WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high-quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation, or view additional materials from 00:00:13.470 --> 00:00:19.300 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.300 --> 00:00:20.090 ocw.mit.edu. 00:00:20.090 --> 00:00:56.110 [MUSIC PLAYING] 00:00:56.110 --> 00:00:58.740 PROFESSOR: During the course, we've developed a number of 00:00:58.740 --> 00:01:01.220 very powerful and useful tools. 00:01:01.220 --> 00:01:04.519 And we've seen how these can be used in designing and 00:01:04.519 --> 00:01:05.970 analyzing systems. 00:01:05.970 --> 00:01:10.750 For example, for filtering, for modulation, et cetera. 00:01:10.750 --> 00:01:14.380 I'd like to conclude this series of lectures with an 00:01:14.380 --> 00:01:17.670 introduction to one more important topic. 00:01:17.670 --> 00:01:22.320 Namely, the analysis of feedback systems. 00:01:22.320 --> 00:01:27.700 And one of the principle reasons that we have left this 00:01:27.700 --> 00:01:31.900 discussion to the last part of the course is so that we can 00:01:31.900 --> 00:01:36.570 exploit some of the ideas that we've just developed in some 00:01:36.570 --> 00:01:38.490 of the previous lectures. 00:01:38.490 --> 00:01:42.350 Namely, the tools afforded us by Laplace and z-transforms. 00:01:45.600 --> 00:01:52.300 Now, as I had indicated in one of the very first lectures, a 00:01:52.300 --> 00:01:56.080 common example of a feedback system is the problem of, 00:01:56.080 --> 00:01:59.790 let's say balancing a broom, or in the case of that lecture 00:01:59.790 --> 00:02:04.480 balancing my son's horse in the palm of your hand. 00:02:04.480 --> 00:02:09.610 And kind of the idea there is that what that relies on, in 00:02:09.610 --> 00:02:14.320 order to make that a stable system, is feedback. 00:02:14.320 --> 00:02:17.680 In that particular case, visual feedback. 00:02:17.680 --> 00:02:21.470 That specific problem, the one of balancing something, let's 00:02:21.470 --> 00:02:26.230 say in the palm of your hand, is an example of a problem, 00:02:26.230 --> 00:02:29.880 which is commonly referred to as the inverted pendulum. 00:02:29.880 --> 00:02:33.250 And it's one that we will actually be analyzing in a 00:02:33.250 --> 00:02:35.400 fair amount of detail not in this lecture, 00:02:35.400 --> 00:02:36.820 but in the next lecture. 00:02:36.820 --> 00:02:39.510 But let me just kind of indicate what some of the 00:02:39.510 --> 00:02:41.670 issues are. 00:02:41.670 --> 00:02:45.030 Let me describe this in the context not of balancing a 00:02:45.030 --> 00:02:48.410 broom on your hand, but let's say that we have a mechanical 00:02:48.410 --> 00:02:50.730 system which consists of a cart. 00:02:50.730 --> 00:02:55.770 And the cart can move, let's say in one dimension, and it 00:02:55.770 --> 00:03:01.460 has mounted on it a bar, a rod with a weight on the top, and 00:03:01.460 --> 00:03:02.880 it pivots around the base. 00:03:02.880 --> 00:03:06.510 So that essentially represents the inverted pendulum. 00:03:06.510 --> 00:03:10.510 So that system can be, more or less, depicted as I've 00:03:10.510 --> 00:03:12.210 indicated here. 00:03:12.210 --> 00:03:16.650 And this is the cart that can move along the x-axis. 00:03:16.650 --> 00:03:20.660 And here we have a pivot point, a rod, a 00:03:20.660 --> 00:03:22.000 weight at the top. 00:03:22.000 --> 00:03:25.100 And then, of course, there are several forces acting on this. 00:03:25.100 --> 00:03:28.740 There is an acceleration that can be applied to the cart, 00:03:28.740 --> 00:03:33.290 and that will be thought of as the external input. 00:03:33.290 --> 00:03:37.130 And then on the pendulum itself, on the weight, there 00:03:37.130 --> 00:03:39.080 is the force of gravity. 00:03:39.080 --> 00:03:43.130 And then typically, a set of external disturbances that 00:03:43.130 --> 00:03:46.840 might represent, for example, air currents, or wind, or 00:03:46.840 --> 00:03:53.630 whatever, that will attempt to destabilize the system. 00:03:53.630 --> 00:03:58.440 Specifically, to have the pendulum fall down. 00:03:58.440 --> 00:04:02.470 Now, if we look at this system in, more or less, a 00:04:02.470 --> 00:04:07.150 straightforward way, what we have then are the system 00:04:07.150 --> 00:04:14.200 dynamics and several inputs, one of which is the external 00:04:14.200 --> 00:04:19.570 disturbances and a second is the acceleration, which is the 00:04:19.570 --> 00:04:21.990 external acceleration that's applied. 00:04:21.990 --> 00:04:25.510 And the output of the system can be thought of as the 00:04:25.510 --> 00:04:27.590 angular displacement of the pendulum. 00:04:27.590 --> 00:04:31.330 Which if we want it balanced, we would like that angular 00:04:31.330 --> 00:04:34.430 displacement to be equal to 0. 00:04:34.430 --> 00:04:39.330 Now, if we know exactly what the system dynamics are and if 00:04:39.330 --> 00:04:43.280 we knew exactly what the external disturbances are, 00:04:43.280 --> 00:04:46.500 then in principle, we could design an acceleration. 00:04:46.500 --> 00:04:47.850 Namely, an input. 00:04:47.850 --> 00:04:51.710 That would exactly generate 0 output. 00:04:51.710 --> 00:04:54.440 In other words, the angle would be equal to 0. 00:04:54.440 --> 00:04:58.750 But as you can imagine, just as it's basically impossible 00:04:58.750 --> 00:05:01.830 to balance a broom in the palm of your hand with you eyes 00:05:01.830 --> 00:05:08.030 closed, what is very hard to ascertain in advance are what 00:05:08.030 --> 00:05:11.100 the various dynamics and disturbances are. 00:05:11.100 --> 00:05:15.040 And so more typically what you would think of doing is 00:05:15.040 --> 00:05:20.130 measuring the output angle, and then using that 00:05:20.130 --> 00:05:24.220 measurement to somehow influence the applied 00:05:24.220 --> 00:05:25.920 acceleration or force. 00:05:25.920 --> 00:05:29.670 And that, then, is an example of a feedback system. 00:05:29.670 --> 00:05:37.160 So we would measure the output angle and generate an input 00:05:37.160 --> 00:05:42.960 acceleration, which is some function of what that 00:05:42.960 --> 00:05:44.300 output angle is. 00:05:44.300 --> 00:05:48.310 And if we choose the feedback dynamics correctly, then in 00:05:48.310 --> 00:05:52.430 fact, we can drive this output to 0. 