1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high-quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation, or view additional materials from 6 00:00:13,470 --> 00:00:19,300 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:19,300 --> 00:00:20,090 ocw.mit.edu. 8 00:00:20,090 --> 00:00:56,110 [MUSIC PLAYING] 9 00:00:56,110 --> 00:00:58,740 PROFESSOR: During the course, we've developed a number of 10 00:00:58,740 --> 00:01:01,220 very powerful and useful tools. 11 00:01:01,220 --> 00:01:04,519 And we've seen how these can be used in designing and 12 00:01:04,519 --> 00:01:05,970 analyzing systems. 13 00:01:05,970 --> 00:01:10,750 For example, for filtering, for modulation, et cetera. 14 00:01:10,750 --> 00:01:14,380 I'd like to conclude this series of lectures with an 15 00:01:14,380 --> 00:01:17,670 introduction to one more important topic. 16 00:01:17,670 --> 00:01:22,320 Namely, the analysis of feedback systems. 17 00:01:22,320 --> 00:01:27,700 And one of the principle reasons that we have left this 18 00:01:27,700 --> 00:01:31,900 discussion to the last part of the course is so that we can 19 00:01:31,900 --> 00:01:36,570 exploit some of the ideas that we've just developed in some 20 00:01:36,570 --> 00:01:38,490 of the previous lectures. 21 00:01:38,490 --> 00:01:42,350 Namely, the tools afforded us by Laplace and z-transforms. 22 00:01:45,600 --> 00:01:52,300 Now, as I had indicated in one of the very first lectures, a 23 00:01:52,300 --> 00:01:56,080 common example of a feedback system is the problem of, 24 00:01:56,080 --> 00:01:59,790 let's say balancing a broom, or in the case of that lecture 25 00:01:59,790 --> 00:02:04,480 balancing my son's horse in the palm of your hand. 26 00:02:04,480 --> 00:02:09,610 And kind of the idea there is that what that relies on, in 27 00:02:09,610 --> 00:02:14,320 order to make that a stable system, is feedback. 28 00:02:14,320 --> 00:02:17,680 In that particular case, visual feedback. 29 00:02:17,680 --> 00:02:21,470 That specific problem, the one of balancing something, let's 30 00:02:21,470 --> 00:02:26,230 say in the palm of your hand, is an example of a problem, 31 00:02:26,230 --> 00:02:29,880 which is commonly referred to as the inverted pendulum. 32 00:02:29,880 --> 00:02:33,250 And it's one that we will actually be analyzing in a 33 00:02:33,250 --> 00:02:35,400 fair amount of detail not in this lecture, 34 00:02:35,400 --> 00:02:36,820 but in the next lecture. 35 00:02:36,820 --> 00:02:39,510 But let me just kind of indicate what some of the 36 00:02:39,510 --> 00:02:41,670 issues are. 37 00:02:41,670 --> 00:02:45,030 Let me describe this in the context not of balancing a 38 00:02:45,030 --> 00:02:48,410 broom on your hand, but let's say that we have a mechanical 39 00:02:48,410 --> 00:02:50,730 system which consists of a cart. 40 00:02:50,730 --> 00:02:55,770 And the cart can move, let's say in one dimension, and it 41 00:02:55,770 --> 00:03:01,460 has mounted on it a bar, a rod with a weight on the top, and 42 00:03:01,460 --> 00:03:02,880 it pivots around the base. 43 00:03:02,880 --> 00:03:06,510 So that essentially represents the inverted pendulum. 44 00:03:06,510 --> 00:03:10,510 So that system can be, more or less, depicted as I've 45 00:03:10,510 --> 00:03:12,210 indicated here. 46 00:03:12,210 --> 00:03:16,650 And this is the cart that can move along the x-axis. 47 00:03:16,650 --> 00:03:20,660 And here we have a pivot point, a rod, a 48 00:03:20,660 --> 00:03:22,000 weight at the top. 49 00:03:22,000 --> 00:03:25,100 And then, of course, there are several forces acting on this. 50 00:03:25,100 --> 00:03:28,740 There is an acceleration that can be applied to the cart, 51 00:03:28,740 --> 00:03:33,290 and that will be thought of as the external input. 52 00:03:33,290 --> 00:03:37,130 And then on the pendulum itself, on the weight, there 53 00:03:37,130 --> 00:03:39,080 is the force of gravity. 54 00:03:39,080 --> 00:03:43,130 And then typically, a set of external disturbances that 55 00:03:43,130 --> 00:03:46,840 might represent, for example, air currents, or wind, or 56 00:03:46,840 --> 00:03:53,630 whatever, that will attempt to destabilize the system. 57 00:03:53,630 --> 00:03:58,440 Specifically, to have the pendulum fall down. 58 00:03:58,440 --> 00:04:02,470 Now, if we look at this system in, more or less, a 59 00:04:02,470 --> 00:04:07,150 straightforward way, what we have then are the system 60 00:04:07,150 --> 00:04:14,200 dynamics and several inputs, one of which is the external 61 00:04:14,200 --> 00:04:19,570 disturbances and a second is the acceleration, which is the 62 00:04:19,570 --> 00:04:21,990 external acceleration that's applied. 63 00:04:21,990 --> 00:04:25,510 And the output of the system can be thought of as the 64 00:04:25,510 --> 00:04:27,590 angular displacement of the pendulum. 65 00:04:27,590 --> 00:04:31,330 Which if we want it balanced, we would like that angular 66 00:04:31,330 --> 00:04:34,430 displacement to be equal to 0. 67 00:04:34,430 --> 00:04:39,330 Now, if we know exactly what the system dynamics are and if 68 00:04:39,330 --> 00:04:43,280 we knew exactly what the external disturbances are, 69 00:04:43,280 --> 00:04:46,500 then in principle, we could design an acceleration. 70 00:04:46,500 --> 00:04:47,850 Namely, an input. 71 00:04:47,850 --> 00:04:51,710 That would exactly generate 0 output. 72 00:04:51,710 --> 00:04:54,440 In other words, the angle would be equal to 0. 73 00:04:54,440 --> 00:04:58,750 But as you can imagine, just as it's basically impossible 74 00:04:58,750 --> 00:05:01,830 to balance a broom in the palm of your hand with you eyes 75 00:05:01,830 --> 00:05:08,030 closed, what is very hard to ascertain in advance are what 76 00:05:08,030 --> 00:05:11,100 the various dynamics and disturbances are. 77 00:05:11,100 --> 00:05:15,040 And so more typically what you would think of doing is 78 00:05:15,040 --> 00:05:20,130 measuring the output angle, and then using that 79 00:05:20,130 --> 00:05:24,220 measurement to somehow influence the applied 80 00:05:24,220 --> 00:05:25,920 acceleration or force. 81 00:05:25,920 --> 00:05:29,670 And that, then, is an example of a feedback system. 82 00:05:29,670 --> 00:05:37,160 So we would measure the output angle and generate an input 83 00:05:37,160 --> 00:05:42,960 acceleration, which is some function of what that 84 00:05:42,960 --> 00:05:44,300 output angle is. 85 00:05:44,300 --> 00:05:48,310 And if we choose the feedback dynamics correctly, then in 86 00:05:48,310 --> 00:05:52,430 fact, we can drive this output to 0. 87 00:05:52,430 --> 00:05:57,640 This is one example of a system which is inherently 88 00:05:57,640 --> 00:06:01,680 unstable because if we left it to its own devices, the 89 00:06:01,680 --> 00:06:03,760 pendulum would simply fall down. 90 00:06:03,760 --> 00:06:07,140 And essentially, by applying feedback, what we're trying to 91 00:06:07,140 --> 00:06:10,390 do is stabilize this inherently unstable system. 