1 00:00:06,270 --> 00:00:09,980 KENDRA PUGH: Today I'd like to talk to about poles. 2 00:00:09,980 --> 00:00:12,540 Last time I ended up talking to you about LTI 3 00:00:12,540 --> 00:00:14,180 representations and manipulations. 4 00:00:14,180 --> 00:00:17,570 And, in particular, I want to emphasize the relationship 5 00:00:17,570 --> 00:00:20,870 between feedforward and feedback systems, in order to 6 00:00:20,870 --> 00:00:23,790 segway us into the poles.. 7 00:00:23,790 --> 00:00:27,160 Using what we know about that relationship, we can find the 8 00:00:27,160 --> 00:00:28,640 base of a geometric sequence. 9 00:00:28,640 --> 00:00:31,240 And that geometric sequence actually represents the 10 00:00:31,240 --> 00:00:35,220 long-term response of our system to a unit sample 11 00:00:35,220 --> 00:00:38,690 response or a delta. 12 00:00:38,690 --> 00:00:42,710 That value is also what we refer to the poles. 13 00:00:42,710 --> 00:00:45,450 At this point, I'll go over how to solve for them and very 14 00:00:45,450 --> 00:00:47,080 basic properties of them. 15 00:00:47,080 --> 00:00:49,440 So that we can use the information that we have about 16 00:00:49,440 --> 00:00:52,980 the poles to try to actually predict the future or look at 17 00:00:52,980 --> 00:00:54,681 the long-term behavior of our system. 18 00:00:57,980 --> 00:01:01,200 First a quick review. 19 00:01:01,200 --> 00:01:03,290 Last time we talked about feedforward systems. 20 00:01:03,290 --> 00:01:06,570 And, in particular, I want to emphasize the fact that if you 21 00:01:06,570 --> 00:01:08,690 have a transient input to your feedforward system, you're 22 00:01:08,690 --> 00:01:10,680 going to end up with a transient response. 23 00:01:10,680 --> 00:01:14,550 There's no method by which a feedforward system can retain 24 00:01:14,550 --> 00:01:17,440 information over more than the amount of time steps that you 25 00:01:17,440 --> 00:01:20,170 feed information into it. 26 00:01:20,170 --> 00:01:23,170 Feedback systems, on the other hand, represent a persistent 27 00:01:23,170 --> 00:01:24,940 response to a transient input. 28 00:01:24,940 --> 00:01:27,480 Because you're working with a feedback system information 29 00:01:27,480 --> 00:01:29,690 that you put in can be reflected in more than one 30 00:01:29,690 --> 00:01:32,540 time step and possibly multiple time steps, depending 31 00:01:32,540 --> 00:01:34,410 upon how many delays you have working in 32 00:01:34,410 --> 00:01:37,710 your feedback system. 33 00:01:37,710 --> 00:01:41,020 Last time I also drew out the relationship between 34 00:01:41,020 --> 00:01:43,370 feedforward and feedback systems. 35 00:01:43,370 --> 00:01:46,200 You can actually turn a feedback system or talk about 36 00:01:46,200 --> 00:01:49,790 a feedback system in terms of a feedforward system that 37 00:01:49,790 --> 00:01:55,980 takes infinite samples of the input and feeds them through a 38 00:01:55,980 --> 00:02:00,570 summation that takes infinite delays through your system. 39 00:02:04,240 --> 00:02:05,960 You can represent that translation 40 00:02:05,960 --> 00:02:08,660 using a geometric sequence. 41 00:02:08,660 --> 00:02:11,470 The basis of that geometric sequence is the object that 42 00:02:11,470 --> 00:02:13,630 we're going to use in order to predict the future. 43 00:02:13,630 --> 00:02:15,320 And that's what we're talking about when 44 00:02:15,320 --> 00:02:16,570 we talk about poles. 45 00:02:21,470 --> 00:02:23,550 You can have multiple geometric sequences involved 46 00:02:23,550 --> 00:02:24,740 in actually determining the long-term 47 00:02:24,740 --> 00:02:27,780 behaviors of your system. 48 00:02:27,780 --> 00:02:31,780 If you only have one, then things are pretty simple. 49 00:02:31,780 --> 00:02:33,680 You find your system function. 50 00:02:33,680 --> 00:02:37,770 You find the value associated with p0 in this expression. 51 00:02:37,770 --> 00:02:44,290 It's OK if there's some sort of scalar on the outside of 52 00:02:44,290 --> 00:02:45,400 this expression. 53 00:02:45,400 --> 00:02:47,525 We're working with linear time and variance systems, so that 54 00:02:47,525 --> 00:02:51,730 scalar is going to affect the initial 55 00:02:51,730 --> 00:02:52,950 response to your system. 