00:05:52.430 --> 00:05:57.640 This is one example of a system which is inherently 00:05:57.640 --> 00:06:01.680 unstable because if we left it to its own devices, the 00:06:01.680 --> 00:06:03.760 pendulum would simply fall down. 00:06:03.760 --> 00:06:07.140 And essentially, by applying feedback, what we're trying to 00:06:07.140 --> 00:06:10.390 do is stabilize this inherently unstable system. 00:06:10.390 --> 00:06:13.560 And we'll talk a little bit more about that specific 00:06:13.560 --> 00:06:17.030 application of feedback shortly. 00:06:17.030 --> 00:06:21.400 Another common example of feedback is in positioning or 00:06:21.400 --> 00:06:26.460 tracking systems, and I indicate one here which 00:06:26.460 --> 00:06:31.790 corresponds to the problem of positioning a telescope, which 00:06:31.790 --> 00:06:34.000 is mounted on a rotating platform. 00:06:34.000 --> 00:06:39.110 So in a system of that type, for example, I indicate here 00:06:39.110 --> 00:06:41.950 the rotating platform and the telescope. 00:06:41.950 --> 00:06:44.310 It's driven by a motor. 00:06:44.310 --> 00:06:48.070 And again, we could imagine, in principle, the possibility 00:06:48.070 --> 00:06:53.320 of driving this to the desired angle by choosing an 00:06:53.320 --> 00:06:56.370 appropriate applied input voltage. 00:06:56.370 --> 00:06:59.230 And as long as we know such things as what the 00:06:59.230 --> 00:07:02.540 disturbances are that influence the telescope mount 00:07:02.540 --> 00:07:06.460 and what the characteristics of the motor are, in principle 00:07:06.460 --> 00:07:10.250 we could in fact carry this out in a form which is 00:07:10.250 --> 00:07:11.920 referred to as open loop. 00:07:11.920 --> 00:07:16.280 Namely, we can choose an appropriate input voltage to 00:07:16.280 --> 00:07:22.020 drive the motor to set the platform angle at the desired 00:07:22.020 --> 00:07:24.450 angular position. 00:07:24.450 --> 00:07:28.350 However, again, there are enough unknowns in a problem 00:07:28.350 --> 00:07:33.790 like that, so that one is motivated to employ feedback. 00:07:33.790 --> 00:07:39.160 Namely, to make a measurement of the output angle and use 00:07:39.160 --> 00:07:44.430 that in a feedback loop to influence the drive for the 00:07:44.430 --> 00:07:49.410 motor, so that the telescope platform is positioned 00:07:49.410 --> 00:07:50.610 appropriately. 00:07:50.610 --> 00:07:56.950 So if we look at this in a feedback context, we would 00:07:56.950 --> 00:08:05.020 then take the measured output angle and the measured output 00:08:05.020 --> 00:08:09.930 angle would be fed back and compared 00:08:09.930 --> 00:08:12.810 with the desired angle. 00:08:12.810 --> 00:08:16.340 And the difference between those, which essentially is 00:08:16.340 --> 00:08:19.240 the error between the platform positioning and the desired 00:08:19.240 --> 00:08:23.880 position would be put perhaps through an appropriate gain or 00:08:23.880 --> 00:08:28.870 attenuation and used as the excitation to the motor. 00:08:28.870 --> 00:08:32.659 So in the mechanical or physical system, that would 00:08:32.659 --> 00:08:37.900 correspond to measuring the angle, let's say with a 00:08:37.900 --> 00:08:39.169 potentiometer. 00:08:39.169 --> 00:08:42.770 So here we're measuring the angle and we have an output, 00:08:42.770 --> 00:08:45.700 which is proportional to that measured angle. 00:08:45.700 --> 00:08:52.290 And then we would use feedback, comparing the 00:08:52.290 --> 00:08:57.670 measured angle to some proportionality factor 00:08:57.670 --> 00:09:00.550 multiplying the desired angle. 00:09:00.550 --> 00:09:05.190 So here we have the desired angle, again through some type 00:09:05.190 --> 00:09:06.660 of potentiometer. 00:09:06.660 --> 00:09:08.710 The two are compared. 00:09:08.710 --> 00:09:12.160 Out of the comparator, we basically have an indication 00:09:12.160 --> 00:09:14.890 of what the difference is, and that represents an error 00:09:14.890 --> 00:09:17.230 between the desired and the true angle. 00:09:17.230 --> 00:09:22.160 And then that is used through perhaps an amplifier to 00:09:22.160 --> 00:09:23.630 control the motor. 00:09:23.630 --> 00:09:28.240 And in that case, of course, when the error goes to 0, that 00:09:28.240 --> 00:09:29.960 means that the actual angle and the 00:09:29.960 --> 00:09:32.010 desired angle are equal. 00:09:32.010 --> 00:09:35.930 And in fact, in that case also with this system, the input to 00:09:35.930 --> 00:09:40.210 the motor is, likewise, equal to 0. 00:09:40.210 --> 00:09:44.520 Now, as I've illustrated it here, it tends to be in the 00:09:44.520 --> 00:09:48.840 context of a continuous time or analog system. 00:09:48.840 --> 00:09:51.770 And in fact, another very common way of doing 00:09:51.770 --> 00:09:58.890 positioning or tracking is to instead implement the feedback 00:09:58.890 --> 00:10:02.300 using a discrete-time or digital system. 00:10:02.300 --> 00:10:07.760 And so in that case, we would basically take the position 00:10:07.760 --> 00:10:13.310 output as it's measured, sample it, essentially convert 00:10:13.310 --> 00:10:18.460 that to a digital discrete-time signal. 00:10:18.460 --> 00:10:23.330 And then that is used in conjunction with the desired 00:10:23.330 --> 00:10:30.060 angle, which both form inputs to this processor. 00:10:30.060 --> 00:10:34.770 And the output of that is converted, let's say, back to 00:10:34.770 --> 00:10:38.120 an analog or continuous-time voltage and used 00:10:38.120 --> 00:10:40.100 to drive the motor. 00:10:40.100 --> 00:10:45.040 Now, you could ask, why would you go to a digital or 00:10:45.040 --> 00:10:49.150 discrete-time measurement rather than doing it the way I 00:10:49.150 --> 00:10:51.340 showed on the previous overlay which seemed relatively 00:10:51.340 --> 00:10:52.530 straightforward? 00:10:52.530 --> 00:10:55.710 And the reason, principally, is that in the context of a 00:10:55.710 --> 00:11:00.360 digital implementation of the feedback process, often you 00:11:00.360 --> 00:11:05.220 can implement a better controlled and often also, 00:11:05.220 --> 00:11:10.140 more sophisticated algorithm for the feedback dynamics. 00:11:10.140 --> 00:11:13.310 So that you can take a count, perhaps not only of the angle 00:11:13.310 --> 00:11:16.970 itself, but also of the rate of change of angle. 00:11:16.970 --> 00:11:18.720 And in fact, the rate of change of the rate 00:11:18.720 --> 00:11:21.270 of change of angle. 00:11:21.270 --> 00:11:27.230 So the system, as it's shown there then, basically has a 00:11:27.230 --> 00:11:30.350 discrete-time or digital feedback loop around a 00:11:30.350 --> 00:11:33.070 continuous time system. 