ARTICLE IN PRESS
Signal Processing 89 (2009) 2171–2177
Contents lists available at ScienceDirect
Signal Processing
journal homepage: www.elsevier.com/locate/sigpro
A novel interacting multiple model algorithm$
Qu HongQuana,, Pang LiPingb, Li ShaoHongb
a
b
College of Information Engineering, North China University of Technology, Beijing 100144, China
School of Electronics and Information Engineering, Beihang University, Beijing 100191, China
a r t i c l e in fo
abstract
Article history:
Received 1 August 2008
Received in revised form
28 April 2009
Accepted 28 April 2009
Available online 6 May 2009
For maneuvering target tracking, the interacting multiple model (IMM) algorithm
employs a fixed model set. The performance of this algorithm depends on the model set
adopted. The result of using too many models is as bad as the case of too few models.
Therefore, a variable structure IMM (VSIMM) was presented and applied to ground
target tracking. This algorithm improves performance and reduces computational load
with using auxiliary information. But it is difficult to extend the VSIMM to other
scenario (for example, aerial target), where there is not auxiliary information such as a
map. A novel interacting multiple model (Novel-IMM) algorithm was presented to solve
the problem of model set adaptation without auxiliary information. The Novel-IMM
algorithm consists of N independent IMM filters operating in parallel, and each
independent IMM filter also consists of multiple sub-filters, which operate interactively.
In every time index, only one IMM output of a certain model set is used; but for a long
time, the algorithm will alternatively choose an output of the model set to be the
optimum final output. The Novel-IMM approach was illustrated in detail with an aerial
complex maneuvering target tracking example.
& 2009 Elsevier B.V. All rights reserved.
Keywords:
Interacting multiple model
Variable structure IMM
Model set adaptation
Novel-IMM
1. Introduction
Computing the optimal state estimate for a jump
Markov system requires exponential complexity, and
hence, practical filtering algorithms are necessarily suboptimal. In target tracking literatures, suboptimal multiple model filtering algorithms, such as the interacting
multiple model (IMM) method, are widely used for state
estimation of such systems [1]. The IMM estimation has
received a great deal of attention in recent years due to its
unique power and great success in handling problems
with both structural and parametric uncertainties and/or
changes, and in decomposing a complex problem into
simpler sub-problems, ranging from target tracking to
fault detection and isolation, and from biomedical signal
$
Supported by National Science Foundation of China (no. 50808007)
and the NCUT Young Major Research Foundation (no. 200802).
Corresponding author. Tel.: +86 10 88801519.
E-mail address: qhqphd@sina.com (H.Q. Qu).
0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.sigpro.2009.04.033
processing to process control [2,3]. The application of IMM
filters for detection and diagnosis of anticipated reaction
wheel failures in the attitude control system has been
described and developed in Refs. [4,5]. The IMM method
for fault detection of railway vehicle from the measurement of the lateral acceleration of bogie and body and
from rate of bogie has been described in Refs. [6,7].
The IMM algorithm, which uses a fixed set of models,
usually performs reasonably well for problems with a
small model set. Many practical problems, however,
involve more than just a small number of models [8].
When these algorithms are applied to solve target
tracking problems, it happens that only using a small
number of models cannot get satisfying result. At any
time, the target trajectory evolves according to one of a
finite number of predetermined model set. This requires
that the model set include as many models as necessary to
handle the varying target motion characteristics. But this
cannot improve the performance successfully. The result
of using too many models is as bad as the case of too few
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models. As demonstrated in Ref. [9], using more models
makes it no better. In fact, the performance will
deteriorate if too many models are used due to the
excessive ‘‘competition’’ from the ‘‘unnecessary’’ (excess)
models. The dilemma is that more models have to be used
to improve the accuracy, but the use of too many models
would degrade the performance [10].
Several modified IMM algorithms were proposed for
improvement of performance or computation efficiency in
recent years. A selected filter IMM (SFIMM) algorithm
which uses a subset of filters with the specific subset
chosen using decision rules was presented to improve
computation efficiency of maneuvering target tracking
[11]. A variable structure IMM (VSIMM) algorithm was
presented to solve the dilemma where the model set not
only differs across targets, but also varies with time for
a given target [2,3]. Not only does VSIMM inherit
the effective cooperation strategies of the IMM and the
superior output processing of the AMM (autonomous
multiple model), but it also adapts to the outside world by
producing new elemental filters if the existing ones are
not good enough and by eliminating those elemental
filters that are harmful. This algorithm works successfully
in ground target tracking where the model set adapts
sequentially according to the target position and the road
network configuration [12–15].