92 00:06:10,390 --> 00:06:13,560 And we'll talk a little bit more about that specific 93 00:06:13,560 --> 00:06:17,030 application of feedback shortly. 94 00:06:17,030 --> 00:06:21,400 Another common example of feedback is in positioning or 95 00:06:21,400 --> 00:06:26,460 tracking systems, and I indicate one here which 96 00:06:26,460 --> 00:06:31,790 corresponds to the problem of positioning a telescope, which 97 00:06:31,790 --> 00:06:34,000 is mounted on a rotating platform. 98 00:06:34,000 --> 00:06:39,110 So in a system of that type, for example, I indicate here 99 00:06:39,110 --> 00:06:41,950 the rotating platform and the telescope. 100 00:06:41,950 --> 00:06:44,310 It's driven by a motor. 101 00:06:44,310 --> 00:06:48,070 And again, we could imagine, in principle, the possibility 102 00:06:48,070 --> 00:06:53,320 of driving this to the desired angle by choosing an 103 00:06:53,320 --> 00:06:56,370 appropriate applied input voltage. 104 00:06:56,370 --> 00:06:59,230 And as long as we know such things as what the 105 00:06:59,230 --> 00:07:02,540 disturbances are that influence the telescope mount 106 00:07:02,540 --> 00:07:06,460 and what the characteristics of the motor are, in principle 107 00:07:06,460 --> 00:07:10,250 we could in fact carry this out in a form which is 108 00:07:10,250 --> 00:07:11,920 referred to as open loop. 109 00:07:11,920 --> 00:07:16,280 Namely, we can choose an appropriate input voltage to 110 00:07:16,280 --> 00:07:22,020 drive the motor to set the platform angle at the desired 111 00:07:22,020 --> 00:07:24,450 angular position. 112 00:07:24,450 --> 00:07:28,350 However, again, there are enough unknowns in a problem 113 00:07:28,350 --> 00:07:33,790 like that, so that one is motivated to employ feedback. 114 00:07:33,790 --> 00:07:39,160 Namely, to make a measurement of the output angle and use 115 00:07:39,160 --> 00:07:44,430 that in a feedback loop to influence the drive for the 116 00:07:44,430 --> 00:07:49,410 motor, so that the telescope platform is positioned 117 00:07:49,410 --> 00:07:50,610 appropriately. 118 00:07:50,610 --> 00:07:56,950 So if we look at this in a feedback context, we would 119 00:07:56,950 --> 00:08:05,020 then take the measured output angle and the measured output 120 00:08:05,020 --> 00:08:09,930 angle would be fed back and compared 121 00:08:09,930 --> 00:08:12,810 with the desired angle. 122 00:08:12,810 --> 00:08:16,340 And the difference between those, which essentially is 123 00:08:16,340 --> 00:08:19,240 the error between the platform positioning and the desired 124 00:08:19,240 --> 00:08:23,880 position would be put perhaps through an appropriate gain or 125 00:08:23,880 --> 00:08:28,870 attenuation and used as the excitation to the motor. 126 00:08:28,870 --> 00:08:32,659 So in the mechanical or physical system, that would 127 00:08:32,659 --> 00:08:37,900 correspond to measuring the angle, let's say with a 128 00:08:37,900 --> 00:08:39,169 potentiometer. 129 00:08:39,169 --> 00:08:42,770 So here we're measuring the angle and we have an output, 130 00:08:42,770 --> 00:08:45,700 which is proportional to that measured angle. 131 00:08:45,700 --> 00:08:52,290 And then we would use feedback, comparing the 132 00:08:52,290 --> 00:08:57,670 measured angle to some proportionality factor 133 00:08:57,670 --> 00:09:00,550 multiplying the desired angle. 134 00:09:00,550 --> 00:09:05,190 So here we have the desired angle, again through some type 135 00:09:05,190 --> 00:09:06,660 of potentiometer. 136 00:09:06,660 --> 00:09:08,710 The two are compared. 137 00:09:08,710 --> 00:09:12,160 Out of the comparator, we basically have an indication 138 00:09:12,160 --> 00:09:14,890 of what the difference is, and that represents an error 139 00:09:14,890 --> 00:09:17,230 between the desired and the true angle. 140 00:09:17,230 --> 00:09:22,160 And then that is used through perhaps an amplifier to 141 00:09:22,160 --> 00:09:23,630 control the motor. 142 00:09:23,630 --> 00:09:28,240 And in that case, of course, when the error goes to 0, that 143 00:09:28,240 --> 00:09:29,960 means that the actual angle and the 144 00:09:29,960 --> 00:09:32,010 desired angle are equal. 145 00:09:32,010 --> 00:09:35,930 And in fact, in that case also with this system, the input to 146 00:09:35,930 --> 00:09:40,210 the motor is, likewise, equal to 0. 147 00:09:40,210 --> 00:09:44,520 Now, as I've illustrated it here, it tends to be in the 148 00:09:44,520 --> 00:09:48,840 context of a continuous time or analog system. 149 00:09:48,840 --> 00:09:51,770 And in fact, another very common way of doing 150 00:09:51,770 --> 00:09:58,890 positioning or tracking is to instead implement the feedback 151 00:09:58,890 --> 00:10:02,300 using a discrete-time or digital system. 152 00:10:02,300 --> 00:10:07,760 And so in that case, we would basically take the position 153 00:10:07,760 --> 00:10:13,310 output as it's measured, sample it, essentially convert 154 00:10:13,310 --> 00:10:18,460 that to a digital discrete-time signal. 155 00:10:18,460 --> 00:10:23,330 And then that is used in conjunction with the desired 156 00:10:23,330 --> 00:10:30,060 angle, which both form inputs to this processor. 157 00:10:30,060 --> 00:10:34,770 And the output of that is converted, let's say, back to 158 00:10:34,770 --> 00:10:38,120 an analog or continuous-time voltage and used 159 00:10:38,120 --> 00:10:40,100 to drive the motor. 160 00:10:40,100 --> 00:10:45,040 Now, you could ask, why would you go to a digital or 161 00:10:45,040 --> 00:10:49,150 discrete-time measurement rather than doing it the way I 162 00:10:49,150 --> 00:10:51,340 showed on the previous overlay which seemed relatively 163 00:10:51,340 --> 00:10:52,530 straightforward? 164 00:10:52,530 --> 00:10:55,710 And the reason, principally, is that in the context of a 165 00:10:55,710 --> 00:11:00,360 digital implementation of the feedback process, often you 166 00:11:00,360 --> 00:11:05,220 can implement a better controlled and often also, 167 00:11:05,220 --> 00:11:10,140 more sophisticated algorithm for the feedback dynamics. 168 00:11:10,140 --> 00:11:13,310 So that you can take a count, perhaps not only of the angle 169 00:11:13,310 --> 00:11:16,970 itself, but also of the rate of change of angle. 170 00:11:16,970 --> 00:11:18,720 And in fact, the rate of change of the rate 171 00:11:18,720 --> 00:11:21,270 of change of angle. 172 00:11:21,270 --> 00:11:27,230 So the system, as it's shown there then, basically has a 173 00:11:27,230 --> 00:11:30,350 discrete-time or digital feedback loop around a 174 00:11:30,350 --> 00:11:33,070 continuous time system. 