INTRODUCTION
Track radars provide continuous range, bearing and elevation data on
one or more specific targets. These radars are equipped with a radar signal
processor and plot extractor. Radar signal processor reduces the noise
interference and decreases the effect of clutter. Plot extractor extracts
the plot position data and sends them to the radar data processor with
azimuth and range sequences. The radar data processor applies tracking
algorithm to the received data and gives a track (Edde, 1993). This algorithm
should be able to track high maneuvering targets with reasonable accuracy.
The KF has been used extensively in target tracking problems. However,
while the target maneuvers the quality of the position and velocity estimation
could be degraded considerably (Lee et al., 2005). To solve this
problem, some techniques have recently been issued to modify the conventional
KF, such as the augmented state Kalman estimator (AUSKE). The AUSKE solves
the problems by including the input parameters as a part of an augmented
state (Haessig and Friedland, 1998; Mookerjee and Reifler, 1999; Mehrotra
and Mahapatra, 1997). The AUSKE suffers from complexity of computational
effort and numerical problems when state dimensions are large. Hsieh and
Chen (1999, 2000) suggested an optimal twostage Kalman estimator (OTSKE)
for a general case to reduce the computational complexity of the AUSKE.
However, OTSKE suffers from two major drawbacks. These drawbacks stem
from assuming constant acceleration and input term assumed to be observable
from the measurement equation (Hsieh and Chen, 1999; Qiu et al.,
2005). Wang and Varshney (1993) proposed a tracking algorithm which was
developed based on constant acceleration (jerk is zero) for low maneuvering
targets. Therefore, the performance of the estimation is reduced when
target moves with nonconstant or high acceleration.
Goodwin and Sin (1984) proposed an adaptive control of time varying systems.
They used a finite data window for error covariance matrix in least square
algorithm and reset that matrix periodically. In this scheme, resetting
times and resetting factor are two important parameters. Any changes in
these two parameters will significantly influences the tracking accuracy.
In Goodwin and Sin (1984) these two parameters are determined offline
and are constant during the simulation. We use a soft computing approach
to determine these two parameters online and adaptively regarding the
target maneuver.
The equivalentnoise approaches assume that the filter correction can
take a simple form of increasing the KF gain equivalent to placing more
weights on the measurements (Cardillo et al., 1999; Li and Jilkov,
2002; Li and BarShalom, 1994). In the other words, the equivalentnoise approaches
assume that the maneuver effect can be modeled by a white or colored noise
process (Li and Jilkov, 2002). Of course, the statistics of these weights
for maneuver compensation in general, are not known. This fundamental
assumption converts the problem of maneuvering target tracking to a state
estimation problem in the presence of nonstationary process noise with
unknown statistics (Li and Jilkov, 2002). This conversion is the most
important drawback of this fundamental assumption. The proposed algorithm
in this paper intends to overcome this major drawback by determining the
matrix covariance resetting level value.
There are some literature about using fuzzy logic in maneuvering target
tracking with intelligent adaptation and capability to add human knowledge
to the system (Duh and Lin, 2004; Lee et al., 2004).
Continuing these efforts, in this research we find some logical rules
for resetting error covariance matrix and add them to the Goodwin and
Sin method with the use of fuzzy logic.
MANEUVERING TARGET TRACKING FORMULATION
Some researches in changepoint detection have been explored in Gustafsson
