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The Optimal Application of Lagrangian Mathematical Equations in Computer Data Analysis

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
Páginas: -
Recibido: 16 Feb 2022
Aceptado: 15 Apr 2022
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Revista
eISSN
2444-8656
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01 Jan 2016
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2 veces al año
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Introduction

The passive detection system can realize the positioning and tracking of multiple targets. It can fully play the advantages of passive detection, such as good concealment and extraction of target attribute information. It can also enhance the system's survivability and effective working ability in electronic warfare [1]. In the passive positioning method, multi-station direction-finding cross positioning is one of the more used ones. It uses high-precision direction-finding equipment to find the direction of the target at more than two observation points. The intersection of each direction finding line is the position of the target. Currently, in the world's existing passive detection systems, Russia's BEÃA85B6-A three-coordinate electronic intelligence station and Israel's EL/L8300G electronic support system adopt a three-station direction-finding cross positioning system. The multi-station direction-finding cross positioning system does not have very high requirements for time synchronization and data transmission between stations. The feature that allows working in a narrow beam scanning model improves the sensitivity of the system. And compared with single-station passive positioning, multi-station passive positioning has the advantages of omnidirectional and fast [2]. Therefore, although multi-station direction-finding cross positioning has shortcomings such as large positioning errors, it still has important research and application value.

In a complex environment, two or more observing stations perform direction finding and cross positioning of multiple targets. The intersection of different direction-finding lines will produce many false positioning points [3]. The number of false positioning points increases sharply with the number of observation stations and targets. At present, people have developed a variety of methods to solve this problem. For example, the minimum distance method, the maximum likelihood algorithm, and the Lagrangian relaxation algorithm. Among them, the minimum distance method has a simple algorithm and a small amount of calculation. Still, in a complex environment, the correct correlation rate of the data is low, and the probability of misjudgment is greater in practical use. The maximum likelihood algorithm and the Lagrangian relaxation algorithm consider correlation from the measurement domain. They require a general trial and comparison of the total effect of each segmentation, so the correct correlation rate is relatively high. But in the case of a large number of sensors and targets, the amount of calculation is relatively large, and it is not suitable for real-time processing. This article adopts a new method to carry on the position association to the target. This method first constructs a test statistic and compares the test statistic with a certain threshold to make a rough correlation. In this way, some false positioning points are eliminated, and the amount of calculation in the future is reduced. On this basis, the maximum likelihood algorithm is used to carry out the good correlation of orientations to find out the most likely combination of orientations from the same target [4]. This can ensure a certain azimuth correct correlation rate. The simulation results show that the method proposed in this paper can quickly and accurately eliminate false positioning points to realize the positioning and tracking of multiple targets.

The basic principle of cross positioning

Assume that eki and esj are the target azimuths measured by the i direction-finding line of observation station k and the j direction-finding line of observing station s, respectively. These two direction-finding lines intersect at point P. The coordinate is set to (x, y). The coordinates of Observation Station k and Observation Station s are set to (xk, yk) and (xs, ys) respectively [5]. Assuming that the measurements between the various sensors are independent, the measurement error obeys the zero-mean Gaussian distribution. Figure 1 is a schematic diagram of the cross positioning of the observation station. tgθki=yykx=xk tg{\theta_{ki}} = {{y - {y_k}} \over {x = {x_k}}} tgθsj=yysxxs tg{\theta_{sj}} = {{y - {y_s}} \over {x - {x_s}}}

Figure 1

Schematic diagram of observing station cross positioning

Through simple mathematical operations, the coordinate (x, y) of intersection P can be obtained as x=xktgθkixstgθsj+ysyktgθkitgθsj x = {{{x_k}tg{\theta_{ki}} - {x_s}tg{\theta_{sj}} + {y_s} - {y_k}} \over {tg{\theta_{ki}} - tg{\theta_{sj}}}} y=ystgθkiyktgθsj+(xkxs)tgθkitgθsjtgθkitgθsj y = {{{y_s}tg{\theta_{ki}} - {y_k}tg{\theta_{sj}} + \left({{x_k} - x_s} \right)tg{\theta_{ki}}tg{\theta_{sj}}} \over {tg{\theta_{ki}} - tg{\theta_{sj}}}}

