00:19:25,630 --> 00:19:27,680 So the point is that we're very sensitive to it. 400 00:19:27,680 --> 00:19:29,630 In fact, 16 isn't even close. 401 00:19:29,630 --> 00:19:31,420 I mean, in a real kind of an amplifier 402 00:19:31,420 --> 00:19:33,670 you would use gains of at least 1,000 403 00:19:33,670 --> 00:19:35,620 in order to make it acceptable. 404 00:19:35,620 --> 00:19:38,900 So the point is that we can use feedback. 405 00:19:38,900 --> 00:19:40,870 So the overall point is to try to illustrate 406 00:19:40,870 --> 00:19:43,090 the kinds of things we can do with feedback. 407 00:19:43,090 --> 00:19:45,700 Here what we're doing is making a system that's intrinsically 408 00:19:45,700 --> 00:19:49,190 non-linear more linear. 409 00:19:52,210 --> 00:19:52,710 Questions? 410 00:19:52,710 --> 00:19:54,510 Comments? 411 00:19:54,510 --> 00:19:57,630 Left turn approaching. 412 00:19:57,630 --> 00:19:59,770 So what I want to close with then 413 00:19:59,770 --> 00:20:02,610 is two more examples that are much more fun. 414 00:20:02,610 --> 00:20:05,220 They're examples of the use of feedback 415 00:20:05,220 --> 00:20:06,690 to stabilize unstable systems. 416 00:20:06,690 --> 00:20:09,390 This is kind of connecting back to the very first lecture 417 00:20:09,390 --> 00:20:11,370 in this series that Russ gave where 418 00:20:11,370 --> 00:20:14,010 he did some very complicated control 419 00:20:14,010 --> 00:20:17,460 problems by doing feedback. 420 00:20:17,460 --> 00:20:19,330 I'm going to do some slightly simpler ones. 421 00:20:19,330 --> 00:20:23,940 But they're hard enough to be clear 422 00:20:23,940 --> 00:20:26,342 that they were a hard problem, and easy enough 423 00:20:26,342 --> 00:20:28,800 that every one of you should be able to write the equations 424 00:20:28,800 --> 00:20:31,789 and figure out what I'm about to do. 425 00:20:31,789 --> 00:20:34,080 If you are interested to do the more complicated things 426 00:20:34,080 --> 00:20:36,570 like having owls come in and swoop, 427 00:20:36,570 --> 00:20:38,400 that's a research project. 428 00:20:38,400 --> 00:20:43,110 So this is supposed to be in the middle ground between research 429 00:20:43,110 --> 00:20:44,670 and trivial, right? 430 00:20:44,670 --> 00:20:48,420 So I'm going to do some examples now of some unstable systems 431 00:20:48,420 --> 00:20:54,200 to illustrate how you would use feedback for stability. 432 00:20:54,200 --> 00:20:56,660 First example, is magnetic levitation. 433 00:20:56,660 --> 00:20:58,700 You've probably heard of magnetic levitation. 434 00:20:58,700 --> 00:21:00,170 Magnetic levitation is hard. 435 00:21:02,840 --> 00:21:05,390 We've talked about magnetic levitation since the 70s 436 00:21:05,390 --> 00:21:07,760 at least, perhaps the 60s. 437 00:21:07,760 --> 00:21:10,580 It's hard because magnetic systems tend 438 00:21:10,580 --> 00:21:12,090 to be intrinsically unstable. 439 00:21:12,090 --> 00:21:15,080 Not always, but the easy ways you 440 00:21:15,080 --> 00:21:18,260 can imagine using a magnet to levitate things. 441 00:21:18,260 --> 00:21:20,870 We'd all like to make anti-grav right? 442 00:21:20,870 --> 00:21:23,810 Anti-grav would be a really good thing. 443 00:21:23,810 --> 00:21:25,760 And it seems like you ought to be able to do 444 00:21:25,760 --> 00:21:27,050 that with a magnet, right? 445 00:21:27,050 --> 00:21:31,010 So the idea is what I would like to do is levitate 446 00:21:31,010 --> 00:21:35,470 a mass with a magnet. 447 00:21:35,470 --> 00:21:36,707 Anti-grav, right? 448 00:21:36,707 --> 00:21:38,290 We've had that since Star Trek, right? 449 00:21:38,290 --> 00:21:39,460 It's time somebody built it. 450 00:21:39,460 --> 00:21:44,872 But it's hard even though we can make magnets. 451 00:21:44,872 --> 00:21:46,330 Even though we can make things that 452 00:21:46,330 --> 00:21:49,450 seemed like they would do it. 453 00:21:49,450 --> 00:21:53,110 It's hard because it's intrinsically unstable. 454 00:21:53,110 --> 00:21:54,850 This particular system is intrinsically 455 00:21:54,850 --> 00:21:58,360 unstable in the sense that if the mass, let's 456 00:21:58,360 --> 00:22:02,170 say that it was at equilibrium by which I mean, 457 00:22:02,170 --> 00:22:04,270 in the absence of somebody touching it, 458 00:22:04,270 --> 00:22:06,100 it would stay exactly the way it is. 459 00:22:06,100 --> 00:22:08,050 That's what I mean by equilibrium. 460 00:22:08,050 --> 00:22:10,390 Let's say it was at equilibrium so 461 00:22:10,390 --> 00:22:13,450 that the magnetic force I found was just sufficient to hold up 462 00:22:13,450 --> 00:22:17,640 the weight, Mg. 463 00:22:17,640 --> 00:22:21,360 If I came over and just very gently bumped 464 00:22:21,360 --> 00:22:26,045 the mass up a little, what would happen to the force? 465 00:22:29,680 --> 00:22:30,577 Force will go up. 466 00:22:30,577 --> 00:22:32,785 So you're supposed to remember this from some subject 467 00:22:32,785 --> 00:22:34,960 that you took before, right? 468 00:22:34,960 --> 00:22:36,290 Yes, no, yes? 469 00:22:36,290 --> 00:22:39,190 It started with an 8, yes? 470 00:22:39,190 --> 00:22:42,250 So the idea is something to do with if you 471 00:22:42,250 --> 00:22:47,440 have a magnet of that type, you generate magnetic fields. 