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 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.750 ocw.mit.edu. 00:00:20.750 --> 00:00:55.770 [MUSIC PLAYING] 00:00:55.770 --> 00:00:59.530 PROFESSOR: Last time, we began a discussion of feedback. 00:00:59.530 --> 00:01:02.240 And I briefly introduced a number of applications. 00:01:02.240 --> 00:01:05.970 For example, the use of feedback in compensating for 00:01:05.970 --> 00:01:08.400 non-ideal elements in a system. 00:01:08.400 --> 00:01:12.900 And also, as another example, the use of feedback in 00:01:12.900 --> 00:01:16.050 stabilizing unstable systems. 00:01:16.050 --> 00:01:19.680 In this final lecture, what I'd like to do is focus in 00:01:19.680 --> 00:01:22.980 more detail on the use of feedback to 00:01:22.980 --> 00:01:25.930 stabilize unstable systems. 00:01:25.930 --> 00:01:29.610 And the specific context in which I'd like to do that is 00:01:29.610 --> 00:01:32.640 in the context of the inverted pendulum. 00:01:32.640 --> 00:01:36.230 Now, I've referred to the inverted pendulum in several 00:01:36.230 --> 00:01:37.610 past lectures. 00:01:37.610 --> 00:01:42.130 And, basically, as everyone realizes, a pendulum is 00:01:42.130 --> 00:01:45.460 essentially a rod with a weight on the bottom. 00:01:45.460 --> 00:01:48.130 And, naturally, an inverted pendulum is just 00:01:48.130 --> 00:01:49.700 that, upside down. 00:01:49.700 --> 00:01:53.740 And so it's essentially a rod that's top-heavy. 00:01:53.740 --> 00:01:58.840 And the idea with the inverted pendulum is, although by 00:01:58.840 --> 00:02:03.270 itself it's unstable around a pivot point at the bottom, the 00:02:03.270 --> 00:02:07.840 idea is to apply an external input, an acceleration, 00:02:07.840 --> 00:02:09.509 essentially to keep it balanced. 00:02:09.509 --> 00:02:14.300 And we've all probably done this type of thing at some 00:02:14.300 --> 00:02:17.970 point in our lives, either childhood or not. 00:02:17.970 --> 00:02:22.930 And so that's the system that I'd like to analyze and then 00:02:22.930 --> 00:02:25.240 illustrate in this lecture. 00:02:25.240 --> 00:02:29.630 Now, the context in which we'll do that is not with my 00:02:29.630 --> 00:02:34.360 son's horse, but actually with an inverted pendulum which is 00:02:34.360 --> 00:02:35.830 mounted on a cart. 00:02:35.830 --> 00:02:39.220 And first I'd like to go through an analysis of it, and 00:02:39.220 --> 00:02:43.110 then we'll actually see how the system works. 00:02:43.110 --> 00:02:47.950 So basically, then, the system that we're talking about is 00:02:47.950 --> 00:02:51.940 going to be a system which is a movable cart. 00:02:51.940 --> 00:02:56.140 Mounted on the cart is a rod with a weight on the top. 00:02:56.140 --> 00:02:58.910 And this then becomes the inverted pendulum. 00:02:58.910 --> 00:03:05.140 And the cart has an external acceleration applied to it, 00:03:05.140 --> 00:03:09.370 which is the input that we can apply to the system. 00:03:09.370 --> 00:03:13.220 And then, in general, we can expect some disturbances. 00:03:13.220 --> 00:03:17.670 And I'm going to represent the disturbances in terms of an 00:03:17.670 --> 00:03:21.860 angular acceleration, which shows up around the weight at 00:03:21.860 --> 00:03:24.550 the top of the rod. 00:03:24.550 --> 00:03:28.070 So if we look at the system, then, as I've kind of 00:03:28.070 --> 00:03:33.840 indicated it here, basically we have two inputs. 00:03:33.840 --> 00:03:37.580 We have an input, which are the external disturbances, 00:03:37.580 --> 00:03:41.580 which we can assume that we have no control over. 00:03:41.580 --> 00:03:45.760 And then there is the acceleration that's apply to 00:03:45.760 --> 00:03:50.620 the cart externally, or in the case of balancing my son's 00:03:50.620 --> 00:03:53.480 horse, it's the movement of my hand. 00:03:53.480 --> 00:03:57.760 And then, through the system dynamics, that ends up 00:03:57.760 --> 00:04:02.680 influencing the angle of the rod. 00:04:02.680 --> 00:04:06.680 And we'll think of the angle of the rod 00:04:06.680 --> 00:04:08.760 as the system output. 00:04:08.760 --> 00:04:14.250 And basically the idea, then, is to try to keep that angle 00:04:14.250 --> 00:04:19.680 at 0 by applying the appropriate acceleration. 00:04:19.680 --> 00:04:24.360 Now, as I've mentioned several times previously, if we know 00:04:24.360 --> 00:04:29.000 exactly what the system dynamics are, and if we know 00:04:29.000 --> 00:04:31.910 exactly what the external disturbances are, then 00:04:31.910 --> 00:04:36.030 theoretically we can choose an acceleration for the cart 00:04:36.030 --> 00:04:40.430 which will exactly balance the system, or balance the rod. 00:04:40.430 --> 00:04:44.390 However, since the system is inherently unstable, any 00:04:44.390 --> 00:04:48.620 deviation from that model-- in particular, any unanticipated 00:04:48.620 --> 00:04:50.580 external disturbances-- 00:04:50.580 --> 00:04:52.250 will excite the instability. 