WEBVTT 00:00:00.058 --> 00:00:02.787 You already know what a feedback controller does. 00:00:02.984 --> 00:00:05.049 This lecture is about a different question, 00:00:05.132 --> 00:00:09.324 and it is the question most of classical control is actually built on. 00:00:09.427 --> 00:00:13.013 Not what a plant does to a step, but what it does to a sine wave, 00:00:13.096 --> 00:00:14.978 one frequency at a time. 00:00:15.174 --> 00:00:16.834 Answer that for every frequency 00:00:16.834 --> 00:00:20.469 and you have said everything there is to say about a linear system. 00:00:21.552 --> 00:00:23.155 So here is the setup. 00:00:23.352 --> 00:00:24.581 Take a plant that is linear 00:00:24.664 --> 00:00:29.295 and time invariant, and drive it with a pure sine wave. Because it is linear, 00:00:29.378 --> 00:00:33.953 whatever comes out is also a sine wave, at exactly the same frequency. 00:00:34.150 --> 00:00:36.344 Nothing else is possible, so only two 00:00:36.344 --> 00:00:41.685 things can differ between what goes in and what comes out: the size, and the timing. 00:00:42.900 --> 00:00:46.868 Let me make that concrete with the simplest plant there is, a single lag, 00:00:46.951 --> 00:00:48.938 one over s plus one. 00:00:49.134 --> 00:00:52.664 Drive it slowly, at a fifth of a radian per second. 00:00:52.768 --> 00:00:54.763 The grey curve is what we push in, 00:00:54.846 --> 00:00:59.514 the blue curve is what comes back out, and time runs along the bottom in seconds. 00:01:00.566 --> 00:01:03.572 Honestly, the two waves are almost the same wave. 00:01:03.655 --> 00:01:06.730 The output stands ninety eight percent as tall as the input, 00:01:06.813 --> 00:01:09.681 and it trails behind it by about eleven degrees. 00:01:09.878 --> 00:01:12.918 At this frequency the plant is barely doing anything at all. 00:01:13.001 --> 00:01:16.427 Ask it to move slowly, and it simply follows. 00:01:17.339 --> 00:01:19.813 One thing I am quietly assuming here. 00:01:19.917 --> 00:01:22.158 These are steady state pictures. 00:01:22.471 --> 00:01:26.593 Switch the drive on and there is a transient while the plant settles. 00:01:27.045 --> 00:01:31.225 Frequency response is what is left after that transient has died away. 00:01:32.022 --> 00:01:35.157 Now speed the drive up to one radian per second. 00:01:35.354 --> 00:01:38.130 Same plant, same amplitude going in. 00:01:38.326 --> 00:01:39.325 The output has dropped 00:01:39.325 --> 00:01:44.108 to seventy one percent of it, and it is now a full forty five degrees behind. 00:01:44.305 --> 00:01:46.918 That is an eighth of a cycle of lag. 00:01:47.715 --> 00:01:51.930 And at five radians per second the plant has more or less given up. 00:01:52.242 --> 00:01:55.375 The output is only two tenths of the input amplitude now, 00:01:55.458 --> 00:01:59.279 and it lags by seventy nine degrees, closing in on ninety. 00:01:59.475 --> 00:02:01.659 Notice the pattern that is forming. 00:02:01.856 --> 00:02:03.293 As the frequency climbs, 00:02:03.376 --> 00:02:07.290 the output shrinks, and it falls further and further behind. 00:02:08.504 --> 00:02:11.243 Both of those numbers come out of one piece of algebra, 00:02:11.326 --> 00:02:13.079 and you have seen it before. 00:02:13.276 --> 00:02:16.911 Put s equal to j omega in the transfer function. 00:02:17.107 --> 00:02:21.369 What comes back, for each frequency, is a single complex number. 00:02:22.072 --> 00:02:24.718 Its magnitude is the amplitude ratio: 00:02:24.801 --> 00:02:27.890 how much taller or shorter the output is than the input. 00:02:28.086 --> 00:02:29.698 Its argument is the phase shift, 00:02:29.781 --> 00:02:33.323 in degrees, and it is negative when the output lags. 00:02:33.520 --> 00:02:35.445 Two real numbers per frequency, 00:02:35.528 --> 00:02:38.652 and that pair is the whole of the frequency response. 00:02:39.449 --> 00:02:42.201 One convention before we plot any of it. 00:02:42.514 --> 00:02:45.080 We do not plot the magnitude directly. 00:02:45.184 --> 00:02:48.784 We plot twenty times its logarithm, in decibels. 00:02:48.980 --> 00:02:51.138 The reason is pure arithmetic. 00:02:51.221 --> 00:02:55.192 A transfer function is a product of factors, and taking logarithms turns 00:02:55.192 --> 00:02:58.710 that product into a sum of curves you can add up by eye. 00:02:59.251 --> 00:03:02.387 And the frequency axis is logarithmic as well. 00:03:02.699 --> 00:03:07.263 Every unit along it is one decade, a factor of ten in frequency. 00:03:07.367 --> 00:03:10.130 So the axis we are about to use runs from a tenth 00:03:10.130 --> 00:03:15.587 of a radian per second on the left to a hundred on the right, in three even steps. 