175 00:11:33,070 --> 00:11:38,950 Now, this is an example of, in fact, a more general way in 176 00:11:38,950 --> 00:11:42,590 which discrete-time feedback is used 177 00:11:42,590 --> 00:11:44,950 with continuous systems. 178 00:11:44,950 --> 00:11:50,950 And let me indicate, in general, what the character or 179 00:11:50,950 --> 00:11:53,510 block diagram of such a system might be. 180 00:11:53,510 --> 00:11:57,860 Typically, if we abstract away from the telescope positioning 181 00:11:57,860 --> 00:12:01,420 system, we might have a more general 182 00:12:01,420 --> 00:12:03,730 continuous-time system. 183 00:12:03,730 --> 00:12:07,980 And around which we want to apply some feedback, which we 184 00:12:07,980 --> 00:12:10,920 could do with a continuous-time system or with 185 00:12:10,920 --> 00:12:16,150 a discrete-time system by first converting these signals 186 00:12:16,150 --> 00:12:20,310 to discrete-time signals. 187 00:12:20,310 --> 00:12:24,620 Then, processing that with a discrete-time system. 188 00:12:24,620 --> 00:12:27,440 And then, through an appropriate interpolation 189 00:12:27,440 --> 00:12:32,670 algorithm, we would then convert that back to a 190 00:12:32,670 --> 00:12:34,580 continuous-time signal. 191 00:12:34,580 --> 00:12:39,030 And the difference between the input signal and this 192 00:12:39,030 --> 00:12:42,270 continuous-time signal which is fed back, then forms the 193 00:12:42,270 --> 00:12:45,220 excitation to the system that essentially 194 00:12:45,220 --> 00:12:46,870 we're trying to control. 195 00:12:46,870 --> 00:12:52,280 And in many systems of this type, the advantage is that 196 00:12:52,280 --> 00:12:56,710 this system can be implemented in a very reproducible way, 197 00:12:56,710 --> 00:13:01,240 either with a digital computer or with a microprocessor. 198 00:13:01,240 --> 00:13:06,110 And although we're not going to go into this in any detail 199 00:13:06,110 --> 00:13:08,850 in this lecture, there is some discussion 200 00:13:08,850 --> 00:13:10,580 of this in the text. 201 00:13:10,580 --> 00:13:14,370 Essentially, if we make certain assumptions about this 202 00:13:14,370 --> 00:13:19,980 particular feedback system, we can move the continuous to 203 00:13:19,980 --> 00:13:25,340 discrete-time converter up to this point and to this point, 204 00:13:25,340 --> 00:13:30,060 and we can move the interpolating system outside 205 00:13:30,060 --> 00:13:31,500 the summer. 206 00:13:31,500 --> 00:13:37,220 And what happens in that case is that we end up with what 207 00:13:37,220 --> 00:13:42,730 looks like an inherently discrete-time feedback system. 208 00:13:42,730 --> 00:13:46,280 So, in fact, if we take those steps, then what we'll end up 209 00:13:46,280 --> 00:13:52,260 with for a feedback system is a system that essentially can 210 00:13:52,260 --> 00:13:55,890 be analyzed as a discrete-time system. 211 00:13:55,890 --> 00:14:00,820 Here we have what is, in the forward path, is basically the 212 00:14:00,820 --> 00:14:04,550 continuous-time system with the interpolator at one end 213 00:14:04,550 --> 00:14:06,450 and the continuous to discrete-time converter 214 00:14:06,450 --> 00:14:07,700 at the other end. 215 00:14:07,700 --> 00:14:11,640 And then we have whatever system it was in the feedback 216 00:14:11,640 --> 00:14:13,090 loop-- discrete-time-- 217 00:14:13,090 --> 00:14:16,900 that shows up in this feedback loop. 218 00:14:16,900 --> 00:14:21,410 Well, I show this mainly to emphasize the fact, although 219 00:14:21,410 --> 00:14:24,500 there are some steps there that we obviously left out. 220 00:14:24,500 --> 00:14:27,080 I show that mainly to emphasize the fact that 221 00:14:27,080 --> 00:14:30,890 feedback arises not just in the context of continuous-time 222 00:14:30,890 --> 00:14:36,140 systems, but also the analysis of discrete-time feedback 223 00:14:36,140 --> 00:14:38,240 systems becomes important. 224 00:14:38,240 --> 00:14:42,600 Perhaps because we have used discrete-time feedback around 225 00:14:42,600 --> 00:14:44,540 a continuous-time system. 226 00:14:44,540 --> 00:14:49,850 But also perhaps because the feedback system is inherently 227 00:14:49,850 --> 00:14:50,660 discrete-time. 228 00:14:50,660 --> 00:14:54,980 And let me just illustrate one, or indicate one example 229 00:14:54,980 --> 00:14:57,850 in which that might arise. 230 00:14:57,850 --> 00:15:03,290 This is an example which is also discussed in somewhat 231 00:15:03,290 --> 00:15:05,270 more detail in the text. 232 00:15:05,270 --> 00:15:09,990 But basically, population studies, for example, 233 00:15:09,990 --> 00:15:12,480 represent examples of 234 00:15:12,480 --> 00:15:14,840 discrete-time feedback systems. 235 00:15:14,840 --> 00:15:19,590 Where let's say that we have some type of model for 236 00:15:19,590 --> 00:15:21,200 population growth. 237 00:15:21,200 --> 00:15:25,180 And since people come in integer amounts that 238 00:15:25,180 --> 00:15:30,950 represents essentially the output of any population 239 00:15:30,950 --> 00:15:33,080 model, essentially or inherently 240 00:15:33,080 --> 00:15:34,330 represents a sequence. 241 00:15:34,330 --> 00:15:38,630 Namely, it's indexed on an integer variable. 242 00:15:38,630 --> 00:15:42,810 And typically, models for population growth 243 00:15:42,810 --> 00:15:44,720 are unstable systems. 244 00:15:44,720 --> 00:15:48,240 You can kind of imagine that because if you take these 245 00:15:48,240 --> 00:15:52,950 simple models of population, what happens is that in any 246 00:15:52,950 --> 00:15:59,050 generation, the number of people, or animals, or 247 00:15:59,050 --> 00:16:02,590 whatever it is that this is modeling, grows essentially 248 00:16:02,590 --> 00:16:07,750 exponentially with the size of the previous generation. 249 00:16:07,750 --> 00:16:09,360 Now, where does the feedback come in? 250 00:16:09,360 --> 00:16:14,370 Well, the feedback typically comes in, in incorporating in 251 00:16:14,370 --> 00:16:17,090 the overall model various retarding factors. 252 00:16:17,090 --> 00:16:20,920 For example, as the population increases, the food supply 253 00:16:20,920 --> 00:16:22,360 becomes more limited. 254 00:16:22,360 --> 00:16:30,030 And that essentially is a feedback process that acts to 255 00:16:30,030 --> 00:16:32,510 retard the population growth. 256 00:16:32,510 --> 00:16:35,360 And so an overall model-- 257 00:16:35,360 --> 00:16:36,610 somewhat simplified-- 258 00:16:38,940 --> 00:16:44,170 for a population system is the open loop model in the absence 259 00:16:44,170 --> 00:16:46,270 of retarding factors. 260 00:16:46,270 --> 00:16:50,200 And then, very often the retarding factors can be 261 00:16:50,200 --> 00:16:55,490 described as being related to the size of the population. 262 00:16:55,490 --> 00:17:02,210 And those essentially act to reduce the overall input to 263 00:17:02,210 --> 00:17:04,060 the population model. 