00:04:52.250 --> 00:04:55.460 And what that means is that the rod will fall. 00:04:55.460 --> 00:05:01.710 So the idea, then, is, as we'll see, to stabilize the 00:05:01.710 --> 00:05:04.950 system by placing feedback around it. 00:05:04.950 --> 00:05:09.190 In particular through a measurement of the output 00:05:09.190 --> 00:05:14.500 angle, processed through some appropriate feedback dynamics, 00:05:14.500 --> 00:05:18.580 using that to control the acceleration of the cart. 00:05:18.580 --> 00:05:22.160 And if we choose the feedback dynamics correctly, then, in 00:05:22.160 --> 00:05:27.030 fact, we can end up with a stable system, even though the 00:05:27.030 --> 00:05:29.760 open-loop system is unstable. 00:05:29.760 --> 00:05:34.450 Well, as a first step, let's analyze the system in its 00:05:34.450 --> 00:05:35.880 open-loop form. 00:05:35.880 --> 00:05:40.090 In particular, let's analyze the system without feedback, 00:05:40.090 --> 00:05:44.390 demonstrate that, indeed, it is unstable, and then see how 00:05:44.390 --> 00:05:47.640 we can stabilize it using feedback. 00:05:47.640 --> 00:05:50.770 So the system, then, is-- 00:05:50.770 --> 00:05:55.610 as we've indicated, the variables involved are the 00:05:55.610 --> 00:06:00.460 measured angle, which represents the output, the 00:06:00.460 --> 00:06:04.680 angular acceleration due to external disturbances, and 00:06:04.680 --> 00:06:07.260 then there is the applied external 00:06:07.260 --> 00:06:09.720 acceleration on the cart. 00:06:09.720 --> 00:06:13.530 So these represent the variables, with L representing 00:06:13.530 --> 00:06:17.300 the length of the rod, and s of t I indicate here, which we 00:06:17.300 --> 00:06:20.430 won't really be paying attention to, as the position 00:06:20.430 --> 00:06:22.230 of the cart. 00:06:22.230 --> 00:06:28.040 And without going into the details specifically, what 00:06:28.040 --> 00:06:30.830 we'll do is set up the equation, or write the 00:06:30.830 --> 00:06:34.290 equation, in terms of balancing the acceleration. 00:06:34.290 --> 00:06:38.080 And if you go through that process, then the basic 00:06:38.080 --> 00:06:41.890 equation that you end up with is the equation that I 00:06:41.890 --> 00:06:43.730 indicate here. 00:06:43.730 --> 00:06:48.710 And this equation, then, tells us how the basic forces, or 00:06:48.710 --> 00:06:52.330 accelerations, are balanced, where this is the second 00:06:52.330 --> 00:06:55.900 derivative of the angle. 00:06:55.900 --> 00:06:59.590 And then we have the acceleration due to gravity, 00:06:59.590 --> 00:07:05.080 reflected through the angular acceleration times the sine of 00:07:05.080 --> 00:07:09.190 the angle, the angular acceleration due to the 00:07:09.190 --> 00:07:12.960 external disturbances, and finally, the angular 00:07:12.960 --> 00:07:16.060 acceleration due to the motion of the cart. 00:07:16.060 --> 00:07:19.730 Now, this equation, as it's written here, is a nonlinear 00:07:19.730 --> 00:07:23.830 equation in the angle theta of t. 00:07:23.830 --> 00:07:28.100 And what we'd like to do is linearize the equation. 00:07:28.100 --> 00:07:31.160 And we'll linearize the equation by making the 00:07:31.160 --> 00:07:35.670 assumption that the angle is very small, close to 0. 00:07:35.670 --> 00:07:38.200 In other words, what we'll assume is that we're able to 00:07:38.200 --> 00:07:40.970 keep the rod relatively vertical. 00:07:40.970 --> 00:07:44.180 And our analysis, because we're linearizing the 00:07:44.180 --> 00:07:46.850 equations, will obviously depend on that. 00:07:46.850 --> 00:07:51.770 So making the assumption that the angle is small, we then 00:07:51.770 --> 00:07:56.070 assume that the sine of the angle is approximately equal 00:07:56.070 --> 00:08:01.050 to the angle, and the cosine of the angle is approximately 00:08:01.050 --> 00:08:02.480 equal to 1. 00:08:02.480 --> 00:08:04.980 That will then linearize this equation. 00:08:04.980 --> 00:08:10.230 And the resulting equation, then, in its linearized form, 00:08:10.230 --> 00:08:13.380 is the one that I indicate here. 00:08:13.380 --> 00:08:20.000 So we have an equation of motion, which linearizes the 00:08:20.000 --> 00:08:22.260 balance of the accelerations. 00:08:22.260 --> 00:08:24.730 And what we see on the right-hand side of the 00:08:24.730 --> 00:08:29.970 equation is the combined inputs due to the angular 00:08:29.970 --> 00:08:33.659 acceleration of the rod and the acceleration of the cart. 00:08:33.659 --> 00:08:37.950 And on the left-hand side of the equation, we have the 00:08:37.950 --> 00:08:42.419 other forces due to the change in the angle. 00:08:42.419 --> 00:08:44.990 So this is, then, the differential equation 00:08:44.990 --> 00:08:49.700 associated with the open-loop system, the basic dynamics of 00:08:49.700 --> 00:08:50.900 the system. 00:08:50.900 --> 00:08:55.280 And if we apply the Laplace transform to this equation and 00:08:55.280 --> 00:09:00.350 solve for the system function, then the basic equation that 00:09:00.350 --> 00:09:03.870 we're left with expresses the Laplace transform of the 00:09:03.870 --> 00:09:09.080 angle, equal to the system function, the open-loop system 00:09:09.080 --> 00:09:13.220 function, times the Laplace transform of 00:09:13.220 --> 00:09:14.750 the combined inputs. 