00:11:33.070 --> 00:11:38.950 Now, this is an example of, in fact, a more general way in 00:11:38.950 --> 00:11:42.590 which discrete-time feedback is used 00:11:42.590 --> 00:11:44.950 with continuous systems. 00:11:44.950 --> 00:11:50.950 And let me indicate, in general, what the character or 00:11:50.950 --> 00:11:53.510 block diagram of such a system might be. 00:11:53.510 --> 00:11:57.860 Typically, if we abstract away from the telescope positioning 00:11:57.860 --> 00:12:01.420 system, we might have a more general 00:12:01.420 --> 00:12:03.730 continuous-time system. 00:12:03.730 --> 00:12:07.980 And around which we want to apply some feedback, which we 00:12:07.980 --> 00:12:10.920 could do with a continuous-time system or with 00:12:10.920 --> 00:12:16.150 a discrete-time system by first converting these signals 00:12:16.150 --> 00:12:20.310 to discrete-time signals. 00:12:20.310 --> 00:12:24.620 Then, processing that with a discrete-time system. 00:12:24.620 --> 00:12:27.440 And then, through an appropriate interpolation 00:12:27.440 --> 00:12:32.670 algorithm, we would then convert that back to a 00:12:32.670 --> 00:12:34.580 continuous-time signal. 00:12:34.580 --> 00:12:39.030 And the difference between the input signal and this 00:12:39.030 --> 00:12:42.270 continuous-time signal which is fed back, then forms the 00:12:42.270 --> 00:12:45.220 excitation to the system that essentially 00:12:45.220 --> 00:12:46.870 we're trying to control. 00:12:46.870 --> 00:12:52.280 And in many systems of this type, the advantage is that 00:12:52.280 --> 00:12:56.710 this system can be implemented in a very reproducible way, 00:12:56.710 --> 00:13:01.240 either with a digital computer or with a microprocessor. 00:13:01.240 --> 00:13:06.110 And although we're not going to go into this in any detail 00:13:06.110 --> 00:13:08.850 in this lecture, there is some discussion 00:13:08.850 --> 00:13:10.580 of this in the text. 00:13:10.580 --> 00:13:14.370 Essentially, if we make certain assumptions about this 00:13:14.370 --> 00:13:19.980 particular feedback system, we can move the continuous to 00:13:19.980 --> 00:13:25.340 discrete-time converter up to this point and to this point, 00:13:25.340 --> 00:13:30.060 and we can move the interpolating system outside 00:13:30.060 --> 00:13:31.500 the summer. 00:13:31.500 --> 00:13:37.220 And what happens in that case is that we end up with what 00:13:37.220 --> 00:13:42.730 looks like an inherently discrete-time feedback system. 00:13:42.730 --> 00:13:46.280 So, in fact, if we take those steps, then what we'll end up 00:13:46.280 --> 00:13:52.260 with for a feedback system is a system that essentially can 00:13:52.260 --> 00:13:55.890 be analyzed as a discrete-time system. 00:13:55.890 --> 00:14:00.820 Here we have what is, in the forward path, is basically the 00:14:00.820 --> 00:14:04.550 continuous-time system with the interpolator at one end 00:14:04.550 --> 00:14:06.450 and the continuous to discrete-time converter 00:14:06.450 --> 00:14:07.700 at the other end. 00:14:07.700 --> 00:14:11.640 And then we have whatever system it was in the feedback 00:14:11.640 --> 00:14:13.090 loop-- discrete-time-- 00:14:13.090 --> 00:14:16.900 that shows up in this feedback loop. 00:14:16.900 --> 00:14:21.410 Well, I show this mainly to emphasize the fact, although 00:14:21.410 --> 00:14:24.500 there are some steps there that we obviously left out. 00:14:24.500 --> 00:14:27.080 I show that mainly to emphasize the fact that 00:14:27.080 --> 00:14:30.890 feedback arises not just in the context of continuous-time 00:14:30.890 --> 00:14:36.140 systems, but also the analysis of discrete-time feedback 00:14:36.140 --> 00:14:38.240 systems becomes important. 00:14:38.240 --> 00:14:42.600 Perhaps because we have used discrete-time feedback around 00:14:42.600 --> 00:14:44.540 a continuous-time system. 00:14:44.540 --> 00:14:49.850 But also perhaps because the feedback system is inherently 00:14:49.850 --> 00:14:50.660 discrete-time. 00:14:50.660 --> 00:14:54.980 And let me just illustrate one, or indicate one example 00:14:54.980 --> 00:14:57.850 in which that might arise. 00:14:57.850 --> 00:15:03.290 This is an example which is also discussed in somewhat 00:15:03.290 --> 00:15:05.270 more detail in the text. 00:15:05.270 --> 00:15:09.990 But basically, population studies, for example, 00:15:09.990 --> 00:15:12.480 represent examples of 00:15:12.480 --> 00:15:14.840 discrete-time feedback systems. 00:15:14.840 --> 00:15:19.590 Where let's say that we have some type of model for 00:15:19.590 --> 00:15:21.200 population growth. 00:15:21.200 --> 00:15:25.180 And since people come in integer amounts that 00:15:25.180 --> 00:15:30.950 represents essentially the output of any population 00:15:30.950 --> 00:15:33.080 model, essentially or inherently 00:15:33.080 --> 00:15:34.330 represents a sequence. 00:15:34.330 --> 00:15:38.630 Namely, it's indexed on an integer variable. 00:15:38.630 --> 00:15:42.810 And typically, models for population growth 00:15:42.810 --> 00:15:44.720 are unstable systems. 00:15:44.720 --> 00:15:48.240 You can kind of imagine that because if you take these 00:15:48.240 --> 00:15:52.950 simple models of population, what happens is that in any 00:15:52.950 --> 00:15:59.050 generation, the number of people, or animals, or 00:15:59.050 --> 00:16:02.590 whatever it is that this is modeling, grows essentially 00:16:02.590 --> 00:16:07.750 exponentially with the size of the previous generation. 00:16:07.750 --> 00:16:09.360 Now, where does the feedback come in? 00:16:09.360 --> 00:16:14.370 Well, the feedback typically comes in, in incorporating in 00:16:14.370 --> 00:16:17.090 the overall model various retarding factors. 00:16:17.090 --> 00:16:20.920 For example, as the population increases, the food supply 00:16:20.920 --> 00:16:22.360 becomes more limited. 00:16:22.360 --> 00:16:30.030 And that essentially is a feedback process that acts to 00:16:30.030 --> 00:16:32.510 retard the population growth. 00:16:32.510 --> 00:16:35.360 And so an overall model-- 00:16:35.360 --> 00:16:36.610 somewhat simplified-- 00:16:38.940 --> 00:16:44.170 for a population system is the open loop model in the absence 00:16:44.170 --> 00:16:46.270 of retarding factors. 00:16:46.270 --> 00:16:50.200 And then, very often the retarding factors can be 00:16:50.200 --> 00:16:55.490 described as being related to the size of the population. 00:16:55.490 --> 00:17:02.210 And those essentially act to reduce the overall input to 00:17:02.210 --> 00:17:04.060 the population model. 