By deriving equations (3) and (4), the positioning error dx, dy of the xy plane can be obtained, and then the variance σx2 \sigma_x^2 , σy2 \sigma_y^2 and the cross-covariance σxy can be obtained. It can be expressed in matrix form as [σx2σxyσxyσy2]=[xθkxθsyθkyθs][σθk200σθk2][xθkxθsyθkyθs]T \left[ {\matrix{{\sigma_x^2} & {{\sigma_{xy}}} \cr {{\sigma_{xy}}} & {\sigma_y^2} \cr}} \right] = \left[ {\matrix{{{{\partial x} \over {\partial {\theta_k}}}} & {{{\partial x} \over {\partial {\theta_s}}}} \cr {{{\partial y} \over {\partial {\theta_k}}}} & {{{\partial y} \over {\partial {\theta_s}}}} \cr}} \right]\left[ {\matrix{{\sigma_{{\theta_k}}^2} & 0 \cr 0 & {\sigma_{{\theta_k}}^2} \cr}} \right]{\left[ {\matrix{{{{\partial x} \over {\partial {\theta_k}}}} & {{{\partial x} \over {\partial {\theta_s}}}} \cr {{{\partial y} \over {\partial {\theta_k}}}} & {{{\partial y} \over {\partial {\theta_s}}}} \cr}} \right]^T} σθk and σθs represent the standard deviation of the angle measurement errors of the two sensors.

Location data association

Bearing data association is a key issue in multi-sensor multi-target direction-finding cross-location technology [6]. This problem is relatively simple when there is only a single target in the surveillance area. This only involves positioning issues. The target's position can be obtained by formulas (3) and (4) through simple mathematical operations. However, if there are multiple targets in the monitoring area, the situation becomes more complicated. At this time, it is necessary to distinguish which position measurement data of each sensor comes from the same target. At the same time, we need to combine the position data belonging to the same target to locate the target [7]. This process is also the process of location data association. Here, a two-level azimuth correlation algorithm is used for azimuth data correlation. First, use certain criteria to perform a rough correlation of azimuths and eliminate some false azimuth combinations to reduce the amount of calculation in the future. Then use the Maximum Likelihood Algorithm to carry out the good correlation of azimuths and find the combination of azimuths most likely to come from the same target. This realizes the positioning and tracking of multiple targets.

Rough correlation of azimuth

Assume that 3 sensors each perform direction finding on N targets. The measured target azimuth angle set is denoted as Zs = {θs1, θs2, θsN}, s = 1, 2, 3. Taking any one azimuth measurement value from set Z1, Z2, Z3 can constitute a possible association combination. For example, Zi1i2i3 = {θ1i1, θ2i2, θ3i3} is a combination of orientations [8]. The combination may be the correct combination from the same target, or it may be wrong. We combine these 3 azimuth angles θ1i1, θ2i2 and θ3i3 in pairs. From formula (3) to formula (5), the coordinates and positioning error of the intersection of the direction-finding line can be obtained. The following assumptions are made here.

(1) (x1, y1) is the anchor point determined by the azimuth angles θ1i1 and θ2i2, measured by sensor 1 and sensor 2. σx12 \sigma_{{x_1}}^2 , σy12 \sigma_{{y_1}}^2 , σx1x1 represents the corresponding positioning error and cross-covariance. (2) (x2, y2) is the anchor point determined by the azimuth angles θ1i1 and θ2i2 measured by sensor 1 and sensor 3. σx22 \sigma_{{x_2}}^2 , σy22 \sigma_{{y_2}}^2 , σx2x2 represents the corresponding positioning error and cross-covariance [9]. Since the azimuth angle measurement errors of different sensors are independent of each other, E{[dθ1i1dθ2i2][dθ1i1dθ3i3]}=0[σθ12000] E\left\{{\left[ {\matrix{{d{\theta_{1{i_1}}}} \cr {d{\theta_{2{i_2}}}} \cr}} \right]\left[ {\matrix{{d{\theta_{1{i_1}}}} & {d{\theta_{3{i_3}}}} \cr}} \right]} \right\} = 0\left[ {\matrix{{\sigma_{{\theta_1}}^2} & 0 \cr 0 & 0 \cr}} \right]

