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44
FEEDBACK
ON
A
HIGHLY
MANEUVERABLE
VESSEL
185
44
Feedback
on
a
Highly
Maneuverable
Vessel
The
directional
behavior
of
a
highly
maneuverable
vessel
is
modeled
by
θ
0
.
1
s
+1
=
δ
s
2
+0
.
6
s
−
0
.
05
where
the
rudder
deﬂection
is
δ
and
the
heading
is
θ
.
You
note
that
because
the
transfer
function
denominator
has
a
negative
coeﬃcient,
the
system
is
unstable
without
a
control
sys
tem.
Also,
the
numerator
shows
that
the
hydrodynamic
lift
on
the
rudder
has
a
component
that
scales
rudder
position
as
well
as
one
that
scales
rudder
rotation
rate
.
1.
Find
the
poles
(the
roots
of
the
denominator
polynomial)
of
this
system.
Are
the
roots
complex
(indicating
an
oscillatory
behavior),
or
real
(indicating
two
exponential
growth/decay
modes)?
At
what
rate
does
the
system
go
unstable
 that
is,
over
what
time
does
the
response
grow
by
a
factor
of
e
?
(You
can
conﬁrm
this
with
a
simulation
of
the
system
behavior
from
some
nonzero
initial
condition.)
The
poles
are
given
by
the
roots
of
the
denominator
polynomial
equation
in
s
,
i.e.,
the
roots
of
s
2
+0
.
6
s
−
0
.
05
=
0
.
These
are
0.674
and
0.074,
both
real.
The
negative
one
pertains
to
a
stable
mode
whereas
the
positive
one
indicates
an
unstable
mode.
The
time
scale
(or
time
constant)
of
the
unstable
mode
is
1/0.074
or
about
13.5
seconds.
This
is
the
amount
of
time
it
takes
for
the
output
to
increase
by
a
factor
of
e
=2
.
718
.
It
is
conﬁrmed
in
an
impulse
response,
for
example,
after
the
transient
response
of
the
ﬁrst,
stable
mode
has
died
away.
2.
Design
a
proportionalderivative
controller
for
the
boat
(PD),
so
as
to
achieve
a
closed
loop
bandwidth
of
ω
c
=
1
rad/s
and
a
damping
ratio
of
ζ
=0
.
5
.
Theideahereis
to
choose
k
p
and
k
d
so
as
to
put
the
closedloop
poles
 that
is,
the
roots
of
1+
P
(
s
)
C
(
s
)=0
√
 at
the
speciﬁc
locations
ω
c
ζ
±
jω
c
1
−
ζ
2
.
The
easiest
way
to
carry
this
out
is
to
write
out
the
polynomial
with
k
d
and
k
p
as
free
variables,
and
recognize
that
this
same
polynomial
can
be
written
as
s
2
+2
ζω
c
s
+
ω
c
2
=
0.
Some
algebra
will
give
you
a
set
of
two
equations
in
two
unknowns.
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 Spring '05
 GeorgeKocur

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