ECE 147A FEEDBACK CONTROL SYSTEMS  THEORY AND DESIGN F04
Final Exam
Closed book and notes. No calculators. Show all work.
Name:
Solution
Problem 1:
/20
Problem 2:
/20
Problem 3:
/20
Problem 4:
/20
Problem 5:
/20
Total:
/100
Miscellaneous:
tan

1
(1
/
3)
≈
18 degrees,
tan

1
(1) = 45 degrees.
tan

1
(2)
≈
63 degrees,
tan

1
(5)
≈
78 degrees,
tan

1
(10)
≈
84 degrees,
tan

1
(20)
≈
87 degrees,
sin(45
◦
) =
√
2
/
2
≈
0
.
7,
sin(50
◦
)
≈
0
.
77,
sin(55
◦
)
≈
0
.
82,
sin(60
◦
) =
√
3
/
2
≈
0
.
87.
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Name:
Problem 1
1. For the unity negative feedback configuration, describe the three properties that
typically characterize a good choice for the function
ω
7→ 
C
(
ω
)
P
(
ω
)

and explain why these
three properties are important.
(a) The function should have large values for
ω
small. This is important for good disturbance
attenuation and reference tracking.
(b) The function should have small values for
ω
large. This is important to keep measurement
noise from affecting the feedback system significantly, and it is important for stability
robustness with respect to unmodeled dynamics.
(c) The function should decrease slowly in the region where its magnitude is near one. This
is important to guarantee that the unity feedback configuration will be stable.
2. What are the typical motivations for using a pole at the origin in a feedback controller?
A pole at the origin in the feedback controller is used to guarantee that the output of the plant
exactly reproduces step references asymptotically, even in the presence of constant disturbances
at the plant input and unmodeled or uncertain plant dynamics.
3. The loop gain for a given unity negative feedback system is shown as the solid curve in the
figure below. (The dashed curve is a circle of radius one.)
(a) Label on the plot the crossover frequency of the openloop.
The crossover frequency corresponds to the frequency at which the solid curve pierces the
dashed circle.
(b) Estimate the phase margin.
The phase margin is given by the angle on the dashed circle between the point
(

1
,
0)
and
the point where the solid curve pierces the dashed circle. It is roughly
30
degrees.
(c) Estimate the gain margin.
It is about 2, since the gain can be increased by 2 before the number of encirclements of
(

1
,
0)
changes.
(d) Label on the plot the frequency range over which the feedback system attenuates the effect
of input additive disturbances. Explain your answer.
To determine this frequency range, we should draw another circle of radius one, this one
centered at
(

1
,
0)
. Disturbance attenuation occurs at the frequnecies for which the solid
curve is outside of this new circle. This is because disturbance attenuation occurs where

S
(
ω
)

<
1
, i.e.,

C
(
ω
)
P
(
ω
) + 1

>
1
, i.e.,

C
(
ω
)
P
(
ω
)

(

1
,
0)

>
1
.
The latter
condition is exactly the condition that
C
(
ω
)
P
(
ω
)
is outside of this new circle.
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 Spring '10
 VolkanRodoplu
 Crossover Frequency

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