472 00:22:47,440 --> 00:22:49,660 You remember they're not supposed to cross, right? 473 00:22:49,660 --> 00:22:51,201 One of the rules is they don't cross. 474 00:22:51,201 --> 00:22:53,170 [LAUGHTER] 475 00:22:53,170 --> 00:22:57,610 But the idea is that the density of flux lines 476 00:22:57,610 --> 00:23:00,850 decreases as you leave the magnet, 477 00:23:00,850 --> 00:23:02,830 and the force that's generated has 478 00:23:02,830 --> 00:23:06,460 something to do with the change in the density of flux lines. 479 00:23:06,460 --> 00:23:08,140 If you move an object through a region 480 00:23:08,140 --> 00:23:10,056 where the number of flux lines doesn't change, 481 00:23:10,056 --> 00:23:11,320 it takes no force. 482 00:23:11,320 --> 00:23:13,300 If you move an object into a region that 483 00:23:13,300 --> 00:23:16,151 has fewer or more flux lines, it requires force 484 00:23:16,151 --> 00:23:16,900 to make that move. 485 00:23:16,900 --> 00:23:18,400 That's the idea. 486 00:23:18,400 --> 00:23:24,250 So, since the flux lines are thinning as you move away, 487 00:23:24,250 --> 00:23:26,837 if I push up it will generate-- 488 00:23:26,837 --> 00:23:28,420 I'm moving the mass into a place where 489 00:23:28,420 --> 00:23:32,110 there is a greater density of flux lines, that will generate 490 00:23:32,110 --> 00:23:37,150 force up on the mass. 491 00:23:37,150 --> 00:23:40,240 That's bad, right? 492 00:23:40,240 --> 00:23:43,060 I had the force exactly where I needed it to be in order 493 00:23:43,060 --> 00:23:44,782 to levitate the mass. 494 00:23:44,782 --> 00:23:46,240 If the force goes up, what happens? 495 00:23:49,460 --> 00:23:50,870 It accelerates up. 496 00:23:50,870 --> 00:23:53,730 Well, OK, so it accelerates up, then what? 497 00:23:53,730 --> 00:23:55,660 Well, it gets closer. 498 00:23:55,660 --> 00:23:57,510 Then what? 499 00:23:57,510 --> 00:24:00,640 It accelerates faster. 500 00:24:00,640 --> 00:24:05,950 So, the problem is that the position is unstable, 501 00:24:05,950 --> 00:24:09,370 or we might say quasi stable. 502 00:24:09,370 --> 00:24:11,410 So what we can do is make a model of this 503 00:24:11,410 --> 00:24:15,670 to try to understand how we could fix it with feedback. 504 00:24:15,670 --> 00:24:17,560 Imagine that I represent the magnet 505 00:24:17,560 --> 00:24:20,770 by a function that maps for every value 506 00:24:20,770 --> 00:24:26,890 of the position of the mass why it generates a force, f m. 507 00:24:26,890 --> 00:24:29,270 Then, because of the shape of the field lines, 508 00:24:29,270 --> 00:24:31,805 I'm expecting that the force increases 509 00:24:31,805 --> 00:24:33,430 on the right of the curve and decreases 510 00:24:33,430 --> 00:24:36,010 on the left of the curve. 511 00:24:36,010 --> 00:24:40,630 If I'm sitting for y zero at equilibrium what matters 512 00:24:40,630 --> 00:24:46,320 is the difference between y and the equilibrium place. 513 00:24:46,320 --> 00:24:48,390 And the important force is the difference 514 00:24:48,390 --> 00:24:53,220 between the magnetic force and Mg, the weight. 515 00:24:53,220 --> 00:24:54,870 So I get a simple relationship that 516 00:24:54,870 --> 00:24:59,010 says that this force, this vertical line, f, 517 00:24:59,010 --> 00:25:02,040 should be m a. 518 00:25:02,040 --> 00:25:07,320 F equals m a which is m y double dot. 519 00:25:07,320 --> 00:25:11,960 So having generated that kind of a model 520 00:25:11,960 --> 00:25:15,780 I can now imagine putting that into a feedback system. 521 00:25:15,780 --> 00:25:19,400 So now I put the model into the feedback system and this junk 522 00:25:19,400 --> 00:25:23,400 over here, this is f equals m a. 523 00:25:23,400 --> 00:25:27,195 f gets multiplied by one over m, that a. 524 00:25:27,195 --> 00:25:29,050 a is y double dot. 525 00:25:29,050 --> 00:25:30,500 Integrate it to get y dot. 526 00:25:30,500 --> 00:25:32,150 Integrate it again to get y. 527 00:25:32,150 --> 00:25:34,400 But y was the thing that controlled the magnet 528 00:25:34,400 --> 00:25:37,170 in the first place. 529 00:25:37,170 --> 00:25:40,560 So I'm trying to cast magnetic levitation into a frame 530 00:25:40,560 --> 00:25:42,030 for understanding feedback. 531 00:25:45,270 --> 00:25:48,690 So now if I want to further simplify things 532 00:25:48,690 --> 00:25:54,330 I can replace the magnet model, at least for small deviations 533 00:25:54,330 --> 00:25:58,059 in position, by a linear gain. 534 00:25:58,059 --> 00:25:59,850 So I'm making a straight line approximation 535 00:25:59,850 --> 00:26:02,234 to this curvy line. 536 00:26:02,234 --> 00:26:03,650 And then we have something that we 537 00:26:03,650 --> 00:26:06,140 analyze all the time, simple feedback 538 00:26:06,140 --> 00:26:11,160 system that has integrators, gains, that kind of stuff. 539 00:26:11,160 --> 00:26:15,590 It's surprisingly similar to the mass spring dashpot. 540 00:26:15,590 --> 00:26:23,100 Springs, by contrast to magnets, springs levitate stably. 541 00:26:23,100 --> 00:26:24,964 What happens if I push-- so I'm suspending 542 00:26:24,964 --> 00:26:26,880 a mass from a spring, if I push up on the mass 543 00:26:26,880 --> 00:26:29,700 how does the force in the spring change? 