00:03:16.384 --> 00:03:20.063 So here is the magnitude of that single lag, in decibels, 00:03:20.146 --> 00:03:22.561 against the logarithm of frequency. 00:03:22.665 --> 00:03:27.168 And here are the three experiments we just ran, sitting on it as three green dots: 00:03:27.251 --> 00:03:29.678 a fifth, one, and five. 00:03:30.475 --> 00:03:34.945 Flat and unbothered on the left, where the plant follows whatever it is given. 00:03:35.142 --> 00:03:38.858 Falling away steadily on the right, where it cannot keep up. 00:03:39.055 --> 00:03:42.596 The table beside it is those same three readings written out. 00:03:43.393 --> 00:03:47.841 Underneath goes the phase, in degrees, on the same frequency axis. 00:03:48.037 --> 00:03:52.809 Zero at low frequency, minus ninety at high frequency, and passing through minus 00:03:52.809 --> 00:03:57.732 forty five right at one radian per second, which is exactly where the pole is. 00:03:58.529 --> 00:04:01.069 Now look hard at what those two curves nearly are. 00:04:01.152 --> 00:04:03.740 The magnitude is nearly two straight lines: 00:04:03.823 --> 00:04:08.549 flat, then falling at twenty decibels per decade, and they meet at the pole. 00:04:09.163 --> 00:04:13.425 The phase is nearly three straight lines, hinged a decade either side of it. 00:04:14.222 --> 00:04:16.832 That is not a coincidence and it is not a drawing trick. 00:04:16.915 --> 00:04:20.851 It falls out of the algebra, and in the next part we work out exactly where it 00:04:20.851 --> 00:04:25.310 comes from, factor by factor, until sketching one of these is arithmetic. 00:04:25.793 --> 00:04:27.127 Then we close the loop. 00:04:27.210 --> 00:04:29.599 And I will show you why one particular frequency, 00:04:29.682 --> 00:04:33.001 the one where the phase reaches minus one hundred and eighty degrees, 00:04:33.084 --> 00:04:35.894 decides whether the closed loop is stable at all. 00:04:35.998 --> 00:04:39.967 Gain margin and phase margin are the two numbers that live at that frequency, 00:04:40.050 --> 00:04:41.339 and by the end you will read both 00:04:41.339 --> 00:04:43.267 of them straight off these curves. 00:04:44.361 --> 00:04:48.844 So why decibels, and why a logarithmic frequency axis. 00:04:49.040 --> 00:04:53.592 Here is the answer, and it is the only reason those two conventions exist. 00:04:53.696 --> 00:04:55.829 A transfer function is a product: 00:04:55.913 --> 00:05:00.836 a constant out front, some factors upstairs, some factors downstairs. 00:05:01.540 --> 00:05:06.580 Take twenty times the logarithm of the magnitude, and that product becomes a sum. 00:05:06.776 --> 00:05:10.155 Every factor on top adds its own decibel curve. 00:05:10.352 --> 00:05:12.845 Every factor on the bottom subtracts one. 00:05:12.929 --> 00:05:14.486 Nothing else happens. 00:05:15.190 --> 00:05:18.159 And the angle of a product is the sum of the angles anyway, 00:05:18.243 --> 00:05:22.760 so the phase adds in exactly the same pattern, with no logarithm needed. 00:05:23.212 --> 00:05:25.848 Two sums, one for each curve. 00:05:26.389 --> 00:05:31.382 So if I can draw four simple curves, I can draw any Bode plot by stacking them up. 00:05:31.579 --> 00:05:33.890 Let me take those four one at a time. 00:05:34.686 --> 00:05:39.539 The first one is trivial. A constant gain K. Its magnitude does not depend 00:05:39.539 --> 00:05:44.637 on frequency at all, so in decibels it is a horizontal line at twenty log K. 00:05:44.833 --> 00:05:48.515 With K equal to two, that sits at six decibels. 00:05:48.618 --> 00:05:51.216 And a positive real number has no angle, 00:05:51.300 --> 00:05:54.320 so it contributes nothing whatsoever to the phase. 00:05:55.116 --> 00:05:58.995 The second one is an integrator, one over j omega. 00:05:59.191 --> 00:06:01.371 Its magnitude is one over omega, 00:06:01.455 --> 00:06:05.497 so in decibels it is minus twenty times the logarithm of omega. 00:06:05.693 --> 00:06:09.990 That is a straight line, exactly, at every frequency. 00:06:10.786 --> 00:06:13.338 And you can read the slope off it by arithmetic. 00:06:13.422 --> 00:06:18.620 At one radian per second it is zero decibels. At ten, it is minus twenty. 00:06:18.704 --> 00:06:24.057 At a hundred, minus forty. Every decade costs twenty decibels, forever. 00:06:25.272 --> 00:06:27.374 Its phase is easier still. 00:06:27.478 --> 00:06:30.973 One over j is minus j, which points straight down. 00:06:31.170 --> 00:06:35.571 Minus ninety degrees, at every frequency, with no bend in it anywhere. 00:06:36.367 --> 00:06:39.464 The third block is the one that actually earns its keep: 00:06:39.548 --> 00:06:44.146 a real pole, written so that it equals one at zero frequency. 