00:09:14.750 --> 00:09:18.190 And I remind you again that our assumption is that this is 00:09:18.190 --> 00:09:21.080 an input that we have no control over. 00:09:21.080 --> 00:09:24.180 This is the input that we can control. 00:09:24.180 --> 00:09:26.250 And the two together, of course, form 00:09:26.250 --> 00:09:27.990 the combined input. 00:09:27.990 --> 00:09:31.650 Now, we can, of course, solve for the poles and 0's. 00:09:31.650 --> 00:09:33.190 There are no 0's. 00:09:33.190 --> 00:09:35.310 And there are two poles, since this is a second-order 00:09:35.310 --> 00:09:36.760 denominator. 00:09:36.760 --> 00:09:40.800 And looking at the poles in the s-plane then, we see that 00:09:40.800 --> 00:09:44.750 we have a pair of poles, one at minus the square root of g 00:09:44.750 --> 00:09:49.860 over L, and one at plus the square root of g over L. 00:09:49.860 --> 00:09:54.390 And the important observation, then, is that while this pole 00:09:54.390 --> 00:09:59.100 represents a stable pole, the right half-plane pole 00:09:59.100 --> 00:10:01.470 represents an unstable pole. 00:10:01.470 --> 00:10:06.620 And so this system, in fact, is an unstable system. 00:10:06.620 --> 00:10:11.170 It's unstable because what we have is a pole in the right 00:10:11.170 --> 00:10:15.630 half-plane for the open-loop system. 00:10:15.630 --> 00:10:19.750 Now, before we see how to stabilize the system, let's, 00:10:19.750 --> 00:10:25.650 in fact, look at the mechanical setup, which we 00:10:25.650 --> 00:10:30.540 won't turn on for now, and just see essentially what this 00:10:30.540 --> 00:10:34.830 instability means and what the physical orientation of the 00:10:34.830 --> 00:10:36.670 equipment is. 00:10:36.670 --> 00:10:40.090 So what we have, which you just saw a caricature of in 00:10:40.090 --> 00:10:45.310 the view graph, is an inverted pendulum. 00:10:45.310 --> 00:10:48.270 This represents the pendulum. 00:10:48.270 --> 00:10:52.550 And it's mounted with the pivot point at the bottom. 00:10:52.550 --> 00:10:54.340 It's mounted on a cart. 00:10:54.340 --> 00:10:57.020 And here we have the cart. 00:10:57.020 --> 00:11:00.830 And this is a pivot point. 00:11:00.830 --> 00:11:04.100 And, as you can see, the cart can move back 00:11:04.100 --> 00:11:06.110 and forth on a track. 00:11:06.110 --> 00:11:10.860 And the external acceleration, which is applied to the cart, 00:11:10.860 --> 00:11:14.120 is applied through this cable. 00:11:14.120 --> 00:11:18.480 And that cable is controlled by a motor. 00:11:18.480 --> 00:11:23.300 And we have the motor at this end of the track. 00:11:23.300 --> 00:11:27.650 And so as the motor turns, then the cart will 00:11:27.650 --> 00:11:29.760 move back and forth. 00:11:29.760 --> 00:11:30.250 OK. 00:11:30.250 --> 00:11:34.350 Now, since the system is not turned on 00:11:34.350 --> 00:11:36.080 and there's no feedback-- 00:11:36.080 --> 00:11:40.480 in fact, the system, as we just saw in the transparency, 00:11:40.480 --> 00:11:42.710 is an unstable system. 00:11:42.710 --> 00:11:47.050 And what that instability means is that, for example, if 00:11:47.050 --> 00:11:53.190 I set the angle to 0 and then let it go, then as soon as 00:11:53.190 --> 00:11:55.310 there's a slight external disturbance, it 00:11:55.310 --> 00:11:57.090 will start to fall. 00:11:57.090 --> 00:12:02.370 Now, you can imagine that not only is the system unstable as 00:12:02.370 --> 00:12:06.530 it is, but certainly can't accommodate changes in the 00:12:06.530 --> 00:12:07.690 system dynamics. 00:12:07.690 --> 00:12:09.660 For example, if we change, let's say, the 00:12:09.660 --> 00:12:10.990 weight of the pendulum. 00:12:10.990 --> 00:12:15.960 So if we thought, for example, of changing the weight by, 00:12:15.960 --> 00:12:19.350 let's say, putting something like a glass with something in 00:12:19.350 --> 00:12:23.190 it on top, and I think about trying to balance it by 00:12:23.190 --> 00:12:25.760 itself, obviously that's hard. 00:12:25.760 --> 00:12:29.030 In fact, I would say, without turning the system on, 00:12:29.030 --> 00:12:31.710 guaranteed to be impossible. 00:12:31.710 --> 00:12:36.430 So the basic system, as we've just analyzed it, is 00:12:36.430 --> 00:12:38.630 inherently an unstable system. 00:12:38.630 --> 00:12:42.860 The instability reflecting in the fact that the pendulum, 00:12:42.860 --> 00:12:46.090 following its own natural forces, will tend to fall. 00:12:46.090 --> 00:12:50.160 And now what we want to look at is how, through the use of 00:12:50.160 --> 00:12:55.820 feedback, we can, in fact, stabilize the system. 00:12:55.820 --> 00:13:05.160 So let's first, once again, look at the open-loop system. 00:13:05.160 --> 00:13:09.410 And the open-loop system function that we saw was a 00:13:09.410 --> 00:13:14.580 system function of this form, 1 over Ls squared minus g, and 00:13:14.580 --> 00:13:18.930 that represents the system function associated with the 00:13:18.930 --> 00:13:22.390 two inputs, one being the external disturbances, the 00:13:22.390 --> 00:13:25.290 other being the externally applied acceleration. 