544 00:26:29,700 --> 00:26:30,780 Gets less. 545 00:26:30,780 --> 00:26:33,000 That's what you want. 546 00:26:33,000 --> 00:26:35,970 It's no longer sufficient to hold the mass up, 547 00:26:35,970 --> 00:26:38,040 so the mass falls. 548 00:26:38,040 --> 00:26:39,990 Push up, the mass tries to fall. 549 00:26:39,990 --> 00:26:43,880 Pull down on the mass, it tries to pull up. 550 00:26:43,880 --> 00:26:46,620 Spring levitates stably. 551 00:26:46,620 --> 00:26:50,300 Magnet levitates unstably. 552 00:26:50,300 --> 00:26:50,840 OK? 553 00:26:50,840 --> 00:26:52,700 Almost the same system though. 554 00:26:52,700 --> 00:26:55,580 Here's a way of thinking about the magnetic levitation 555 00:26:55,580 --> 00:26:56,720 like we just saw. 556 00:26:56,720 --> 00:26:59,210 Here's an equivalent diagram for the spring levitation. 557 00:26:59,210 --> 00:27:01,340 Almost the same thing. 558 00:27:01,340 --> 00:27:05,600 The critical thing is that minus sign and that plus sign. 559 00:27:05,600 --> 00:27:11,300 So what's the relationship between the poles of the spring 560 00:27:11,300 --> 00:27:15,600 and mass system and the poles of the levitation system? 561 00:27:15,600 --> 00:27:17,730 Look at your neighbor. 562 00:27:17,730 --> 00:27:19,950 Say the word poles. 563 00:27:19,950 --> 00:27:22,780 We say that a lot in this class. 564 00:27:22,780 --> 00:27:24,750 How do the poles relate in those two systems? 565 00:27:24,750 --> 00:27:26,718 [AUDIENCE CHATTER] 566 00:27:54,800 --> 00:27:57,340 OK, so draw the poles in the air for the two systems. 567 00:27:57,340 --> 00:27:59,530 How do they relate? 568 00:27:59,530 --> 00:28:00,030 Excellent. 569 00:28:00,030 --> 00:28:00,529 Excellent. 570 00:28:00,529 --> 00:28:01,630 I saw one right answer. 571 00:28:01,630 --> 00:28:05,350 Somebody else draw some-- 572 00:28:05,350 --> 00:28:08,896 I'm not sure if I caught that one. 573 00:28:08,896 --> 00:28:10,020 So how do the poles relate? 574 00:28:10,020 --> 00:28:11,040 Come on, somebody drawn it in the air. 575 00:28:11,040 --> 00:28:12,123 I've got one right answer. 576 00:28:14,700 --> 00:28:18,360 OK, your air drawings are a little cryptic. 577 00:28:18,360 --> 00:28:20,490 Here's a not so air drawing. 578 00:28:20,490 --> 00:28:23,430 So here's a picture of what they look like for a spring and mass 579 00:28:23,430 --> 00:28:24,810 system. 580 00:28:24,810 --> 00:28:28,670 And all you do is think about the differential equation, 581 00:28:28,670 --> 00:28:33,030 turn it into system function, there's two poles, 582 00:28:33,030 --> 00:28:35,610 factor the denominator, you find that the two poles 583 00:28:35,610 --> 00:28:38,160 are on the j omega axis. 584 00:28:38,160 --> 00:28:39,810 j omega axis, that sounds bad. 585 00:28:39,810 --> 00:28:41,550 Yeah, it is bad, right? what happens 586 00:28:41,550 --> 00:28:44,450 if they're on the j omega axis? 587 00:28:44,450 --> 00:28:45,250 Oscillate. 588 00:28:45,250 --> 00:28:47,750 If they're right on the j omega axis, it oscillates forever. 589 00:28:47,750 --> 00:28:49,458 That's exactly what a mass spring dashpot 590 00:28:49,458 --> 00:28:52,220 does if you have no damping. 591 00:28:52,220 --> 00:28:53,610 OK, it's bad. 592 00:28:53,610 --> 00:28:57,480 But it's not nearly as bad as magnetic levitation. 593 00:28:57,480 --> 00:29:00,970 Magnetic levitation, you get the same distance from the origin, 594 00:29:00,970 --> 00:29:06,380 but it's turned to plus and minus something. 595 00:29:06,380 --> 00:29:07,170 That one. 596 00:29:07,170 --> 00:29:08,090 I like. 597 00:29:08,090 --> 00:29:10,700 That one I don't like. 598 00:29:10,700 --> 00:29:13,384 That one is unstable. 599 00:29:13,384 --> 00:29:15,800 So the question is, how can you make a magnetic levitation 600 00:29:15,800 --> 00:29:16,299 system? 601 00:29:16,299 --> 00:29:18,200 How do you make a system that we use 602 00:29:18,200 --> 00:29:23,250 a magnet to levitate a mass when the physics are intrinsically 603 00:29:23,250 --> 00:29:23,910 stable? 604 00:29:23,910 --> 00:29:26,980 And the answer is-- 605 00:29:26,980 --> 00:29:29,170 feedback, of course, yes. 606 00:29:29,170 --> 00:29:31,900 So the idea is that here is our system that 607 00:29:31,900 --> 00:29:34,090 was intrinsically unstable. 608 00:29:34,090 --> 00:29:37,992 What we'd like to do is somehow change the physics. 609 00:29:37,992 --> 00:29:39,700 One of the ways we can change the physics 610 00:29:39,700 --> 00:29:43,656 is by modifying the current that runs through the magnet. 611 00:29:43,656 --> 00:29:45,280 If we changed the current from i naught 612 00:29:45,280 --> 00:29:47,225 to 1.1 times i naught, then the forces 613 00:29:47,225 --> 00:29:48,350 everywhere would be bigger. 614 00:29:50,970 --> 00:29:54,690 We could use that idea to introduce a second feedback 615 00:29:54,690 --> 00:29:55,540 path. 616 00:29:55,540 --> 00:29:57,480 There is the intrinsic one that comes just 617 00:29:57,480 --> 00:29:58,950 from the physical layout. 618 00:29:58,950 --> 00:30:03,720 But I'm going to add a path that senses the position of the mass 619 00:30:03,720 --> 00:30:06,100 and feeds back a change in current. 