264 00:17:04,060 --> 00:17:09,560 And so population studies are one very common example of 265 00:17:09,560 --> 00:17:11,205 discrete-time feedback systems. 266 00:17:14,839 --> 00:17:19,720 Well, what we want to look at and understand are the basic 267 00:17:19,720 --> 00:17:23,030 properties of feedback systems. 268 00:17:23,030 --> 00:17:27,910 And to do that, let's look at the basic block diagram and 269 00:17:27,910 --> 00:17:30,530 equations for feedback systems, either 270 00:17:30,530 --> 00:17:33,820 continuous-time or discrete-time. 271 00:17:33,820 --> 00:17:38,540 Let's begin with the continuous-time case. 272 00:17:38,540 --> 00:17:44,670 And now what we've done is simply abstract out any of the 273 00:17:44,670 --> 00:17:49,150 applications to a fairly general system, in which we 274 00:17:49,150 --> 00:17:54,980 have a system H of s in what's referred to as the forward 275 00:17:54,980 --> 00:17:59,590 path, and a system G of s in the feedback path. 276 00:18:02,440 --> 00:18:07,600 The input to the system H of s is the difference between the 277 00:18:07,600 --> 00:18:11,320 input to the overall system and the output of 278 00:18:11,320 --> 00:18:13,020 the feedback loop. 279 00:18:13,020 --> 00:18:17,020 And I draw your attention to the fact that what we 280 00:18:17,020 --> 00:18:18,440 illustrate here and what we're 281 00:18:18,440 --> 00:18:20,710 analyzing is negative feedback. 282 00:18:20,710 --> 00:18:25,010 Namely, this output is subtracted from the input. 283 00:18:25,010 --> 00:18:29,170 And that's done more for reasons of convention then for 284 00:18:29,170 --> 00:18:30,370 any other reasons. 285 00:18:30,370 --> 00:18:34,770 It's typical to do that and appropriate certainly in some 286 00:18:34,770 --> 00:18:37,040 feedback systems, but not all. 287 00:18:37,040 --> 00:18:41,350 And the output of the adder is commonly referred to as the 288 00:18:41,350 --> 00:18:44,400 error signal, indicating that it's the difference between 289 00:18:44,400 --> 00:18:49,280 the signal fed back and the input to the overall system. 290 00:18:49,280 --> 00:18:53,310 Now, if we want to analyze the feedback system, we would do 291 00:18:53,310 --> 00:18:55,540 that essentially by writing the appropriate equations. 292 00:18:58,460 --> 00:19:01,220 In generating the equivalent system function for the 293 00:19:01,220 --> 00:19:06,780 overall system, it's best done in the frequency or Laplace 294 00:19:06,780 --> 00:19:10,290 transform domain rather than in the time domain. 295 00:19:10,290 --> 00:19:12,330 And let me just indicate what the steps 296 00:19:12,330 --> 00:19:13,420 are that are involved. 297 00:19:13,420 --> 00:19:17,260 And there are a few steps of algebra that I'll leave in 298 00:19:17,260 --> 00:19:18,480 your hands. 299 00:19:18,480 --> 00:19:23,440 But basically, if we look at this feedback system, we can 300 00:19:23,440 --> 00:19:27,610 label, of course-- since the output is y of t, we can label 301 00:19:27,610 --> 00:19:31,080 the Laplace transform of the output as Y of s. 302 00:19:31,080 --> 00:19:35,330 And we also have Y of s as the input here. 303 00:19:35,330 --> 00:19:38,520 Because this is the system function, the Laplace 304 00:19:38,520 --> 00:19:42,620 transform of r of t is simply the Laplace transform of this 305 00:19:42,620 --> 00:19:45,330 input, which is Y of s times G of s. 306 00:19:45,330 --> 00:19:50,410 So here we have Y of s times G of s. 307 00:19:50,410 --> 00:19:55,940 At the adder, the input here is x of s. 308 00:19:55,940 --> 00:20:01,110 And so the Laplace transform of the error signal is simply 309 00:20:01,110 --> 00:20:08,070 x of s minus r of s, which is Y of s G of s. 310 00:20:08,070 --> 00:20:12,440 So this is minus Y of s G of s. 311 00:20:12,440 --> 00:20:15,580 That's the Laplace transform of the error signal. 312 00:20:15,580 --> 00:20:20,400 The Laplace transform of the output of this system is 313 00:20:20,400 --> 00:20:25,970 simply this expression times H of s. 314 00:20:25,970 --> 00:20:27,690 So that's what we have here. 315 00:20:27,690 --> 00:20:30,820 But what we have here we already called Y of s. 316 00:20:30,820 --> 00:20:35,600 So in fact, we can simply say that these two expressions 317 00:20:35,600 --> 00:20:37,180 have to be equal. 318 00:20:37,180 --> 00:20:41,020 And so we've essentially done the analysis, saying that 319 00:20:41,020 --> 00:20:43,150 those two expressions are equal. 320 00:20:43,150 --> 00:20:47,440 Let's solve for Y of s over x of s, which is the overall 321 00:20:47,440 --> 00:20:48,940 system function. 322 00:20:48,940 --> 00:20:52,970 And if we do that, what we end up with for the overall system 323 00:20:52,970 --> 00:20:55,820 function is the algebraic expression 324 00:20:55,820 --> 00:20:56,910 that I indicate here. 325 00:20:56,910 --> 00:21:02,840 It's H of s divided by 1 plus G of s H of s. 326 00:21:02,840 --> 00:21:07,470 Said another way, it's the system function in the open 327 00:21:07,470 --> 00:21:14,070 loop forward path divided by 1 plus, what's referred to as 328 00:21:14,070 --> 00:21:17,150 the loop gain, G of s times H of s. 329 00:21:17,150 --> 00:21:19,990 Let's just look back up at the block diagram. 330 00:21:19,990 --> 00:21:26,660 G of s times H of s is simply the gain around the entire 331 00:21:26,660 --> 00:21:30,760 loop from this point around to this point. 332 00:21:30,760 --> 00:21:34,820 So the overall system function is the gain in the forward 333 00:21:34,820 --> 00:21:39,770 path divided by 1 plus the loop gain, which is H of s 334 00:21:39,770 --> 00:21:42,430 times G of s. 335 00:21:42,430 --> 00:21:46,540 Now, none of the equations that we wrote had relied 336 00:21:46,540 --> 00:21:48,490 specifically on this being continuous-time. 337 00:21:48,490 --> 00:21:52,120 We just did some algebra and we used the system function 338 00:21:52,120 --> 00:21:53,770 property of the systems. 339 00:21:53,770 --> 00:21:58,620 And so, pretty obviously, the same kind of algebraic 340 00:21:58,620 --> 00:22:01,990 procedure would work in discrete-time. 341 00:22:01,990 --> 00:22:07,250 And so, in fact, if we carried out a discrete-time analysis 342 00:22:07,250 --> 00:22:11,290 rather than a continuous-time analysis, we would simply end 343 00:22:11,290 --> 00:22:18,600 up with exactly the same system and exactly the same 344 00:22:18,600 --> 00:22:24,600 equation for the overall system function. 