The major objective of the MM is to achieve best
modeling accuracy with a minimum number of models.
Here VSIMM estimation has certain advantages. In a
general setting, the problem of efficient model set design
for MM estimation is still open, Researches in the field of
VSIMM focus on the efficient model set design [16–18]
and model set adaptation algorithms [1,3,19,20], which
includes the decision for activating a candidate model as
well as terminating the model in effect. In general, those
decisions should consist of a set of complex rules based
on both a priori and a posteriori information about the
current system mode in effect, but there are not unified
theories and practical methods to guide us how to use
these rules. These theories about model adaptation
is much more challenging [21,22]. So the advantages in
all presented model set adaptation methods based on
the statistical hypothesis testing have been surprisingly
limited. In addition, it is very difficult to realize the
VSIMM without the auxiliary information.
A novel interacting multiple model (Novel-IMM)
algorithm has been presented in this paper to solve the
problem of model set adaptation without auxiliary
information. This method adopts independent parallel
model set method but not a serial model set adaptation
which is adopted in VSIMM. It consists of N independent
IMM filters operating in parallel, and each independent
IMM filter also consists of multiple sub-filters, which
operate interactively. In every time index, only one IMM
output of a certain model set is used; but for a long time,
the algorithm alternatively chooses an output of the
model set to be the optimum final output. The method
does not use the decision for model activation and
the decision for termination of the model in effect. The
computer simulations illustrate the Novel-IMM could
improve the performance of target tracking.
2. The Novel-IMM algorithm
2.1. The IMM algorithm
Multiple model (MM) estimation is a powerful
approach to adaptive estimation. It is particularly good
for systems subject to structural as well as parametric
changes. In this approach a model set is selected to
represent (or ‘‘cover’’) the possible system behavior
patterns and the overall estimate is obtained by a certain
combination of the estimates based on these models.
The multiple model approach is best described in terms
of stochastic hybrid systems.
The IMM algorithm consists of r interacting filters
operating in parallel. In the IMM approach, at time k the
state estimate is computed under each possible current
model using r filters, with each filter using a different
combination of the previous model-conditioned estimates
(mixed initial condition).
Fig. 1 describes the IMM algorithm, which consists of r
interacting filters operating in parallel, one cycle of the
algorithm consists of the following [23–25]:
(1) Calculation of mixing probabilities:
lk ðijjÞ ¼
cj ¼
r
X
1
p mi ;
cj ij k
i; j ¼ 1; 2; . . . ; r
pij mik
(1)
i¼1
where lk(i|j) is mixing probability, [pij] is the model
transition probability matrix, mik is the model probability, and cj is normalizing constant.
Fig. 1. IMM algorithm (r ¼ 2).
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(2) Interaction:
0j
X^ kjk ¼
r
X
2.2. The VSIMM algorithm
i
X^ kjk lk ðijjÞ;
j ¼ 1; . . . ; r
(2)
i¼1
0j
X^ kjk
is mixed initial condition for the filter
where
matched to model j. The covariance corresponding to
the above is
¼
P 0j
kjk
r
X
i
j
i
j
lk ðijjÞfP ikjk þ ðX^ kjk X^ kjk ÞðX^ kjk X^ kjk Þ0 g
i¼1
j ¼ 1; :::; r
(3)
1
(3) Model-matched filtering: Compute X^ kþ1jkþ1 P1kþ1jkþ1
2
2
^
X kþ1jkþ1 P kþ1jkþ1 using Kalman filter. The likelihood
function of each model is
expððd
j
Þ2 =2Þ
kþ1
Ljkþ1 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
;
ð2pÞM Sjkþ1 j ¼ 1; . . . ; r
(4)
j
where ðdkþ1 Þ2 ¼ vjkþ1 ðSjkþ1 Þ1 ðvjkþ1 Þ0 , vjkþ1 , and Sjkþ1 are
the innovation vector and its variance matrix, and M is
the measurement dimension.