00:13:25.290 --> 00:13:29.470 And the system function here 00:13:29.470 --> 00:13:31.200 represents the system dynamics. 00:13:31.200 --> 00:13:35.670 And the resulting output is the angle. 00:13:35.670 --> 00:13:40.230 Now, the feedback, the basic strategy behind the feedback, 00:13:40.230 --> 00:13:45.110 is to in some way use the measured angle, make a 00:13:45.110 --> 00:13:49.000 measurement of the angle, and use that through appropriate 00:13:49.000 --> 00:13:52.160 feedback dynamics to stabilize the system. 00:13:52.160 --> 00:13:56.920 And so with feedback applied around the system then, there 00:13:56.920 --> 00:14:00.770 would be some measurement of the angle through some 00:14:00.770 --> 00:14:03.550 appropriately chosen feedback dynamics. 00:14:03.550 --> 00:14:09.490 And that then would determine for us the applied external 00:14:09.490 --> 00:14:13.420 acceleration corresponding to what the motor does by pulling 00:14:13.420 --> 00:14:15.210 the cable back and forth. 00:14:15.210 --> 00:14:19.070 And we would like to choose G of s so that the overall 00:14:19.070 --> 00:14:21.270 system is stable. 00:14:21.270 --> 00:14:24.980 Now, as you recall from the lecture last time, with the 00:14:24.980 --> 00:14:28.890 feedback around the system and expressed here in terms of 00:14:28.890 --> 00:14:34.480 negative feedback, the overall transfer function then is the 00:14:34.480 --> 00:14:38.030 transfer function associated with the basic feedback loop, 00:14:38.030 --> 00:14:43.420 where H of s is the open-loop system and G of s corresponds 00:14:43.420 --> 00:14:46.160 to the feedback dynamics. 00:14:46.160 --> 00:14:49.070 Now, that's the basic strategy. 00:14:49.070 --> 00:14:52.490 We haven't decided yet how to choose the feedback dynamics. 00:14:52.490 --> 00:14:54.710 And that becomes the next step. 00:14:54.710 --> 00:14:57.500 And we want to choose the dynamics in such a way that 00:14:57.500 --> 00:15:01.840 the system ends up being stabilized. 00:15:01.840 --> 00:15:06.290 Well, I think the thing to do is just simply begin with what 00:15:06.290 --> 00:15:10.060 would seem to be the most obvious, which is a simple 00:15:10.060 --> 00:15:11.750 measurement of the angle. 00:15:11.750 --> 00:15:16.910 Let's take the angular measurements, feed that back, 00:15:16.910 --> 00:15:19.770 let's say through a potentiometer and perhaps an 00:15:19.770 --> 00:15:22.850 amplifier, so that there's some gain, and see if we can 00:15:22.850 --> 00:15:27.410 stabilize the system simply through feedback which is 00:15:27.410 --> 00:15:30.160 proportional to a measurement of the angle, what. 00:15:30.160 --> 00:15:36.710 Is typically referred to as "proportional feedback." Well, 00:15:36.710 --> 00:15:40.200 let's analyze the results of doing that. 00:15:40.200 --> 00:15:46.380 Once again we have the basic feedback equation, where the 00:15:46.380 --> 00:15:50.730 open-loop transfer function is what we had developed 00:15:50.730 --> 00:15:54.670 previously, given by the second-order expression. 00:15:54.670 --> 00:15:59.570 Now, using proportional feedback, we choose the 00:15:59.570 --> 00:16:05.145 acceleration a of t to be directly proportional through 00:16:05.145 --> 00:16:09.260 a gain constant or attenuation constant K1, directly 00:16:09.260 --> 00:16:12.690 proportional to the measured angle theta of t. 00:16:12.690 --> 00:16:16.970 And consequently the system function in the feedback path 00:16:16.970 --> 00:16:20.400 is just simply a gain or attenuate K1. 00:16:20.400 --> 00:16:23.270 So this, then, is the system function for 00:16:23.270 --> 00:16:25.270 the feedback path. 00:16:25.270 --> 00:16:31.720 Well, substituting this into the closed-loop expression, 00:16:31.720 --> 00:16:36.970 then the overall expression for the Laplace transform of 00:16:36.970 --> 00:16:42.750 the output angle is, as I indicate here, namely that 00:16:42.750 --> 00:16:48.170 theta of s is proportional through this system function. 00:16:48.170 --> 00:16:52.570 So the Laplace transform of the input X of t-- and let me 00:16:52.570 --> 00:16:56.930 remind you that X of t, the external disturbances, now 00:16:56.930 --> 00:17:00.040 represent the only input, since the other input 00:17:00.040 --> 00:17:03.300 corresponding to the applied acceleration to the cart is 00:17:03.300 --> 00:17:06.930 now controlled only through the feedback loop. 00:17:06.930 --> 00:17:09.950 So we have this system function, then, for the 00:17:09.950 --> 00:17:11.579 closed-loop system. 00:17:11.579 --> 00:17:15.990 We recognize this once again as a second-order system. 00:17:15.990 --> 00:17:20.960 And the poles of this second-order system are then 00:17:20.960 --> 00:17:25.339 given by plus and minus the square root of g minus K1, 00:17:25.339 --> 00:17:31.550 divided by L. So K1, the feedback constant, clearly 00:17:31.550 --> 00:17:34.320 influences the position of the poles. 00:17:34.320 --> 00:17:37.960 And let's look, in fact, at where the 00:17:37.960 --> 00:17:42.200 poles are in the s-plane. 