00:06:44.250 --> 00:06:48.140 One over one plus j omega over omega p. 00:06:48.244 --> 00:06:51.437 Take it to its two limits and everything falls out. 00:06:52.141 --> 00:06:56.051 Well below the corner, omega over omega p is tiny, 00:06:56.135 --> 00:06:58.725 so the denominator is essentially one. 00:06:59.038 --> 00:07:00.629 The factor is one. 00:07:00.826 --> 00:07:02.948 Zero decibels, and no phase. 00:07:03.032 --> 00:07:04.867 That is the flat piece. 00:07:05.780 --> 00:07:07.066 Well above the corner, 00:07:07.150 --> 00:07:11.423 the one is negligible and the factor is omega p over j omega. 00:07:11.526 --> 00:07:14.592 That is an integrator again, scaled. 00:07:14.789 --> 00:07:19.910 So it falls at twenty decibels per decade, and its phase is minus ninety. 00:07:20.521 --> 00:07:22.262 So we have a flat line at zero 00:07:22.262 --> 00:07:26.079 on the left, a falling line at minus twenty on the right, 00:07:26.163 --> 00:07:31.783 and they cross where omega equals omega p, which here is two radians per second. 00:07:31.980 --> 00:07:36.253 That crossing is what everybody calls the corner, or the break frequency. 00:07:37.050 --> 00:07:39.245 Now, how wrong is that sketch. 00:07:39.557 --> 00:07:42.144 Right at the corner the denominator is one plus j, 00:07:42.228 --> 00:07:47.615 whose magnitude is root two, and twenty log of root two is minus three decibels. 00:07:47.812 --> 00:07:50.119 Three decibels below the corner of the sketch, 00:07:50.203 --> 00:07:52.654 and that is the worst it ever gets. 00:07:53.450 --> 00:07:57.058 Half a decade either side of the corner the error is down to one decibel, 00:07:57.142 --> 00:08:00.066 and a decade out it is under a quarter of one. 00:08:00.150 --> 00:08:02.550 So the straight lines are not a cartoon. 00:08:02.634 --> 00:08:06.257 They are the truth to within three decibels, everywhere. 00:08:07.054 --> 00:08:11.571 The phase needs one more rule, and it is a convention rather than a theorem. 00:08:11.884 --> 00:08:16.861 Zero below a decade under the corner, minus ninety above a decade over it, 00:08:16.945 --> 00:08:20.928 and a straight ramp of minus forty five degrees per decade joining them. 00:08:21.632 --> 00:08:26.018 At the corner itself the ramp is halfway down, at minus forty five degrees, 00:08:26.102 --> 00:08:27.891 and that one is exact: 00:08:28.087 --> 00:08:32.419 one over one plus j has an angle of minus forty five, precisely. 00:08:32.871 --> 00:08:37.516 The worst error in the ramp is about six degrees, out near its two hinges. 00:08:38.312 --> 00:08:41.363 And the fourth block is the same numbers with every sign flipped, 00:08:41.447 --> 00:08:44.269 because a zero is a pole moved upstairs. 00:08:44.466 --> 00:08:47.853 Flat, then rising at twenty decibels per decade. 00:08:47.937 --> 00:08:51.084 Zero degrees, then a ramp up to plus ninety. 00:08:51.509 --> 00:08:53.681 That is the whole vocabulary. 00:08:53.878 --> 00:08:57.488 Four shapes, and a Bode plot is what you get when you add up however many 00:08:57.488 --> 00:08:59.520 of them your plant happens to have. 00:09:00.662 --> 00:09:03.333 Written out as a table, that is it. 00:09:03.529 --> 00:09:08.116 A gain moves the whole magnitude curve up or down and touches nothing else. 00:09:08.313 --> 00:09:12.687 An integrator tilts it by twenty decibels per decade and drops the phase by ninety, 00:09:12.771 --> 00:09:14.478 at every frequency. 00:09:15.275 --> 00:09:19.127 A zero puts a bend upward in the magnitude at its own frequency, 00:09:19.211 --> 00:09:23.623 and hands you ninety degrees of phase lead spread across two decades. 00:09:23.820 --> 00:09:29.150 A pole does the mirror image: a bend downward, and ninety degrees of lag. 00:09:29.946 --> 00:09:32.672 And the two error facts are worth memorising, 00:09:32.756 --> 00:09:36.414 because they are the whole difference between the sketch and the truth. 00:09:36.610 --> 00:09:41.278 Three decibels at each corner, and a decade either side for the phase. 00:09:41.730 --> 00:09:43.589 Nothing else is approximate. 00:09:44.292 --> 00:09:46.437 So the straight lines are not magic. 00:09:46.521 --> 00:09:48.902 They are two limits and a hinge. 00:09:49.099 --> 00:09:50.213 Now let me stack them, 00:09:50.213 --> 00:09:53.999 on a plant with a pole, another pole, and an integrator in it. 00:09:55.145 --> 00:09:58.281 Here is the plant we will carry for the rest of the lecture. 