(4) Model probability update:
Ljkþ1 cj
mjkþ1 ¼ Pr
j
j¼1 Lkþ1 cj
;
j ¼ 1; . . . ; r
(5)
(5) Estimate and covariance combination:
X^ kþ1jkþ1 ¼
r
X
j
X^ kþ1jkþ1 mjkþ1 ;
j ¼ 1; . . . ; r
(6)
j¼1
P kþ1jkþ1 ¼
r
X
2173
j
mjkþ1 fPjkþ1jkþ1 þ ½X^ kþ1jkþ1 X^ kþ1jkþ1 j¼1
j
½X^ kþ1jkþ1 X^ kþ1jkþ1 0 g
(7)
Note that this combination is only for output purposes
and it is not part of the algorithm recursions. It can be got
from above equations that the performance will deteriorate if too many models are used due to the excessive
competition from the unnecessary models.
The IMM approach computes the state estimation that
accounts for each possible model using a suitable mixing
of the model-conditioned estimations depending on the
model probability. These algorithms are decision free
(no maneuver detection decision is needed) and undergo
a soft switching according to the latest updated mode
probabilities.
When target motion is complex, small number of
models is insufficient, worse still, using more models will
result in excessive competition from the unnecessary
models. In Ref. [9] the conclusion indicates that two
models is the best solution in normal cases when model
transition matrix is effective. Considering model match,
three models are often used in IMM algorithms. One
model set cannot solve the diploma, so the multiple model
set algorithms are proposed and applied to complex target
motion tracking.
Most of the work on IMM estimator has considered
only fixed model sets, i.e. it is assumed that, at any time,
the target trajectory evolves according to one of a finite
number of predetermined model set. This requires that
the model set include as many models as necessary to
handle the varying target motion characteristics.
All the models in the estimator are kept throughout the
entire tracking period and, when it is necessary for a large
number of models, it brings extra computational load, still
more, it may degrade estimation accuracy. One remedy
for the above shortcomings is to vary the model set in
the IMM estimator based on some criteria to yield better
estimation. This results in a variable structure IMM
(VSIMM) estimator where the model set not only differs
across targets, but also varies with time for a given target.
In [10], a general framework for variable structure hybrid
estimation based on a graph theoretic formulation was
proposed and applied to ground target tracking [12–15]. It
assumes that the total models can be covered by a number
of model set, and a particular model set is running at any
given time determined by a hard decision. One cycle of the
simple and practical algorithm is described in Fig. 2.
Model set adaptation is done by using additive
auxiliary information. The targets under track are moving
along a constrained path, for example, a highway, with
varying obscuration due to changing terrain conditions. In
addition, the roads on which the targets travel can branch,
merge, or cross. Some of the targets can also move in an
open field. Because of the varying terrain conditions, the
possible ways (models) in which a target trajectory can
evolve also vary. For example, at junctions, target motion
uncertainty increases—the target can move along any
of the roads meeting at the junction. Another example
of terrain-conditioned motion is that of a target along a
highway or a bridge, where the motion of the target is
orthogonal to the road or the bridge is restricted. In this
case, the motion of a target is highly directional. Similarly,
another constraint is imposed when the targets are
moving in a file, where the target motion is unidirectional
and the target velocity is such that ‘‘passing’’ is not
allowed.
Kirubarajan presents a VS-IMM estimator, where filter
model set is adaptively modified depend on the terrain
topography (auxiliary information) [10]. For example, the
added uncertainty at junctions is handled with model set
Model Set 1
Model Set 2
...
Model Set r
Model Set Selection (Auxiliary Information)
Model Set n
General IMM
Fig. 2. VSIMM algorithm.
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which represents motion along the possible roads
(hard decision using additive auxiliary information).
These additional model set is replaced by the primitive
model set after the target passes through the junction.
At each scan, the structure of the estimator for every
target is individually modified based on the known
topography. This enables the estimator to handle the
variation in the possible motion modes across targets as
well as with time for each target.
The VSIMM algorithm eliminates the need for carrying
all the possible models throughout the entire tracking
period, significantly improving performance. But the
model set adaptation is very difficult to extend to aerial
target without auxiliary information.
2.3. The Novel-IMM algorithm
Although much effort has been made and several ways
have been tried, the IMM and VSIMM are not very
successful for complex motion track without prior
information. The objective of this paper is to develop a
better method to solve the dilemma about how many
models to be used and how to interact with them.