In the formula, σθ12 \sigma_{{\theta_1}}^2 represents the covariance of sensor 1 angle measurement error. Then the cross-covariance σx1x2, σy1y2 can be obtained. σy1x2 And σx1y2 satisfy the following formula [σx1x2σx1y2σy1x2σy1y2]=[x1θ1i1x1θ2i2y1θ1i1y1θ2i2][σθi2000][x2θ1i1x2θ3i3y2θ1i1y2θ3i3]T \left[ {\matrix{{{\sigma_{{x_1}{x_2}}}} & {{\sigma_{{x_1}{y_2}}}} \cr {{\sigma_{{y_1}{x_2}}}} & {{\sigma_{{y_1}{y_2}}}} \cr}} \right] = \left[ {\matrix{{{{\partial {x_1}} \over {\partial {\theta_{1{i_1}}}}}} & {{{\partial {x_1}} \over {\partial {\theta_{2{i_2}}}}}} \cr {{{\partial {y_1}} \over {\partial {\theta_{1{i_1}}}}}} & {{{\partial {y_1}} \over {\partial {\theta_{2{i_2}}}}}} \cr}} \right]\left[ {\matrix{{\sigma_{{\theta_i}}^2} & 0 \cr 0 & 0 \cr}} \right]{\left[ {\matrix{{{{\partial {x_2}} \over {\partial {\theta_{1{i_1}}}}}} & {{{\partial {x_2}} \over {\partial {\theta_{3{i_3}}}}}} \cr {{{\partial {y_2}} \over {\partial {\theta_{1{i_1}}}}}} & {{{\partial {y_2}} \over {\partial {\theta_{3{i_3}}}}}} \cr}} \right]^T}

Define the distance difference between anchor point (x1, y1) and (x2, y2) as Di1i2i3=[Δx,Δy]T=[(x1,y1)T(x2,y2)T] {D_{{i_1}{i_2}{i_3}}} = {\left[ {\Delta x,\,\Delta y} \right]^T} = \left[ {{{\left({{x_1},\,{y_1}} \right)}^T} - {{\left({{x_2},\,{y_2}} \right)}^T}} \right]

Then the mean and variance of the distance difference [Ax, Ay]T can be obtained, which are respectively E[ΔxΔy]=[00] E\left[ {\matrix{{\Delta x} \cr {\Delta y} \cr}} \right] = \left[ {\matrix{0 \cr 0 \cr}} \right] P=[σΔx2σΔxΔyσΔxΔyσΔy2] P = \left[ {\matrix{{\sigma_{\Delta x}^2} & {{\sigma_{\Delta x\Delta y}}} \cr {{\sigma_{\Delta x\Delta y}}} & {\sigma_{\Delta y}^2} \cr}} \right]

Where σΔx2=E[(ΔxE(Δx))2]=σx12+σx222σx1x2 \sigma_{\Delta x}^2 = E\left[ {{{\left({\Delta x - E\left({\Delta x} \right)} \right)}^2}} \right] = \sigma_{{x_1}}^2 + \sigma_{{x_2}}^2 - 2{\sigma_{{x_1}{x_2}}} σΔy2=E[(ΔyE(Δy))2]=σy12+σy222σy1y2 \sigma_{\Delta y}^2 = E\left[ {{{\left({\Delta y - E\left({\Delta y} \right)} \right)}^2}} \right] = \sigma_{{y_1}}^2 + \sigma_{{y_2}}^2 - 2{\sigma_{{y_1}{y_2}}} σΔxΔy=E[(ΔxE(Δx))(ΔyE(Δy))]=σx1y1σy1x2σx1y2+σx2y2 {\sigma_{\Delta x\Delta y}} = E\left[ {\left({\Delta x - E\left({\Delta x} \right)} \right)\left({\Delta y - E\left({\Delta y} \right)} \right)} \right] = {\sigma_{{x_1}{y_1}}} - {\sigma_{{y_1}{x_2}}} - {\sigma_{{x_1}{y_2}}} + {\sigma_{{x_2}{y_2}}}

