MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
Department
of
Mechanical
Engineering
2.004
Dynamics
and
Control
II
Fall
2007
Problem
Set
#7
Solution
Posted:
Friday,
Nov.
2,
’07
1.
Nise
problem
5
from
chapter
8,
page
476.
Answer:
The
open
loop
transfer
function
is
K
(
s
+ 1)(
s
+ 2)
G
(
s
) =
.
(
s
+ 5)(
s
+ 6)
Root Locus
−8
−7
−6
−5
−4
−3
−2
−1
0
1
Real Axis
To
find
the
break–in
and
breakaway
points,
we
apply
rule
6
as
follows:
−3
−2
−1
0
1
2
3
Imaginary Axis
(
σ
+ 5)(
σ
+ 6)
σ
2
+ 11
σ
+ 30
K
(
σ
) =
−
=
−
.
(
σ
+ 1)(
σ
+ 2)
σ
2
+ 3
σ
+ 2
Taking
the
derivative,
dK
−
8
σ
2
−
56
σ
−
68
−
2
σ
2
−
14
σ
−
17
=
−
=
−
4
.
dσ
(
σ
2
+ 3
σ
+ 2)
2
(
σ
2
+ 3
σ
+ 2)
2
and
setting
dK/dσ
=
0,
we
find
σ
1
=
−
1
.
5635
and
σ
2
=
−
5
.
4365.
2.
Nise
problem
7
from
chapter
8,
page
477.
Answer:
From
rule
3
about
the
real–axis
segment,
we
know
that
the
root
locus
should
exist
between
the
two
zeros
in
the
right–hand
plane
as
well
as
the
pole
and
zero
in
the
left–hand
plane.
Next
step
is
to
deal
with
the
two
poles
with
1
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1
Imaginary Axis
imaginary
parts.
The
root
locus
should
depart
from
the
two
poles.
It
cannot
go
to
the
infinity
because
we
already
have
the
same
number
of
poles
and
zeros.
Therefore
it
has
to
be
joined
one
of
the
two
real–axis
segment.
The
root
locus
departing
from
the
two
poles
eventually
should
arrive
at
the
two
zeros
in
the
right–hand
plane.
Here
two
possibilities
arise:
1)
The
root
locus
directly
is
joined
to
the
real–axis
segment
between
the
two
zeros
in
right–hand
plane,
or
2)
It
first
cuts
through
the
real–axis
segment
between
the
pole
and
zero
in
the
left–hand
plane,
and
comes
back
to
the
right–hand
plane
to
join
the
real–axis
segment
between
the
two
zeros.
Which
option
turns
out
to
be
the
actual
root
locus
depends
on
the
relative
location
of
the
open–loop
poles
and
zeros.
The
two
plots
below
show
two
examples
obtained
using
Matlab
.
Root Locus
Root Locus
3
6
4
2
2
Imaginary Axis
0
−1
−2
−2
−4
−3
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
−25
−20
−15
−10
−5
0
5
Real Axis
Real Axis
3.
Nise
problem
9
from
chapter
8,
page
477.
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 Fall '08
 DerekRowell
 Mechanical Engineering, Root Locus, Complex number, real axis

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