The IMM algorithm consists of r interacting filters
operating in parallel. In the IMM approach, at time k the
state estimation is computed under each possible current
model using r filters, with each filter using a different
combination of the previous model-conditioned estimates
(mixed initial condition).
The Novel-IMM algorithm consists of N independent
IMM filters operating in parallel, and these multiple IMM
filters are independent. But each independent IMM filter
also consists of multiple sub-filters, which operate
interactively. At time k, the state estimation is computed
with all possible models in the N model sets, and in these
possible compute, every IMM filter uses their own
independent initial condition without being combined.
One of outputs of the IMM filter of the N model sets is
used to match target motion, but the chosen model set
would vary with time. Therefore, it is very important to
set up a method to choose the optimum output in these
independent IMM filters.
To determine the optimum final output, the Novel-IMM
algorithm performs a continuous MSPT (model set probability test) for N hypotheses (H1: model set 1 is chosen to
match target motion, H2: model set 2 is chosen, y, HN:
model set N is chosen), and only one IMM output with its
corresponding model set is chosen as the optimum output
or final estimation. So model set adaptation is done by
using MSPT without any auxiliary information.
Independence between different IMM filters can solve
the excessive competition from the unnecessary models.
The MSPT algorithm helps to choose the model set which
matches the target motion best. When target motion
changed, the MSPT algorithm will choose another model
set accordingly. For each IMM, the model set does not
change with time.
The flow chart for one cycle of the Novel-IMM logic is
shown in Fig. 3. We consider two model sets.
A recursion of the Novel-IMM algorithm consists of
four fundamental steps:
(1) Parallel independent IMM: X^ kþ1jkþ1 ðiÞ, Pkþ1jkþ1 ðiÞ, mkþ1 ðiÞ,
and Lkþ1 ðiÞ are computed for each model set using
1
1
general IMM (see Fig. 1). X^ kjk ð1Þ and P^ kjk ð1Þ are target
state and its covariance estimation of model 1 in
2
2
model set 1 at k time index, while X^ kjk ð1Þ and P^ kjk ð1Þ
are target state and its covariance estimation of model
2 in model set 1. PT(1) and mk(1) are transition matrix
and model probability vector of model set 1.
1
1
2
2
X^ kjk ð2Þ,P^ kjk ð2Þ,X^ kjk ð2Þ,P^ kjk ð2Þ,P T ð2Þ, and mk(2) are corresponding terms of model set 2. PT(1) and PT(2) are as
same as Pij in Eq. (1). mk(1) and mk(2) are model
probability.
Fig. 3. Novel-IMM algorithm (two model sets and two models in each model set).
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(2) The likelihood function value of every model set:
C kþ1 ðnÞ ¼
r
X
Ljkþ1 ðnÞmjk ðnÞ;
n ¼ 1; . . . ; N
multiple model set algorithm and its model set is adaptive
not using any additional auxiliary information.
(8)
j¼1
3. Simulations
(3) Normalized likelihood function value of every model set:
The weight probability of model set kk+1(n) is updated
from Ck+1(n) and kk(n):
C
2175
kkþ1 ðnÞ ¼ PN kþ1
ðnÞkk ðnÞ
n¼1 C kþ1 ðnÞkk ðnÞ
;
n ¼ 1; . . . ; N
The new approach is illustrated in detail with two
examples of complex aerial maneuvering target tracking.
The sensor sampling period T is 1 s.
The trajectory 1 is a target flying in the (x,y) plane,
starting with an initial position [10 km, 40 km]0 and an
initial velocity [300 m/s, 0 m/s]0 , Fig. 4(a) shows trajectory
1 that executes a 5-motion sequences (CV–CA–CV–
CT–CV):
(9)
(4) Estimate and covariance: According to MSPT, the
output of IMM, whose sequential likelihood function
value is maximum, is chosen as target state output. So
the final output, X^ kþ1jkþ1 and P kþ1jkjþ1 , are computed as
follows:
kkþ1 ðnm Þ ¼ max fkkþ1 ðnÞg
(10)
X^ kþ1jkþ1 ¼ X^ kþ1jkþ1 ðnm Þ
(11)
n¼1;...;N
P kþ1jkjþ1 ¼ P kþ1jkþ1 ðnm Þ
(1) CV motion in 30 s;
(2) CA motion in 30 s, its acceleration is (10 m/s2,
10 m/s2);
(3) CV motion in 30 s;
(4) coordinated turn motion in 8 s, its radius is 1.5 km, its
acceleration is 60 m/s2; and
(5) CV motion in 30 s.