345 00:22:24,600 --> 00:22:28,130 The only difference being that here things are a function of 346 00:22:28,130 --> 00:22:31,840 z, whereas if I just flip back the other 347 00:22:31,840 --> 00:22:34,880 overlay, we simply have-- 348 00:22:34,880 --> 00:22:39,070 previously everything is function of t and in the 349 00:22:39,070 --> 00:22:41,050 frequency domain s. 350 00:22:41,050 --> 00:22:45,980 In the discrete-time case, we've simply replaced in the 351 00:22:45,980 --> 00:22:48,920 time domain, the independent variable by n. 352 00:22:48,920 --> 00:22:54,830 And in the frequency domain, the independent variable by z. 353 00:22:54,830 --> 00:23:02,440 So what we see is that we have a basic feedback equation, and 354 00:23:02,440 --> 00:23:04,990 that feedback equation is exactly the same for 355 00:23:04,990 --> 00:23:07,590 continuous-time and discrete-time. 356 00:23:07,590 --> 00:23:10,880 Although we have to be careful about what implications we 357 00:23:10,880 --> 00:23:14,150 draw, depending on whether we're talking about 358 00:23:14,150 --> 00:23:15,950 continuous-time or discrete-time. 359 00:23:18,870 --> 00:23:24,660 Now, to illustrate the importance of feedback, let's 360 00:23:24,660 --> 00:23:28,750 look at a number of common applications. 361 00:23:28,750 --> 00:23:31,820 And also, as we talk about these applications, what will 362 00:23:31,820 --> 00:23:36,580 see is that while these applications and context in 363 00:23:36,580 --> 00:23:41,800 which feedback is used are extremely useful and powerful, 364 00:23:41,800 --> 00:23:47,090 they fall out in an almost straightforward way from this 365 00:23:47,090 --> 00:23:52,090 very simple feedback equation that we've just derrived. 366 00:23:52,090 --> 00:23:57,590 Well, the examples that I want to just talk about are, first 367 00:23:57,590 --> 00:24:01,930 of all, the use of feedback in amplifier design. 368 00:24:01,930 --> 00:24:05,980 And we're not going to design amplifiers in detail, but what 369 00:24:05,980 --> 00:24:11,050 I'd like to illustrate is the basic principle behind why 370 00:24:11,050 --> 00:24:14,740 feedback is useful in designing amplifiers. 371 00:24:14,740 --> 00:24:19,840 In particular, how it plays a role in compensating for a 372 00:24:19,840 --> 00:24:22,640 non-constant frequency response. 373 00:24:22,640 --> 00:24:25,880 So that's one context that we'll talk about. 374 00:24:25,880 --> 00:24:30,390 A second that I'll indicate is the use of feedback for 375 00:24:30,390 --> 00:24:33,890 implementing inverse systems. 376 00:24:33,890 --> 00:24:38,340 And the third, which we indicated in the case of the 377 00:24:38,340 --> 00:24:42,150 inverted pendulum, is an important context in which 378 00:24:42,150 --> 00:24:45,860 feedback is used is in stabilizing unstable systems. 379 00:24:45,860 --> 00:24:52,960 And what we want to see is why or how a feedback system or 380 00:24:52,960 --> 00:24:56,720 the basic feedback equation, in fact, let's us do each of 381 00:24:56,720 --> 00:24:59,090 these various things. 382 00:24:59,090 --> 00:25:03,140 Well, let's begin with amplifier design. 383 00:25:03,140 --> 00:25:07,960 And let's suppose that we've built somehow without 384 00:25:07,960 --> 00:25:13,380 feedback, an amplifier that is terrific in terms of its gain, 385 00:25:13,380 --> 00:25:17,690 but has the problem that whereas we might like the 386 00:25:17,690 --> 00:25:21,760 amplifier to have a very flat frequency response, in fact 387 00:25:21,760 --> 00:25:26,200 the frequency response of this amplifier is not constant. 388 00:25:26,200 --> 00:25:29,290 And what we'd like to do is compensate for that. 389 00:25:29,290 --> 00:25:32,430 Well, it turns out, interestingly, that if we 390 00:25:32,430 --> 00:25:38,250 embed the amplifier in a feedback loop where in the 391 00:25:38,250 --> 00:25:42,640 feedback path we incorporate an attenuator, then in fact, 392 00:25:42,640 --> 00:25:44,250 we can compensate for that 393 00:25:44,250 --> 00:25:46,500 non-constant frequency response. 394 00:25:46,500 --> 00:25:47,910 Well, let's see how that works out from 395 00:25:47,910 --> 00:25:49,990 the feedback equation. 396 00:25:49,990 --> 00:25:53,060 We have the basic feedback equation that we derived. 397 00:25:53,060 --> 00:25:55,900 And we want to look at frequency response, so we'll 398 00:25:55,900 --> 00:25:58,620 look specifically at the Fourier transform. 399 00:25:58,620 --> 00:26:02,450 And of course, the frequency response of the overall system 400 00:26:02,450 --> 00:26:05,920 is the frequency response of the Fourier transform of the 401 00:26:05,920 --> 00:26:08,260 output divided by the input. 402 00:26:08,260 --> 00:26:11,960 Using the feedback equation that we had just arrived, that 403 00:26:11,960 --> 00:26:15,430 has, in the numerator, the frequency response in the 404 00:26:15,430 --> 00:26:21,470 forward path divided by 1 plus the loop gain, which is H of j 405 00:26:21,470 --> 00:26:23,430 omega times k. 406 00:26:23,430 --> 00:26:24,920 And this is the key. 407 00:26:24,920 --> 00:26:32,830 Because here, if we choose k times H of j omega to be very 408 00:26:32,830 --> 00:26:38,170 large, much larger than 1, then what happens is that 409 00:26:38,170 --> 00:26:41,720 these two cancel out. 410 00:26:41,720 --> 00:26:46,380 H of j omega here and in the denominator will cancel out as 411 00:26:46,380 --> 00:26:48,590 long as this term dominates. 412 00:26:48,590 --> 00:26:52,240 And in that case, under that assumption, the overall system 413 00:26:52,240 --> 00:26:54,650 function is approximately 1/k. 414 00:26:57,260 --> 00:27:04,360 Well, if k is constant as a function of frequency, then we 415 00:27:04,360 --> 00:27:08,320 somehow magically have ended up with an amplifier that has 416 00:27:08,320 --> 00:27:11,210 a flat frequency response. 417 00:27:11,210 --> 00:27:14,190 Well, it seems like we're getting something for nothing. 418 00:27:14,190 --> 00:27:15,840 And actually, we're not. 419 00:27:15,840 --> 00:27:17,850 There's a price that we pay for that. 420 00:27:17,850 --> 00:27:24,110 Because notice the fact that in order to get gain out of 421 00:27:24,110 --> 00:27:29,390 the overall system, k must be less than 1. 422 00:27:29,390 --> 00:27:32,660 So this has to correspond to attenuator. 423 00:27:32,660 --> 00:27:37,680 And we also require that k, which is less than 1, times 424 00:27:37,680 --> 00:27:40,940 the gain of the original amplifier, that that product 425 00:27:40,940 --> 00:27:42,340 be greater than 1. 426 00:27:42,340 --> 00:27:45,370 And the implication of this, without tracking it in detail 427 00:27:45,370 --> 00:27:49,930 right now, the implication in this is that whereas we 428 00:27:49,930 --> 00:27:53,990 flatten the frequency response, we have in fact paid 429 00:27:53,990 --> 00:27:54,680 a price for that. 430 00:27:54,680 --> 00:27:58,030 The price that we've paid is that the gain is somewhat 431 00:27:58,030 --> 00:28:01,880 reduced from the gain that we had before the feedback. 432 00:28:01,880 --> 00:28:07,710 Because k times h must be much larger than 1, but the gain is 433 00:28:07,710 --> 00:28:09,250 proportional to 1/k. 434 00:28:09,250 --> 00:28:12,640 Now, one last point to make related to that. 435 00:28:12,640 --> 00:28:17,080 One could ask, well, why is it any easier to make k flat with 436 00:28:17,080 --> 00:28:21,920 frequency than to build an amplifier with a flat 437 00:28:21,920 --> 00:28:23,350 frequency response? 