00:09:58.593 --> 00:10:03.656 Forty, over s times s plus two times s plus ten. 00:10:03.853 --> 00:10:09.461 Third order, one integrator, two real poles, and nothing exotic anywhere in it. 00:10:10.165 --> 00:10:13.891 Before we can stack anything we have to rewrite it, because our four 00:10:13.891 --> 00:10:18.617 building blocks all equal one at low frequency and these brackets do not. 00:10:19.069 --> 00:10:23.110 So pull a two out of the first bracket, and a ten out of the second. 00:10:23.651 --> 00:10:28.354 Forty over twenty leaves two, and now every factor is in the right shape. 00:10:28.458 --> 00:10:29.536 Read them off. 00:10:29.619 --> 00:10:32.499 A gain of two, worth six decibels. 00:10:32.811 --> 00:10:34.194 One integrator. 00:10:34.390 --> 00:10:37.828 A pole at two radians per second, and a pole at ten. 00:10:38.024 --> 00:10:42.994 Those four items are the entire plant, and they are the entire Bode plot. 00:10:43.791 --> 00:10:45.405 So let us draw it. 00:10:45.602 --> 00:10:49.292 I will lay the magnitude down first, one segment at a time, 00:10:49.375 --> 00:10:51.674 and check a number at every break. 00:10:52.471 --> 00:10:57.545 Start at the far left of the magnitude plot, at a tenth of a radian per second. 00:10:58.160 --> 00:11:03.141 Down there both poles are still asleep, because we are well below both corners. 00:11:03.453 --> 00:11:07.227 So only the gain and the integrator are doing anything at all. 00:11:08.024 --> 00:11:10.091 Six decibels from the gain. 00:11:10.195 --> 00:11:13.934 And the integrator is worth plus twenty down there, because it costs twenty per 00:11:13.934 --> 00:11:17.801 decade and we are one decade below one radian per second. 00:11:18.114 --> 00:11:19.946 Six plus twenty is twenty six, 00:11:20.029 --> 00:11:23.896 and from there the line comes down at twenty decibels per decade. 00:11:24.693 --> 00:11:29.477 That segment runs until the first pole wakes up, at omega equal to two. 00:11:29.673 --> 00:11:31.752 Where has the line got to by then? 00:11:31.949 --> 00:11:36.442 Six from the gain, minus twenty log of two, which is another six. 00:11:36.639 --> 00:11:39.600 It arrives at exactly zero decibels. 00:11:40.397 --> 00:11:43.578 And past that corner the pole adds its own twenty per decade 00:11:43.578 --> 00:11:45.413 on top of the integrator's. 00:11:45.610 --> 00:11:48.139 So the slope steepens to minus forty, 00:11:48.222 --> 00:11:51.473 and the second segment falls twice as fast as the first. 00:11:52.386 --> 00:11:54.242 It runs from two up to ten, 00:11:54.325 --> 00:11:58.435 which is log ten of five, about seven tenths of a decade. 00:11:58.632 --> 00:12:02.798 At forty decibels a decade, that is twenty eight decibels lost, 00:12:02.881 --> 00:12:06.121 so we reach the second corner at minus twenty eight. 00:12:07.034 --> 00:12:11.502 Beyond ten the second pole joins in as well, so the slope steepens again, 00:12:11.585 --> 00:12:15.974 to minus sixty decibels per decade, and it never comes back up. 00:12:16.078 --> 00:12:20.885 Three segments, three slopes, and every number in them was arithmetic. 00:12:21.682 --> 00:12:25.058 Now here is the exact curve, drawn over the top. 00:12:25.141 --> 00:12:28.949 It hugs the skeleton everywhere, except that it rounds each corner off 00:12:28.949 --> 00:12:33.049 and sits about three decibels below it, exactly as we worked out. 00:12:34.263 --> 00:12:37.979 The phase is built the same way, and it starts lower down. 00:12:38.431 --> 00:12:43.006 The integrator hands us minus ninety degrees at every frequency, so the phase 00:12:43.006 --> 00:12:48.045 begins at minus ninety on the left and does not move until the first ramp starts. 00:12:48.842 --> 00:12:53.670 The pole at two begins bending a decade below itself, at nought point two, 00:12:53.753 --> 00:12:56.610 and finishes a decade above, at twenty. 00:12:56.806 --> 00:13:00.464 So this second piece falls at forty five degrees per decade. 00:13:01.377 --> 00:13:05.473 But at one radian per second the pole at ten begins its own ramp, 00:13:05.556 --> 00:13:07.786 and now both are running together. 00:13:08.099 --> 00:13:11.510 Two ramps at forty five each make ninety degrees per decade, 00:13:11.593 --> 00:13:14.903 so this middle piece is the steepest part of the whole curve. 00:13:15.700 --> 00:13:20.272 The first ramp finishes at twenty, and after that only the second is left, 00:13:20.355 --> 00:13:24.779 so we go back to forty five per decade and settle at minus two hundred and seventy. 00:13:24.976 --> 00:13:28.262 Ninety from the integrator, and ninety from each pole. 00:13:28.966 --> 00:13:31.855 And there is the exact phase over the top of that, 00:13:31.938 --> 00:13:34.493 never more than a few degrees from the hinges. 00:13:34.689 --> 00:13:38.637 Two curves, built out of nothing but four factors and a ruler. 