1
00:00:04,710 --> 00:00:06,880
PROFESSOR: Last time we
introduced poles.

2
00:00:06,880 --> 00:00:09,270
And in particular, we introduced
how to move from

3
00:00:09,270 --> 00:00:11,590
the manipulation of feed-forward
and feedback

4
00:00:11,590 --> 00:00:15,370
systems and the geometric
sequence that fell out into

5
00:00:15,370 --> 00:00:18,240
using the base of that geometric
sequence to attempt

6
00:00:18,240 --> 00:00:21,010
to predict the long-term
behavior of the system.

7
00:00:21,010 --> 00:00:23,600
When we're solving for poles and
we're only interested in

8
00:00:23,600 --> 00:00:26,320
long term behavior, one of the
easiest ways to do so to solve

9
00:00:26,320 --> 00:00:29,770
for the roots of z, where z is a
substitution for 1 over R in

10
00:00:29,770 --> 00:00:33,230
the denominator of the
system function.

11
00:00:33,230 --> 00:00:36,380
Once we've done that, we
have a list of poles.

12
00:00:36,380 --> 00:00:38,990
From that list of poles, we
would like to select, the

13
00:00:38,990 --> 00:00:41,400
dominant pole, or the pole with
the greatest magnitude,

14
00:00:41,400 --> 00:00:43,820
and then based on the magnitude
and period of that

15
00:00:43,820 --> 00:00:46,540
pole we can determine what the
long term behavior of our

16
00:00:46,540 --> 00:00:47,790
system looks like.

17
00:00:52,380 --> 00:00:55,620
Today I'd like to mention
to you some notable

18
00:00:55,620 --> 00:00:57,150
things about poles.

19
00:00:57,150 --> 00:00:59,730
If you are interested in this
information or feedback and

20
00:00:59,730 --> 00:01:03,130
controls in the general sense,
I highly recommend 6.003.

21
00:01:03,130 --> 00:01:05,340
But here's some information you
should at least be aware

22
00:01:05,340 --> 00:01:08,400
of as a consequence of 6.01.

23
00:01:08,400 --> 00:01:10,130
The other thing I would like to
do is just walk through a

24
00:01:10,130 --> 00:01:12,940
couple of pole problems to
familiarize you, or get you

25
00:01:12,940 --> 00:01:15,740
more comfortable with the idea
of solving for the poles of a

26
00:01:15,740 --> 00:01:19,450
system function, or looking at
the unit sample response of a

27
00:01:19,450 --> 00:01:23,670
system function and then
graphing the poles.

28
00:01:23,670 --> 00:01:25,450
The first thing that I want
to mention is pole-zero

29
00:01:25,450 --> 00:01:26,450
cancellation.

30
00:01:26,450 --> 00:01:27,990
And what do I mean
when I say that?

31
00:01:27,990 --> 00:01:40,950
I mean that if both the
numerator and the denominator

32
00:01:40,950 --> 00:01:43,970
have a degree of R in them, then
you're going to have both

33
00:01:43,970 --> 00:01:46,420
a zero and a pole.

34
00:01:46,420 --> 00:01:50,740
If the zero and the pole have
the same value associated with

35
00:01:50,740 --> 00:01:54,590
them, you may be tempted
to cancel them out.

36
00:01:54,590 --> 00:01:58,740
Unless both the zero and the
pole are equal to 0 --

37
00:01:58,740 --> 00:02:00,230
don't do it.

38
00:02:00,230 --> 00:02:03,720
The reason why is that when you
get to a implementation of

39
00:02:03,720 --> 00:02:07,300
a real system, it is highly
unlikely that both the zero

40
00:02:07,300 --> 00:02:12,170
and the pole will be implemented
to a degree of

41
00:02:12,170 --> 00:02:15,320
accuracy that you will actually
see those two things

42
00:02:15,320 --> 00:02:16,190
cancel out.

43
00:02:16,190 --> 00:02:18,640
The only exception to this is
when both the pole and the

44
00:02:18,640 --> 00:02:21,140
zero are equal to 0,
or in this case.

45
00:02:34,520 --> 00:02:37,990
This you should feel free
to convert to this.

46
00:02:37,990 --> 00:02:40,880
In almost any other situation,
don't factor.

47
00:02:43,400 --> 00:02:45,280
The other thing I want to talk
about is repeated roots.

48
00:02:57,530 --> 00:03:01,540
If you have a repeated root,
you'll have repeated poles.

49
00:03:01,540 --> 00:03:06,370
This does get tricky when you're
talking about how to

50
00:03:06,370 --> 00:03:09,290
add the unit response
of those poles.