56 00:02:52,950 --> 00:02:56,150 But in terms of a long term behavior, it 57 00:02:56,150 --> 00:02:57,080 doesn't matter as much. 58 00:02:57,080 --> 00:02:59,750 So don't worry about it right now. 59 00:02:59,750 --> 00:03:04,450 Relatedly, if you're solving for these expressions in 60 00:03:04,450 --> 00:03:07,880 second or higher order systems, you're going to end 61 00:03:07,880 --> 00:03:10,450 up having to solve partial fractions. 62 00:03:10,450 --> 00:03:11,480 You can do this. 63 00:03:11,480 --> 00:03:13,220 And in part, one of the reasons that you would want to 64 00:03:13,220 --> 00:03:15,800 do this is so that you can get out those scalars, if you're 65 00:03:15,800 --> 00:03:18,960 going to be talking about the very short-term response to 66 00:03:18,960 --> 00:03:22,780 something, like a transient input. 67 00:03:22,780 --> 00:03:24,515 We're not going to be too interested in 68 00:03:24,515 --> 00:03:25,220 those in this course. 69 00:03:25,220 --> 00:03:26,180 We're mostly going to be talking 70 00:03:26,180 --> 00:03:27,860 about long-term response. 71 00:03:27,860 --> 00:03:32,450 So we can get around the fact the we're dealing with a 72 00:03:32,450 --> 00:03:37,390 higher order of systems and not solving partial fractions 73 00:03:37,390 --> 00:03:42,960 by substituting in for an expression called z, which 74 00:03:42,960 --> 00:03:46,990 actually represents the inverse power of R and then 75 00:03:46,990 --> 00:03:50,760 solving for the roots of that equation. 76 00:03:50,760 --> 00:03:56,600 If you substitute z in for 1 over R in this denominator and 77 00:03:56,600 --> 00:04:00,040 then solve for the root associated with that 78 00:04:00,040 --> 00:04:03,470 expression, you'll get the same result. 79 00:04:03,470 --> 00:04:05,290 You'll actually end up out with p0. 80 00:04:09,080 --> 00:04:09,640 All right. 81 00:04:09,640 --> 00:04:14,530 So now we know how to find the pole or multiple poles, if 82 00:04:14,530 --> 00:04:16,660 we're interested in multiple poles. 83 00:04:16,660 --> 00:04:17,310 What do we do now? 84 00:04:17,310 --> 00:04:19,529 I still haven't gone over how to figure out the long-term 85 00:04:19,529 --> 00:04:20,779 behavior of your system. 86 00:04:24,020 --> 00:04:27,250 The first thing you do is look at the magnitude of all the 87 00:04:27,250 --> 00:04:31,140 poles that you've solved for and select the poles with the 88 00:04:31,140 --> 00:04:34,580 largest magnitude. 89 00:04:34,580 --> 00:04:39,000 If there are multiple poles with the same magnitude, then 90 00:04:39,000 --> 00:04:40,250 you'll end up looking at all of them. 91 00:04:44,680 --> 00:04:48,870 If you have different properties than the ones here, 92 00:04:48,870 --> 00:04:52,400 you can end up with some, you know, complex behavior. 93 00:04:52,400 --> 00:04:53,960 I would not worry about that too much. 94 00:04:53,960 --> 00:04:57,960 Or I would ask a professor or TA when that happens. 95 00:04:57,960 --> 00:05:02,890 But in the general sense, if your dominant pole has a 96 00:05:02,890 --> 00:05:06,790 magnitude greater than 1, then you're going to see long-term 97 00:05:06,790 --> 00:05:09,130 divergence in your system. 98 00:05:09,130 --> 00:05:10,930 This make sense if you think about it. 99 00:05:10,930 --> 00:05:17,570 If at every time step your unit sample response is 100 00:05:17,570 --> 00:05:20,810 multiplied by a value that is greater than 1, then it's 101 00:05:20,810 --> 00:05:21,730 going to increase. 102 00:05:21,730 --> 00:05:26,890 And, in fact, the extent to which the magnitude of your 103 00:05:26,890 --> 00:05:29,280 dominant pole is greater than 1 is going to determine your 104 00:05:29,280 --> 00:05:34,150 rate of increase and also determine how fast your 105 00:05:34,150 --> 00:05:35,400 envelope explodes. 106 00:05:40,040 --> 00:05:42,380 Similarly, if the magnitude of your dominant pole is less 107 00:05:42,380 --> 00:05:48,070 than 1, then in response to a unit sample input or a delta, 108 00:05:48,070 --> 00:05:49,320 your system's going to converge. 109 00:05:52,110 --> 00:05:53,470 This also makes sense intuitively. 110 00:05:53,470 --> 00:05:56,260 If you are progressively multiplying the values in your 111 00:05:56,260 --> 00:06:00,670 system by a scalar that is less than 1, then eventually 112 00:06:00,670 --> 00:06:02,030 you're going to end up converging to 0. 