00:17:04.060 --> 00:17:09.560 And so population studies are one very common example of 00:17:09.560 --> 00:17:11.205 discrete-time feedback systems. 00:17:14.839 --> 00:17:19.720 Well, what we want to look at and understand are the basic 00:17:19.720 --> 00:17:23.030 properties of feedback systems. 00:17:23.030 --> 00:17:27.910 And to do that, let's look at the basic block diagram and 00:17:27.910 --> 00:17:30.530 equations for feedback systems, either 00:17:30.530 --> 00:17:33.820 continuous-time or discrete-time. 00:17:33.820 --> 00:17:38.540 Let's begin with the continuous-time case. 00:17:38.540 --> 00:17:44.670 And now what we've done is simply abstract out any of the 00:17:44.670 --> 00:17:49.150 applications to a fairly general system, in which we 00:17:49.150 --> 00:17:54.980 have a system H of s in what's referred to as the forward 00:17:54.980 --> 00:17:59.590 path, and a system G of s in the feedback path. 00:18:02.440 --> 00:18:07.600 The input to the system H of s is the difference between the 00:18:07.600 --> 00:18:11.320 input to the overall system and the output of 00:18:11.320 --> 00:18:13.020 the feedback loop. 00:18:13.020 --> 00:18:17.020 And I draw your attention to the fact that what we 00:18:17.020 --> 00:18:18.440 illustrate here and what we're 00:18:18.440 --> 00:18:20.710 analyzing is negative feedback. 00:18:20.710 --> 00:18:25.010 Namely, this output is subtracted from the input. 00:18:25.010 --> 00:18:29.170 And that's done more for reasons of convention then for 00:18:29.170 --> 00:18:30.370 any other reasons. 00:18:30.370 --> 00:18:34.770 It's typical to do that and appropriate certainly in some 00:18:34.770 --> 00:18:37.040 feedback systems, but not all. 00:18:37.040 --> 00:18:41.350 And the output of the adder is commonly referred to as the 00:18:41.350 --> 00:18:44.400 error signal, indicating that it's the difference between 00:18:44.400 --> 00:18:49.280 the signal fed back and the input to the overall system. 00:18:49.280 --> 00:18:53.310 Now, if we want to analyze the feedback system, we would do 00:18:53.310 --> 00:18:55.540 that essentially by writing the appropriate equations. 00:18:58.460 --> 00:19:01.220 In generating the equivalent system function for the 00:19:01.220 --> 00:19:06.780 overall system, it's best done in the frequency or Laplace 00:19:06.780 --> 00:19:10.290 transform domain rather than in the time domain. 00:19:10.290 --> 00:19:12.330 And let me just indicate what the steps 00:19:12.330 --> 00:19:13.420 are that are involved. 00:19:13.420 --> 00:19:17.260 And there are a few steps of algebra that I'll leave in 00:19:17.260 --> 00:19:18.480 your hands. 00:19:18.480 --> 00:19:23.440 But basically, if we look at this feedback system, we can 00:19:23.440 --> 00:19:27.610 label, of course-- since the output is y of t, we can label 00:19:27.610 --> 00:19:31.080 the Laplace transform of the output as Y of s. 00:19:31.080 --> 00:19:35.330 And we also have Y of s as the input here. 00:19:35.330 --> 00:19:38.520 Because this is the system function, the Laplace 00:19:38.520 --> 00:19:42.620 transform of r of t is simply the Laplace transform of this 00:19:42.620 --> 00:19:45.330 input, which is Y of s times G of s. 00:19:45.330 --> 00:19:50.410 So here we have Y of s times G of s. 00:19:50.410 --> 00:19:55.940 At the adder, the input here is x of s. 00:19:55.940 --> 00:20:01.110 And so the Laplace transform of the error signal is simply 00:20:01.110 --> 00:20:08.070 x of s minus r of s, which is Y of s G of s. 00:20:08.070 --> 00:20:12.440 So this is minus Y of s G of s. 00:20:12.440 --> 00:20:15.580 That's the Laplace transform of the error signal. 00:20:15.580 --> 00:20:20.400 The Laplace transform of the output of this system is 00:20:20.400 --> 00:20:25.970 simply this expression times H of s. 00:20:25.970 --> 00:20:27.690 So that's what we have here. 00:20:27.690 --> 00:20:30.820 But what we have here we already called Y of s. 00:20:30.820 --> 00:20:35.600 So in fact, we can simply say that these two expressions 00:20:35.600 --> 00:20:37.180 have to be equal. 00:20:37.180 --> 00:20:41.020 And so we've essentially done the analysis, saying that 00:20:41.020 --> 00:20:43.150 those two expressions are equal. 00:20:43.150 --> 00:20:47.440 Let's solve for Y of s over x of s, which is the overall 00:20:47.440 --> 00:20:48.940 system function. 00:20:48.940 --> 00:20:52.970 And if we do that, what we end up with for the overall system 00:20:52.970 --> 00:20:55.820 function is the algebraic expression 00:20:55.820 --> 00:20:56.910 that I indicate here. 00:20:56.910 --> 00:21:02.840 It's H of s divided by 1 plus G of s H of s. 00:21:02.840 --> 00:21:07.470 Said another way, it's the system function in the open 00:21:07.470 --> 00:21:14.070 loop forward path divided by 1 plus, what's referred to as 00:21:14.070 --> 00:21:17.150 the loop gain, G of s times H of s. 00:21:17.150 --> 00:21:19.990 Let's just look back up at the block diagram. 00:21:19.990 --> 00:21:26.660 G of s times H of s is simply the gain around the entire 00:21:26.660 --> 00:21:30.760 loop from this point around to this point. 00:21:30.760 --> 00:21:34.820 So the overall system function is the gain in the forward 00:21:34.820 --> 00:21:39.770 path divided by 1 plus the loop gain, which is H of s 00:21:39.770 --> 00:21:42.430 times G of s. 00:21:42.430 --> 00:21:46.540 Now, none of the equations that we wrote had relied 00:21:46.540 --> 00:21:48.490 specifically on this being continuous-time. 00:21:48.490 --> 00:21:52.120 We just did some algebra and we used the system function 00:21:52.120 --> 00:21:53.770 property of the systems. 00:21:53.770 --> 00:21:58.620 And so, pretty obviously, the same kind of algebraic 00:21:58.620 --> 00:22:01.990 procedure would work in discrete-time. 00:22:01.990 --> 00:22:07.250 And so, in fact, if we carried out a discrete-time analysis 00:22:07.250 --> 00:22:11.290 rather than a continuous-time analysis, we would simply end 00:22:11.290 --> 00:22:18.600 up with exactly the same system and exactly the same 00:22:18.600 --> 00:22:24.600 equation for the overall system function. 