The trajectory 2 is a target flying in the (x,y) plane,
starting with an initial position [10 km, 40 km]0 and an
initial velocity [0 m/s, 300 m/s]0 , Fig. 4(b) shows trajectory 2 that executes a 9-motion sequences (CV–CA–CV–
CA–CV–CT–CV–CT–CV):
(12)
From the above interpretation, we insure the independence of all IMM filters, so the deterioration of performance due to the excessive competition from the
unnecessary models is solved. While in every IMM filter,
filter results of every model in every model set are
interacted, so the good performance of general IMM is
preserved. Therefore, the Novel-IMM algorithm not only
inherits the merit of IMM method, but also overcomes the
weakness of IMM.
The VSIMM is a series of multiple model set adaptation
algorithm and its model set selection depends on the
auxiliary information. While the Novel-IMM is a parallel
45
(1) CV motion in 50 s;
(2) CA motion in 15 s, its acceleration is (20 m/s2,
20 m/s2);
(3) CV motion in 50 s;
(4) CA motion in 10 s, its acceleration is (30 m/s2,
30 m/s2);
(5) CV motion in 50 s;
(6) coordinated turn motion in 31 s, its radius is 1.5 km, its
acceleration is 60 m/s2 (trajectory 2);
45
40
40
2 CA
30
Y (km)
Y (km)
35
3 CV
30
5 CV
25
1 CV
35
1 CV
2 CA
25
15
7 CV
9 CV
5 CV
5
0
20
4 CA
8 CT
20
10
4 CT
3 CV
5
10
15
X (km)
20
25
-5
6 CT
0
10
20
X (km)
Fig. 4. Target trajectory: (a) trajectory 1 and (b) trajectory 2.
30
40
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(7) CV motion in 50 s;
(8) coordinated turn motion in 31 s, its radius is 1.5 km, its
acceleration is 60 m/s2 (trajectory 2); and
(9) CV motion in 50 s.
probabilities of model set CVCA2 and CVCT2, respectively.
The transitions between the model sets are quite clear.
During the CA portion, the probability of model set CVCA2
is much higher than CVCT2, but during the coordinated
turn motion portion, the probability of CVCT2 is much
higher than CVCA2. The RMSE of IMM and Novel-IMM
are given in Fig. 6. During the CA maneuvering, two
algorithms get the same performance, but during the
coordinated turn motion, Novel-IMM algorithm has better
performance than IMM.
In addition, for a ground target track with the map
auxiliary information, the Novel-IMM algorithm can also
use the map auxiliary information for the hard decision,
now only one model set is used for Novel-IMM algorithm
at a given time. In this case, Novel-IMM algorithm equals
to VSIMM algorithm. Therefore, Novel-IMM algorithm can
be used in more situations with or without auxiliary
information. But VSIMM algorithm cannot be used in
above simulation case without any prior auxiliary information.
To validate the performance of the method, the RMSE
of VSIMM and Novel-IMM are given in Fig. 7. The
trajectory of the target is as shown in Fig. 4(b). In VSIMM
method the maneuvering onsets are known as the prior
auxiliary condition while there is not prior information
in Novel-IMM method. Performance of the Novel-IMM
is nearly same as VSIMM, but VSIMM could only be used
The Novel-IMM algorithm includes two model sets
(CVCA2 and CVCT2).
The first model set CVCA2 consists of three models: CV
model, CA model with large process noise covariance [26]
and CA model with small process noise covariance [26].
The second model set CVCT2 consists of three models:
CV model, clockwise CT (coordinated turn) model with
known turn rate [27], counter clockwise CT model with
known turn rate [27].
The standard covariance of models’ process noise is
sv ¼ 2 m/s, while the standard deviation of measurement
noise is sx ¼ sy ¼ 50 m. The matrix of the probabilities
should have a large diagonal element, and it is as follows:
2
3
0:95 0:025 0:025
6 0:025 0:95 0:025 7
P¼4
5
0:025 0:025 0:95
1
CVCA2
CVCT2
0.5
0
30
40
50
60
70 80 90
Time (s)
100 110 120 130
Weight Probability
Weight Probability
The IMM algorithm includes 1 model set CVCA2 as the
first model set in the Novel-IMM algorithms.