113 00:06:07,730 --> 00:06:10,630 To cover the only category we haven't talked about, if your 114 00:06:10,630 --> 00:06:14,580 dominant pole is actually equal in magnitude to 1, then 115 00:06:14,580 --> 00:06:17,350 you're not going to see convergence or divergence. 116 00:06:17,350 --> 00:06:21,710 And this is one of the places in which the magnitude of the 117 00:06:21,710 --> 00:06:23,640 scalar that you end up multiplying your system by can 118 00:06:23,640 --> 00:06:24,990 become relevant. 119 00:06:24,990 --> 00:06:28,170 We're not going to focus on this situation too much. 120 00:06:28,170 --> 00:06:30,640 But it's good to know what actually happens when the 121 00:06:30,640 --> 00:06:32,220 magnitude of your dominant pole is equal to 1. 122 00:06:35,760 --> 00:06:39,770 The other feature that we're interested in when we're 123 00:06:39,770 --> 00:06:44,380 looking at the dominant pole of a system is if we were to 124 00:06:44,380 --> 00:06:48,240 represent the dominant pole in this form, what the angle 125 00:06:48,240 --> 00:06:54,240 associated with that pole is, if you were to graph that pole 126 00:06:54,240 --> 00:06:56,710 on the complex plane using polar coordinates. 127 00:07:00,590 --> 00:07:03,720 If your pole stays on the real axis, or if your pole does not 128 00:07:03,720 --> 00:07:09,180 have a complex component, then you'll see one of two things. 129 00:07:09,180 --> 00:07:11,490 The first thing that it's possible for you to see is 130 00:07:11,490 --> 00:07:14,570 that you'll get absolutely non-alternating behavior. 131 00:07:14,570 --> 00:07:17,660 Your system response stays on one side of the x-axis and 132 00:07:17,660 --> 00:07:24,540 either converges, diverges, or remains constant as a 133 00:07:24,540 --> 00:07:29,220 consequence of input of the unit sample. 134 00:07:29,220 --> 00:07:30,855 And you won't see any sort of alternating 135 00:07:30,855 --> 00:07:32,950 or oscillating behavior. 136 00:07:32,950 --> 00:07:34,730 This only happens when your dominant 137 00:07:34,730 --> 00:07:37,640 pole is real and positive. 138 00:07:37,640 --> 00:07:40,910 If you're dominant pole is real and negative, this also 139 00:07:40,910 --> 00:07:42,980 means that it's still on the real axis, but 140 00:07:42,980 --> 00:07:44,350 its value is negative. 141 00:07:44,350 --> 00:07:46,600 So if you're looking at polar coordinates, it's going to 142 00:07:46,600 --> 00:07:48,800 have an angle pi associated with it. 143 00:07:48,800 --> 00:07:51,550 This means you get alternating behavior. 144 00:07:51,550 --> 00:07:53,970 And what I mean when I say alternating behavior is that 145 00:07:53,970 --> 00:07:58,770 your unit sample response is going to jump across the 146 00:07:58,770 --> 00:08:02,690 x-axis at every time step. 147 00:08:02,690 --> 00:08:04,790 This is also equivalent to having a period of 2. 148 00:08:07,610 --> 00:08:10,880 The other situation you can run into is that this angle is 149 00:08:10,880 --> 00:08:12,530 neither 0 nor pi. 150 00:08:12,530 --> 00:08:14,270 And at that point you're going to be talking about 151 00:08:14,270 --> 00:08:19,060 oscillatory behavior or a sinusoidal response that 152 00:08:19,060 --> 00:08:21,270 retains its edges at the envelope of your function. 153 00:08:24,030 --> 00:08:27,070 In order to find the period, or in order to find the amount 154 00:08:27,070 --> 00:08:31,020 of time it takes for your unit sample response to complete 155 00:08:31,020 --> 00:08:34,960 one period, you're going to take the angle associated with 156 00:08:34,960 --> 00:08:39,070 your dominant pole and divide 2pi by it. 157 00:08:39,070 --> 00:08:43,049 This is the general equation for a period. 158 00:08:43,049 --> 00:08:47,880 This covers the basics of what you want to do once you 159 00:08:47,880 --> 00:08:49,270 already have your poles. 160 00:08:49,270 --> 00:08:52,430 Next time I'm actually going to solve a pole problem and 161 00:08:52,430 --> 00:08:55,370 show you what the long-term response looks like and also 162 00:08:55,370 --> 00:08:58,100 talk about some things about poles that I've pretty much 163 00:08:58,100 --> 00:09:00,300 skimmed over. 164 00:09:00,300 --> 00:09:03,330 And at that point you should be able to solve and look at 165 00:09:03,330 --> 00:09:04,580 poles for yourself.