620 00:30:06,100 --> 00:30:10,350 And if I do this carefully, I can arrange this path, 621 00:30:10,350 --> 00:30:14,310 the intrinsic one has positive feedback that was bad, 622 00:30:14,310 --> 00:30:19,600 but the one I add has negative feedback, which is good. 623 00:30:19,600 --> 00:30:23,920 Then the overall system looks more like this where I've got 624 00:30:23,920 --> 00:30:27,930 k, which I cannot affect, that's physics. 625 00:30:27,930 --> 00:30:29,990 But I've got k two that I can affect. 626 00:30:32,910 --> 00:30:35,160 And what's showed here is a root locus. 627 00:30:35,160 --> 00:30:39,550 As I change k two, what happens to the poles? 628 00:30:39,550 --> 00:30:41,890 And what you can see is since they were equally disposed 629 00:30:41,890 --> 00:30:45,220 on the two sides of the j omega axis, as I turned up the gain-- 630 00:30:45,220 --> 00:30:47,140 you can do this just by doing-- 631 00:30:47,140 --> 00:30:48,560 factor the polynomial. 632 00:30:48,560 --> 00:30:49,060 Right? 633 00:30:49,060 --> 00:30:55,330 So I factored this to generate this plot. 634 00:30:55,330 --> 00:30:57,610 As you increase k two the poles go toward each other 635 00:30:57,610 --> 00:31:00,010 and then they split, and they become complex. 636 00:31:00,010 --> 00:31:02,545 Complex is good. 637 00:31:02,545 --> 00:31:05,450 But this still has the problem that the mass spring 638 00:31:05,450 --> 00:31:08,640 dashpot had if it didn't have any damping. 639 00:31:08,640 --> 00:31:11,166 So it's better because at least it's stable-- 640 00:31:11,166 --> 00:31:12,290 Oh well no it's not stable. 641 00:31:12,290 --> 00:31:13,640 It's marginally stable. 642 00:31:13,640 --> 00:31:16,160 It's now oscillatory. 643 00:31:16,160 --> 00:31:20,320 It no longer has eigenfunctions that grow with time. 644 00:31:20,320 --> 00:31:23,000 The problem is they don't shrink either. 645 00:31:23,000 --> 00:31:26,500 They state constant in amplitude. 646 00:31:26,500 --> 00:31:28,580 But you can fix that, and what I'll do now 647 00:31:28,580 --> 00:31:32,810 is show a demo that was done by Professor James Roberge where 648 00:31:32,810 --> 00:31:38,320 the idea was you don't have to just put the extra gain in. 649 00:31:38,320 --> 00:31:43,090 What Jim did was put a zero too. 650 00:31:43,090 --> 00:31:46,930 So these are the intrinsic poles from the magnetic levitation. 651 00:31:46,930 --> 00:31:50,400 This is a zero introduced by electronics. 652 00:31:50,400 --> 00:31:52,740 The zero has the effect of bending the poles, 653 00:31:52,740 --> 00:31:55,540 the closed loop poles around, so that you can then 654 00:31:55,540 --> 00:31:56,890 make it stable. 655 00:31:56,890 --> 00:31:58,840 So [INAUDIBLE]. 656 00:31:58,840 --> 00:32:00,070 So here's Jim's device. 657 00:32:02,950 --> 00:32:05,920 And the goal now is to smash fingers 658 00:32:05,920 --> 00:32:08,830 because it will try to levitate. 659 00:32:08,830 --> 00:32:11,980 So this is a hunky magnet. 660 00:32:11,980 --> 00:32:15,560 This is a hunky power supply. 661 00:32:15,560 --> 00:32:17,980 And that is-- well this is the power supply. 662 00:32:17,980 --> 00:32:21,970 This generates DC voltage that drives the amplifier that's 663 00:32:21,970 --> 00:32:23,680 inside this box. 664 00:32:23,680 --> 00:32:27,160 So the idea then is we have this metal ball. 665 00:32:27,160 --> 00:32:29,840 So we have this metal ball, and it's kind of heavy, right? 666 00:32:29,840 --> 00:32:33,250 You know, it's not like some nuclear-- 667 00:32:33,250 --> 00:32:36,350 it's not uranium or anything, but it's heavy. 668 00:32:36,350 --> 00:32:40,374 And the idea is going to be to use the hunky magnet. 669 00:32:40,374 --> 00:32:41,290 Let's see, can we see? 670 00:32:41,290 --> 00:32:43,660 Oh yeah, OK, so you can see it up there. 671 00:32:43,660 --> 00:32:47,440 So you can see maybe that here is 672 00:32:47,440 --> 00:32:49,840 the pole of the magnet, which is right there. 673 00:32:52,362 --> 00:32:54,320 So there is the pole, a different kind of pole. 674 00:32:54,320 --> 00:32:55,160 [LAUGHTER] 675 00:32:55,160 --> 00:32:55,955 Not to be confused. 676 00:32:55,955 --> 00:32:57,614 [LAUGHTER] 677 00:32:59,370 --> 00:33:03,110 So that's the iron core of the magnet. 678 00:33:03,110 --> 00:33:05,570 And you can see the turns up here. 679 00:33:05,570 --> 00:33:08,030 And here you can see some light. 680 00:33:08,030 --> 00:33:11,090 So what's going on here is a source of light 681 00:33:11,090 --> 00:33:14,270 is being aimed toward a photo sensor 682 00:33:14,270 --> 00:33:19,070 so that when the mass, the ball, is levitating at exactly 683 00:33:19,070 --> 00:33:22,250 the right place, there's a signal that gets fed back 684 00:33:22,250 --> 00:33:25,370 into the control box to do the thing that I have illustrated 685 00:33:25,370 --> 00:33:26,700 here. 686 00:33:26,700 --> 00:33:29,510 OK, everybody knows what's going on? 687 00:33:29,510 --> 00:33:31,340 I'll smash my fingers now and then 688 00:33:31,340 --> 00:33:33,644 ask Lorenzo to do it correctly. 689 00:33:33,644 --> 00:33:35,060 So what I'm going to do now is try 690 00:33:35,060 --> 00:33:39,290 to move the mass into the air so it levitates. 