438 00:28:23,350 --> 00:28:28,040 The reason is that the gain in the feedback path is an 439 00:28:28,040 --> 00:28:30,280 attenuator, not an amplifier. 440 00:28:30,280 --> 00:28:34,930 And generally, attenuation with a flat frequency response 441 00:28:34,930 --> 00:28:37,460 is much easier to get than gain is. 442 00:28:37,460 --> 00:28:43,820 For example, a resistor, which attenuates, would generally 443 00:28:43,820 --> 00:28:45,510 have a flatter frequency response than a 444 00:28:45,510 --> 00:28:48,650 very high-gain amplifier. 445 00:28:48,650 --> 00:28:52,390 So that's one common example of feedback. 446 00:28:52,390 --> 00:28:57,310 And feedback, in fact, is very often used in high-quality 447 00:28:57,310 --> 00:28:59,210 amplifier systems. 448 00:28:59,210 --> 00:29:04,660 Another very common example in which feedback is used is in 449 00:29:04,660 --> 00:29:08,230 implementing inverse systems. 450 00:29:08,230 --> 00:29:13,220 Now, what I mean by that is, suppose that we have a system, 451 00:29:13,220 --> 00:29:15,400 which I indicate here, P of s-- 452 00:29:15,400 --> 00:29:17,250 input and output. 453 00:29:17,250 --> 00:29:21,710 And what we would like to do is implement a system which is 454 00:29:21,710 --> 00:29:23,610 the inverse of this system. 455 00:29:23,610 --> 00:29:28,630 Namely, has a Laplace transform or system function 456 00:29:28,630 --> 00:29:31,980 which is 1 over P of s. 457 00:29:31,980 --> 00:29:36,640 For example, we may have measured a particular system 458 00:29:36,640 --> 00:29:38,590 and what we would like to design is a 459 00:29:38,590 --> 00:29:40,140 compensator for it. 460 00:29:40,140 --> 00:29:42,930 And the question is, by putting this in a feedback 461 00:29:42,930 --> 00:29:46,660 loop, can we, in fact, implement the 462 00:29:46,660 --> 00:29:48,360 inverse of this system? 463 00:29:48,360 --> 00:29:50,580 The answer to that is yes. 464 00:29:50,580 --> 00:29:53,780 And the feedback system, in that case, is 465 00:29:53,780 --> 00:29:55,900 as I indicate here. 466 00:29:55,900 --> 00:30:01,570 So here what we choose to do is to put the system whose 467 00:30:01,570 --> 00:30:06,050 inverse we're trying to generate in the feedback loop. 468 00:30:06,050 --> 00:30:10,960 And in this case, a high-gain in the forward path. 469 00:30:10,960 --> 00:30:14,430 Now for this situation, k is, again, a constant. 470 00:30:14,430 --> 00:30:16,970 But in fact, it's a high-gain constant. 471 00:30:16,970 --> 00:30:20,460 And now if we look at the feedback equation, then what 472 00:30:20,460 --> 00:30:24,210 we see is an equation of this form. 473 00:30:24,210 --> 00:30:30,950 And notice that if k times P of s is large compared with 474 00:30:30,950 --> 00:30:35,340 one, then this term dominates. 475 00:30:35,340 --> 00:30:38,700 The gain in the forward path cancels out. 476 00:30:38,700 --> 00:30:44,360 And what we're left with is a system function, which is just 477 00:30:44,360 --> 00:30:46,960 1 over P of s. 478 00:30:46,960 --> 00:30:51,552 And a system of this type is used in a 479 00:30:51,552 --> 00:30:52,970 whole variety of contexts. 480 00:30:52,970 --> 00:30:56,910 One very common one is in building what are called 481 00:30:56,910 --> 00:31:00,460 logarithmic devices or logarithmic amplifiers. 482 00:31:00,460 --> 00:31:04,730 Ones in which the input-output characteristic is logarithmic. 483 00:31:04,730 --> 00:31:09,260 It's common to do that with a diode that has an exponential 484 00:31:09,260 --> 00:31:10,630 characteristic. 485 00:31:10,630 --> 00:31:14,580 And using that with feedback-- 486 00:31:14,580 --> 00:31:17,800 as feedback around a high-gain operational amplifier. 487 00:31:17,800 --> 00:31:20,570 And by the way, the logarithmic 488 00:31:20,570 --> 00:31:22,170 amplifier is nonlinear. 489 00:31:22,170 --> 00:31:24,440 What I've said here is linear, or the 490 00:31:24,440 --> 00:31:26,170 analysis here was linear. 491 00:31:26,170 --> 00:31:29,790 But that example, in fact, suggests something which is 492 00:31:29,790 --> 00:31:35,750 true, which is that same basic idea, in fact, can be used 493 00:31:35,750 --> 00:31:38,520 often in the context of nonlinear feedback and 494 00:31:38,520 --> 00:31:39,845 nonlinear feedback systems. 495 00:31:42,580 --> 00:31:48,760 Well, as a final example, what I'd like to analyze is the 496 00:31:48,760 --> 00:31:52,520 context in which we would consider 497 00:31:52,520 --> 00:31:56,230 stabilizing unstable systems. 498 00:31:56,230 --> 00:32:00,360 And I had indicated that one context in which that arises 499 00:32:00,360 --> 00:32:06,250 and which we will be analyzing in the next lecture is the 500 00:32:06,250 --> 00:32:07,930 inverted pendulum. 501 00:32:07,930 --> 00:32:12,230 And in that situation, or in a situation where we're 502 00:32:12,230 --> 00:32:16,430 attempting to stabilize an unstable system, we have now 503 00:32:16,430 --> 00:32:22,120 in the forward path a system which is unstable. 504 00:32:22,120 --> 00:32:27,420 And in the feedback path, we've put an appropriate 505 00:32:27,420 --> 00:32:33,000 system so that the overall system, in fact, is stable. 506 00:32:33,000 --> 00:32:38,480 Now, how can stability arise out of having an initially 507 00:32:38,480 --> 00:32:40,020 unstable system? 508 00:32:40,020 --> 00:32:45,110 Well, again, if we look at the basic feedback equation, the 509 00:32:45,110 --> 00:32:48,690 overall system function is the system function for the 510 00:32:48,690 --> 00:32:52,840 forward path divided by 1 plus the loop gain, G of 511 00:32:52,840 --> 00:32:55,070 s times H of s. 512 00:32:55,070 --> 00:32:59,790 And for stability what we want to examine are the roots of 1 513 00:32:59,790 --> 00:33:02,030 plus G of s times H of s. 514 00:33:02,030 --> 00:33:08,880 And in particular, the poles are the zeroes of that factor. 515 00:33:08,880 --> 00:33:12,720 And as long as we choose G of s, so that the 516 00:33:12,720 --> 00:33:15,490 poles of this term-- 517 00:33:15,490 --> 00:33:21,700 I'm sorry, so that the zeroes of this term are in the left 518 00:33:21,700 --> 00:33:24,100 half of the s-plane, then what we'll 519 00:33:24,100 --> 00:33:25,730 end up with is stability. 520 00:33:25,730 --> 00:33:29,530 So stability is dependent not just on h of s for the 521 00:33:29,530 --> 00:33:34,310 closed-loop system, but on 1 plus G of s times H of s. 522 00:33:34,310 --> 00:33:40,860 And this kind of notion is used in lots of situations. 523 00:33:40,860 --> 00:33:42,610 I indicated the inverted pendulum. 524 00:33:42,610 --> 00:33:47,780 Another very common example is in some very high-performance 525 00:33:47,780 --> 00:33:52,910 aircraft where the basic aircraft system 526 00:33:52,910 --> 00:33:55,060 is an unstable system. 