MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
Department
of
Mechanical
Engineering
2.004
Dynamics
and
Control
II
Fall
2007
Problem
Set
#8
Posted:
Friday,
Nov.
2,
’07
Due:
Nov.
9,
’07
1.
Consider
the
complex
number
s
1
=
−
1+
j
and
its
geometrical
representation
on
the
complex
s
–plane
shown
below.
Do
not
use
Matlab
until
you
reach
question
(d)
of
this
problem.
ω
j
s
1
σ
1
1
0
a)
Using
geometrical
arguments
and
calculations
on
the
s
–plane,
compute
the
phase
of
the
complex
number
(
s
1
+ 2)(
s
1
+
0).
Use
complex
number
algebra
to
verify
your
geometrical
computation.
b)
Using
geometrical
arguments
and
calculations
on
the
s
–plane,
compute
the
value
of
the
real
number
K
such
that
K

s
1
+2

s
1
+0

=1
.
Use
complex
number
algebra
to
verify
your
geometrical
computation.
c)
Based
on
the
answers
to
the
previous
two
questions,
do
you
expect
s
1
to
be
a
closed–loop
pole
of
a
feedback
system
which
has
two
open–loop
poles
located
at
0,
−
2
and
no
open–loop
zeros?
Justify
your
answer.
If
yes,
what
is
the
value
of
the
feedback
gain
that
would
yield
a
closed–loop
pole
at
location
s
1
?
d)
Verify
numerically,
using
Matlab
,
your
answers
to
all
the
above
questions.
Include
corroborating
plots
in
your
argument.
1
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Consider
a
feedback
system
with
open–loop
poles
at
0,
−
1,
−
2
and
no
open–loop
zeros.
The
geometrical
representation
of
this
system’s
open–loop
poles
on
the
complex
s
–plane
is
shown
below.
Do
not
use
Matlab
until
you
reach
question
(f)
of
this
problem.
j
ω
2
σ
2
1
0
2
a)
Using
geometrical
arguments
and
calculations
on
the
s
–plane,
show
that
√
±
j
2
belongs
to
the
root
locus
of
this
system.
b)
Using
geometrical
arguments
and
calculations
on
the
s
–plane,
compute
the
feedback
gain
K
that
would
be
required
to
drive
two
closed–loop
poles
of
√
this
system
to
the
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 Fall '08
 DerekRowell
 Mechanical Engineering, Massachusetts Institute of Technology, geometrical arguments

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