691 00:33:39,290 --> 00:33:41,420 so successful would be you're watching up there 692 00:33:41,420 --> 00:33:45,030 and there's going to be a ball up here, and it will levitate. 693 00:33:45,030 --> 00:33:47,080 Unsuccessful is I'll have a black and white-- 694 00:33:47,080 --> 00:33:49,680 a black and blue fingernail. 695 00:33:49,680 --> 00:33:50,540 So, let's see-- 696 00:33:55,267 --> 00:33:56,350 Lorenzo is laughing at me. 697 00:33:56,350 --> 00:33:58,306 [LAUGHTER] 698 00:33:59,284 --> 00:34:00,751 LORENZO: See your hand can move. 699 00:34:00,751 --> 00:34:02,220 OK. 700 00:34:02,220 --> 00:34:04,224 DENNIS FREEMAN: Nope, that's not going to work. 701 00:34:04,224 --> 00:34:06,177 LORENZO: You don't have a way to move? 702 00:34:06,177 --> 00:34:07,010 DENNIS FREEMAN: Yes. 703 00:34:07,010 --> 00:34:08,280 LORENZO: Yes. 704 00:34:08,280 --> 00:34:10,110 DENNIS FREEMAN: OK, this will work better. 705 00:34:10,110 --> 00:34:14,310 If I get coached enough, I can do things. 706 00:34:14,310 --> 00:34:15,640 LORENZO: Yeah! 707 00:34:15,640 --> 00:34:16,139 Oh! 708 00:34:16,139 --> 00:34:16,562 DENNIS FREEMAN: Almost! 709 00:34:16,562 --> 00:34:17,409 [LAUGHTER] 710 00:34:17,409 --> 00:34:18,310 OK, it was close. 711 00:34:22,900 --> 00:34:24,100 No, no, no. 712 00:34:24,100 --> 00:34:25,300 Failing once is fine. 713 00:34:25,300 --> 00:34:28,350 Failing twice, no that's not acceptable. 714 00:34:28,350 --> 00:34:30,060 Wonderful. 715 00:34:30,060 --> 00:34:33,449 OK, so that's magnetic levitation made possible 716 00:34:33,449 --> 00:34:34,120 by feedback. 717 00:34:34,120 --> 00:34:35,190 That's the point, OK? 718 00:34:35,190 --> 00:34:36,856 Everybody sort of knows what's going on? 719 00:34:41,000 --> 00:34:43,832 OK, so to redeem myself we'll leave the demo set up 720 00:34:43,832 --> 00:34:45,290 and after lecture you can come down 721 00:34:45,290 --> 00:34:46,998 and you can do it and smash your fingers. 722 00:34:50,230 --> 00:34:53,949 Questions, comments, everybody knows the point? 723 00:34:53,949 --> 00:34:56,184 So we'll leave it there just for entertainment value. 724 00:34:56,184 --> 00:34:57,225 LORENZO: It will get hot. 725 00:34:57,225 --> 00:34:58,360 DENNIS FREEMAN: Oh, OK. 726 00:34:58,360 --> 00:34:59,020 OK. 727 00:34:59,020 --> 00:35:00,436 You don't want to burn it up, huh? 728 00:35:03,951 --> 00:35:04,450 Good. 729 00:35:04,450 --> 00:35:04,960 Thank you. 730 00:35:04,960 --> 00:35:06,198 Thank you. 731 00:35:06,198 --> 00:35:08,114 [APPLAUSE] 732 00:35:10,990 --> 00:35:15,040 So the next demo is the famous inverted pendulum. 733 00:35:15,040 --> 00:35:18,360 The idea is that if you have a pendulum-- 734 00:35:18,360 --> 00:35:18,860 Yes? 735 00:35:18,860 --> 00:35:21,141 AUDIENCE: Was k oscillating also? 736 00:35:21,141 --> 00:35:21,640 With the-- 737 00:35:21,640 --> 00:35:22,640 DENNIS FREEMAN: Correct. 738 00:35:22,640 --> 00:35:23,770 Correct. 739 00:35:23,770 --> 00:35:25,390 So, that's a good question. 740 00:35:25,390 --> 00:35:28,010 OK, so did I screw up? 741 00:35:28,010 --> 00:35:30,460 Not a chance, right? 742 00:35:30,460 --> 00:35:33,340 Or for those of you who know me-- 743 00:35:33,340 --> 00:35:35,957 what would you conclude if it was oscillating 744 00:35:35,957 --> 00:35:37,540 and I told you I thought I built this? 745 00:35:41,020 --> 00:35:44,030 I screw up the circuit. 746 00:35:44,030 --> 00:35:48,000 I screwed up the root locus diagram. 747 00:35:48,000 --> 00:35:51,310 Everything's hunky dory, and I did everything right. 748 00:35:51,310 --> 00:35:52,540 None of the above. 749 00:35:52,540 --> 00:35:53,332 Yes? 750 00:35:53,332 --> 00:35:55,657 AUDIENCE: [INAUDIBLE] 751 00:35:55,657 --> 00:35:56,740 DENNIS FREEMAN: Positions. 752 00:35:56,740 --> 00:35:58,570 OK, so flaky hardware. 753 00:35:58,570 --> 00:36:00,540 Universal solution right? 754 00:36:00,540 --> 00:36:01,140 Could be. 755 00:36:01,140 --> 00:36:01,639 Could be. 756 00:36:01,639 --> 00:36:03,320 Could be. 757 00:36:03,320 --> 00:36:07,360 There's a different explanation, which is possible, 758 00:36:07,360 --> 00:36:09,130 but your explanation may be right. 759 00:36:09,130 --> 00:36:12,639 What's another explanation besides flaky hardware? 760 00:36:12,639 --> 00:36:13,138 Yes? 761 00:36:13,138 --> 00:36:14,952 AUDIENCE: [INAUDIBLE] complex poles. 762 00:36:14,952 --> 00:36:17,410 DENNIS FREEMAN: Complex poles, but didn't I stabilize them? 763 00:36:20,340 --> 00:36:23,160 Leading question. 764 00:36:23,160 --> 00:36:24,720 Didn't I stabilize them? 765 00:36:24,720 --> 00:36:27,220 Leading question. 766 00:36:27,220 --> 00:36:29,500 The answer is, didn't I stabilize them? 767 00:36:32,270 --> 00:36:32,770 Yes? 768 00:36:32,770 --> 00:36:35,320 No? 769 00:36:35,320 --> 00:36:37,455 Completely cryptic he is. 770 00:36:37,455 --> 00:36:40,904 AUDIENCE: I mean they can still have complex values. 771 00:36:40,904 --> 00:36:42,820 DENNIS FREEMAN: They could have complex-- how? 772 00:36:42,820 --> 00:36:44,360 AUDIENCE: [INAUDIBLE] don't know where they are. 773 00:36:44,360 --> 00:36:45,901 DENNIS FREEMAN: And so what depends-- 774 00:36:45,901 --> 00:36:47,520 what determines where they are? 775 00:36:47,520 --> 00:36:48,330 AUDIENCE: Sorry? 