527 00:33:55,060 --> 00:33:59,800 But in fact, it's stabilized by putting the right kind of 528 00:33:59,800 --> 00:34:02,530 feedback dynamics around it. 529 00:34:02,530 --> 00:34:05,200 And those feedback dynamics might, in fact, involve the 530 00:34:05,200 --> 00:34:07,870 pilot as well. 531 00:34:07,870 --> 00:34:17,790 Now, for the system that we just talked about, the 532 00:34:17,790 --> 00:34:22,520 stability was described in terms of a 533 00:34:22,520 --> 00:34:24,449 continuous-time system. 534 00:34:24,449 --> 00:34:27,540 And the stability condition that we end up with, of 535 00:34:27,540 --> 00:34:32,710 course, relates to the zeroes of this denominator term. 536 00:34:32,710 --> 00:34:36,850 And we require for stability that the real parts of the 537 00:34:36,850 --> 00:34:41,600 associated roots be in the left half of the s-plane. 538 00:34:41,600 --> 00:34:44,610 Exactly the same kind of analysis, in terms of 539 00:34:44,610 --> 00:34:48,690 stability, applies in discrete-time. 540 00:34:48,690 --> 00:34:55,580 That is, in discrete-time, as we saw previously, the basic 541 00:34:55,580 --> 00:34:58,630 discrete-time feedback system is exactly the same, except 542 00:34:58,630 --> 00:35:02,060 that the independent variable is now an integer variable 543 00:35:02,060 --> 00:35:04,320 rather than a continuous variable. 544 00:35:04,320 --> 00:35:07,460 The feedback equation is exactly the same. 545 00:35:07,460 --> 00:35:11,750 So to analyze stability of the feedback system, we would want 546 00:35:11,750 --> 00:35:17,080 to look at the zeroes of 1 plus G of z times H of z. 547 00:35:17,080 --> 00:35:21,250 So again, it's those zeroes that affect stability. 548 00:35:21,250 --> 00:35:25,580 And the principal difference between the continuous-time 549 00:35:25,580 --> 00:35:30,550 and discrete-time cases is the fact that the stability 550 00:35:30,550 --> 00:35:35,220 condition in discrete-time is different than it is in 551 00:35:35,220 --> 00:35:36,030 continuous-time. 552 00:35:36,030 --> 00:35:42,270 Namely, in continuous-time, we care for stability about poles 553 00:35:42,270 --> 00:35:46,960 of the overall system being in the left half of the s-plane 554 00:35:46,960 --> 00:35:48,690 or the right half of the s-plane. 555 00:35:48,690 --> 00:35:53,060 In discrete-time, what we care about is whether the poles are 556 00:35:53,060 --> 00:35:55,420 inside or outside the unit circle. 557 00:35:55,420 --> 00:36:00,530 So in the discrete-time case, what we would impose for 558 00:36:00,530 --> 00:36:04,390 stability is that the zeroes have a magnitude which 559 00:36:04,390 --> 00:36:06,060 is less than 1. 560 00:36:06,060 --> 00:36:12,090 So the basic analysis is the same, but the details of the 561 00:36:12,090 --> 00:36:15,700 stability condition, of course, are different. 562 00:36:15,700 --> 00:36:22,660 Now, what I've just indicated is that feedback can be used 563 00:36:22,660 --> 00:36:27,200 to stabilize an unstable system. 564 00:36:27,200 --> 00:36:30,660 And as you can imagine there's the other side of the coin. 565 00:36:30,660 --> 00:36:37,120 Namely, if you start with a stable system and put feedback 566 00:36:37,120 --> 00:36:40,300 around it, if you're not careful what can happen, in 567 00:36:40,300 --> 00:36:43,170 fact, is that you can destabilize the system. 568 00:36:43,170 --> 00:36:46,380 So there's always the potential hazard, unless it's 569 00:36:46,380 --> 00:36:50,610 something you want to have happen, that feedback around 570 00:36:50,610 --> 00:36:54,140 what used to be a stable system now generates a system 571 00:36:54,140 --> 00:36:55,520 which is unstable. 572 00:36:55,520 --> 00:36:57,860 And there are lots of examples of that. 573 00:36:57,860 --> 00:37:01,910 One very common example is in audio systems. 574 00:37:01,910 --> 00:37:04,800 And this is probably an example that you're somewhat 575 00:37:04,800 --> 00:37:06,880 familiar with. 576 00:37:06,880 --> 00:37:12,550 Basically, an audio system, if you have the speaker and the 577 00:37:12,550 --> 00:37:15,980 microphone in any kind of proximity to each other is, in 578 00:37:15,980 --> 00:37:18,190 fact, a feedback system. 579 00:37:18,190 --> 00:37:23,590 Well, first of all, the audio input to the microphone 580 00:37:23,590 --> 00:37:31,360 consists of the external audio inputs, and the external audio 581 00:37:31,360 --> 00:37:35,360 inputs might, for example, be my voice. 582 00:37:35,360 --> 00:37:37,580 It might be the room noise. 583 00:37:37,580 --> 00:37:40,140 And in fact, as we'll illustrate shortly if I'm not 584 00:37:40,140 --> 00:37:44,490 careful, might in fact be the output from a speaker, which 585 00:37:44,490 --> 00:37:46,360 represents feedback. 586 00:37:46,360 --> 00:37:52,410 That audio, of course, after appropriate amplification 587 00:37:52,410 --> 00:37:56,680 drives a speaker. 588 00:37:56,680 --> 00:38:02,730 And if, in fact, the speaker is, let's say has any 589 00:38:02,730 --> 00:38:07,580 proximity to the microphone, then there can be a certain 590 00:38:07,580 --> 00:38:13,200 amount of the output of the speaker that feeds back around 591 00:38:13,200 --> 00:38:18,560 and is fed back into the microphone. 592 00:38:18,560 --> 00:38:22,200 Now, the system function associated with the feedback I 593 00:38:22,200 --> 00:38:26,280 indicate here as a constant times e to the minus s times 594 00:38:26,280 --> 00:38:30,940 capital T. The e to the minus s times capital T represents 595 00:38:30,940 --> 00:38:36,690 the fact that there is, in general, some delay between 596 00:38:36,690 --> 00:38:41,110 the time delay between the speaker output and the input 597 00:38:41,110 --> 00:38:43,730 that it generates to the microphone. 598 00:38:43,730 --> 00:38:47,410 The reason for that delay of course, being that there may 599 00:38:47,410 --> 00:38:50,570 be some distance between the speaker and the microphone. 600 00:38:50,570 --> 00:38:54,850 And then the constant K2 that I have in the feedback path 601 00:38:54,850 --> 00:38:57,580 represents the fact that between the speaker and the 602 00:38:57,580 --> 00:39:01,050 microphone, there may be some attenuation. 603 00:39:01,050 --> 00:39:04,970 So if I have, for example, a speaker as I happen to have 604 00:39:04,970 --> 00:39:13,970 here, and I were to have that speaker putting out what in 605 00:39:13,970 --> 00:39:16,120 fact I'm putting into the microphone, or the output of 606 00:39:16,120 --> 00:39:19,640 the microphone, then what we have is a feedback path. 607 00:39:19,640 --> 00:39:23,050 And the feedback path is from the microphone, through the 608 00:39:23,050 --> 00:39:26,990 speaker, out of the speaker, back into the microphone. 609 00:39:26,990 --> 00:39:32,700 And the feedback path is from here to the microphone. 