00:13:39.341 --> 00:13:42.509 Two frequencies on these curves matter more than all the others, 00:13:42.592 --> 00:13:44.984 and the rest of the lecture is about them. 00:13:45.181 --> 00:13:48.316 The first is where the phase passes through minus one hundred 00:13:48.316 --> 00:13:49.698 and eighty degrees. 00:13:50.123 --> 00:13:51.912 You can find it exactly. 00:13:52.108 --> 00:13:54.846 Set the phase equal to minus one eighty. 00:13:54.929 --> 00:13:57.122 The integrator has already spent the ninety, 00:13:57.205 --> 00:14:01.385 so the two arctangents have to add up to ninety degrees between them. 00:14:02.089 --> 00:14:05.953 Two angles summing to ninety means their tangents multiply to one, 00:14:06.036 --> 00:14:10.146 and running that through the addition formula leaves one minus omega squared 00:14:10.146 --> 00:14:12.341 over twenty equal to zero. 00:14:13.138 --> 00:14:16.991 So the phase crosses minus one eighty at the square root of twenty, 00:14:17.074 --> 00:14:20.128 about four point four seven radians per second. 00:14:20.441 --> 00:14:23.100 Call it the phase crossover frequency. 00:14:23.297 --> 00:14:25.062 There it is on the curve. 00:14:25.766 --> 00:14:29.886 The second frequency is where the magnitude passes through zero decibels, 00:14:29.969 --> 00:14:33.104 where the loop neither amplifies nor attenuates. 00:14:33.301 --> 00:14:35.934 The skeleton puts that at two radians per second, 00:14:36.017 --> 00:14:38.828 where its two segments straddle the axis. 00:14:39.880 --> 00:14:44.023 The exact curve crosses lower, at about one point five six, 00:14:44.106 --> 00:14:48.309 and the gap is the three decibel corner error we already know about. 00:14:48.506 --> 00:14:52.963 Call that one the gain crossover frequency, and take the exact value, 00:14:53.046 --> 00:14:56.309 because we are about to measure a stability margin with it. 00:14:57.105 --> 00:15:00.961 Two frequencies, both read off curves we built by hand. 00:15:01.157 --> 00:15:04.351 Now let us close the loop and find out what they are worth. 00:15:05.459 --> 00:15:08.719 Close the loop around that plant with unity negative feedback, 00:15:08.802 --> 00:15:12.936 and the closed loop transfer function is L over one plus L. 00:15:13.249 --> 00:15:16.791 Everything about stability is hiding in that denominator. 00:15:17.587 --> 00:15:18.609 The loop is on the edge 00:15:18.609 --> 00:15:22.639 of trouble when the denominator vanishes at some real frequency. 00:15:22.836 --> 00:15:26.410 That is one plus L of j omega equal to zero, 00:15:26.493 --> 00:15:30.209 which is L of j omega equal to minus one. 00:15:31.005 --> 00:15:36.196 Now read minus one in polar form, because that is what a Bode plot shows you. 00:15:36.508 --> 00:15:41.827 Its magnitude is one, and its angle is minus one hundred and eighty degrees. 00:15:42.023 --> 00:15:45.763 Two conditions, and they have to hold at the same frequency. 00:15:46.559 --> 00:15:51.076 Let me show you what that actually means physically, because the algebra hides it. 00:15:51.389 --> 00:15:56.020 Break the loop, and inject a sine wave at the phase crossover frequency, 00:15:56.103 --> 00:15:58.785 four point four seven radians per second. 00:15:58.982 --> 00:16:00.840 That is the grey wave. 00:16:01.544 --> 00:16:03.112 Send it round the plant. 00:16:03.309 --> 00:16:05.759 What comes back is the blue wave. 00:16:05.956 --> 00:16:10.006 It is one sixth the size, because we measured that magnitude, 00:16:10.089 --> 00:16:14.815 and it is upside down, because the phase there is minus one eighty. 00:16:15.128 --> 00:16:18.496 Half a cycle of delay is exactly an inversion. 00:16:19.199 --> 00:16:21.928 And now the summing junction does its job. 00:16:22.543 --> 00:16:27.246 Negative feedback subtracts, which flips that wave over one more time. 00:16:27.442 --> 00:16:32.119 So what actually gets added back in is the green wave, and look where it sits. 00:16:32.202 --> 00:16:34.688 It is in phase with what we injected. 00:16:34.791 --> 00:16:37.381 Perfectly, exactly in phase. 00:16:38.178 --> 00:16:40.907 That is the whole danger of this frequency. 00:16:41.104 --> 00:16:45.142 Everywhere else the returning signal partly cancels what is already there. 00:16:45.225 --> 00:16:47.304 Here it reinforces it. 00:16:47.501 --> 00:16:49.986 The only thing saving us is the size. 00:16:50.597 --> 00:16:52.801 It comes back one sixth as tall, 00:16:52.884 --> 00:16:57.065 so each trip round the loop it shrinks, and the ringing dies away. 00:16:57.261 --> 00:17:00.048 But suppose we multiplied the gain by six. 00:17:00.245 --> 00:17:03.810 Then it would come back exactly the same size it went in. 00:17:04.723 --> 00:17:08.822 And there it is, landing precisely on top of the wave we injected. 