776 00:36:48,330 --> 00:36:52,085 DENNIS FREEMAN: What determines what they are? 777 00:36:52,085 --> 00:36:55,965 AUDIENCE: Whatever your-- whatever you set [INAUDIBLE] 778 00:36:55,965 --> 00:36:56,940 k or gain. 779 00:36:56,940 --> 00:36:57,886 DENNIS FREEMAN: k 780 00:36:57,886 --> 00:36:58,800 AUDIENCE: Yeah. 781 00:36:58,800 --> 00:36:59,799 DENNIS FREEMAN: Exactly. 782 00:36:59,799 --> 00:37:03,150 So I get to control this k two thing. 783 00:37:03,150 --> 00:37:07,660 And as I change k two, the poles start out here, 784 00:37:07,660 --> 00:37:09,980 they go toward each other. 785 00:37:09,980 --> 00:37:12,070 So if k equals zero they're here. 786 00:37:12,070 --> 00:37:13,615 As k gets bigger they do this. 787 00:37:16,810 --> 00:37:20,550 So it's possible that they're just simply here. 788 00:37:20,550 --> 00:37:21,370 Right? 789 00:37:21,370 --> 00:37:23,180 The poles could still be complex. 790 00:37:25,810 --> 00:37:28,300 If that were true, you would expect the oscillations 791 00:37:28,300 --> 00:37:30,250 to die with time. 792 00:37:30,250 --> 00:37:30,760 Right? 793 00:37:30,760 --> 00:37:33,160 If the oscillations didn't die with time, 794 00:37:33,160 --> 00:37:36,050 it's still possible-- well, if they don't die with time, 795 00:37:36,050 --> 00:37:38,591 then I guess I screwed something up. 796 00:37:38,591 --> 00:37:39,090 Right? 797 00:37:39,090 --> 00:37:40,590 Because even if they just come here, 798 00:37:40,590 --> 00:37:44,400 they're oscillating but they're on the left part of the axis so 799 00:37:44,400 --> 00:37:46,800 they should decay eventually. 800 00:37:46,800 --> 00:37:48,570 So I guess to the extent that they 801 00:37:48,570 --> 00:37:53,460 didn't die eventually, something screwed up with the analysis, 802 00:37:53,460 --> 00:37:55,500 the hardware is screwed up, something. 803 00:37:55,500 --> 00:37:56,238 Yes? 804 00:37:56,238 --> 00:37:58,680 AUDIENCE: So is root locus due to k or k two. 805 00:37:58,680 --> 00:38:00,130 DENNIS FREEMAN: k two. 806 00:38:00,130 --> 00:38:03,340 You can draw a root locus for anything. 807 00:38:03,340 --> 00:38:05,680 You can change any parameter and sketch 808 00:38:05,680 --> 00:38:08,750 the locus of points where the roots move to, 809 00:38:08,750 --> 00:38:10,870 hence the name, root locus. 810 00:38:10,870 --> 00:38:12,970 This is for k two. 811 00:38:12,970 --> 00:38:19,550 So z zero was fixed here, held constant, and k two 812 00:38:19,550 --> 00:38:21,160 was modified to trace out that loop. 813 00:38:26,150 --> 00:38:28,950 The next example is pendulum. 814 00:38:28,950 --> 00:38:32,720 So just like spring mass system, spring 815 00:38:32,720 --> 00:38:35,990 levitated mass in a stable fashion, 816 00:38:35,990 --> 00:38:39,020 although perhaps oscillatory. 817 00:38:39,020 --> 00:38:44,420 So this pendulum is also stable in that same sense. 818 00:38:44,420 --> 00:38:47,780 Might be oscillatory, but it will eventually stop. 819 00:38:47,780 --> 00:38:49,910 Of course pendula are not very interesting. 820 00:38:49,910 --> 00:38:51,670 What's interesting is inverted pendula 821 00:38:51,670 --> 00:38:54,260 where the rule is what I have to do 822 00:38:54,260 --> 00:39:01,580 is stabilize this by moving my hand to compensate. 823 00:39:01,580 --> 00:39:05,270 So let's say that it starts to move this way. 824 00:39:05,270 --> 00:39:09,470 I've got positive feedback that tends to make it continue. 825 00:39:09,470 --> 00:39:15,710 So if I started at equilibrium, by which I mean a state 826 00:39:15,710 --> 00:39:19,530 that would persist if nothing changed. 827 00:39:19,530 --> 00:39:22,610 So if I started at equilibrium, but then 828 00:39:22,610 --> 00:39:25,780 perturbed it by breathing, or whatever, 829 00:39:25,780 --> 00:39:29,060 or by just not being quite at equilibrium, it'll move 830 00:39:29,060 --> 00:39:31,280 and then I have to do something to compensate, 831 00:39:31,280 --> 00:39:33,590 otherwise it'll fall down. 832 00:39:33,590 --> 00:39:37,130 And there are people who can do this better than me. 833 00:39:37,130 --> 00:39:39,049 But we don't need to have-- 834 00:39:39,049 --> 00:39:40,340 I don't need to be coordinated. 835 00:39:40,340 --> 00:39:42,890 I can build a system that's coordinated instead. 836 00:39:42,890 --> 00:39:44,570 That's the value of feedback. 837 00:39:44,570 --> 00:39:46,790 So I can be uncoordinated. 838 00:39:46,790 --> 00:39:50,110 Before I do that-- 839 00:39:50,110 --> 00:39:52,050 no, let's do this first. 840 00:39:52,050 --> 00:39:56,120 So, here is a mechanical system that 841 00:39:56,120 --> 00:39:58,090 uses feedback to stabilize-- 842 00:39:58,090 --> 00:39:59,750 I'm not going to read anything right? 843 00:39:59,750 --> 00:40:04,490 So here is a pendulum, an inverted pendulum. 844 00:40:04,490 --> 00:40:05,810 There's some stuff down here. 845 00:40:05,810 --> 00:40:08,480 There's no active parts. 846 00:40:08,480 --> 00:40:10,190 This stuff is a potentiometer, just 847 00:40:10,190 --> 00:40:12,148 like the potentiometer on the back of the motor 848 00:40:12,148 --> 00:40:13,130 that I demonstrated. 