21
LOOP
TRANSFER
RECOVERY
21.1
Introduction
The
Linear
Quadratic
Regulator(LQR)
and
Kalman
Filter
(KF)
provide
practical
solutions
to
the
fullstate
feedback
and
state
estimation
problems,
respectively.
If
the
sensor
noise
and
disturbance
properties
of
the
plant
are
indeed
wellknown,
then
an
LQG
design
approach,
that
is,
combining
the
LQR
and
KF
into
an
output
feedback
compensator,
may
yield
good
results.
The
LQR
tuning
matrices
Q
and
R
would
be
picked
heuristically
to
give
a
reasonable
closedloop
response.
There
are
two
reasons
to
avoid
this
kind
of
direct
LQG
design
procedure,
however.
First,
although
the
LQR
and
KF
each
possess
good
robustness
properties,
there
do
exist
plants
for
which
there
is
no
robustness
guarantee
for
an
LQG
compensator.
Even
if
one
could
steer
clear
of
such
pathological
cases,
a
second
problem
is
that
this
design
technique
has
no
clear
equivalent
in
frequency
space.
It
cannot
be
directly
mapped
to
the
intuitive
ideas
of
loopshaping
and
the
Nyquist
plot,
which
are
at
the
root
of
feedback
control.
We
now
reconsider
just
the
feedback
loop
of
the
Kalman
±lter.
The
KF
has
openloop
transfer
function
L
(
s
)
=
Cδ
(
s
)
H
,
where
δ
(
s
)
=
(
sI
−
A
)
−
1
.
This
follows
from
the
estimator
evolution
equation
(Continued on next page)
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21
LOOP
TRANSFER
RECOVERY
^
φ
(s)
x
C
y
H
ˆ
˙
x
+
Bu
+
H
(
y
−
C
ˆ
x
=
A
ˆ
x
)
and
the
fgure.
Note
that
we
have
not
included
the
Factor
Bu
as
part
oF
the
fgure,
since
it
does
not
aﬀect
the
error
dynamics
oF
the
flter.
As
noted
previously,
the
K±
loop
has
good
robustness
properties,
specifcally
to
perturbations
at
the
output
ˆ
y
,
and
Further
is
amenable
to
output
tracking.
In
short,
the
K±
loop
is
an
ideal
candidate
For
a
loopshaping
design.
Supposing
that
we
have
an
estimator
gain
H
which
creates
an
attractive
loop
Function
L
(
s
),
we
would
like
to
fnd
the
compensator
C
(
s
)
that
establishes
P
(
s
)
C
(
s
)
Cδ
(
s
)
H
,
or
(259)
⇐
Cδ
(
s
)
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 Fall '04
 MichaelTriantafyllou
 Kalman filter, LQR, Loop Transfer Recovery, A1 BCD

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