00:17:09.018 --> 00:17:11.747 At that point the loop no longer needs us. 00:17:12.060 --> 00:17:16.264 It can sustain that oscillation with no input at all, forever. 00:17:17.061 --> 00:17:21.055 So the factor six is a real quantity, and it has a name. 00:17:21.368 --> 00:17:23.122 It is the gain margin: 00:17:23.225 --> 00:17:25.002 how much you may multiply the loop gain 00:17:25.002 --> 00:17:28.114 by before the returning wave comes back full size. 00:17:29.166 --> 00:17:31.779 You read it off the plots in two steps. 00:17:32.231 --> 00:17:36.319 Find the frequency where the phase curve crosses minus one eighty. 00:17:36.422 --> 00:17:38.966 Drop straight up to the magnitude curve. 00:17:39.162 --> 00:17:43.807 Whatever gap you find between there and zero decibels is the gain margin. 00:17:44.720 --> 00:17:48.145 Here that gap measures fifteen point six decibels. 00:17:48.760 --> 00:17:52.067 And fifteen point six decibels is a factor of six, 00:17:52.150 --> 00:17:55.436 which is exactly the number we found by hand a moment ago. 00:17:56.233 --> 00:17:59.612 The other margin runs the same procedure the other way round. 00:18:00.227 --> 00:18:01.862 Start on the magnitude curve, 00:18:01.945 --> 00:18:06.450 at the frequency where it crosses zero decibels, one point five six. 00:18:06.902 --> 00:18:09.225 Drop down to the phase curve there. 00:18:09.708 --> 00:18:13.062 The phase reads minus one hundred and thirty seven degrees. 00:18:13.145 --> 00:18:16.661 So we are forty three degrees short of minus one eighty, 00:18:16.744 --> 00:18:18.881 and that is the phase margin: 00:18:18.985 --> 00:18:22.840 how much extra lag you could pour into this loop before it sang. 00:18:23.637 --> 00:18:27.304 And that second one is usually the number engineers care about more, 00:18:27.387 --> 00:18:32.400 because extra lag arrives whether you asked for it or not. A transport delay, 00:18:32.483 --> 00:18:37.348 a filter you forgot, an actuator that is slower than the model says. 00:18:37.545 --> 00:18:41.528 All of it eats phase margin, and none of it touches gain margin. 00:18:42.325 --> 00:18:45.379 Forty three degrees and a factor of six. 00:18:45.691 --> 00:18:47.492 Now let me spend them. 00:18:48.289 --> 00:18:51.294 First, what a gain change does to these two curves, 00:18:51.377 --> 00:18:53.978 because it is the simplest thing in the subject. 00:18:54.291 --> 00:18:58.759 Multiplying by a constant adds a constant number of decibels at every frequency, 00:18:58.842 --> 00:19:01.908 so the magnitude curve moves bodily upward. 00:19:02.105 --> 00:19:04.265 Its shape does not change at all. 00:19:05.062 --> 00:19:07.277 And a positive constant has no angle, 00:19:07.360 --> 00:19:10.252 so the phase curve does not move by one degree. 00:19:10.449 --> 00:19:11.541 Watch it. 00:19:11.644 --> 00:19:14.534 It will sit perfectly still through everything that follows, 00:19:14.617 --> 00:19:18.077 which is why the phase crossover frequency never moves either. 00:19:19.291 --> 00:19:24.807 Now lift the gain by our whole margin, a factor of six, fifteen point six decibels. 00:19:25.004 --> 00:19:28.859 Up goes the magnitude curve, and it keeps its shape exactly. 00:19:29.563 --> 00:19:32.628 And look what has happened at the phase crossover. 00:19:32.825 --> 00:19:34.357 The point that was fifteen point 00:19:34.357 --> 00:19:38.375 six decibels down is now sitting exactly on zero. 00:19:38.688 --> 00:19:43.007 The gain crossover and the phase crossover have arrived at the same frequency. 00:19:43.804 --> 00:19:48.042 Both conditions for minus one now hold together, and here is what 00:19:48.042 --> 00:19:49.935 that does in the time domain. 00:19:50.131 --> 00:19:54.765 This is the closed loop answering a step, at the original gain of forty. 00:19:55.217 --> 00:19:59.699 Some overshoot, a couple of swings, settled inside five seconds. 00:20:00.310 --> 00:20:03.433 And this is the same loop with the gain multiplied by six, 00:20:03.433 --> 00:20:05.034 at two hundred and forty. 00:20:05.117 --> 00:20:06.578 It never settles. 00:20:06.661 --> 00:20:12.478 It never diverges either. It just rings, with the same amplitude, indefinitely. 00:20:13.275 --> 00:20:15.316 Measure the period of that ringing. 00:20:15.399 --> 00:20:17.151 It is one point four seconds, 00:20:17.234 --> 00:20:21.287 and two pi over four point four seven is one point four. 00:20:21.390 --> 00:20:26.323 The loop is oscillating at precisely the frequency we identified on the phase plot, 00:20:26.406 --> 00:20:28.543 using nothing but arctangents. 