MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
Department
of
Aeronautics
and
Astronautics
16.060:
Principles
of
Automatic
Control
Fall
2003
PROBLEM
SET
9
Due:
12/04/2003
Problem
1:
Representing
a
Pure
Delay
in
the
Frequency
Domain
One
situation
in
which
frequency
response
techniques
are
useful
is
when
a
transfer
function
can
not
be
represented
easily
in
terms
of
poles
and
zeros.
A
very
common
example
is
a
pure
delay,
which
is
always
present
in
any
control
system
implemented
on
a
digital
computer
(e.g.
Simulink).
For
a
pure
delay
of
time
τ
,
the
output
y
(
t
)
in
terms
of
the
input
u
(
t
)
is
given
by:
y
(
t
)
=
u
(
t
−
τ
)
1.
Using
the
definition
of
the
Laplace
transform,
find
the
transfer
function
G
(
s
)
=
Y
(
s
)
U
(
s
)
corresponding
to
a
pure
delay.
2.
Draw
the
polar
plot
for
G
(
s
)
for
a
time
delay
of
1
second.
(The
polar
plot
is
the
part
of
the
Nyquist
plot
corresponding
to
0
< ω <
∞
.)
Problem
2:
Nyquist
Plots
with
Negative
Gain
For
each
of
the
following
openloop
transfer
functions,
draw
one
Nyquist
plot
for
K >
0,
and
then
another
Nyquist
plot
for
K <
0.
In
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '03
 willcox
 Aeronautics, Astronautics, Signal Processing, Phase margin, Nyquist plot

Click to edit the document details