(1996) and Malladi and Speyer (1999). It is assumed that the target moves
in a plane, which is the twodimensional case, such as a helicopter with
fixed elevation. The state equation for the nonmaneuvering model is given
by 1.
Where:
X(.) 
= 
State vector 
x(.) 
= 
White system driving uncertainty 
X(0) 
= 
Initial condition which may be uncertain 
Z(.) 
= 
Observation vector 
v(.) 
= 
White observation uncertainty 
where, R(.), Q(.) and ψ denote the measurement, process and initial
state covariance matrices, respectively. The expression for G(n), F(n)
and H(n) as functions of the update time T (T is the time interval between
two consecutive measurements) are:
The standard KF which is an efficient and unbiased filter (Schweppe,
1973) is summarized in the following relations.
K(n) is the Kalman gain and notation
denotes the prediction at the (n+1)^{th} sample point given the
measurement up to and including the (n)^{th} whilst
denotes the estimation at the (n)^{th} sample point given the
measurement up to and including the (n)^{th}. Σ(nn) is the
error covariance matrix and Σ(n+1n) is the error covariance matrix
of the onestep prediction.
The maneuvering model treats the acceleration as an additive term:
Where:
U(n) 
= 
[u_{x}(n) u_{y}(n)]^{T} 
U(n) 
= 
Target acceleration which is modeled as an unknown variable 
ERROR COVARIANCE MATRIX RESETTING
Kalman filter is known to provide extremely rapid initial convergence rate
and optimal tracking in nonmaneuvering problem. However, the algorithm was
developed with some assumptions. The most important one is constant speed of
target movement. To be more precise, while the target maneuvers, the quality
of the position and velocity estimation could be decreased significantly. Therefore,
using KF is suitable until target starts to maneuver.
As we know, in this algorithm the error covariance matrix (Σ(nn))
becomes small after a few iterations (Goodwin and Sin, 1984). Consequently,
when the target begins to maneuver with high acceleration, tracker which
uses KF would not be functionally accurate. This motivates a related scheme
in which Σ(nn) is reset at various times. In other words, old data
is discarded to keep the algorithm alive. The main idea of resetting Σ(nn)
is to retain the fast initial convergence of KF and track a target maneuver
immediately.
The trace norm is usually used as a matrix measure. Therefore, after
some iterations the error covariance matrix norm (Trace[Σ(nn)])
becomes smaller than specified value called Trace Limit (TL), the following
presetting procedure is performed:
where, K_{p} is resetting factor.
The main drawbacks of the conventional resetting scheme are:
• 
The TL of the algorithm is determined offline and remains constant
during the simulation. This drawback may cause a delay in resetting
Σ(nn) when the target maneuvers. Furthermore, it can lead to
unwanted resetting when the target does not maneuver. 
• 
The Resetting Factor (K_{p}) is determined offline and
remains constant during the simulation. This drawback may produce
a small Kalman gain when the target maneuvers (incomplete compensation)
or a large Kalman gain when the target does not maneuver. 
To overcome these drawbacks the Intelligent Error Covariance Matrix Resetting
Algorithm is suggested in this study.
INTELLIGENT ERROR COVARIANCE MATRIX RESETTING
Target maneuver value plays a critical role in determining the resetting
factor and the TL value in each step. Since target maneuver is unknown
for the tracker, its estimation would be essential to determine these
two parameters.
Fuzzy maneuver detector: Radar output signal has no explicit mathematical
relationship with target maneuver. However, there exists a complex nonlinear
mapping between them. Finding effective input elements is essential to
map the input vector to target acceleration vector. In this research,
two features are used as inputs of fuzzy acceleration estimator system.

