ECE 3085
Spring 2008
Exam 3
ClosedBook, ClosedNotes, 4 Problems.
Name:
1. A control system has the following structure, where
L
(
s
) is a ratio of two polynomials;
L
(
s
) has more
poles than zeros, and all its poles are located in the open left half plane. Let
H
(
s
) denote the overall transfer
function satisfying
Y
(
s
) =
H
(
s
)
R
(
s
).
L
y
r
+

(a) The loop transfer function
L
(
s
) is characterized in the diagram to the left and the overall transfer
function
H
(
s
) is characterized in the diagram to the right; the labeling of these diagrams establishes
a unique correspondence between them.
Precisely mark the diagram to the right with an asterisk
corresponding to the asterisk found on the diagram to the left.
1.5
1
0.5
0
0.5
1
1.5
1.5
1
0.5
0
0.5
1
1.5
Re{L(j
ω
)}
Im{L(j
ω
)}
1
0.5
0
0.5
1
1
0.5
0
0.5
1
Re{H(j
ω
)}
Im{H(j
ω
)}
L
(
jω
) =

j
⇒
H
(
jω
) =
L
(
jω
)
1 +
L
(
jω
)
=

j
1

j
=

j
1

j
·
1 +
j
1 +
j
=
1

j
2
(b) Prove that this system is internally stable.
The given plot of
L
(
jω
) is at least a portion of the Nyquist plot of
L
(
s
). Since
L
(
s
) is strictly proper,
the point
L
(
jω
) = 0 corresponds to
ω
=
±∞
. Since the coefficients of
L
(
s
) are real, the point
L
(
jω
) = 1
corresponds to
ω
= 0. As further confirmation that the entire
jω
axis has been accounted for, note
that the given plot is a closed curve symmetric about the real axis. Finally, since
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 Spring '08
 Taylor
 Nyquist plot, Nichols plot

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