51
00:03:09,290 --> 00:03:11,330
But the long term behavior
of your system is

52
00:03:11,330 --> 00:03:12,640
going to look the same.

53
00:03:12,640 --> 00:03:16,090
So if both of these poles are
the dominant pole, then the

54
00:03:16,090 --> 00:03:19,240
characteristics of both, which
are the same, are going to

55
00:03:19,240 --> 00:03:22,060
determine what your long term
behavior looks like.

56
00:03:22,060 --> 00:03:24,930
If they're not, then the
dominant pole is going to

57
00:03:24,930 --> 00:03:28,630
determine what your long term
behavior looks like.

58
00:03:28,630 --> 00:03:29,910
The last thing I want to mention
is superposition.

59
00:03:35,270 --> 00:03:38,490
So far we've only talked about
the unit sample response of a

60
00:03:38,490 --> 00:03:42,430
system function and how we use
poles to determine what the

61
00:03:42,430 --> 00:03:45,500
long term behavior of our
system's going to be.

62
00:03:45,500 --> 00:03:50,730
We can look at the response to
more complicated inputs than

63
00:03:50,730 --> 00:03:53,160
the unit sample response,
or the delta.

64
00:03:53,160 --> 00:03:54,850
In fact, one of the things we'
probably end up looking at

65
00:03:54,850 --> 00:03:58,380
some point is the
step function.

66
00:03:58,380 --> 00:04:00,570
The thing that you need to know
to go from talking about

67
00:04:00,570 --> 00:04:03,940
unit sample response to any
other sort of response, is

68
00:04:03,940 --> 00:04:05,820
that we're still working
with an LTI system.

69
00:04:09,350 --> 00:04:17,589
What that means is if you take
the summation of your inputs,

70
00:04:17,589 --> 00:04:20,649
and apply the system function
to that summation, it is the

71
00:04:20,649 --> 00:04:27,300
same as the output that would
result from inputting all

72
00:04:27,300 --> 00:04:28,550
those values at once.

73
00:04:39,540 --> 00:04:41,500
The best way I would like to
explain it is by referring

74
00:04:41,500 --> 00:05:04,990
again to if your function was
a system function, the same

75
00:05:04,990 --> 00:05:06,240
property applies.

76
00:05:12,190 --> 00:05:14,730
Now let's walk through
a pole problem.

77
00:05:14,730 --> 00:05:17,170
Here I have a second order
system set up.

78
00:05:17,170 --> 00:05:25,700
We've got two degrees of R. I
have feedback, and I can solve

79
00:05:25,700 --> 00:05:29,540
for an expression of
y in terms of x.

80
00:05:29,540 --> 00:05:32,320
In fact, let's do
that right now.

81
00:05:32,320 --> 00:05:38,960
y is the result of the
summation or a linear

82
00:05:38,960 --> 00:05:48,900
combination of x plus a delayed
signal of y scaled by

83
00:05:48,900 --> 00:05:58,710
1.6 in a linear combination with
a delayed value of the

84
00:05:58,710 --> 00:06:10,020
delayed value of y scaled
by negative 0.63.

85
00:06:10,020 --> 00:06:11,270
There's my first degree.

86
00:06:21,070 --> 00:06:24,150
For consistency's sake, there's
my second degree.

87
00:06:27,610 --> 00:06:29,380
Let's first solve for
the system function.

88
00:07:02,460 --> 00:07:05,600
If you're confused, I recommend
doing the algebra

89
00:07:05,600 --> 00:07:09,520
from here to this expression.

90
00:07:09,520 --> 00:07:10,770
You should get this
fraction out.

91
00:07:13,450 --> 00:07:16,510
Our second step is to solve
for the roots of z.

92
00:07:16,510 --> 00:07:27,210
Remember that z is equal to 1
over R in the denominator of

93
00:07:27,210 --> 00:07:28,460
the system function.

94
00:07:39,520 --> 00:07:44,085
In this case, we'll
be working with--

95
00:07:52,780 --> 00:07:56,885
all I've done there is taken
every degree of R, substitute

96
00:07:56,885 --> 00:07:58,740
it in for 1/z.

97
00:07:58,740 --> 00:08:02,120
and then multiply it out, so
that I'm not working with z in

98
00:08:02,120 --> 00:08:03,370
the denominator anymore.

99
00:08:03,370 --> 00:08:06,730
I'm actually just working with
everything in the numerator.

100
00:08:23,810 --> 00:08:25,930
If I follow this back out,
I get this expression.

101
00:08:31,720 --> 00:08:37,150
And my poles are going
to be 0.7 and 0.9.

102
00:08:37,150 --> 00:08:42,460
All right, based on my poles,
what are the properties of the

103
00:08:42,460 --> 00:08:45,380
unit sample response
in the long term?