00:17:42.200 --> 00:17:48.090 Well, first of all, with K1 equal to 0, which, of course, 00:17:48.090 --> 00:17:52.290 is no feedback and corresponds to the open-loop system, we 00:17:52.290 --> 00:17:56.480 have the poles where they were previously. 00:17:56.480 --> 00:18:03.510 And if we now choose K1, let's say less than 0, then what 00:18:03.510 --> 00:18:09.690 will happen as K1 becomes more and more negative, so that 00:18:09.690 --> 00:18:12.965 this term is more and more positive, is that the left 00:18:12.965 --> 00:18:16.425 half-plane pole will move further into the left 00:18:16.425 --> 00:18:20.150 half-plane, the right half-plane pole will move 00:18:20.150 --> 00:18:22.560 further into the right half-plane. 00:18:22.560 --> 00:18:28.960 So clearly with K1 negative, this pole, which represents an 00:18:28.960 --> 00:18:34.010 instability, becomes even more unstable. 00:18:34.010 --> 00:18:38.450 Well, instead of K1 negative, let's try K1 positive and see 00:18:38.450 --> 00:18:39.740 what happens. 00:18:39.740 --> 00:18:47.890 And with K1 positive, what happens in this case is that 00:18:47.890 --> 00:18:53.900 the left half-plane pole moves closer to the origin, the 00:18:53.900 --> 00:18:57.420 right half-plane pole moves closer to the origin. 00:18:57.420 --> 00:19:00.360 What one would hope is that they both end up in the left 00:19:00.360 --> 00:19:01.680 half-plane at some point. 00:19:01.680 --> 00:19:03.450 But, in fact, they don't. 00:19:03.450 --> 00:19:06.370 What happens is that eventually they both reach the 00:19:06.370 --> 00:19:11.010 origin, split at that point, and travel along the 00:19:11.010 --> 00:19:13.360 j-omega-axis. 00:19:13.360 --> 00:19:19.020 Now, this movement of the poles, as we vary K1 either 00:19:19.020 --> 00:19:22.350 positive or negative, what we see is that with the open-loop 00:19:22.350 --> 00:19:26.760 system, as we introduce feedback, those basic poles 00:19:26.760 --> 00:19:31.080 move in the s-plane, either this way, as K1 becomes more 00:19:31.080 --> 00:19:35.010 negative, or for this particular case, if K1 is 00:19:35.010 --> 00:19:37.300 positive, they move in together, split, and move 00:19:37.300 --> 00:19:39.330 along the j-omega-axis. 00:19:39.330 --> 00:19:44.740 And that locus of the poles, in fact, is referred to in 00:19:44.740 --> 00:19:48.540 feedback terminology as the "root locus." And as is 00:19:48.540 --> 00:19:52.510 discussed in much more detail in the text, there are lots of 00:19:52.510 --> 00:19:57.580 ways of determining the root locus for feedback systems 00:19:57.580 --> 00:20:00.470 without explicitly solving for the roots. 00:20:00.470 --> 00:20:03.700 For this particular example, in fact, we can determine the 00:20:03.700 --> 00:20:07.420 root locus in the most straightforward way simply by 00:20:07.420 --> 00:20:10.060 solving for the roots of the denominator 00:20:10.060 --> 00:20:11.810 of the system function. 00:20:11.810 --> 00:20:17.380 Now, notice that in this case, these poles, with K1 positive, 00:20:17.380 --> 00:20:18.760 have moved together. 00:20:18.760 --> 00:20:21.570 They move along the j-omega-axis. 00:20:21.570 --> 00:20:24.850 And before they come together, the 00:20:24.850 --> 00:20:27.460 system is clearly unstable. 00:20:27.460 --> 00:20:31.510 Even when they come together and split, the system is 00:20:31.510 --> 00:20:35.190 marginally stable, because, in fact, the system 00:20:35.190 --> 00:20:36.660 would tend to oscillate. 00:20:36.660 --> 00:20:39.420 And what that oscillation means is that with the 00:20:39.420 --> 00:20:43.020 measurement of the angles, essentially if the poles are 00:20:43.020 --> 00:20:48.230 operating on the j-omega-axis, what will happen is that 00:20:48.230 --> 00:20:50.460 things will oscillate back and forth. 00:20:50.460 --> 00:20:54.210 Perhaps with the cart moving back and forth trying to 00:20:54.210 --> 00:20:56.750 compensate, and the rod sort of moving in 00:20:56.750 --> 00:20:58.870 the opposite direction. 00:20:58.870 --> 00:21:02.770 In any case, what's happened is, with just proportional 00:21:02.770 --> 00:21:07.030 feedback, we apparently are unable to 00:21:07.030 --> 00:21:10.250 stabilize the system. 00:21:10.250 --> 00:21:13.140 Well, you could think that the reason, perhaps, is that we're 00:21:13.140 --> 00:21:14.490 not responding fast enough. 00:21:14.490 --> 00:21:19.310 For example, if the angle starts to change, perhaps, in 00:21:19.310 --> 00:21:23.540 fact, we should make the feedback proportional to the 00:21:23.540 --> 00:21:25.780 rate of change of angles, rather than 00:21:25.780 --> 00:21:27.280 to the angle itself. 00:21:27.280 --> 00:21:30.710 And so we could examine the possibility of using what's 00:21:30.710 --> 00:21:34.700 referred to as "derivative feedback." 00:21:34.700 --> 00:21:42.770 In derivative feedback, what we would do is to choose, for 00:21:42.770 --> 00:21:49.390 the feedback equation, or for the feedback system function, 00:21:49.390 --> 00:21:53.020 something reflecting a measurement of the derivative 00:21:53.020 --> 00:21:54.410 of the angle. 