4
Kalman filter
1
basedona,
r’
I
Hypothesis
conditional probability
computation
I
I’ I I
PZ
1
PK
r.
INPUTS
Rg.
1.
MMAC.
c
w-
SYSTEM SYSTEM
CGT
COMMAND
STATES
MODEL CONTROLLER
rc
r
_I
MEASUREMENTS
One expects the residuals
of
the Kalman filter
based upon the “best” model to have mean squared
value most in consonance with its
own
computed
&(ti),
while “mismatched” filters will have larger
residuals than anticipated through
Ak(fi).
Therefore,
(2),
(3),
and (6)-(12) will most heavily weight the
filter based upon the most correct assumed parameter
value. However, the performance
of
the algorithm
depends on there being significant differences in
the characteristics
of
residuals in correct versus
mismatched filters. Each filter should be tuned for
best performance when the “true” values
of
the
uncertain parameters are identical to its assumed
value for these parameters. One should specifically
avoid the “conservative” philosophy
of
adding
considerable dynamics pseudonoise, often
used
to
open
the
bandwidth
of
a single Kalman fdter to
guard against divergence, since this tends to mask the
differences between
good
and bad models. Specifically
for
this
reason,
LTR
tuning [12]
was
not employed for
robustness enhancement
of
each elemental controller
within the MMAC algorithm.
T
J\
f\
472
f\
f\
111.
ELEMENTAL CONTROLLER DESIGN
A
command generator
tracker/proportional-plus-
integralEdman filter (CGT/PI/KF)
form
of
controller
[S,
8-10]
was chosen for each
of
the
elemental
controllers within the MMAC algorithm, and each
was
designed via
LQG
synthesis
to
provide desirable
vehicle behavior for a particular failure status
of
sensors
and actuators. The CGT
portion
of
such
a controller
is
a
form
of
explicit
model follower
that
forces the plant
to
mimic the behavior
of
a
preferred response model, in order to incorporate
desired handling qualities
[13]
into the whicle.
This
precompensator
is
combined with a PI feedback
controller rather than a simple regulator, in order
to
achieve
“type-1”
properties: able to track a
nonzero
step input with
zero
steady-state error, despite
unmodeled constant disturbances, as particularly due
to the aircraft being at a different operating point
than
used
to
generate the
linear
perturbation model
upon which its controller is based. Implicit model
following [8-10, 121
is
embedded into the quadratic
cost
used
in the
LQG
synthesis
of
the feedback gains
to penalize deviations in achieved transient response
from the desired, robust characteristics
of
the implicit
design modeL
A
Kalman filter
is
inserted into the
loop to accept noise-corrupted sensor signals and
provide estimates
of
plant states
needed
by the CGT/PI
law. The fdter also generates the residuals required
eventually by the MMAC adaptation mechanism for
elemental controller selection.
The result
of
this elemental controller design
process
is
as depicted in Fig.
2
It accepts command
inputs, as from the pilot’s stick, and generates a
feedforward control through the command generator
explicit model and the CGT compensator. The actual
system
is
then driven to follow the CGT-generated
reference inputs
by
the PI feedback controller,
using
state estimates from the Kalman filter.
I
DISTURBANCE STATE ESTIMATES FILTER
r
7
KALMAN
CONTROLLER
L
FEEDBACK
INPUTS
I
SYSTEM
STATE
ESTIMATES
I
Fig.
2
CGT/PI/KF
elemental
controller.
IEEE TRANSACTIONS ON AEROSPACE
AND
ELECTRONIC
SYSTEMS
VOL.
27,
NO.
3
MAY
1991