INTRODUCTION
With the development of information war, the active radar is threatened strongly
for its defects, such as bad abilities of anti-reconnaissance and anti-interference.
By contrast, passive detection system has long ranging, high concealing and
survival ability, due to not emitting electromagnetic wave (Sun
and Guo, 2008). In fact, the key of single observer passive localization
is the typical nonlinear filtering. Traditional EKF (Extended Kalman Filter)
method and its derivative algorithms performed low calculation precision and
significant ill-posed feature which lead to bad stability and divergence of
filter in strong nonlinear conditions (Tawfeig et al.,
2011). However, UKF produces several sigma points which are gotten by the
Unscented Transformation (UT) and deal them with nonlinear transformation (Chouraqui
and Benyettou, 2009). It can avoid the issues which are introduced by the
linearization process of extended Kalman filter, while its performance is superior
to EKF (Juliers and Uhlmann, 2004; Hassanzadeh
and Fallah, 2008). But in the system of single observer passive localization,
classic UKF algorithm is effected by the rounding error of calculator, weak
observability and large observational noise, which lead to the problem of poor
stability, slow convergence speed and low precision. Therefore, SRUKF (square
root unscented Kalman filtering) is put forward under this great background
in some references and its performance is superior to the ordinary UKF (Juliers
and Uhlmann, 2004). However, the above algorithms can only be used in the
environment with Gaussian noise (Kaawaase et al.,
2011). Due to scattering characteristic of target, Radar observation noise
is not the only white Gaussian noise, but regularly glint noise with “the
long tail” under the actual condition (Hu et al.,
2004; Zhu, 2007). To solve the problem, this study
focuses on the method of moment matching, combined with SSUT (Yong
et al., 2010), principle of adaptive iteration (Zhan
and Wan, 2007; Barzamini et al., 2009; Xinchao
et al., 2011) and square root filter, a unite algorithm of SISRUKF
(Simplified Iteration Square Root Unscented Kalman filter) is proposed (Farivar
et al., 2009; Tong et al., 2007),
this new method is more effective in approximating the state estimation and
evidently decreases calculational amount, finally provides an effective way
to solve the filtering problem under Non-Gaussian situation.
PRINCIPLE AND PROCESS
Locating model: Locating and tracking of the target are always in certain
coordinate system (Yedjour et al., 2011), a two-dimensional
model was built with observation point as the origin of coordinate, meanwhile,
radiation signal reaching direction Angle, Angle variation, Doppler frequency
variation and Doppler frequency were used as observational parameters. As shown
in the first diagram
is the state vectors of target at k point in time and β is the azimuth.
Figure 1 showed Geometrical relation between observer and
target in 2-D plane.
|
| Fig. 1: |
Geometrical relation between observer and target in 2-D plane |
Then, state equation and observed equation of positioning system can be established accordingly:
where:
and:
denote state and noise transition matrixes, I3 is 3-D unit matrix. T is measurement period. Wk is the state noise. Vk is measurement noise. Given Vk and Wk irrelevantly. It means:
Based on the theory of kinematics, we get formulas as follows:
when observation point and target have relative radial velocity, Doppler frequency is obtained by observation station. Assuming target signal frequency is constant, there is:
where, f denotes the received signal frequency of observation station, fT
and fd are frequency of target radiation and Doppler frequency (Gong,
2004). The expression is:
Where, c is the speed of electromagnetic wave, the expression of Doppler frequency variation is:
Spherical simplex unscented transformation (SSUT): Computation efficiency
of UKF algorithm depends on the number of sampling points in UT. For n-D random
vector, the classic UT needs 2n+1 sampling points and calculated amount is increased
with the increase of the dimension of vector. SSUT has a good performance in
approximating the probability distribution of the state by n+1 points with equal
weight values. These points distribute in the hypersphere with the mean of random
state as the centre. Then, n+2 sampling points of UT are composed of n+1 sampling
points of hyperspheres distribution and the state mean-value point (Liu
et al., 2010). The selection procedure of SSUT is as follows:
| Step 1: |
Given weight value w0 and 0≤w0≤1
|
| Step 2: |
Set weight value wi: |
| Step 3: |
Initialization of vector sequences: |
| Step 4: |
Extend the vector sequences (j = 2,..., n): |
where,
denote the j dimension random variables at i sampling point. 0j is
zero vector of j dimensions.
For N dimensions random variables, with the mean
and mean-square deviation Pxx, sampling points of hyperspheres distribution
can be obtained by:
SSUT with n+2 sampling points takes the place of the classic UT with 2n+1 sampling points. It can reduce nearly the half of the sampling points and greatly lessen the amount of calculation system.
SRUKF algorithm: Considering the robustness of the nonlinear filter, a square root version of the UKF was introduced, SSRUKF replaces the covariance matrix with the square root of the covariance matrix in recursive operation and it can avoid the trouble of negative definite covariance matrix.