104
00:08:45,380 --> 00:08:48,190
First thing I'm going to do is
look for the dominant pole

105
00:08:48,190 --> 00:08:49,990
among the poles that I found.

106
00:08:49,990 --> 00:08:52,530
In this case, I don't even have
to worry about finding

107
00:08:52,530 --> 00:08:55,680
the length of the distance from
the origin for poles in

108
00:08:55,680 --> 00:08:56,940
the complex plane.

109
00:08:56,940 --> 00:08:58,890
All I have to worry about
is the magnitude of

110
00:08:58,890 --> 00:09:02,070
poles on the real axis.

111
00:09:02,070 --> 00:09:04,700
0.9 is my dominant pole, because
it's the largest pole.

112
00:09:10,500 --> 00:09:13,010
0.9 is less than 1.

113
00:09:13,010 --> 00:09:15,670
So I'm going to end up
with convergence.

114
00:09:15,670 --> 00:09:18,630
Eventually, my system is
going to converge, or

115
00:09:18,630 --> 00:09:19,880
tend towards 0.

116
00:09:23,250 --> 00:09:26,180
The other interesting property
of my system is what is its

117
00:09:26,180 --> 00:09:30,780
period, how does that relate
to what my function's

118
00:09:30,780 --> 00:09:32,400
going to look like.

119
00:09:32,400 --> 00:09:34,430
In this case, we're only working
on the positive real

120
00:09:34,430 --> 00:09:38,390
axis, so the angle associated
with graphing this pole on the

121
00:09:38,390 --> 00:09:42,470
complex plane is 0, so there is
no period for our system.

122
00:09:42,470 --> 00:09:44,140
This means that our system
is going to converge

123
00:09:44,140 --> 00:09:45,390
monotonically.

124
00:10:08,030 --> 00:10:11,080
Now let's walk through some unit
sample responses and then

125
00:10:11,080 --> 00:10:15,620
graph the poles that generated
those unit sample responses on

126
00:10:15,620 --> 00:10:17,922
the unit circle, where this
is the complex plane.

127
00:10:25,010 --> 00:10:28,190
Let's look at this
graph first.

128
00:10:28,190 --> 00:10:30,670
The first thing that I notice
about this graph is that, like

129
00:10:30,670 --> 00:10:33,500
in the previous example, we have
monotonic convergence.

130
00:10:33,500 --> 00:10:36,260
We're tending towards 0, and
we're not alternating or

131
00:10:36,260 --> 00:10:37,510
oscillating about the x-axis.

132
00:10:41,640 --> 00:10:43,870
So I know I'm going to be
working somewhere along this

133
00:10:43,870 --> 00:10:47,240
line before the edge
of the unit circle.

134
00:10:47,240 --> 00:10:48,990
Because at the edge of the unit
circle, the distance from

135
00:10:48,990 --> 00:10:52,070
the origin is equal to 1.

136
00:10:52,070 --> 00:10:55,010
If you made me guess, then I
would look at the distance

137
00:10:55,010 --> 00:10:57,485
here and compare it
to the distance at

138
00:10:57,485 --> 00:10:58,735
the next time step.

139
00:11:02,080 --> 00:11:03,920
I realize this is
a blackboard.

140
00:11:03,920 --> 00:11:06,320
It's not entirely to scale.

141
00:11:06,320 --> 00:11:08,440
But for the purposes of this
demonstration, I'd like to say

142
00:11:08,440 --> 00:11:12,220
that the signal at this time
step is 0.5 the signal from

143
00:11:12,220 --> 00:11:14,670
the previous times.

144
00:11:14,670 --> 00:11:18,240
Likewise at the next time step,
I would like to say that

145
00:11:18,240 --> 00:11:22,600
this signal is 0.5 the signal
from the previous time step,

146
00:11:22,600 --> 00:11:26,130
and so on and so forth.

147
00:11:26,130 --> 00:11:29,050
Therefore, I'm going to graph
my pole right here.

148
00:11:36,310 --> 00:11:38,620
Let's take a look
at this graph.

149
00:11:38,620 --> 00:11:43,470
I've drawn these squiggles to
indicate that the unit sample

150
00:11:43,470 --> 00:11:46,420
response exceeds the bounds
of the space that I

151
00:11:46,420 --> 00:11:47,610
gave for this graph.

152
00:11:47,610 --> 00:11:51,630
So just assume that these values
are much larger than

153
00:11:51,630 --> 00:11:52,880
I've drawn them.