The average model set probabilities for the Novel-IMM
are given in Fig. 5, where CVCA2 and CVCT2 denote the
1
CVCA2
CVCT2
0.5
0
0
50
100
150
200
Time (s)
250
300
350
Fig. 5. Model set weight probability: (a) trajectory 1 and (b) trajectory 2.
50
30
40
50
60
70 80 90
Time (s)
150
50
0
30
40
50
60
70
80 90
Time (s)
100 110 120 130
IMM
Novel-IMM
50
0
100 110 120 130
IMM
Novel-IMM
100
100
RMSEy (m)
100
0
RMSEVx (m/s)
IMM
Novel-IMM
RMSEVy (m/s)
RMSEx (m)
150
0
50
100
150
200
Time (s)
250
80
300
350
IMM
Novel-IMM
60
40
20
0
0
50
100
150
200
Time (s)
Fig. 6. Position RMSE for IMM and Novel-IMM: (a) trajectory 1 and (b) trajectory 2.
250
300
350
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RMSEy (m)
80
Novel-IMM
VSIMM
60
40
20
0
0
50
100
150
200 250
Time (s)
300
350
400
RMSEVy (m/s)
H.Q. Qu et al. / Signal Processing 89 (2009) 2171–2177
2177
60
Novel-IMM
VSIMM
40
20
0
0
50
100
150
200 250
Time (s)
300
350
400
Fig. 7. Position RMSE for VSIMM and Novel-IMM (trajectory 2).
under the assumption that the true model sequence is
known while Novel-IMM could be used without the true
model sequence.
Indeed, the Novel-IMM algorithm must cost more
computational resources than IMM for real time tracking
system, but this problem can be solved easily with the
development of high speed DSP and computer.
4. Conclusion
A Novel-IMM algorithm was presented which leads to
a systematic treatment of model set adaptation without
additional auxiliary information. The Novel-IMM algorithm consists of N independent IMM filters operating in
parallel, and these multiple IMM filters are independent.
The Novel-IMM uses MSPT algorithm to choose the
outputing model set, which matches the target motion
better, according to the target motion. For each IMM, the
model set does not change with time. The Novel-IMM
algorithm solves the deterioration of performance due to
the excessive competition from the unnecessary models,
and inherits the merit of the IMM method and overcomes
weakness of the IMM. It is a new thought for the study
and design of the MM estimation algorithms.
References
[1] L.A. Johnston, V. Krishnamurthy, An improvement to the interacting
multiple model (IMM) algorithm, IEEE Trans. Signal Process. 49 (12)
(2001) 2909–2923.
[2] X.R. Li, A survey of maneuvering target tracking—part II: model set
adaptation, IEEE Trans. Autom. Control 45 (11) (2000) 2047–2060.
[3] X.R. Li, X.R. Zhi, Y. Zhang, A survey of maneuvering target
tracking—part III: model-group switching algorithm, IEEE Trans.
Aerospace Electron. Syst. 35 (1) (1999) 225–241.
[4] N. Tudoroiu, K. Khorasani, Fault detection and diagnosis for
satellite’s attitude control system (ACS) using an interactive multiple model (IMM) approach, in: Proceedings of the 2005 IEEE
Conference on Control Applications, Toronto, Canada, August 28–31,
2005, pp. 1287–1292.
[5] N. Tudoroiu, K. Khorasani, Satellite fault diagnosis using a bank of
interacting Kalman filters, IEEE Trans. Aerospace Electron. Syst. 43
(4) (2007) 1334–1350.
[6] Y. Hayashi, H. Tsunashima, Y. Marumo, Detection of railway vehicles
using multiple model approach, in: SICE-ICASE International Joint
Conference 2006, Bexco, Busan, Korea October 18–21, 2006, pp.
2812–2817.
[7] Y. Hayashi, H. Tsunashima, Y. Marumo, Fault detection of railway
vehicle suspensions using multiple model approach, in: SICE
Annual Conference 2007, Kagawa University, Japan, September
17–20, 2007, pp. 1539–1543.