In the formula, σxi2 \sigma_{{x_i}}^2 , σyi2 \sigma_{{y_i}}^2 and σxiyi (i = 1, 2) can be obtained by formula (5), and σx1x2, σy1y2, σy1x2, and σx1y2, can be obtained by formula (7). We use the distance difference obtained by equation (8) and the variance obtained by equation (10). We can construct the following test statistics αi1i2i3=Di1i2i3TP1Di1i2i3=1(1ρ2)[Δx2σΔx22ρΔxΔyσΔxσΔy+Δy2σΔy2] {\alpha_{{i_1}{i_2}{i_3}}} = D_{{i_1}{i_2}{i_3}}^T{P^{- 1}}{D_{{i_1}{i_2}{i_3}}} = {1 \over {\left({1 - {\rho^2}} \right)}}\left[ {{{\Delta {x^2}} \over {\sigma_{\Delta x}^2}} - {{2\rho \Delta x\Delta y} \over {{\sigma_{\Delta x}}{\sigma_{\Delta y}}}} + {{\Delta {y^2}} \over {\sigma_{\Delta y}^2}}} \right]

Where n% represents the correlation coefficient between Ax and Ay, namely ρ=σΔxΔyσΔxσΔy \rho = {{{\sigma_{\Delta x\Delta y}}} \over {{\sigma_{\Delta x}}{\sigma_{\Delta y}}}}

The test statistic obtained by formula (14) can be regarded as approximately following the χ2 distribution. When the test statistic αi1i2i3 is lower than a certain detection threshold obtained from the χ2 distribution, it is considered that the azimuth angles θ1i1, θ2i2 and θ3i2 corresponding to the positioning points (x1, y1) and (x2, y2) may belong to the same target. This combination of orientation may be correct, so we keep it; otherwise, it is considered a wrong combination, and we delete it.

Directional fine correlation

Assume that the combination of the positions retained after the rough correlation of the positions is represented by the set ZT. Each position combination in ZT maybe the correct combination from the same target, but it may also be a wrong combination [10]. It is necessary to use the maximum likelihood algorithm to carry out the good correlation of the bearings to find out the combination of the bearings most likely to come from the target. Assuming that Zi1i2i3 = {θ1i1, θ2i2, θ3i3} is any combination of azimuths inset ZT, the joint probability density function of the three azimuth measurement values in the combination from the same target can be expressed as p(Zi1i2i3)=s=13p(θsis) p\left({{Z_{{i_1}{i_2}{i_3}}}} \right) = \prod\limits_{s = 1}^3 {p\left({{\theta_{s{i_s}}}} \right)}

We combine the set positions ZT that may come from N different targets to form a target source group. We denote it as a feasible partition γ, namely γ=(Zi1i2i3,,Zi1'i2'i3') \gamma = \left({{Z_{{i_1}{i_2}{i_3}}},\female ,{Z_{i_1^{'}i_2^{'}i_3^{'}}}} \right)

Where Zi1i2i3,Zi1'i2'i3'ZT(isis's=1,2,3) {Z_{{i_1}{i_2}{i_3}}},\female {Z_{i_1^{'}i_2^{'}i_3^{'}}} \in {Z_T}\left({{i_s} \ne i_s^{'}\,\,s = 1,\,2,\,3} \right)

The above formula is the position combination corresponding to N different targets and Zi1i2i3LZi1'i2'i3'=Φ {Z_{{i_1}{i_2}{i_3}}} \cap L \cap {Z_{i_1^{'}i_2^{'}i_3^{'}}} = \Phi

Then the likelihood function of feasible partition γ can be obtained p(γ)=p(Zi1i2i3)××p(Zi1'i2'i3') p\left(\gamma \right) = p\left({{Z_{{i_1}{i_2}{i_3}}}} \right) \times \female \times p\left({{Z_{i_1^{'}i_2^{'}i_3^{'}}}} \right)

In the same way, other feasible partitions inset ZT and their likelihood functions can be obtained. We compare the likelihood functions of all feasible partitions. The division with the largest likelihood function is the correct division. The position combination under this feasible division is the correct combination from N different targets.