00:21:54.410 --> 00:21:58.910 And so here again, once again, we have the open-loop system 00:21:58.910 --> 00:22:01.490 function with derivative feedback. 00:22:01.490 --> 00:22:03.820 We'll attempt to-- 00:22:03.820 --> 00:22:06.040 or we will use this feedback. 00:22:06.040 --> 00:22:09.040 And acceleration, which instead of being proportional 00:22:09.040 --> 00:22:12.540 to the angle, as it was in the previous case, an acceleration 00:22:12.540 --> 00:22:16.840 which is proportional to the derivative of the angle. 00:22:16.840 --> 00:22:20.470 And so the basic feedback dynamics, or feedback system 00:22:20.470 --> 00:22:25.060 function, is then G of s is equal to K2 times s. 00:22:25.060 --> 00:22:28.110 The multiplication by s reflecting the fact that in 00:22:28.110 --> 00:22:33.130 the time domain it's measuring the derivative of the angle. 00:22:33.130 --> 00:22:38.670 The associated system function is then indicated here. 00:22:38.670 --> 00:22:42.960 And if we solve again this second-order equation for its 00:22:42.960 --> 00:22:47.730 roots, that tells us that the location of the poles are 00:22:47.730 --> 00:22:51.810 given by this equation. 00:22:51.810 --> 00:22:57.100 And so now what we would want to look at is how these poles 00:22:57.100 --> 00:23:03.950 move as we vary the derivative feedback constant K2. 00:23:03.950 --> 00:23:08.810 So let's look at the root locus for that case. 00:23:08.810 --> 00:23:17.860 And first, once again, we have the basic open-loop poles. 00:23:17.860 --> 00:23:20.915 And the open-loop poles consist of a pole on the left 00:23:20.915 --> 00:23:23.880 half-plane and a pole on the right half-plane, 00:23:23.880 --> 00:23:28.400 corresponding in this equation to K2 equal to 0. 00:23:28.400 --> 00:23:32.700 That is, no feedback in the system. 00:23:32.700 --> 00:23:37.840 If K2 is negative, then what you can see is that this real 00:23:37.840 --> 00:23:41.700 part will become more negative. 00:23:41.700 --> 00:23:46.220 Since K2 is squared here, this is still a positive quantity. 00:23:46.220 --> 00:23:51.280 And, in fact, then with K2 less than 0, the root locus 00:23:51.280 --> 00:23:55.050 that we get is indicated by this. 00:23:55.050 --> 00:23:59.990 Now, notice, then, that this right half-plane pole is 00:23:59.990 --> 00:24:02.020 becoming more unstable. 00:24:02.020 --> 00:24:04.460 And the left half-plane pole likewise is 00:24:04.460 --> 00:24:06.250 becoming more unstable. 00:24:06.250 --> 00:24:09.800 And this point, by the way, corresponds to where this pole 00:24:09.800 --> 00:24:13.930 ends up, or where the root locus ends up, when K2 00:24:13.930 --> 00:24:15.680 eventually becomes infinite. 00:24:15.680 --> 00:24:20.450 So clearly K2 negative is not going to stabilize the system. 00:24:20.450 --> 00:24:24.700 Let's try K2 positive. 00:24:24.700 --> 00:24:29.960 And the root locus dictated by this equation, then, is what I 00:24:29.960 --> 00:24:31.810 indicate here. 00:24:31.810 --> 00:24:35.130 The left half-plane pole moves further into the left 00:24:35.130 --> 00:24:35.760 half-plane. 00:24:35.760 --> 00:24:36.340 That's good. 00:24:36.340 --> 00:24:38.430 That's getting more stable. 00:24:38.430 --> 00:24:42.600 The right half-plane pole is moving closer to the left 00:24:42.600 --> 00:24:46.630 half-plane, but unfortunately never gets there. 00:24:46.630 --> 00:24:50.840 And, in fact, it's when K2 eventually becomes infinite 00:24:50.840 --> 00:24:54.600 that this pole just gets to the point where the system 00:24:54.600 --> 00:24:57.870 becomes marginally stable. 00:24:57.870 --> 00:25:02.010 So what we found is that with proportional feedback, we 00:25:02.010 --> 00:25:05.640 can't stabilize the system, with the derivative feedback, 00:25:05.640 --> 00:25:09.930 we can't stabilize the system, by themselves. 00:25:09.930 --> 00:25:13.890 And a logical next choice is to see if we can stabilize the 00:25:13.890 --> 00:25:19.790 system by both measuring the angle and at the same time 00:25:19.790 --> 00:25:23.440 being careful to be responsive to how fast that angle is 00:25:23.440 --> 00:25:27.340 changing, so that if it's changing too fast, we can move 00:25:27.340 --> 00:25:30.630 the cart or our hand under the inverted 00:25:30.630 --> 00:25:32.950 pendulum more quickly. 00:25:32.950 --> 00:25:37.370 Well, now then what we want to examine is the use of 00:25:37.370 --> 00:25:40.290 proportional plus derivative feedback. 00:25:40.290 --> 00:25:49.610 And in that case, we then have a choice for the acceleration, 00:25:49.610 --> 00:25:54.330 which is proportional with one constant to the angle and 00:25:54.330 --> 00:25:57.360 proportional with another constant to the 00:25:57.360 --> 00:25:59.350 derivative of the angle. 00:25:59.350 --> 00:26:04.480 And so the basic system function, then, with 00:26:04.480 --> 00:26:07.850 proportional plus derivative feedback, is a system function 00:26:07.850 --> 00:26:12.330 which is K1 plus K2 times s. 00:26:12.330 --> 00:26:17.280 We then have an overall closed-loop system function, 00:26:17.280 --> 00:26:23.870 theta of s, which is given by this equation. 