610 00:39:32,700 --> 00:39:36,230 And the characteristics or frequency response or system 611 00:39:36,230 --> 00:39:39,390 function is associated with the characteristics of 612 00:39:39,390 --> 00:39:41,580 propagation or transmission. 613 00:39:41,580 --> 00:39:44,440 If I were to move closer to the speaker and I, by the way, 614 00:39:44,440 --> 00:39:46,825 don't have the speaker on right now. 615 00:39:46,825 --> 00:39:50,690 And I'm sure you all understand why. 616 00:39:50,690 --> 00:39:58,480 If I move closer, then the constant K2 gets what? 617 00:39:58,480 --> 00:39:59,750 Gets larger. 618 00:39:59,750 --> 00:40:04,040 And if I move further away the constant K2 gets smaller. 619 00:40:04,040 --> 00:40:08,120 Well, let's look at an analysis of this and see what 620 00:40:08,120 --> 00:40:11,020 it is, or why it is, that in fact we get an 621 00:40:11,020 --> 00:40:12,970 instability in terms-- 622 00:40:12,970 --> 00:40:15,420 or that an instability is predicted by the basic 623 00:40:15,420 --> 00:40:17,550 feedback equation. 624 00:40:17,550 --> 00:40:20,760 Now, notice first of all, that we're talking about positive 625 00:40:20,760 --> 00:40:22,160 feedback here. 626 00:40:22,160 --> 00:40:27,210 And just simply substituting the appropriate system 627 00:40:27,210 --> 00:40:31,690 functions into our basic feedback equation, we have an 628 00:40:31,690 --> 00:40:35,180 equation that says that the overall system function is 629 00:40:35,180 --> 00:40:38,260 given by the forward gain, which is the gain of the 630 00:40:38,260 --> 00:40:42,040 amplifier between the microphone and the speaker, 631 00:40:42,040 --> 00:40:45,190 divided by 1 minus-- 632 00:40:45,190 --> 00:40:48,120 and the minus because we have positive feedback-- 633 00:40:48,120 --> 00:40:52,350 the overall loop gain, which is K1, K2, e to the minus s 634 00:40:52,350 --> 00:40:58,000 capital T. And these two gains, K1 and K2 are assumed 635 00:40:58,000 --> 00:41:01,030 to be positive, and generally are positive. 636 00:41:01,030 --> 00:41:06,280 So in order for us to-- 637 00:41:06,280 --> 00:41:09,620 well, if we want to look at the poles of the system, then 638 00:41:09,620 --> 00:41:12,770 we want to look at the zeroes of this denominator. 639 00:41:12,770 --> 00:41:16,190 And the zeroes of this denominator occur at values of 640 00:41:16,190 --> 00:41:21,000 s such that e to the minus s capital T is equal to 1 over 641 00:41:21,000 --> 00:41:23,020 K1 times K2. 642 00:41:23,020 --> 00:41:28,940 And equivalently that says that the poles of the closed 643 00:41:28,940 --> 00:41:35,260 loop system occur at 1 over capital T, and capital T is 644 00:41:35,260 --> 00:41:37,220 related to the time delay. 645 00:41:37,220 --> 00:41:43,460 1 over capital T times the log to the base e of K1 times K2. 646 00:41:43,460 --> 00:41:47,880 Well, for stability we want these poles to all be in the 647 00:41:47,880 --> 00:41:49,800 left half of the s-plane. 648 00:41:49,800 --> 00:41:54,040 And what that means then is that for stability what we 649 00:41:54,040 --> 00:41:59,010 require is that K1 times K2 be less than 1. 650 00:41:59,010 --> 00:42:02,840 In other words, we require that the overall loop gain be 651 00:42:02,840 --> 00:42:06,100 less-- the magnitude of the loop gain be less than 1. 652 00:42:06,100 --> 00:42:09,620 If it's not, then what we generate is an instability. 653 00:42:09,620 --> 00:42:13,990 And just to illustrate that, let's turn the speaker on. 654 00:42:13,990 --> 00:42:18,090 And what we'll demonstrate is feedback. 655 00:42:18,090 --> 00:42:20,340 Right now the system is stable. 656 00:42:20,340 --> 00:42:23,320 And I'm being careful to keep my distance from the speaker. 657 00:42:23,320 --> 00:42:29,090 As I get closer, K2 will increase. 658 00:42:29,090 --> 00:42:33,280 And as K2 increases, eventually the poles will move 659 00:42:33,280 --> 00:42:36,640 into the right half of the s-plane, or they'll try to. 660 00:42:36,640 --> 00:42:39,060 What will happen is that the system will start to oscillate 661 00:42:39,060 --> 00:42:41,230 and go into nonlinear distortion. 662 00:42:41,230 --> 00:42:46,350 So as I get closer, you can hear that we get feedback, we 663 00:42:46,350 --> 00:42:48,010 get oscillation. 664 00:42:48,010 --> 00:42:53,250 And I guess neither you nor I can take too much of that. 665 00:42:53,250 --> 00:42:58,120 But you can see that what's happening-- 666 00:42:58,120 --> 00:43:00,680 if we can just turn the speaker off now. 667 00:43:00,680 --> 00:43:05,310 You can see that what's happening is that as K2 668 00:43:05,310 --> 00:43:10,660 increases, the poles are moving on to the j omega axis, 669 00:43:10,660 --> 00:43:12,490 the system starts to oscillate. 670 00:43:12,490 --> 00:43:15,450 They won't actually move into the right half plane because 671 00:43:15,450 --> 00:43:19,650 there are nonlinearities that inherently control the system. 672 00:43:22,800 --> 00:43:31,010 OK, so what we've seen in today's lecture is the basic 673 00:43:31,010 --> 00:43:34,950 analysis equation and a few of the applications. 674 00:43:34,950 --> 00:43:39,960 And one application, or one both application and hazard 675 00:43:39,960 --> 00:43:44,870 that we've talked about, is the application in which we 676 00:43:44,870 --> 00:43:47,170 may stabilize unstable systems. 677 00:43:47,170 --> 00:43:51,580 Or if we're not careful, destabilize stable systems. 678 00:43:51,580 --> 00:43:55,560 As I've indicated at several times during the lecture, one 679 00:43:55,560 --> 00:44:01,210 common example of an unstable system which feedback can be 680 00:44:01,210 --> 00:44:05,970 used to stabilize is the inverted pendulum, which I've 681 00:44:05,970 --> 00:44:08,090 referred to several times. 682 00:44:08,090 --> 00:44:13,210 And in the next lecture, what I'd like to do is focus in on 683 00:44:13,210 --> 00:44:15,410 a more detailed analysis of this. 684 00:44:15,410 --> 00:44:20,860 And what we'll see, in fact, is that the feedback dynamics, 685 00:44:20,860 --> 00:44:24,760 the form of the feedback dynamics are important with 686 00:44:24,760 --> 00:44:27,750 regard to whether you can and can't stabilize the system. 687 00:44:27,750 --> 00:44:30,630 Interestingly enough, for this particular system, as we'll 688 00:44:30,630 --> 00:44:34,520 see in the next lecture, if you simply try to measure the 689 00:44:34,520 --> 00:44:40,030 angle and feed that back that, in fact, you can't stabilize 690 00:44:40,030 --> 00:44:40,780 the system. 691 00:44:40,780 --> 00:44:45,140 What it requires is not only the angle, but some 692 00:44:45,140 --> 00:44:47,870 information about the rate of change of angle. 693 00:44:47,870 --> 00:44:49,760 But we'll see that in much more 694 00:44:49,760 --> 00:44:51,120 detail in the next lecture. 695 00:44:51,120 --> 00:44:52,370 Thank you.