00:20:29.339 --> 00:20:33.587 Push a little further, to two hundred and eighty, which is past the margin, 00:20:33.670 --> 00:20:36.445 and the oscillation grows instead of holding. 00:20:36.758 --> 00:20:41.077 The closed loop poles have crossed into the right half plane, and the frequency 00:20:41.077 --> 00:20:45.281 response of the open loop told us the exact moment it would happen. 00:20:46.193 --> 00:20:48.560 So the margins are not bookkeeping. 00:20:48.643 --> 00:20:52.264 They are the distance, measured in decibels and in degrees, 00:20:52.347 --> 00:20:56.039 between the loop you have and a loop that sings on its own. 00:20:56.491 --> 00:20:59.534 What is left is to buy some of that distance back. 00:21:00.577 --> 00:21:03.294 Everything so far has been diagnosis. 00:21:03.398 --> 00:21:07.869 We took a plant apart, drew its two curves, and read two margins off them. 00:21:08.065 --> 00:21:12.489 Now let us change those curves on purpose, with the two simplest tools there are. 00:21:13.193 --> 00:21:17.000 Both of them are one zero and one pole. That is all. 00:21:17.083 --> 00:21:17.849 And the only thing 00:21:17.849 --> 00:21:21.983 that distinguishes them is which of the two sits at the lower frequency. 00:21:22.687 --> 00:21:27.726 Put the pole first, below the zero, and you have a lag compensator. 00:21:27.830 --> 00:21:32.126 At zero frequency the two brackets are both one, so it does nothing. 00:21:32.323 --> 00:21:37.525 Above the zero, the denominator has grown more than the numerator, so it attenuates. 00:21:38.437 --> 00:21:43.651 Put the zero first instead, below the pole, and you have a lead compensator. 00:21:44.103 --> 00:21:46.563 Again it is worth one at zero frequency, 00:21:46.646 --> 00:21:50.582 but now above the pole the numerator has won, so it amplifies. 00:21:50.686 --> 00:21:55.063 And in between it hands you phase lead, which is the thing you actually wanted. 00:21:55.860 --> 00:21:59.123 So one of them cuts the high end and one of them lifts it. 00:21:59.227 --> 00:22:02.456 Let us watch what each does to a loop we already know. 00:22:03.508 --> 00:22:07.467 Here is our plant again, in blue, on both plots. 00:22:07.919 --> 00:22:12.109 I have widened the frequency axis down to a hundredth of a radian per second, 00:22:12.192 --> 00:22:17.057 two decades lower than before, because that is where a lag compensator lives. 00:22:17.970 --> 00:22:20.850 Now the lag section on its own, in yellow. 00:22:21.046 --> 00:22:24.133 Its pole is at a hundred and twenty five ten thousandths, 00:22:24.216 --> 00:22:26.388 its zero at five hundredths. 00:22:26.492 --> 00:22:28.800 So it starts flat at zero decibels, 00:22:28.883 --> 00:22:33.029 falls at twenty per decade between the two, and then goes flat again. 00:22:33.825 --> 00:22:35.567 How far does it fall? 00:22:35.764 --> 00:22:40.699 The ratio of the pole to the zero, a quarter, which is twelve decibels. 00:22:41.012 --> 00:22:43.764 That is the whole of what this section is for: 00:22:43.961 --> 00:22:48.501 a twelve decibel cut, delivered at every frequency above five hundredths. 00:22:49.205 --> 00:22:50.947 And here is its phase. 00:22:51.050 --> 00:22:55.962 It dips down about thirty degrees in the middle, and comes back to zero afterwards. 00:22:56.159 --> 00:22:59.547 That dip is the price you pay, and look where it is: 00:22:59.630 --> 00:23:02.475 two decades below anything we care about. 00:23:02.579 --> 00:23:06.294 By the time we reach the interesting frequencies the section has given all 00:23:06.294 --> 00:23:07.502 of it back. 00:23:08.415 --> 00:23:12.943 Add the yellow to the blue, and the green curves are the compensated loop. 00:23:13.140 --> 00:23:18.516 Identical to the plant at low frequency, twelve decibels below it everywhere else. 00:23:19.731 --> 00:23:21.935 So the magnitude curve has dropped, 00:23:22.018 --> 00:23:26.430 and the point where it crosses zero decibels has slid a long way to the left. 00:23:26.627 --> 00:23:30.946 From one point five six down to about nought point four nine. 00:23:31.650 --> 00:23:33.508 And that is the whole trick. 00:23:33.612 --> 00:23:37.955 Down at nought point four nine, the plant had barely started lagging. 00:23:38.152 --> 00:23:41.680 The phase there is about minus one hundred and eleven degrees, 00:23:41.763 --> 00:23:45.804 so the phase margin is sixty nine, up from forty three. 00:23:46.600 --> 00:23:51.222 The gain margin improves too, from six to twenty four, because the whole magnitude 00:23:51.222 --> 00:23:55.030 curve dropped and the phase crossover frequency hardly moved. 00:23:55.343 --> 00:23:57.318 But notice what it cost. 00:23:57.421 --> 00:24:00.043 The crossover frequency is the loop's bandwidth, 00:24:00.126 --> 00:24:02.449 and we just divided it by three. 