Journal
of
Process
Control
23 (2013) 1555–
1561
Contents
lists
available
at
ScienceDirect
Journal
of
Process
Control
jo
u
r
nal
homep
age:
www.elsevier.com/locate/jprocont
Short
communication
Robust
derivative-free
Kalman
filter
based
on
Huber’s
M-estimation
methodology
Lubin
Chang
a,∗
,
Baiqing
Hu
a
,
Guobin
Chang
a,b
,
An
Li
a
a
Department
of
Navigation
Engineering,
Naval
University
of
Engineering,
Wuhan,
People’s
Republic
of
China
b
Tianjin
Institute
of
Hydrographic
Surveying
and
Charting,
Tianjin
300000,
People’s
Republic
of
China
a
r
t
i
c
l
e
i
n
f
o
Article
history:
Received
28
November
2012
Received
in
revised
form
8
April
2013
Accepted
23
May
2013
Available online 28 June 2013
Keywords:
Huber’s
M-estimation
Nonlinear
filtering
Robust
regression
Unscented
Kalman
filter
a
b
s
t
r
a
c
t
In
this
study,
a
discrete-time
robust
nonlinear
filtering
algorithm
is
proposed
to
deal
with
the
contami-
nated
Gaussian
noise
in
the
measurement,
which
is
based
on
a
robust
modification
of
the
derivative-free
Kalman
filter.
By
interpreting
the
Kalman
type
filter
(KTF)
as
the
recursive
Bayesian
approximation,
the
innovation
is
reformulated
capitalizing
on
the
Huber’s
M-estimation
methodology.
The
proposed
algo-
rithm
achieves
not
only
the
robustness
of
the
M-estimation
but
also
the
accuracy
and
flexibility
of
the
derivative-free
Kalman
filter
for
the
nonlinear
problems.
The
reliability
and
accuracy
of
the
proposed
algorithm
are
tested
in
the
Univariate
Nonstationary
Growth
Model.
© 2013 Elsevier Ltd. All rights reserved.
1.
Introduction
The
Kalman
filter
(KF)
[1]
is
concerned
with
estimation
of
the
dynamic
state
from
noisy
measurements
in
the
class
of
estima-
tion
problems
where
the
dynamic
and
measurement
processes
can
be
approximated
by
linear
Gaussian
state
space
models.
Unfor-
tunately,
the
performance
of
the
KF
can
be
severely
degraded
when
the
distribution
of
the
true
noise
deviates
from
the
assumed
Gaussian
model,
often
being
characterized
by
heavier
tails
and
gen-
erating
high-intensity
noise
realizations,
named
outliers
[2].
The
contaminated
Gaussian
noise
and
outliers
arise
naturally
in
many
areas
of
engineering.
Heavy
tailed
densities
occur
in
applications
related
to
glint
noise,
air
turbulence,
and
asset
returns
[3].
Outliers
can
appear
due
to
clutter,
glint
noise,
and
model
mismatch
caused
by
e.g.,
linearization
[4].
Therefore,
there
appears
to
be
consider-
able
motivation
for
considering
filters
that
are
robust
to
perform
fairly
well
in
non-Gaussian
environments.
To
handle
the
non-Gaussian
noise,
many
methods
have
been
proposed.
By
minimizing
the
worst-case
estimation
error
averaged
over
all
samples,
the
H
∞
based
KF
can
be
used
to
treat
model-
ing
errors
and
uncertainties
as
unknown-but-bounded
noise
[5,6].
However,
it
breaks
down
in
the
presence
of
randomly
occurring
outliers
since
the
design
matrices
of
the
H
∞
filter,
just
like
the
covariance
matrix
in
the
classical
KF,
cannot
accommodate
well
the
outliers
induced
by
the
thick
tails
of
a
noise
distribution
[13].
∗
Corresponding
author.
E-mail
address:
changlubin@163.com
(L.
Chang).
The
Gaussian
sum
methodology
can
also
yield
a
robust
estimate,
but
at
an
extreme
cost
in
computation.
Specifically,
the
number
of
terms
kept
in
the
Gaussian
sum
grows
exponentially
with
time
[7].
A
more
general
and
versatile
method
is
the
particle
filtering
(PF)
which
approximates
the
minimum
variance
estimate
using
stochastic
simulation
methods.
However,
the
PF
is
also
very
compu-
tationally
intensive
since
it
requires
Monte
Carlo
integration
[8,9].
In
addition,
it
typically
requires
a
delicate
tuning
of
proposal
den-
sities
to
improve
its
convergence
rate.
The
extension
of
the
concept
of
Huber’s
M-estimation
method-
ology
to
the
problem
of
robust
Kalman
filtering
has
been
widely
researched
and
emphasized
over
the
other
ones,
since
it
is
moti-
vated
by
the
maximum
likelihood
estimation,
which
makes
it
more
natural
and
rather
easy
[10–12].
The
Huber’s
M-estimation
methodology
is
essentially
based
on
modifying
the
quadratic
cost
function
in
KF
by
the
robust
convex
Huber
-function,
which
exhibits
l
2
-norm
properties
for
small
residuals
and
l
1
-norm
prop-
erties
for
large
residuals.
Possessing
a
moderate
breakdown
point
(BP),
high
efficiency,
and
a
manageable
computational
complexity,
the
Huber’s
M-estimation
methodology
is
still
a
subject
of
consider-
able
research
interest
in
recent
years
[13–17].
However,
most
of
the
previous
work
is
limited
to
the
linear
case.
Although
these
methods
can
be
directly
extended
to
the
nonlinear
problems
making
use
of
the
extended
Kalman
filter
(EKF),
the
crude
approximation
and
the
cumbersome
derivation
and
evaluation
of
Jacobian
matrices
in
the
EKF
may
degrade
their
performance.
In
recent
years,
there
has
been
renewed
interest
in
the
topic
of
nonlinear
filtering
with
the
discovery
of
the
“sigma-point”
Kalman
filters
(SPKF)
[18].
The
various
sigma
point
algorithms
differ
from
0959-1524/$
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front
matter ©
2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jprocont.2013.05.004
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