EE 428
PROBLEM SET 8
DUE: 10 December 2007
Reading assignment: Ch 6, sections 6.1 through 6.4
Laboratory sections meet during the week of December 3.
Problem 34:
(17 points)
The closedloop system in Figure 1 contains a plant with transfer function
G
p
(
s
) =
100
(
s
+ 1)(
s
2
+ 10
s
+ 100)
and uses proportional control with
G
c
(
s
) =
K,
where
K
≥
0. In this problem you will use the RouthHurwitz criterion and the concept of
gain margin
to determine
the range of
K
for which the closedloop system is stable.
Figure 1: Feedback control system using cascade compensation.
1. (3 points) Using the RouthHurwitz criterion, determine the largest positive gain
K
for which the closedloop
system is stable.
2. (6 points) Using MATLAB, generate the Bode magnitude and phase plots of the openloop transfer function
B
(
s
)
E
(
s
)
=
G
c
(
s
)
G
p
(
s
)
for unity gain feedback,
G
c
(
s
) =
K
= 1.
3. (2 points) Using the Bode phase plot, determine the frequency at which the phase of the openloop transfer
function is 180
◦
. This point is called the
phase crossover frequency
and is denoted by
ω
pc
.
4. (2 points) The
gain margin
GM
is the number of decibels which must be added to the magnitude curve in
order to make

G
c
(
ω
pc
)
G
p
(
ω
pc
)

= 0 dB. With
K
= 1, determine the gain margin of the system shown in
Figure 1.
5. (2 points) Verify your answers in parts (4) and (5) using the MATLAB command
margin(num,den)
, where
num
and
den
are the numerator and denominator polynomials of
G
c
(
s
)
G
p
(
s
), respectively.
6. (2 points) Using the gain margin measured in part (5), determine the largest positive value of the gain
K
for which the system is closedloop stable.
Compare your result to that obtained using the RouthHurwitz
criterion.
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Problem 35:
(17 points)
In this problem you will investigate the effect of a time delay on the stability of a closedloop system. Time delays
occur frequently in industrial applications. For example, in temperature control systems the control engineer must
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 Fall '07
 SCHIANO
 Signal Processing, GC

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