00:26:23.870 --> 00:26:28.900 And so the roots of this equation then represent the 00:26:28.900 --> 00:26:32.470 poles of the closed-loop system. 00:26:32.470 --> 00:26:37.980 And those poles involve two parameters. 00:26:37.980 --> 00:26:42.120 They involve the parameter K2, which is proportional to the 00:26:42.120 --> 00:26:45.740 derivative of the angle, and the constant K1, which is 00:26:45.740 --> 00:26:47.700 proportional to the angle. 00:26:47.700 --> 00:26:55.160 And we'll, first of all, examine this just with K2 00:26:55.160 --> 00:26:59.560 positive, because what we can see is that as we vary K1, if 00:26:59.560 --> 00:27:05.120 K2 were negative, that would, more or less immediately, put 00:27:05.120 --> 00:27:07.270 poles into the right half-plane. 00:27:07.270 --> 00:27:11.280 And the more negative K2 got, the larger this 00:27:11.280 --> 00:27:12.910 term is going to get. 00:27:12.910 --> 00:27:15.790 So, in fact, as it will turn out, we can stabilize the 00:27:15.790 --> 00:27:20.350 system, provided that we choose K2 to be 00:27:20.350 --> 00:27:23.160 greater than 0. 00:27:23.160 --> 00:27:23.410 All right. 00:27:23.410 --> 00:27:30.960 Now, with K2 greater than 0, what happens in the location 00:27:30.960 --> 00:27:37.640 of the poles is that if K2 is greater than 0, and we choose 00:27:37.640 --> 00:27:39.330 K1 greater than 0-- 00:27:39.330 --> 00:27:39.780 I'm sorry. 00:27:39.780 --> 00:27:45.870 We choose K1 equal to 0, then in effect the influence of K2 00:27:45.870 --> 00:27:51.220 is to shift the poles of the open-loop system slightly. 00:27:51.220 --> 00:27:58.450 And so with K2 greater than 0 and K1 equal to 0, we have a 00:27:58.450 --> 00:28:06.250 set of poles, which are indicated here, and so this is 00:28:06.250 --> 00:28:08.800 just a shift, slight shift, to the left, depending on the 00:28:08.800 --> 00:28:13.900 value of K2, a shift to the left of the open-loop poles. 00:28:13.900 --> 00:28:17.190 Now, as we vary K1, and in particular we're going to 00:28:17.190 --> 00:28:24.700 choose K1 greater than 0, what happens is that the poles will 00:28:24.700 --> 00:28:31.000 begin to move together, as they did previously when we 00:28:31.000 --> 00:28:33.230 looked at the variation of K1. 00:28:33.230 --> 00:28:36.500 The poles will move together, reach a point where we have a 00:28:36.500 --> 00:28:41.820 second-order pole, and then those poles will split and 00:28:41.820 --> 00:28:43.900 move parallel to the j-omega-axis. 00:28:43.900 --> 00:28:48.100 So what's indicated here is the root locus. 00:28:48.100 --> 00:28:52.990 And what this represents, then, as long as we make K1 00:28:52.990 --> 00:28:56.075 large enough so that this poles moves into the left 00:28:56.075 --> 00:28:59.570 half-plane, is that it represents 00:28:59.570 --> 00:29:02.050 now a stable system. 00:29:02.050 --> 00:29:02.330 OK. 00:29:02.330 --> 00:29:05.840 So what we've seen is that proportional feedback by 00:29:05.840 --> 00:29:10.330 itself or derivative feedback by itself won't stabilize the 00:29:10.330 --> 00:29:14.820 system, whereas with the right choice of feedback constants, 00:29:14.820 --> 00:29:16.900 proportional plus derivative feedback will. 00:29:16.900 --> 00:29:20.620 And we saw that basically by examining the root locus in 00:29:20.620 --> 00:29:22.080 the s-plane. 00:29:22.080 --> 00:29:22.350 All right. 00:29:22.350 --> 00:29:26.200 Well, let's actually watch the system in action. 00:29:26.200 --> 00:29:29.820 And I described it to you previously. 00:29:29.820 --> 00:29:33.730 Basically, an inverted pendulum on a cart. 00:29:33.730 --> 00:29:36.630 And so I still have it off. 00:29:36.630 --> 00:29:38.870 And, of course, we have the pendulum. 00:29:38.870 --> 00:29:43.310 And as I indicated, it's pivoted at the base. 00:29:43.310 --> 00:29:50.260 And the angle is measured by a potentiometer that we have 00:29:50.260 --> 00:29:52.240 attached to the pivot point. 00:29:52.240 --> 00:29:57.040 And the measurement of the angle is fed back through this 00:29:57.040 --> 00:30:02.370 wire to a motor that we have at the other end of the table. 00:30:02.370 --> 00:30:06.350 And then that motor basically is used to provide the 00:30:06.350 --> 00:30:09.660 acceleration to drive the cart. 00:30:09.660 --> 00:30:09.950 OK. 00:30:09.950 --> 00:30:12.330 Well, let's turn it on. 00:30:12.330 --> 00:30:15.120 And when we turn it on, it'll take just 00:30:15.120 --> 00:30:16.370 an instant to stabilize. 00:30:19.030 --> 00:30:24.200 And fortunately we have the constants set right, and there 00:30:24.200 --> 00:30:28.710 we have now the stabilization of an unstable system. 00:30:28.710 --> 00:30:31.460 Remember that with the feedback off, the system is 00:30:31.460 --> 00:30:34.610 unstable because the pendulum will fall, whereas now it's 00:30:34.610 --> 00:30:35.760 stabilized. 00:30:35.760 --> 00:30:41.680 Now, also, as you can see, not only have we stabilized it, 00:30:41.680 --> 00:30:45.340 but we're able to compensate through the feedback to 00:30:45.340 --> 00:30:47.540 changes in the external disturbances. 