00:22:24.600 --> 00:22:28.130 The only difference being that here things are a function of 00:22:28.130 --> 00:22:31.840 z, whereas if I just flip back the other 00:22:31.840 --> 00:22:34.880 overlay, we simply have-- 00:22:34.880 --> 00:22:39.070 previously everything is function of t and in the 00:22:39.070 --> 00:22:41.050 frequency domain s. 00:22:41.050 --> 00:22:45.980 In the discrete-time case, we've simply replaced in the 00:22:45.980 --> 00:22:48.920 time domain, the independent variable by n. 00:22:48.920 --> 00:22:54.830 And in the frequency domain, the independent variable by z. 00:22:54.830 --> 00:23:02.440 So what we see is that we have a basic feedback equation, and 00:23:02.440 --> 00:23:04.990 that feedback equation is exactly the same for 00:23:04.990 --> 00:23:07.590 continuous-time and discrete-time. 00:23:07.590 --> 00:23:10.880 Although we have to be careful about what implications we 00:23:10.880 --> 00:23:14.150 draw, depending on whether we're talking about 00:23:14.150 --> 00:23:15.950 continuous-time or discrete-time. 00:23:18.870 --> 00:23:24.660 Now, to illustrate the importance of feedback, let's 00:23:24.660 --> 00:23:28.750 look at a number of common applications. 00:23:28.750 --> 00:23:31.820 And also, as we talk about these applications, what will 00:23:31.820 --> 00:23:36.580 see is that while these applications and context in 00:23:36.580 --> 00:23:41.800 which feedback is used are extremely useful and powerful, 00:23:41.800 --> 00:23:47.090 they fall out in an almost straightforward way from this 00:23:47.090 --> 00:23:52.090 very simple feedback equation that we've just derrived. 00:23:52.090 --> 00:23:57.590 Well, the examples that I want to just talk about are, first 00:23:57.590 --> 00:24:01.930 of all, the use of feedback in amplifier design. 00:24:01.930 --> 00:24:05.980 And we're not going to design amplifiers in detail, but what 00:24:05.980 --> 00:24:11.050 I'd like to illustrate is the basic principle behind why 00:24:11.050 --> 00:24:14.740 feedback is useful in designing amplifiers. 00:24:14.740 --> 00:24:19.840 In particular, how it plays a role in compensating for a 00:24:19.840 --> 00:24:22.640 non-constant frequency response. 00:24:22.640 --> 00:24:25.880 So that's one context that we'll talk about. 00:24:25.880 --> 00:24:30.390 A second that I'll indicate is the use of feedback for 00:24:30.390 --> 00:24:33.890 implementing inverse systems. 00:24:33.890 --> 00:24:38.340 And the third, which we indicated in the case of the 00:24:38.340 --> 00:24:42.150 inverted pendulum, is an important context in which 00:24:42.150 --> 00:24:45.860 feedback is used is in stabilizing unstable systems. 00:24:45.860 --> 00:24:52.960 And what we want to see is why or how a feedback system or 00:24:52.960 --> 00:24:56.720 the basic feedback equation, in fact, let's us do each of 00:24:56.720 --> 00:24:59.090 these various things. 00:24:59.090 --> 00:25:03.140 Well, let's begin with amplifier design. 00:25:03.140 --> 00:25:07.960 And let's suppose that we've built somehow without 00:25:07.960 --> 00:25:13.380 feedback, an amplifier that is terrific in terms of its gain, 00:25:13.380 --> 00:25:17.690 but has the problem that whereas we might like the 00:25:17.690 --> 00:25:21.760 amplifier to have a very flat frequency response, in fact 00:25:21.760 --> 00:25:26.200 the frequency response of this amplifier is not constant. 00:25:26.200 --> 00:25:29.290 And what we'd like to do is compensate for that. 00:25:29.290 --> 00:25:32.430 Well, it turns out, interestingly, that if we 00:25:32.430 --> 00:25:38.250 embed the amplifier in a feedback loop where in the 00:25:38.250 --> 00:25:42.640 feedback path we incorporate an attenuator, then in fact, 00:25:42.640 --> 00:25:44.250 we can compensate for that 00:25:44.250 --> 00:25:46.500 non-constant frequency response. 00:25:46.500 --> 00:25:47.910 Well, let's see how that works out from 00:25:47.910 --> 00:25:49.990 the feedback equation. 00:25:49.990 --> 00:25:53.060 We have the basic feedback equation that we derived. 00:25:53.060 --> 00:25:55.900 And we want to look at frequency response, so we'll 00:25:55.900 --> 00:25:58.620 look specifically at the Fourier transform. 00:25:58.620 --> 00:26:02.450 And of course, the frequency response of the overall system 00:26:02.450 --> 00:26:05.920 is the frequency response of the Fourier transform of the 00:26:05.920 --> 00:26:08.260 output divided by the input. 00:26:08.260 --> 00:26:11.960 Using the feedback equation that we had just arrived, that 00:26:11.960 --> 00:26:15.430 has, in the numerator, the frequency response in the 00:26:15.430 --> 00:26:21.470 forward path divided by 1 plus the loop gain, which is H of j 00:26:21.470 --> 00:26:23.430 omega times k. 00:26:23.430 --> 00:26:24.920 And this is the key. 00:26:24.920 --> 00:26:32.830 Because here, if we choose k times H of j omega to be very 00:26:32.830 --> 00:26:38.170 large, much larger than 1, then what happens is that 00:26:38.170 --> 00:26:41.720 these two cancel out. 00:26:41.720 --> 00:26:46.380 H of j omega here and in the denominator will cancel out as 00:26:46.380 --> 00:26:48.590 long as this term dominates. 00:26:48.590 --> 00:26:52.240 And in that case, under that assumption, the overall system 00:26:52.240 --> 00:26:54.650 function is approximately 1/k. 00:26:57.260 --> 00:27:04.360 Well, if k is constant as a function of frequency, then we 00:27:04.360 --> 00:27:08.320 somehow magically have ended up with an amplifier that has 00:27:08.320 --> 00:27:11.210 a flat frequency response. 00:27:11.210 --> 00:27:14.190 Well, it seems like we're getting something for nothing. 00:27:14.190 --> 00:27:15.840 And actually, we're not. 00:27:15.840 --> 00:27:17.850 There's a price that we pay for that. 00:27:17.850 --> 00:27:24.110 Because notice the fact that in order to get gain out of 00:27:24.110 --> 00:27:29.390 the overall system, k must be less than 1. 00:27:29.390 --> 00:27:32.660 So this has to correspond to attenuator. 00:27:32.660 --> 00:27:37.680 And we also require that k, which is less than 1, times 00:27:37.680 --> 00:27:40.940 the gain of the original amplifier, that that product 00:27:40.940 --> 00:27:42.340 be greater than 1. 00:27:42.340 --> 00:27:45.370 And the implication of this, without tracking it in detail 00:27:45.370 --> 00:27:49.930 right now, the implication in this is that whereas we 00:27:49.930 --> 00:27:53.990 flatten the frequency response, we have in fact paid 00:27:53.990 --> 00:27:54.680 a price for that. 00:27:54.680 --> 00:27:58.030 The price that we've paid is that the gain is somewhat 00:27:58.030 --> 00:28:01.880 reduced from the gain that we had before the feedback. 00:28:01.880 --> 00:28:07.710 Because k times h must be much larger than 1, but the gain is 00:28:07.710 --> 00:28:09.250 proportional to 1/k. 00:28:09.250 --> 00:28:12.640 Now, one last point to make related to that. 00:28:12.640 --> 00:28:17.080 One could ask, well, why is it any easier to make k flat with 00:28:17.080 --> 00:28:21.920 frequency than to build an amplifier with a flat 00:28:21.920 --> 00:28:23.350 frequency response? 