The procedure for implementing the SSRUKF can be summarized as follows:
| Step 2: |
Sampling points calculation:: |
where,
denotes vector sequences of SSUT
| Step 4: |
Measurement update: |
where, Qw denotes the state noise variance of system Rv
is the measurement noise variance of system qr means QR decomposition chol means
Cholesky first-order update (Van der Merwe and Wan, 2001).
SISRUKF: In order to enhance the stability, convergence and precision
of the algorithm, adaptive iterative method was introduced into SISRUKF algorithm.
When observation information has been obtained, estimate value and prediction
covariance are used to resample. Then, SSUT recalculate around the state estimation.
Finally, the state estimation is updated by observed values and the performance
of the filter is improved (Gui et al., 2009;
Gao et al., 2008). The process of SISRUKF algorithm
are expressed as follows:
| Step 1: |
At k time, state estimation and covariance can be calculated
via the formula 13~28. |
where, d is the iteration time.
| Step 3: |
Set the adaptive factor: |
where, θ is an empirical value, which is always 1~2.5.
| Step 4: |
State estimation and variance update: |
| Step 5: |
Given the following equation: |
Then, set this inequality:
If inequality (41) is workable, back to step 2 again;
Or else, return to the following values:
where,
and S(k) are state measure estimate and the square root of covariance after
iteration updated respectively (Zhao et al., 2011).
THE SIMULATION EXPERIMENT
Glint noise environment: Glint noise is a typical Non-Gaussian noise
in actual application. Glint noise distribution has a “long tail”
which is similar to Gaussian shape. The model creation method of Glint noise
is achieved through weighted sum of Gaussian noise and other noise. In this
paper, the glint noise is obtained by the weighted sum of two kinds of Gaussian
noise with different variances. Probability density function of glint noise
can be expressed as:
where, N(ω; μ1, p1) is a Gaussian probability
density with the mean of μ1 and the variance of p1,
N(ω; μ2, p2) is an another Gaussian probability
density with the mean of μ2 and the variance of p2,
ε denotes the strength of the glint noise which values 0.05 in this study
(Zhu, 2007).
Moments matching method is introduced to deal with the glint noise in EKF and UKF algorithms. So, the first and second order moments can be obtained as:
where, ,
then, the SISRUKF algorithm can be applied in the non-Gaussian environment (Hu
et al., 2004).
SIMULATION
Assume the target is at constant velocity in 2-D plane, the initial position and velocity are [150 km 100 km -250 m/s 150 m/s]T. Observation station is located in the origin of the coordinate. Wx = wy = 1 m/s2 is the system error. Three groups of different observation accuracy are as follows:
where,
and
denote standard deviation of azimuth and their variation in glint noise environment.
EKF, UKF, SRUKF and SISRUKF are tested separately.
In this simulation, T is sampling cycle which values 1s, N denotes the number
of time which values 100, fT denotes target signal frequency which
values 10 GHz.
is observation accuracy which values 1 MHz, RRE (Relative Range Error) is used
as evaluation index.
|
| Fig. 2: |
Statistical average curve in observation accuracy, condition
(a) |
|
| Fig. 3: |
Statistical average curve in observation accuracy, condition
(b) |
| Table 1: |
Comparing robustness in different observation accuracy |
|
|
Each group do Monte Carlo experiment for 100 times, the relative error which
is less than 15% is defined as convergence, positioning accuracy is the average
of RRE at the tracking end time. The simulation results is showed in Table
1 and Fig. 2-4.
Table 1 gives convergence times under different observation accuracy of the four algorithms. First, in high precision conditions, all of the algorithms have fairly good convergence. As the observation noise increases, the stability of various algorithms decreases accordingly, the stability of SISRUKF algorithm declines much more slowly than the other three. Then, in the condition of the same observation accuracy, the stability of the SISRUKF algorithm has the most convergence times than others. For reasons of SSUT and square root filtering, SISRUKF will decrease computation cost. The simulation results have proven that SISRUKF saved computing time through Table 1, compared with UKF and SRUKF.
Figure 2-4 reveal that, as the observation
noise increases, the convergence speed of all filtering algorithms becomes slow,
positioning accuracy and stability also decline, EKF falls the most, followed
by UKF and SRUKF.
|
| Fig. 4: |
Statistical average curve in observation accuracy, condition
(c) |
However, SISRUKF algorithm has the highest positioning accuracy and the best
stability. This is because SISRUKF is built based on SRUKF, the adaptive factor
is used to adjust the effect of state information to SRUKF filter. Adaptive
iteration method makes covariance of state parameter prediction more accurate.
So, the performance of SISRUKF algorithm is superior to other algorithms.
CONCLUSION
In this study, a new adaptive simplified iterative algorithm was applied in single observer passive localization system. The method puts SSUT and adaptive iterative thought into square root filtering to improve the stability of the filter effectively. Simulation results showed that, in glint noise environment, the algorithm convergence, positioning accuracy and stability were improved compared with other similar algorithms. Therefore, the algorithm has certain directive significance in practical single observer location passive system.