00:24:02.646 --> 00:24:05.387 This loop is safer and it is slower. 00:24:06.184 --> 00:24:08.713 Now clear those away and try the other one. 00:24:08.796 --> 00:24:13.067 The lead has its zero at one point two and its pole at seven point two, 00:24:13.150 --> 00:24:17.574 so both of them sit right in the middle of the action, near the old crossover. 00:24:18.370 --> 00:24:21.784 In yellow again, its magnitude is flat, then rises 00:24:21.784 --> 00:24:26.600 at twenty decibels per decade between the zero and the pole, then flat again. 00:24:26.683 --> 00:24:29.619 The ratio of zero to pole is one sixth, 00:24:29.702 --> 00:24:33.023 so it lifts the high end by about sixteen decibels. 00:24:33.564 --> 00:24:36.200 And here is the part we actually came for. 00:24:36.304 --> 00:24:39.007 Its phase rises to a maximum partway up, 00:24:39.090 --> 00:24:43.747 and for a ratio of one sixth that maximum is forty six degrees. 00:24:43.850 --> 00:24:47.367 It happens at the geometric mean of the zero and the pole, 00:24:47.450 --> 00:24:49.470 which is two point nine four. 00:24:50.267 --> 00:24:54.261 So you place the compensator so that its phase peak lands where the loop is going 00:24:54.261 --> 00:24:56.305 to cross zero decibels. 00:24:56.501 --> 00:25:00.658 Add the two together, and here are the compensated curves. 00:25:01.455 --> 00:25:05.008 The magnitude has been lifted at the top end, so crossover moves 00:25:05.008 --> 00:25:09.954 the other way this time, up from one point five six to two point six five. 00:25:10.151 --> 00:25:13.388 And on an uncompensated plant that would be bad news, 00:25:13.471 --> 00:25:15.597 because the phase up there is worse. 00:25:16.509 --> 00:25:18.077 But the lead has bumped the phase up 00:25:18.077 --> 00:25:21.270 by forty five degrees at exactly that frequency. 00:25:21.374 --> 00:25:22.837 So instead of the minus one hundred 00:25:22.837 --> 00:25:26.945 and fifty eight the plant would have given us, we read minus one hundred and twelve, 00:25:27.028 --> 00:25:29.374 and the phase margin is sixty eight. 00:25:30.286 --> 00:25:34.604 The gain margin is essentially unchanged, about sixteen decibels, 00:25:34.687 --> 00:25:36.161 because the lead pushed the phase 00:25:36.161 --> 00:25:40.922 crossover frequency out and lifted the magnitude there by about the same amount. 00:25:41.118 --> 00:25:43.383 Both effects roughly cancel. 00:25:44.087 --> 00:25:45.933 So look at what we bought. 00:25:46.130 --> 00:25:48.323 The same phase margin the lag gave us, 00:25:48.406 --> 00:25:52.572 near enough, but with the crossover frequency up rather than down. 00:25:52.655 --> 00:25:55.488 This loop is safer and it is faster. 00:25:56.285 --> 00:25:57.900 Side by side, then. 00:25:58.212 --> 00:26:02.483 The lag cuts the high end of the magnitude curve. The lead lifts it, 00:26:02.566 --> 00:26:06.178 and that single difference drives everything else about the two of them. 00:26:06.975 --> 00:26:11.013 The lag's own phase contribution is a nuisance to be parked out of the way, 00:26:11.096 --> 00:26:13.303 two decades below the action. 00:26:13.615 --> 00:26:16.423 The lead's phase contribution is the entire point, 00:26:16.506 --> 00:26:20.118 and you place it deliberately on top of the crossover frequency. 00:26:20.462 --> 00:26:23.864 So the lag moves the crossover down and the lead moves it up. 00:26:24.061 --> 00:26:26.276 Both of them end up buying you phase margin, 00:26:26.359 --> 00:26:30.177 and they buy it in opposite currencies: one pays with speed, 00:26:30.260 --> 00:26:34.859 the other pays with amplified high frequency, which means amplified noise. 00:26:35.238 --> 00:26:40.672 And both are one zero and one pole, the same two curves we drew by hand earlier. 00:26:40.776 --> 00:26:43.969 Which one you put first is the whole difference between them. 00:26:44.673 --> 00:26:50.012 That is the arc. A sine wave in gives a sine wave out, and two numbers record it. 00:26:50.095 --> 00:26:53.983 Those two numbers, plotted against a logarithmic frequency axis, 00:26:54.066 --> 00:26:58.037 are sums of straight lines you can draw from the poles and zeros alone. 00:26:58.741 --> 00:27:02.991 Close the loop, and one frequency on those plots decides everything: 00:27:03.095 --> 00:27:06.587 the one where the phase reaches minus one hundred and eighty degrees, 00:27:06.670 --> 00:27:10.166 where the returning signal comes back in step with itself. 00:27:10.362 --> 00:27:14.703 How far the magnitude sits below unity there is your gain margin, 00:27:14.786 --> 00:27:20.150 and how far the phase sits above minus one eighty at unity gain is your phase margin.