Sensors have measurement errors in actual detection. The direction-finding lines corresponding to the same target may not intersect at one point, so at this time, it is necessary to perform position fusion on these intersection positioning points. This minimizes the random error of the system and obtains an estimate closer to the true position of the target [11]. At present, the more commonly used target location fusion algorithms include an average method, least-squares algorithm, iterative method, etc. In the multi-station direction finding and positioning, the bearing data association is the key. We use different methods for position fusion to have a small impact on the tracking results, so here we use the average method with a small amount of calculation for position fusion.

Suppose that after the azimuth data is associated, it is determined that the azimuth angle measurement values correspond to the same target measured by the three sensors. We combine these 3 azimuth angles in pairs by formulas (3) and (4) to obtain the positions of 3 different intersections. Then take the average of the positions of these 3 intersections to get the estimated position of the target. After obtaining the target's location information, it is equivalent to using a single active sensor to track multiple targets. We can track multiple targets using conventional sensor tracking methods.

Simulation Analysis

This article simulates the algorithms described in the previous sections. In the simulation, it is assumed that the angle measurement errors of the three observation stations are the same, and all obey the Gaussian distribution with a mean value of zero. The target is assumed to be a ship [12]. The speed is 22 knots. The target distance is 5km. The longest distance between the target and the sensor is 71.25km. As shown in picture 2. Detection probability Pd=0.95, false alarm rate Pf=0.05. After 30 Monte-Carlo experiments, the azimuth correct correlation rate for different targets with different angle measurement errors is shown in Table 1. Figure 3 shows the estimated trajectory of the target when the angle measurement error is 0.1°.

Correlation rate of azimuth under different angle measurement errors

Angle measurement error 0.1 degree 0.8 degrees 1.5 degrees 2.5 degrees
Goal 1 0.9802 0.9703 0.8812 0.6713
Goal 2 1 0.9802 0.901 0.6931
Goal 3 0.9901 0.9802 0.8218 0.6535
Goal 4 0.9604 0.8218 0.7723 0.6634

Figure 2

Target and sensor location map

Figure 3

Target estimated trajectory graph

It can be seen from Table 1 and Figure 3 that the angle measurement error has a greater influence on the correct correlation rate of the azimuth. When the angle measurement error is small, the correlation rate of position correctness is high, and the positioning error is small. In this way, more accurate positioning and tracking of multiple targets can be achieved. When the angle measurement error is large, there will be more wrong positioning. The location of the target cannot be accurately given at this time.

Conclusion

Multi-station direction-finding cross positioning is one of the most used passive positioning methods. However, this method is prone to produce many false positioning points in a complex environment. The rapid and accurate elimination of false positioning points has always been a difficult problem to solve. Based on the existing methods, this paper adopts a new method to eliminate false positioning points. This method first uses certain criteria to carry out a rough correlation of azimuths. It eliminates a part of false positioning points to reduce the amount of calculation in the future. On this basis, the azimuth is finely correlated to ensure a certain correct correlation rate. The simulation analysis shows that the method proposed in this paper has a high correlation rate of correct azimuth and has a moderate amount of calculation.

Figure 1

Schematic diagram of observing station cross positioning
Schematic diagram of observing station cross positioning

Figure 2

Target and sensor location map
Target and sensor location map

Figure 3

Target estimated trajectory graph
Target estimated trajectory graph

Correlation rate of azimuth under different angle measurement errors

Angle measurement error 0.1 degree 0.8 degrees 1.5 degrees 2.5 degrees
Goal 1 0.9802 0.9703 0.8812 0.6713
Goal 2 1 0.9802 0.901 0.6931
Goal 3 0.9901 0.9802 0.8218 0.6535
Goal 4 0.9604 0.8218 0.7723 0.6634

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