[8] X.R. Li, Multiple-model estimation with variable structure: some
theoretical considerations, in: Proceedings of the Third Conference
on Decision and Control, 1994, pp. 1199–1204.
[9] L. Bloomer, J.E. Gray, Are more models better? The effect of the
model transition matrix on the IMM filter, in: Proceedings of
the 34th Southeastern Symposium on System Theory, IEEE, 2002,
pp. 20–25.
[10] X.R. Li, Y. Bar-Shalom, Multiple-model estimation with variable
structure, IEEE Trans. Autom. Control 41 (9) (1996) 478–493.
[11] H.-J. Lin, D.P. Atherton, An investigation of the SFIMM algorithm
for tracking manoeuvring targets, in: Proceedings of the 32nd
Conference on Decision and Control, San Antonlo, TX, 1993,
pp. 930–935.
[12] B. Paneutier, K. Benameur, U. Nimier, M. Rombaut. VSIMM using
road map information for a ground target tracking, in: Seventh
International Conference on Information fusion, 2005, pp. 24–31.
[13] T. Kirubarajan, Y. Bar-Shalom, K.R. Pattipati, I. Kadar, B. Abrams,
E. Eadan, Tracking ground target with road constraint using an IMM
estimator, in: Proceedings of IEEE Aerospace Conference, (5), 1998,
pp. 5–12.
[14] T. Kirubarajan, Y. Bar-Shalom, K.R. Pattipati, Topography-based VSIMM estimator for large-scale ground target tracking, in: IEE
Colloquium Target Tracking: algorithms and Applications, London,
UK, 1999, pp. 11/1–11/4.
[15] T. Kirubarajan, Y. Bar-Shalom, K.R. Pattipati, I. Kadar, Ground target
tracking with variable structure IMM estimator, IEEE Trans. Aerospace Electron. Syst. 36 (1) (2000) 26–46.
[16] X.R. Li, Model set design for multiple-model estimation—part I, in:
Proceedings of the 2002 International Conference on Information
Fusion, Annapolis, MD, 2002, pp. 26–33.
[17] X.R. Li, Z.-L. Zhao, P. Zhang, C. He, Model set design for multiplemodel estimation—part II: examples, in: Proceedings of the 2002
International Conference on Information Fusion, Annapolis, MD,
2002, pp. 1347–1354.
[18] X.R. Li, Z.-L. Zhao, X.-B. Li, General model set design methods for
multiple-model approach, IEEE Trans. Autom. Control 50 (9) (2005)
1260–1276.
[19] X.R. Li, V.P. Jilkov, R.U. Jifeng, Multiple-model estimation with
variable structure—part VI: expected-mode augmentation, IEEE
Trans. Aerospace Electron. Syst. 41 (3) (2005) 853–867.
[20] X.R. Li, Y. Zhang, Multiple-model estimation with variable structure.
Part V: likely-model set algorithm, IEEE Trans. Aerospace Electron.
Syst. 36 (2) (2000) 448–466.
[21] X.R. Li, V.P. Jilkov, Survey of maneuvering target tracking. Part V:
multiple-model methods, IEEE Trans. Aerospace Electron. Syst. 41
(4) (2005) 1255–1321.
[22] X.R. Li, V.P. Jilkov, A survey of maneuvering target tracking—part IV:
decision-based methods, signal and data processing of small
targets, in: Proceedings of SPIE, vol. 4728, 2002, pp. 511–534.
[23] H.A.P. Blom, Y. Bar-Shalom, The interacting multiple model
algorithm for systems with Markovian switching coefficients, IEEE
Trans. Autom. Control 33 (8) (1988) 780–783.
[24] S. Blackman, R. Popoli, Design and Analysis of Modern Tracking
Systems, Artech House, Boston, 1999.
[25] Y.B. Shalom, X.R. Li, Estimation and Tracking: Principles, Techniques, and Software, Artech House, Boston, 1993.
[26] X.R. Li, V.P. Jilkov, Survey of maneuvering target tracking. Part I:
dynamic models, IEEE Trans. Aerospace Electron. Syst. 39 (4) (2003)
1333–1363.
[27] G.A. Watson, W.D. Blair, IMM algorithm for tracking that maneuver
through coordinated turns, in: SPIE Signal and Data Processing of
Small Targets, vol. 1698, 1992, pp. 236–247.