154
00:11:59,210 --> 00:12:00,830
The first thing that I notice
about this unit sample

155
00:12:00,830 --> 00:12:06,590
response graph is the fact that
not only am I increasing

156
00:12:06,590 --> 00:12:13,610
in a way that does not seem
to change in any way--

157
00:12:13,610 --> 00:12:14,860
we're going to end
up diverging--

158
00:12:18,200 --> 00:12:20,465
is that I'm actually alternating
about the x-axis.

159
00:12:23,210 --> 00:12:26,730
And that particularly, that
if I were to call this an

160
00:12:26,730 --> 00:12:28,160
oscillation, then I
would say it's an

161
00:12:28,160 --> 00:12:29,410
oscillation with period 2.

162
00:12:32,290 --> 00:12:36,840
This means that I'm working
with a negative real pole.

163
00:12:36,840 --> 00:12:38,860
The fact that I'm diverging
means I'm working with a

164
00:12:38,860 --> 00:12:46,170
negative real pole that has
magnitude greater than 1.

165
00:12:46,170 --> 00:12:48,860
If you had to make me guess, I
would look at the distance

166
00:12:48,860 --> 00:12:51,860
associated with this time
step, compare it to the

167
00:12:51,860 --> 00:12:53,520
distance associated with
this time step.

168
00:12:56,290 --> 00:13:08,290
And if you had to ask me, I
would say this is about 1.3

169
00:13:08,290 --> 00:13:11,460
the value at the
previous step.

170
00:13:11,460 --> 00:13:15,010
Likewise, if I were to look at
the next time step, I would

171
00:13:15,010 --> 00:13:19,200
say that this increase
is about 30% of

172
00:13:19,200 --> 00:13:20,450
the previous value.

173
00:13:25,730 --> 00:13:27,150
I'm not even going
to try that one.

174
00:13:27,150 --> 00:13:31,110
But what I'm trying to get
at is that you can use

175
00:13:31,110 --> 00:13:34,180
comparisons of previous and
future time steps in order to

176
00:13:34,180 --> 00:13:37,020
attempt to determine the
magnitude of the pole if

177
00:13:37,020 --> 00:13:38,430
you're working with the
first order system.

178
00:13:38,430 --> 00:13:41,100
If you're working with the
second order system, then it's

179
00:13:41,100 --> 00:13:43,080
possible that you'll see
some really interesting

180
00:13:43,080 --> 00:13:44,790
initialization effects.

181
00:13:44,790 --> 00:13:48,580
And you should probably ask
one of us what's up.

182
00:13:48,580 --> 00:13:51,530
But for this example,
we're going to put

183
00:13:51,530 --> 00:13:52,780
our pole over here.

184
00:14:02,780 --> 00:14:06,010
Here's the last graph I
want to talk about.

185
00:14:06,010 --> 00:14:09,400
The first thing that I notice is
that it doesn't seem to be

186
00:14:09,400 --> 00:14:10,750
diverging, but it doesn't
really seem to

187
00:14:10,750 --> 00:14:12,550
be converging either.

188
00:14:12,550 --> 00:14:14,990
If this is the case, then
I'm going to put

189
00:14:14,990 --> 00:14:17,350
it on the unit circle.

190
00:14:17,350 --> 00:14:20,950
The second thing that I notice
is that it's not monotonic and

191
00:14:20,950 --> 00:14:22,960
it's not alternating.

192
00:14:22,960 --> 00:14:24,030
This is oscillating.

193
00:14:24,030 --> 00:14:29,050
So in order to determine what
angle I'm going to sign to my

194
00:14:29,050 --> 00:14:32,350
unit simple response, I'm going
to count out the time

195
00:14:32,350 --> 00:14:35,260
steps that it takes to cycle
through an entire period and

196
00:14:35,260 --> 00:14:37,390
then from there figure out what
the angle would have to

197
00:14:37,390 --> 00:14:41,620
be in order to determine a
period of that length.

198
00:14:41,620 --> 00:14:43,140
So I start here.

199
00:14:43,140 --> 00:14:44,505
I'm just going to count 1,
2, 3, 4, 5, 6, 7, 8 --

200
00:14:49,040 --> 00:14:51,400
to complete one full
oscillation.

201
00:14:51,400 --> 00:14:53,330
This means that my
period is 8.

202
00:14:53,330 --> 00:14:56,590
If I have to divide 2pi by a
particular angle in order to

203
00:14:56,590 --> 00:15:01,680
get out 8, I want to
divide by pi/4.

204
00:15:01,680 --> 00:15:06,470
So at this point, I'm working
with a magnitude of about 1,

205
00:15:06,470 --> 00:15:08,670
and I want this angle
to be about pi/4.

206
00:15:15,740 --> 00:15:18,380
This concludes my tutorial
on solving poles.

207
00:15:18,380 --> 00:15:20,070
Next time, we'll end up talking
about circuits.