Fig. 1: 
Target movement geometry in Cartesian coordinates

Absolute value of difference between last target course (ψ) and
observation target course (ξ): Figure 1 is target movement geometry
in Cartesian coordinates. Δθ is one of the most useful elements
to detect the target maneuver (Bahari et al., 2007).
When Δθ is low, then the target with high probability is
moving around its last direction and when Δθ High, then the
target with high probability is moving toward sensor`s observation. This
fact was used as a fuzzy rule in fuzzy acceleration estimator.
Δθ, ψ and ξ are calculated with the use of following
equations.
Where:
ψ 
= 
Last target course 
ξ 
= 
Observation target course 
Absolute value of measurement residue (R): The objective here
is to develop a maneuver detection algorithm, which detects the acceleration
and jerk of a maneuvering target. Similar idea of quickest detection and
change detection algorithm only for constant acceleration has been investigated
in the Wang and Varshney (1993). The standard KF is an efficient and unbias
filter with the Measurement residual as follows:
The measurement residue for nonmaneuvering target is a stochastic zero
mean white process i.e.,

Fig.2: 
The proposed scheme block diagram 
where,
denote the measurement covariance matrix.
Therefore, for nonmaneuvering targets, the mean of this sequence
is zero. But, for maneuvering target case, this sequence is no longer
zero and contains more information. In fact, acceleration term leads to
a bias in measurement residue. The amount of this bias, supply some information
about the existence of target acceleration. This fact was used
as another fuzzy rule in fuzzy acceleration estimator system for target
maneuver detection and estimation.
Intelligent error covariance matrix resetting: Block diagram of
proposed system is shown in Fig. 2. In Fig. 2 block 1, calculates Δθ
and R. Block 2 is a fuzzy controller. The fuzzy system has two inputs
and one output. The input variables of fuzzy system are Δθ
and R. Inputs and output fuzzy sets all have three Gaussian membership
functions with the following membership grade u_{i}^{j}(x_{i}).
where, c_{i}^{j}(x_{i}) and σ_{i}
are the center value and the standard deviation of Gaussian membership
function for i^{th} input variable of j^{th} fuzzy rule,
respectively. The output of the fuzzy logic controller determines the
estimated acceleration value of the target (a_{t}) based on Δθ
and R inputs. Fuzzy inference rules support mentioned information.
Block 3 is a simple low pass filter. The main purpose of using low pass
filter is that output of block 2 (a_{t}) is a noisy signal. To
vivify, a_{t} is target acceleration signal added with high frequency
noise.
After passing this noisy signal through a low pass filter, real value
of target acceleration will be achieved. Block 4, which determines K_{p}
is another fuzzy system. This fuzzy system has two inputs and one output.
Inputs are the target acceleration (output of Block 3) and Trace[Σ(n\n)].
Output of this fuzzy controller is optimum value for K_{p} in
each iteration.
Obviously, error covariance matrix should be reset when the target starts
to maneuver and tracker steps are large enough (Σ(nn) is small)
to track the target. In such a situation, fuzzy system in block 4 determines
a proper value for K_{p} proportional to the target acceleration.
While the target does not maneuver or Σ(nn) is large enough to track
the target accurately, output of block 4 remains zero. Inputs and output
fuzzy sets all have two Gaussian membership functions.
Block 5 is error covariance matrix resetting centre. Decision about resetting
Σ(nn) is made in this block of the system. Input of this block is
just K_{p}. In block five if K_{p} is a nonzero value
then system reset Σ(nn) using relation 4 (with the use of determined
K_{p} in block 4). At the other pole, if resetting factor is equal
to zero then system updates the error covariance matrix using (2) as the
conventional KF equations.
RESULTS
Here, simulation was carried out to illustrate the efficiency of the
proposed scheme. Simple KF, method of Wang and Varshney (1993), conventional
resetting scheme and proposed method are compared in the different case
studies.
In experiments reported in this section, the following assumptions and
parameter values are used. In these simulations, the sampling time is
T = 0.015 (sec). Covariance elements generated for R and θ axis are
both Gaussian random variables. In addition, the measurement noise vector
in Cartesian coordinates is related to the measurement noise vector in
polar coordinates by the following equation (Bahari et al., 2007).
Where:
δ_{R} = 200, δ_{θ} = 1
R(0) and θ(0) denote the target initial range and azimuth, respectively.
It is important to mention that in our simulations the initial state
(including initial position and initial speed) of the target is not known
for the trackers.
First case study: The initial position of the target is given
by [x(0), y(0)] = [1532.1(m),1285.6(m)] with an initial speed of [v_{x}(0),
v_{y}(0)] = [38.3(m sec^{1}),32.1(m sec^{1})].

Fig. 3: 
Trajectory of the maneuvering target in Cartesian coordinates
and the tracking result of the proposed method and method of Wang
in the first case study 

Fig. 4: 
Speed of the maneuvering target and the estimation result of the
proposed method and method of Wang in the first case study 
Target moves with constant acceleration [u_{x}(0), u_{y}(0)] = [0.5(m sec^{2}),0.5(m sec^{2})]
until t = 75(sec) (sample time = 5000). Then, it starts to maneuver with
acceleration value [u_{x}(5001), u_{y}(5001)] = [4(m
sec^{2}),4(m sec^{2})]. This acceleration continues
until t = 105(sec) (Sample time = 7000). At t = 105(sec) moment, target
starts to another maneuver with acceleration value [u_{x}(7001),
u_{y}(7001)] = [8(m sec^{2}), 8(m sec^{2})].
Target moves with this acceleration up to end of this simulation at t
= 150(sec).
Figure 3 shows, target trajectory estimation by proposed
method and method of Wang in this Case Study. As can be seen in this figure,
proposed scheme tracks target more accurately in comparison with the other
method.
Table 1: 
Estimation error in the first case study 