00:28:23.350 --> 00:28:28.040 The reason is that the gain in the feedback path is an 00:28:28.040 --> 00:28:30.280 attenuator, not an amplifier. 00:28:30.280 --> 00:28:34.930 And generally, attenuation with a flat frequency response 00:28:34.930 --> 00:28:37.460 is much easier to get than gain is. 00:28:37.460 --> 00:28:43.820 For example, a resistor, which attenuates, would generally 00:28:43.820 --> 00:28:45.510 have a flatter frequency response than a 00:28:45.510 --> 00:28:48.650 very high-gain amplifier. 00:28:48.650 --> 00:28:52.390 So that's one common example of feedback. 00:28:52.390 --> 00:28:57.310 And feedback, in fact, is very often used in high-quality 00:28:57.310 --> 00:28:59.210 amplifier systems. 00:28:59.210 --> 00:29:04.660 Another very common example in which feedback is used is in 00:29:04.660 --> 00:29:08.230 implementing inverse systems. 00:29:08.230 --> 00:29:13.220 Now, what I mean by that is, suppose that we have a system, 00:29:13.220 --> 00:29:15.400 which I indicate here, P of s-- 00:29:15.400 --> 00:29:17.250 input and output. 00:29:17.250 --> 00:29:21.710 And what we would like to do is implement a system which is 00:29:21.710 --> 00:29:23.610 the inverse of this system. 00:29:23.610 --> 00:29:28.630 Namely, has a Laplace transform or system function 00:29:28.630 --> 00:29:31.980 which is 1 over P of s. 00:29:31.980 --> 00:29:36.640 For example, we may have measured a particular system 00:29:36.640 --> 00:29:38.590 and what we would like to design is a 00:29:38.590 --> 00:29:40.140 compensator for it. 00:29:40.140 --> 00:29:42.930 And the question is, by putting this in a feedback 00:29:42.930 --> 00:29:46.660 loop, can we, in fact, implement the 00:29:46.660 --> 00:29:48.360 inverse of this system? 00:29:48.360 --> 00:29:50.580 The answer to that is yes. 00:29:50.580 --> 00:29:53.780 And the feedback system, in that case, is 00:29:53.780 --> 00:29:55.900 as I indicate here. 00:29:55.900 --> 00:30:01.570 So here what we choose to do is to put the system whose 00:30:01.570 --> 00:30:06.050 inverse we're trying to generate in the feedback loop. 00:30:06.050 --> 00:30:10.960 And in this case, a high-gain in the forward path. 00:30:10.960 --> 00:30:14.430 Now for this situation, k is, again, a constant. 00:30:14.430 --> 00:30:16.970 But in fact, it's a high-gain constant. 00:30:16.970 --> 00:30:20.460 And now if we look at the feedback equation, then what 00:30:20.460 --> 00:30:24.210 we see is an equation of this form. 00:30:24.210 --> 00:30:30.950 And notice that if k times P of s is large compared with 00:30:30.950 --> 00:30:35.340 one, then this term dominates. 00:30:35.340 --> 00:30:38.700 The gain in the forward path cancels out. 00:30:38.700 --> 00:30:44.360 And what we're left with is a system function, which is just 00:30:44.360 --> 00:30:46.960 1 over P of s. 00:30:46.960 --> 00:30:51.552 And a system of this type is used in a 00:30:51.552 --> 00:30:52.970 whole variety of contexts. 00:30:52.970 --> 00:30:56.910 One very common one is in building what are called 00:30:56.910 --> 00:31:00.460 logarithmic devices or logarithmic amplifiers. 00:31:00.460 --> 00:31:04.730 Ones in which the input-output characteristic is logarithmic. 00:31:04.730 --> 00:31:09.260 It's common to do that with a diode that has an exponential 00:31:09.260 --> 00:31:10.630 characteristic. 00:31:10.630 --> 00:31:14.580 And using that with feedback-- 00:31:14.580 --> 00:31:17.800 as feedback around a high-gain operational amplifier. 00:31:17.800 --> 00:31:20.570 And by the way, the logarithmic 00:31:20.570 --> 00:31:22.170 amplifier is nonlinear. 00:31:22.170 --> 00:31:24.440 What I've said here is linear, or the 00:31:24.440 --> 00:31:26.170 analysis here was linear. 00:31:26.170 --> 00:31:29.790 But that example, in fact, suggests something which is 00:31:29.790 --> 00:31:35.750 true, which is that same basic idea, in fact, can be used 00:31:35.750 --> 00:31:38.520 often in the context of nonlinear feedback and 00:31:38.520 --> 00:31:39.845 nonlinear feedback systems. 00:31:42.580 --> 00:31:48.760 Well, as a final example, what I'd like to analyze is the 00:31:48.760 --> 00:31:52.520 context in which we would consider 00:31:52.520 --> 00:31:56.230 stabilizing unstable systems. 00:31:56.230 --> 00:32:00.360 And I had indicated that one context in which that arises 00:32:00.360 --> 00:32:06.250 and which we will be analyzing in the next lecture is the 00:32:06.250 --> 00:32:07.930 inverted pendulum. 00:32:07.930 --> 00:32:12.230 And in that situation, or in a situation where we're 00:32:12.230 --> 00:32:16.430 attempting to stabilize an unstable system, we have now 00:32:16.430 --> 00:32:22.120 in the forward path a system which is unstable. 00:32:22.120 --> 00:32:27.420 And in the feedback path, we've put an appropriate 00:32:27.420 --> 00:32:33.000 system so that the overall system, in fact, is stable. 00:32:33.000 --> 00:32:38.480 Now, how can stability arise out of having an initially 00:32:38.480 --> 00:32:40.020 unstable system? 00:32:40.020 --> 00:32:45.110 Well, again, if we look at the basic feedback equation, the 00:32:45.110 --> 00:32:48.690 overall system function is the system function for the 00:32:48.690 --> 00:32:52.840 forward path divided by 1 plus the loop gain, G of 00:32:52.840 --> 00:32:55.070 s times H of s. 00:32:55.070 --> 00:32:59.790 And for stability what we want to examine are the roots of 1 00:32:59.790 --> 00:33:02.030 plus G of s times H of s. 00:33:02.030 --> 00:33:08.880 And in particular, the poles are the zeroes of that factor. 00:33:08.880 --> 00:33:12.720 And as long as we choose G of s, so that the 00:33:12.720 --> 00:33:15.490 poles of this term-- 00:33:15.490 --> 00:33:21.700 I'm sorry, so that the zeroes of this term are in the left 00:33:21.700 --> 00:33:24.100 half of the s-plane, then what we'll 00:33:24.100 --> 00:33:25.730 end up with is stability. 00:33:25.730 --> 00:33:29.530 So stability is dependent not just on h of s for the 00:33:29.530 --> 00:33:34.310 closed-loop system, but on 1 plus G of s times H of s. 00:33:34.310 --> 00:33:40.860 And this kind of notion is used in lots of situations. 00:33:40.860 --> 00:33:42.610 I indicated the inverted pendulum. 00:33:42.610 --> 00:33:47.780 Another very common example is in some very high-performance 00:33:47.780 --> 00:33:52.910 aircraft where the basic aircraft system 00:33:52.910 --> 00:33:55.060 is an unstable system. 