00:30:47.540 --> 00:30:51.490 For example, by tapping it, because of the feedback and 00:30:51.490 --> 00:30:54.200 the measurement of the angle, it will more or less 00:30:54.200 --> 00:30:56.880 automatically stabilize. 00:30:56.880 --> 00:31:00.600 Now, in addition to being stable in the presence of 00:31:00.600 --> 00:31:04.940 external disturbances, it also remains stable and remains 00:31:04.940 --> 00:31:09.210 balanced even if we were to change the system dynamics. 00:31:09.210 --> 00:31:13.950 And let me just illustrate that with the glass that we've 00:31:13.950 --> 00:31:15.510 talked about before. 00:31:15.510 --> 00:31:19.340 Let's first not be too bold, and we'll take the liquid out 00:31:19.340 --> 00:31:20.420 of the glass. 00:31:20.420 --> 00:31:24.440 And presumably if it can adjust to changes in the 00:31:24.440 --> 00:31:28.850 system dynamics, then if I put the glass on, in fact, it will 00:31:28.850 --> 00:31:29.660 remain balanced. 00:31:29.660 --> 00:31:31.220 And indeed it does. 00:31:31.220 --> 00:31:33.890 And let me point out, by the way, that I don't have to be 00:31:33.890 --> 00:31:37.480 very careful about exactly where I position the glass. 00:31:37.480 --> 00:31:41.770 And furthermore, I can change the overall system even 00:31:41.770 --> 00:31:47.480 further by, let's say for example, pouring a liquid in. 00:31:47.480 --> 00:31:51.540 And now let me also comment that I've changed the physics 00:31:51.540 --> 00:31:52.200 of it a little bit. 00:31:52.200 --> 00:31:55.920 Because the liquid can slosh around a little bit, it 00:31:55.920 --> 00:31:58.440 becomes a little more complicated a system. 00:31:58.440 --> 00:32:03.650 But as you can see, it still remains balanced. 00:32:03.650 --> 00:32:08.270 Now, if we really don't want to be too conservative at all, 00:32:08.270 --> 00:32:11.022 we could wonder whether, with the feedback constants we 00:32:11.022 --> 00:32:14.290 have, we could, in fact, balance the 00:32:14.290 --> 00:32:15.450 pitcher on the top. 00:32:15.450 --> 00:32:21.000 And, well, I guess we may as well give that a try. 00:32:21.000 --> 00:32:27.270 And so now we're changing the mass at the top of the 00:32:27.270 --> 00:32:31.200 pendulum by a considerable amount. 00:32:31.200 --> 00:32:34.890 And, again, the system basically can respond to it. 00:32:34.890 --> 00:32:38.090 Now, this is a fairly complicated system. 00:32:38.090 --> 00:32:40.420 The liquid is sloshing around. 00:32:40.420 --> 00:32:44.210 We, in fact, as you can see, have an instability right now, 00:32:44.210 --> 00:32:45.770 although it's controlled. 00:32:45.770 --> 00:32:49.060 And that's because the physics of the dynamics has changed. 00:32:49.060 --> 00:32:53.130 And we can put a little bit more mass into the system, and 00:32:53.130 --> 00:32:55.505 maybe or maybe not that will cut down on the instability. 00:33:02.060 --> 00:33:02.300 OK. 00:33:02.300 --> 00:33:06.050 Well, in fact, what happened there is that we increased the 00:33:06.050 --> 00:33:09.150 mass at the top of the pendulum slightly. 00:33:09.150 --> 00:33:12.020 And that provided just enough damping to 00:33:12.020 --> 00:33:13.270 stabilize the system. 00:33:16.480 --> 00:33:17.030 OK. 00:33:17.030 --> 00:33:22.870 Well, with this lecture and this demonstration, this 00:33:22.870 --> 00:33:25.310 concludes this entire set of lectures. 00:33:25.310 --> 00:33:28.270 It concludes it especially if the pitcher happens to fall. 00:33:28.270 --> 00:33:34.040 But, seriously, this concludes the set of lectures as we have 00:33:34.040 --> 00:33:36.180 put them together. 00:33:36.180 --> 00:33:41.880 And let me just comment that, as a professor of mine once 00:33:41.880 --> 00:33:45.750 said, and which I've never forgotten, the purpose of a 00:33:45.750 --> 00:33:48.470 set of lectures, or of a course, or, for that matter 00:33:48.470 --> 00:33:53.260 anything that you study, is not really to cover a subject, 00:33:53.260 --> 00:33:55.760 but to uncover the subject. 00:33:55.760 --> 00:34:01.980 And I hope that, at least to some degree, we were able to 00:34:01.980 --> 00:34:05.020 uncover the topic of signals and systems through this 00:34:05.020 --> 00:34:07.240 series of lectures. 00:34:07.240 --> 00:34:12.679 There are a lot of topics that we got only a very brief 00:34:12.679 --> 00:34:14.010 glimpse into. 00:34:14.010 --> 00:34:18.290 And I hope that at least what we've been able to do is get 00:34:18.290 --> 00:34:21.280 you interested enough in them so that you'll pursue some of 00:34:21.280 --> 00:34:23.020 these on your own. 00:34:23.020 --> 00:34:27.699 And so I'd like to conclude by thanking you, both for your 00:34:27.699 --> 00:34:29.159 patience and your interest. 00:34:29.159 --> 00:34:33.130 And I hope that you have enough interest to pursue some 00:34:33.130 --> 00:34:34.469 of these topics further. 00:34:34.469 --> 00:34:35.200 Thank you. 00:34:35.200 --> 00:34:36.760 FILMING DIRECTOR: Okay. 00:34:36.760 --> 00:34:38.010 That's a wrap.