Fig. 5: 
Trajectory of the maneuvering target in Cartesian coordinates
and tracking result of the proposed method, conventional resetting
scheme and the method of Wang in the second case study 
The Fig. 4 shows method of Wang tracks maneuvering target speed with lower accuracy in comparison with the proposed
scheme.
In order to compare proposed scheme with method of Wang and simple KF
(with no error covariance matrix resetting), a Mont Carlo (Robert and
Casella, 1999) simulation of 50 runs was performed. The standard deviation
(STD) of estimation error of range, azimuth, course and speed of all three
methods in this case study is compared in Table 1.
Second case study: The initial position of the target is given
by [x(0), y(0)] = [100(m),173.2(m)] with an initial speed of [v_{x}(0),
v_{y}(0)] = [50(m sec^{1}),86.6(m sec^{1})].
During this case study which longs 45 (sec) target moves with the following
jerk value [J_{x},j_{y}] = 4Cos(t/8)(m sec^{3}),
4Cos(t/8)(m sec^{3})].
Tracking result of three methods Method of Wang and Varshney (1993) Conventional
Resetting Scheme and proposed Method is compared in this case study. In
this simulation resetting factor (k_{p}) and Trace Limit of conventional
resetting scheme are assumed to be 1.2 and 100, respectively. Note that
proper values for these two parameters were found using try and error
method.
Figure 5 shows the accuracy of proposed method in tracking
target, which moves with jerk.
Table 2: 
Estimation error in the second case study (STD) 


Fig. 6: 
Range of the maneuvering target and estimation result
of the proposed method, conventional resetting scheme and method of
Wang in the second case study 

Fig. 7: 
Azimuth of the maneuvering target and estimation
result of the proposed method, conventional resetting scheme and
method of Wang in the second case study 
As can be seen in Fig. 5,
very quick maneuvers are detected by proposed maneuver detector system and error covariance matrix is reset when target
maneuvers. Therefore, an accurate estimation result is obtained. Figure
6 and 7 show, target range and azimuth estimation by three methods.

Fig. 8: 
Speed of the maneuvering target and estimation result
of the proposed method, conventional resetting scheme and method of
Wang in the second case study 

Fig. 9: 
Course of the maneuvering target and estimation result
of the proposed method, conventional resetting scheme and method of
Wang in the second case study 
In
fact, proposed scheme is able to track target speed and course much more accurately, as far as Fig. 8 and 9
suggest, in comparison with the other methods when target maneuvers quickly.
Table 2 highlights ability of proposed method to estimate
target trajectory, range, azimuth, course and speed in contrast with method
of (Wang and Vershney, 1993) conventional resetting scheme. Mont Carlo
simulation of 50 runs was performed and STD of estimation error on different
parameters in the second case study obtained.
CONCLUSION
In order to track a maneuvering target, a new algorithm has been introduced
in this paper, which uses an intelligent technique to reset the error
covariance matrix of the KF. The proposed intelligent method is equipped
with a fuzzy acceleration estimator system working based on some information
about target maneuver dynamics. Proposed algorithm uses estimated acceleration
in an innovative architecture in order to add two crucial ability to the
conventional error covariance matrix resetting scheme. First, finding
proper instances to reset the error covariance matrix. Second, determining
optimized value for the resetting factor in each of iterations. Therefore,
KF is used when target velocity is constant and when the target maneuvers
system reset the error covariance matrix intelligently in order to have
an accurate estimation.
Proposed method has been compared with the method of Wang, simple Kalman
filter and the conventional resetting scheme in two different case studies
of target movement including a high maneuvering target (in the first case
study) and jerking target (in the second case study). Target range, azimuth,
course, speed and trajectory estimation accuracy were considered in the
simulations. Results indicate that the proposed method is significantly
superior to three other methods in all target parameters estimation accuracy.