00:33:55.060 --> 00:33:59.800 But in fact, it's stabilized by putting the right kind of 00:33:59.800 --> 00:34:02.530 feedback dynamics around it. 00:34:02.530 --> 00:34:05.200 And those feedback dynamics might, in fact, involve the 00:34:05.200 --> 00:34:07.870 pilot as well. 00:34:07.870 --> 00:34:17.790 Now, for the system that we just talked about, the 00:34:17.790 --> 00:34:22.520 stability was described in terms of a 00:34:22.520 --> 00:34:24.449 continuous-time system. 00:34:24.449 --> 00:34:27.540 And the stability condition that we end up with, of 00:34:27.540 --> 00:34:32.710 course, relates to the zeroes of this denominator term. 00:34:32.710 --> 00:34:36.850 And we require for stability that the real parts of the 00:34:36.850 --> 00:34:41.600 associated roots be in the left half of the s-plane. 00:34:41.600 --> 00:34:44.610 Exactly the same kind of analysis, in terms of 00:34:44.610 --> 00:34:48.690 stability, applies in discrete-time. 00:34:48.690 --> 00:34:55.580 That is, in discrete-time, as we saw previously, the basic 00:34:55.580 --> 00:34:58.630 discrete-time feedback system is exactly the same, except 00:34:58.630 --> 00:35:02.060 that the independent variable is now an integer variable 00:35:02.060 --> 00:35:04.320 rather than a continuous variable. 00:35:04.320 --> 00:35:07.460 The feedback equation is exactly the same. 00:35:07.460 --> 00:35:11.750 So to analyze stability of the feedback system, we would want 00:35:11.750 --> 00:35:17.080 to look at the zeroes of 1 plus G of z times H of z. 00:35:17.080 --> 00:35:21.250 So again, it's those zeroes that affect stability. 00:35:21.250 --> 00:35:25.580 And the principal difference between the continuous-time 00:35:25.580 --> 00:35:30.550 and discrete-time cases is the fact that the stability 00:35:30.550 --> 00:35:35.220 condition in discrete-time is different than it is in 00:35:35.220 --> 00:35:36.030 continuous-time. 00:35:36.030 --> 00:35:42.270 Namely, in continuous-time, we care for stability about poles 00:35:42.270 --> 00:35:46.960 of the overall system being in the left half of the s-plane 00:35:46.960 --> 00:35:48.690 or the right half of the s-plane. 00:35:48.690 --> 00:35:53.060 In discrete-time, what we care about is whether the poles are 00:35:53.060 --> 00:35:55.420 inside or outside the unit circle. 00:35:55.420 --> 00:36:00.530 So in the discrete-time case, what we would impose for 00:36:00.530 --> 00:36:04.390 stability is that the zeroes have a magnitude which 00:36:04.390 --> 00:36:06.060 is less than 1. 00:36:06.060 --> 00:36:12.090 So the basic analysis is the same, but the details of the 00:36:12.090 --> 00:36:15.700 stability condition, of course, are different. 00:36:15.700 --> 00:36:22.660 Now, what I've just indicated is that feedback can be used 00:36:22.660 --> 00:36:27.200 to stabilize an unstable system. 00:36:27.200 --> 00:36:30.660 And as you can imagine there's the other side of the coin. 00:36:30.660 --> 00:36:37.120 Namely, if you start with a stable system and put feedback 00:36:37.120 --> 00:36:40.300 around it, if you're not careful what can happen, in 00:36:40.300 --> 00:36:43.170 fact, is that you can destabilize the system. 00:36:43.170 --> 00:36:46.380 So there's always the potential hazard, unless it's 00:36:46.380 --> 00:36:50.610 something you want to have happen, that feedback around 00:36:50.610 --> 00:36:54.140 what used to be a stable system now generates a system 00:36:54.140 --> 00:36:55.520 which is unstable. 00:36:55.520 --> 00:36:57.860 And there are lots of examples of that. 00:36:57.860 --> 00:37:01.910 One very common example is in audio systems. 00:37:01.910 --> 00:37:04.800 And this is probably an example that you're somewhat 00:37:04.800 --> 00:37:06.880 familiar with. 00:37:06.880 --> 00:37:12.550 Basically, an audio system, if you have the speaker and the 00:37:12.550 --> 00:37:15.980 microphone in any kind of proximity to each other is, in 00:37:15.980 --> 00:37:18.190 fact, a feedback system. 00:37:18.190 --> 00:37:23.590 Well, first of all, the audio input to the microphone 00:37:23.590 --> 00:37:31.360 consists of the external audio inputs, and the external audio 00:37:31.360 --> 00:37:35.360 inputs might, for example, be my voice. 00:37:35.360 --> 00:37:37.580 It might be the room noise. 00:37:37.580 --> 00:37:40.140 And in fact, as we'll illustrate shortly if I'm not 00:37:40.140 --> 00:37:44.490 careful, might in fact be the output from a speaker, which 00:37:44.490 --> 00:37:46.360 represents feedback. 00:37:46.360 --> 00:37:52.410 That audio, of course, after appropriate amplification 00:37:52.410 --> 00:37:56.680 drives a speaker. 00:37:56.680 --> 00:38:02.730 And if, in fact, the speaker is, let's say has any 00:38:02.730 --> 00:38:07.580 proximity to the microphone, then there can be a certain 00:38:07.580 --> 00:38:13.200 amount of the output of the speaker that feeds back around 00:38:13.200 --> 00:38:18.560 and is fed back into the microphone. 00:38:18.560 --> 00:38:22.200 Now, the system function associated with the feedback I 00:38:22.200 --> 00:38:26.280 indicate here as a constant times e to the minus s times 00:38:26.280 --> 00:38:30.940 capital T. The e to the minus s times capital T represents 00:38:30.940 --> 00:38:36.690 the fact that there is, in general, some delay between 00:38:36.690 --> 00:38:41.110 the time delay between the speaker output and the input 00:38:41.110 --> 00:38:43.730 that it generates to the microphone. 00:38:43.730 --> 00:38:47.410 The reason for that delay of course, being that there may 00:38:47.410 --> 00:38:50.570 be some distance between the speaker and the microphone. 00:38:50.570 --> 00:38:54.850 And then the constant K2 that I have in the feedback path 00:38:54.850 --> 00:38:57.580 represents the fact that between the speaker and the 00:38:57.580 --> 00:39:01.050 microphone, there may be some attenuation. 00:39:01.050 --> 00:39:04.970 So if I have, for example, a speaker as I happen to have 00:39:04.970 --> 00:39:13.970 here, and I were to have that speaker putting out what in 00:39:13.970 --> 00:39:16.120 fact I'm putting into the microphone, or the output of 00:39:16.120 --> 00:39:19.640 the microphone, then what we have is a feedback path. 00:39:19.640 --> 00:39:23.050 And the feedback path is from the microphone, through the 00:39:23.050 --> 00:39:26.990 speaker, out of the speaker, back into the microphone. 00:39:26.990 --> 00:39:32.700 And the feedback path is from here to the microphone. 