849 00:40:13,130 --> 00:40:15,650 And what it's doing is providing a signal 850 00:40:15,650 --> 00:40:20,690 to electronics that will let the electronics say, oh, it's 851 00:40:20,690 --> 00:40:23,190 moving this way. 852 00:40:23,190 --> 00:40:26,390 Now, the cart is on wheels, and the wheels 853 00:40:26,390 --> 00:40:29,480 are connected to a hunky motor. 854 00:40:29,480 --> 00:40:31,400 If the weight starts to move this way 855 00:40:31,400 --> 00:40:34,040 what should the hunky motor do? 856 00:40:34,040 --> 00:40:37,440 Push the cart that way. 857 00:40:37,440 --> 00:40:38,700 So that's the idea. 858 00:40:38,700 --> 00:40:44,030 So I've got a one dimensional inverted pendulum 859 00:40:44,030 --> 00:40:48,290 and I'm explicitly modeling the idea that I'm-- that the base 860 00:40:48,290 --> 00:40:49,670 is moving. 861 00:40:49,670 --> 00:40:51,770 So over here is an outline of the way you 862 00:40:51,770 --> 00:40:53,960 think about that-- the way you can think about that. 863 00:40:53,960 --> 00:40:59,100 I got the car and it's in a not invert-- 864 00:40:59,100 --> 00:41:00,700 the car as supporting a pendulum. 865 00:41:00,700 --> 00:41:04,770 The pendulum is in a non inertial frame 866 00:41:04,770 --> 00:41:06,780 because the mass-- 867 00:41:06,780 --> 00:41:09,150 let's say that I'm approximating the pendulum as having 868 00:41:09,150 --> 00:41:10,740 all of its mass at the end. 869 00:41:10,740 --> 00:41:12,900 That's an approximation, but it's not 870 00:41:12,900 --> 00:41:15,190 a critical approximation. 871 00:41:15,190 --> 00:41:17,970 So I can imagine now that I've got an input variable, which 872 00:41:17,970 --> 00:41:21,060 is x, the position of the car. 873 00:41:21,060 --> 00:41:23,760 And I've got a feedback signal that's 874 00:41:23,760 --> 00:41:26,350 trying to feedback to me how much is 875 00:41:26,350 --> 00:41:30,070 theta, the angle that the pendulum is making. 876 00:41:30,070 --> 00:41:31,860 And so what I want you to do now-- 877 00:41:31,860 --> 00:41:36,360 so if you just write 801 kinds of stuff 878 00:41:36,360 --> 00:41:37,950 you can imagine that the if-- 879 00:41:37,950 --> 00:41:41,550 So, first off, you've got-- 880 00:41:41,550 --> 00:41:44,840 instead of f equals m a, you have torque equals i omega. 881 00:41:44,840 --> 00:41:47,520 Everybody's happy about torque equals i omega, right? 882 00:41:47,520 --> 00:41:51,570 And, if I want to be in a non inertial frame, 883 00:41:51,570 --> 00:41:54,450 one way to account for the fact that the car is moving 884 00:41:54,450 --> 00:41:57,930 is to compute torques about the point of insertion. 885 00:41:57,930 --> 00:42:01,110 Then I can represent the effect of the moving car 886 00:42:01,110 --> 00:42:05,620 as a pseudo force on the mass. 887 00:42:05,620 --> 00:42:07,120 That pseudo force is in proportion 888 00:42:07,120 --> 00:42:11,000 to the acceleration of the car. 889 00:42:11,000 --> 00:42:14,750 And I've got the weight of the inverted pendulum trying 890 00:42:14,750 --> 00:42:16,290 to pull it in the other direction. 891 00:42:16,290 --> 00:42:17,270 And if you think about all this stuff 892 00:42:17,270 --> 00:42:18,840 you get this kind of an equation. 893 00:42:18,840 --> 00:42:25,340 So this is i omega dot, i omega dot equals, and then 894 00:42:25,340 --> 00:42:27,110 I have to worry about torques. 895 00:42:27,110 --> 00:42:31,730 There's the torque that's due to the weight, 896 00:42:31,730 --> 00:42:34,620 and there's the torque that's due to the cart movement. 897 00:42:37,460 --> 00:42:42,944 so the question then is, what are the poles of this system? 898 00:42:42,944 --> 00:42:45,360 We want to think about all of these kinds of systems using 899 00:42:45,360 --> 00:42:48,030 003, because it's such a powerful thing. 900 00:42:48,030 --> 00:42:52,480 So what are the poles of the inverted pendulum? 901 00:42:55,640 --> 00:42:56,620 Look at your neighbor. 902 00:42:56,620 --> 00:42:57,410 Say the word pole. 903 00:42:57,410 --> 00:42:59,270 [AUDIENCE CHATTER] 904 00:43:02,007 --> 00:43:02,840 Where do the poles-- 905 00:43:02,840 --> 00:43:04,832 [AUDIENCE CHATTER] 906 00:43:33,716 --> 00:43:35,210 Bump the car. 907 00:43:35,210 --> 00:43:41,260 [AUDIENCE CHATTER] 908 00:43:41,260 --> 00:43:44,770 OK, so where's the poles? 909 00:43:44,770 --> 00:43:45,950 Sketch them in the air. 910 00:43:45,950 --> 00:43:48,480 No that didn't work at all. 911 00:43:48,480 --> 00:43:51,020 So to think about where are the poles you 912 00:43:51,020 --> 00:43:52,700 can think of-- the hard part is just 913 00:43:52,700 --> 00:43:55,630 the sinusoidal trigonometric non-linearity, 914 00:43:55,630 --> 00:43:59,060 so what do I do with that? 915 00:43:59,060 --> 00:44:00,980 AUDIENCE: [INAUDIBLE] 916 00:44:00,980 --> 00:44:03,010 DENNIS FREEMAN: Throw it away, absolutely. 917 00:44:03,010 --> 00:44:06,700 So I'll assume that the sine of theta is-- 918 00:44:06,700 --> 00:44:07,390 AUDIENCE: Theta. 919 00:44:07,390 --> 00:44:07,720 DENNIS FREEMAN: Theta. 920 00:44:07,720 --> 00:44:08,757 And the cos of theta is? 921 00:44:08,757 --> 00:44:09,340 AUDIENCE: One. 922 00:44:09,340 --> 00:44:10,330 DENNIS FREEMAN: One. 923 00:44:10,330 --> 00:44:13,240 For small angular deflections, just make a small angle 924 00:44:13,240 --> 00:44:14,500 approximation. 925 00:44:14,500 --> 00:44:17,830 When you do that it comes out surprisingly 926 00:44:17,830 --> 00:44:23,270 to look just like the levitated mass problem. 927 00:44:23,270 --> 00:44:27,980 You get real poles on each side of the j omega axis. 928 00:44:27,980 --> 00:44:32,690 So in fact, you have to do a similar kind of stability thing 929 00:44:32,690 --> 00:44:34,012 to make this work. 930 00:44:34,012 --> 00:44:35,720 And so that's what we'll demonstrate now. 931 00:44:35,720 --> 00:44:40,190 So the idea is going to be to turn on electronics 932 00:44:40,190 --> 00:44:42,710 to stabilize the inverted pendulum so that I don't 933 00:44:42,710 --> 00:44:45,566 need to do it with my hand. 934 00:44:45,566 --> 00:44:48,518 LORENZO: I'm not sure it's going to work, because I [INAUDIBLE] 935 00:44:48,518 --> 00:44:50,000 the cord. 936 00:44:50,000 --> 00:44:54,127 So if [INAUDIBLE] 937 00:44:54,127 --> 00:44:55,460 DENNIS FREEMAN: It always works. 938 00:44:55,460 --> 00:44:56,454 He does it really well. 939 00:44:59,358 --> 00:45:00,326 LORENZO: That works. 940 00:45:10,962 --> 00:45:13,170 DENNIS FREEMAN: So it's a little more stable than me, 941 00:45:13,170 --> 00:45:14,350 but not much. 942 00:45:14,350 --> 00:45:16,470 I mean, I'm pretty good, right? 943 00:45:16,470 --> 00:45:18,330 [LAUGHTER] 944 00:45:18,330 --> 00:45:20,631 DENNIS FREEMAN: I can keep it going too, right? 945 00:45:20,631 --> 00:45:23,670 Well, sort of. 946 00:45:23,670 --> 00:45:25,020 So is it any better than I am? 947 00:45:34,700 --> 00:45:37,210 So can you share that it's better or worse than me. 948 00:45:37,210 --> 00:45:38,460 LORENZO: It's better than you. 949 00:45:38,460 --> 00:45:40,740 DENNIS FREEMAN: It's better than me? 950 00:45:40,740 --> 00:45:41,240 Prove it. 951 00:45:44,216 --> 00:45:46,696 Go ahead. 952 00:45:46,696 --> 00:45:48,894 LORENZO: The way [INAUDIBLE]. 953 00:45:48,894 --> 00:45:50,560 DENNIS FREEMAN: So you ought to do this. 954 00:45:50,560 --> 00:45:51,934 So this guy. 955 00:45:51,934 --> 00:45:53,350 So can somebody think of something 956 00:45:53,350 --> 00:45:57,990 that's different about this from the magnetic levitation 957 00:45:57,990 --> 00:45:58,490 problem? 958 00:46:04,359 --> 00:46:05,734 AUDIENCE: Like right now it looks 959 00:46:05,734 --> 00:46:07,880 like it's going to converge. 960 00:46:07,880 --> 00:46:10,890 DENNIS FREEMAN: But it's doing a really good thing. 961 00:46:10,890 --> 00:46:16,230 It's converging around here. 962 00:46:16,230 --> 00:46:20,790 What would happen if I was doing my thing 963 00:46:20,790 --> 00:46:23,550 and it started to fall this way, so I started to follow it. 964 00:46:27,680 --> 00:46:29,080 That's actually very complicated. 965 00:46:29,080 --> 00:46:32,750 So there's actually another feedback loop inside here 966 00:46:32,750 --> 00:46:36,090 that's trying to make it stay in the center. 967 00:46:36,090 --> 00:46:38,735 So if I tried to pull that one out. 968 00:46:42,900 --> 00:46:43,770 Wap! 969 00:46:43,770 --> 00:46:49,220 OK, so there's many feedback loops going on here. 970 00:46:49,220 --> 00:46:51,960 There's the one that you can see. 971 00:46:51,960 --> 00:46:54,600 But then there's also a slow feedback loop 972 00:46:54,600 --> 00:46:57,080 trying to keep the whole thing oscillating in the center. 973 00:46:57,080 --> 00:46:59,169 Everybody follow that? 974 00:46:59,169 --> 00:47:00,960 So there's the primary feedback loop, which 975 00:47:00,960 --> 00:47:02,282 is very analogous to that guy. 976 00:47:02,282 --> 00:47:04,740 But then there's a slow feedback loop saying wait a minute, 977 00:47:04,740 --> 00:47:07,320 you're spending too much time to the left. 978 00:47:07,320 --> 00:47:09,660 I'm going to move it a little bit in the wrong way 979 00:47:09,660 --> 00:47:12,960 so that you migrate toward the right. 980 00:47:12,960 --> 00:47:15,320 But the result of that is very stable. 981 00:47:15,320 --> 00:47:16,358 How stable? 982 00:47:16,358 --> 00:47:17,358 LORENZO: We'll find out. 983 00:47:23,810 --> 00:47:26,246 DENNIS FREEMAN: I didn't put a battery on the top of mine. 984 00:47:26,246 --> 00:47:28,190 [LAUGHTER] 985 00:47:28,190 --> 00:47:29,540 I could have! 986 00:47:29,540 --> 00:47:31,040 Well, I could've put it there. 987 00:47:33,990 --> 00:47:35,425 So now we put the-- 988 00:47:35,425 --> 00:47:37,405 [LAUGHTER] 989 00:48:13,540 --> 00:48:16,015 AUDIENCE: I think someone should sell this. 990 00:48:16,015 --> 00:48:17,260 DENNIS FREEMAN: A volunteer! 991 00:48:17,260 --> 00:48:20,270 [LAUGHTER] 992 00:48:20,270 --> 00:48:23,640 Everybody in favor? 993 00:48:23,640 --> 00:48:26,011 It's overwhelming. 994 00:48:26,011 --> 00:48:26,510 OK. 995 00:48:26,510 --> 00:48:27,510 So that's all for today. 996 00:48:27,510 --> 00:48:29,420 The idea is that we can use feedback 997 00:48:29,420 --> 00:48:30,670 to do a whole bunch of things. 998 00:48:30,670 --> 00:48:31,878 And maybe even more tomorrow. 999 00:48:31,878 --> 00:48:34,060 Oh well now that's just a joke. 1000 00:48:34,060 --> 00:48:36,430 See you tomorrow at the exam. 1001 00:48:36,430 --> 00:48:37,547 No recitation. 1002 00:48:37,547 --> 00:48:39,380 We'll stick around for two or three minutes. 1003 00:48:39,380 --> 00:48:42,150 You can look at these more if you want to.