00:39:32.700 --> 00:39:36.230 And the characteristics or frequency response or system 00:39:36.230 --> 00:39:39.390 function is associated with the characteristics of 00:39:39.390 --> 00:39:41.580 propagation or transmission. 00:39:41.580 --> 00:39:44.440 If I were to move closer to the speaker and I, by the way, 00:39:44.440 --> 00:39:46.825 don't have the speaker on right now. 00:39:46.825 --> 00:39:50.690 And I'm sure you all understand why. 00:39:50.690 --> 00:39:58.480 If I move closer, then the constant K2 gets what? 00:39:58.480 --> 00:39:59.750 Gets larger. 00:39:59.750 --> 00:40:04.040 And if I move further away the constant K2 gets smaller. 00:40:04.040 --> 00:40:08.120 Well, let's look at an analysis of this and see what 00:40:08.120 --> 00:40:11.020 it is, or why it is, that in fact we get an 00:40:11.020 --> 00:40:12.970 instability in terms-- 00:40:12.970 --> 00:40:15.420 or that an instability is predicted by the basic 00:40:15.420 --> 00:40:17.550 feedback equation. 00:40:17.550 --> 00:40:20.760 Now, notice first of all, that we're talking about positive 00:40:20.760 --> 00:40:22.160 feedback here. 00:40:22.160 --> 00:40:27.210 And just simply substituting the appropriate system 00:40:27.210 --> 00:40:31.690 functions into our basic feedback equation, we have an 00:40:31.690 --> 00:40:35.180 equation that says that the overall system function is 00:40:35.180 --> 00:40:38.260 given by the forward gain, which is the gain of the 00:40:38.260 --> 00:40:42.040 amplifier between the microphone and the speaker, 00:40:42.040 --> 00:40:45.190 divided by 1 minus-- 00:40:45.190 --> 00:40:48.120 and the minus because we have positive feedback-- 00:40:48.120 --> 00:40:52.350 the overall loop gain, which is K1, K2, e to the minus s 00:40:52.350 --> 00:40:58.000 capital T. And these two gains, K1 and K2 are assumed 00:40:58.000 --> 00:41:01.030 to be positive, and generally are positive. 00:41:01.030 --> 00:41:06.280 So in order for us to-- 00:41:06.280 --> 00:41:09.620 well, if we want to look at the poles of the system, then 00:41:09.620 --> 00:41:12.770 we want to look at the zeroes of this denominator. 00:41:12.770 --> 00:41:16.190 And the zeroes of this denominator occur at values of 00:41:16.190 --> 00:41:21.000 s such that e to the minus s capital T is equal to 1 over 00:41:21.000 --> 00:41:23.020 K1 times K2. 00:41:23.020 --> 00:41:28.940 And equivalently that says that the poles of the closed 00:41:28.940 --> 00:41:35.260 loop system occur at 1 over capital T, and capital T is 00:41:35.260 --> 00:41:37.220 related to the time delay. 00:41:37.220 --> 00:41:43.460 1 over capital T times the log to the base e of K1 times K2. 00:41:43.460 --> 00:41:47.880 Well, for stability we want these poles to all be in the 00:41:47.880 --> 00:41:49.800 left half of the s-plane. 00:41:49.800 --> 00:41:54.040 And what that means then is that for stability what we 00:41:54.040 --> 00:41:59.010 require is that K1 times K2 be less than 1. 00:41:59.010 --> 00:42:02.840 In other words, we require that the overall loop gain be 00:42:02.840 --> 00:42:06.100 less-- the magnitude of the loop gain be less than 1. 00:42:06.100 --> 00:42:09.620 If it's not, then what we generate is an instability. 00:42:09.620 --> 00:42:13.990 And just to illustrate that, let's turn the speaker on. 00:42:13.990 --> 00:42:18.090 And what we'll demonstrate is feedback. 00:42:18.090 --> 00:42:20.340 Right now the system is stable. 00:42:20.340 --> 00:42:23.320 And I'm being careful to keep my distance from the speaker. 00:42:23.320 --> 00:42:29.090 As I get closer, K2 will increase. 00:42:29.090 --> 00:42:33.280 And as K2 increases, eventually the poles will move 00:42:33.280 --> 00:42:36.640 into the right half of the s-plane, or they'll try to. 00:42:36.640 --> 00:42:39.060 What will happen is that the system will start to oscillate 00:42:39.060 --> 00:42:41.230 and go into nonlinear distortion. 00:42:41.230 --> 00:42:46.350 So as I get closer, you can hear that we get feedback, we 00:42:46.350 --> 00:42:48.010 get oscillation. 00:42:48.010 --> 00:42:53.250 And I guess neither you nor I can take too much of that. 00:42:53.250 --> 00:42:58.120 But you can see that what's happening-- 00:42:58.120 --> 00:43:00.680 if we can just turn the speaker off now. 00:43:00.680 --> 00:43:05.310 You can see that what's happening is that as K2 00:43:05.310 --> 00:43:10.660 increases, the poles are moving on to the j omega axis, 00:43:10.660 --> 00:43:12.490 the system starts to oscillate. 00:43:12.490 --> 00:43:15.450 They won't actually move into the right half plane because 00:43:15.450 --> 00:43:19.650 there are nonlinearities that inherently control the system. 00:43:22.800 --> 00:43:31.010 OK, so what we've seen in today's lecture is the basic 00:43:31.010 --> 00:43:34.950 analysis equation and a few of the applications. 00:43:34.950 --> 00:43:39.960 And one application, or one both application and hazard 00:43:39.960 --> 00:43:44.870 that we've talked about, is the application in which we 00:43:44.870 --> 00:43:47.170 may stabilize unstable systems. 00:43:47.170 --> 00:43:51.580 Or if we're not careful, destabilize stable systems. 00:43:51.580 --> 00:43:55.560 As I've indicated at several times during the lecture, one 00:43:55.560 --> 00:44:01.210 common example of an unstable system which feedback can be 00:44:01.210 --> 00:44:05.970 used to stabilize is the inverted pendulum, which I've 00:44:05.970 --> 00:44:08.090 referred to several times. 00:44:08.090 --> 00:44:13.210 And in the next lecture, what I'd like to do is focus in on 00:44:13.210 --> 00:44:15.410 a more detailed analysis of this. 00:44:15.410 --> 00:44:20.860 And what we'll see, in fact, is that the feedback dynamics, 00:44:20.860 --> 00:44:24.760 the form of the feedback dynamics are important with 00:44:24.760 --> 00:44:27.750 regard to whether you can and can't stabilize the system. 00:44:27.750 --> 00:44:30.630 Interestingly enough, for this particular system, as we'll 00:44:30.630 --> 00:44:34.520 see in the next lecture, if you simply try to measure the 00:44:34.520 --> 00:44:40.030 angle and feed that back that, in fact, you can't stabilize 00:44:40.030 --> 00:44:40.780 the system. 00:44:40.780 --> 00:44:45.140 What it requires is not only the angle, but some 00:44:45.140 --> 00:44:47.870 information about the rate of change of angle. 00:44:47.870 --> 00:44:49.760 But we'll see that in much more 00:44:49.760 --> 00:44:51.120 detail in the next lecture. 00:44:51.120 --> 00:44:52.370 Thank you.