﻿ 基于状态和属性的多目标联合关联算法
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 应用科技  2020, Vol. 47 Issue (3): 74-79  DOI: 10.11991/yykj.202004005 0

### 引用本文

SUN Hantao. Multi-objective joint association algorithm based on state and attribute[J]. Applied Science and Technology, 2020, 47(3): 74-79. DOI: 10.11991/yykj.202004005.

### 文章历史

Multi-objective joint association algorithm based on state and attribute
SUN Hantao
No.92493 Unit of PLA, Huludao 125000, China
Abstract: In the process of multi-target passive positioning, the correctness of the target batch division affects accuracy of the positioning solution results. In response to this problem, this paper studies a multi-objective data joint association algorithm. First, the target feature data is correlated, and the similarity between the targets is judged by using the gray system theory and system cluster analysis method. For the uncertain set, the orientation data association is performed again. It can be seen from the simulation results that the joint association of the two algorithms can make up for the shortcomings brought by a single method and improve the accuracy of target association.
Keywords: orientation association    feature association    multi-target    multiple node    gray correlation    adaptive entropy weight    orientation-feature association    cluster analysis

1 方位数据关联 1.1 方位粗关联

 ${r_{P{s_i}}} = {r_{{s_i}{s_j}}}\sin ({\theta _j} + {\theta _{ij}})/\sin({\theta _i} - {\theta _j})$ (1)

 $\sigma _{rP{s_i}}^2 = {\left(\frac{{\partial {r_{P{s_i}}}}}{{\partial {\theta _i}}}\right)^2}\sigma _{{\theta _i}}^2 + {\left(\frac{{\partial {r_{P{s_i}}}}}{{\partial {\theta _j}}}\right)^2}\sigma _{{\theta _j}}^2$

 ${r_{Q{s_i}}} = {r_{{s_i}{s_k}}}\sin({\theta _k} + {\theta _{ik}})/\sin({\theta _i} - {\theta _k})$

 $\sigma _{rQ{s_i}}^2 = {\left(\frac{{\partial {r_{Q{s_i}}}}}{{\partial {\theta _i}}}\right)^2}\sigma _{{\theta _i}}^2 + {\left(\frac{{\partial {r_{Q{s_i}}}}}{{\partial {\theta _k}}}\right)^2}\sigma _{{\theta _k}}^2$

 ${D_{PQ}} = \left| {{r_{P{s_i}}} - {r_{Q{s_i}}}} \right|$

 $G = 3\sqrt {\sigma _{rP{s_i}}^2 + \sigma _{rQ{s_i}}^2}$ (2)

 $\left\{ \begin{array}{l} \sigma _{rP{s_i}}^2 = \left(\dfrac{{{r_{{s_i}{s_j}}}\sin ({\theta _j} + {\theta _{ij}})\cos({\theta _i} - {\theta _j})}}{{{{\sin }^2}({\theta _i} - {\theta _j})}}\right)\sigma _{{\theta _i}}^2 + \\ \quad \quad \quad \left(\dfrac{{{r_{{s_i}{s_j}}}\sin ({\theta _i} + {\theta _{ij}})}}{{{{\sin }^2}({\theta _i} - {\theta _j})}}\right)\sigma _{{\theta _j}}^2 \\ \sigma _{rQ{s_i}}^2 = \left(\dfrac{{{r_{{s_i}{s_k}}}\sin ({\theta _k} + {\theta _{ik}})\cos({\theta _i} - {\theta _k})}}{{{{\sin }^2}({\theta _i} - {\theta _k})}}\right)\sigma _{{\theta _i}}^2 + \\ \quad \quad \quad \left(\dfrac{{{r_{{s_i}{s_k}}}\sin ({\theta _i} + {\theta _{ik}})}}{{{{\sin }^2}({\theta _i} - {\theta _k})}}\right)\sigma _{{\theta _k}}^2 \\ \end{array} \right.$

1)假设每个声呐阵都能测量到所有目标，针对 $M$ 个目标，每个声呐阵测得 $M$ 条测向线。对声呐阵1的每条测向线进行编号，记为 ${L_{1j}}(j = 1,2, \cdots ,M)$ ，分别统计声呐阵1与声呐阵2、声呐阵1与声呐阵3各条测向线交点。以声呐阵1的第 $j$ 条测向线 ${L_{1j}}$ 为基准，声呐阵2各条测向线与其交叉形成的定位点集合记为 $\{ {d_{1j,2l}} = ({x_{1j,2l}},{y_{1j,2l}})\}$ 。类似地，声呐阵3与声呐阵1的交叉定位点集合为 $\{ {d_{1j,3k}} = ({x_{1j,3k}},{y_{1j,3k}})\}$

2)以声呐阵1测量1号目标的测向线 ${L_{11}}$ 为基准，搜索所有可能的候选关联组合。与 ${L_{11}}$ 有关的交叉定位点集合是 $\{ {d_{11,2l}}\}$ $\{ {d_{11,3k}}\}$ ，由于角度测量误差的存在，每个集合里只有1个点是与目标真实定位点相关联的。遍历2个定位点集合，计算每2个点之间几何距离。记为集合{ ${D_{lk}} = \left| {{d_{11,2l}} - {d_{11,3k}}} \right|$ }。

3)利用最小距离法，针对测向线 ${L_{11}}$ 上的关联集合 ${A_1}$ ，对每个候选组合的 ${d_{11,2l}}$ ${d_{11,3k}}$ 几何距离由小到大排序，认为几何距离最小的关联组合是正确组合，记为 $R = \{ 1,l',k'\}$ 。由于1条测向线仅针对1个目标，与 ${L_{11}}$ 上的定位点关联后， ${L_{2{l^{'}}}}$ 与其他测向线形成的交叉定位点即为虚假定位点。如图2所示， ${p_1}$ ${L_{11}}$ 上与声呐阵2的 ${L_{21}}$ 测向线确定的真实目标交叉点，则 ${L_{21}}$ ${L_{12}}$ 测向线的交点 ${q_1}$ ${L_{21}}$ ${L_{13}}$ 测向线交点 ${q_2}$ 均被排除。

4)确定了 ${L_{11}}$ 上针对目标1的候选关联集合 ${A_1}$ 后，重复步骤2)和3)，确定 ${L_{12}}$ 上针对目标2的候选关联集合 ${A_2}$ 。由于之前计算测向线组合 $R\{ 1,l',k'\}$ 时，对测向线 ${L_{12}}$ 与其他测向线的交叉点做过排除处理，可能会导致 ${A_2}$ 是一个空集，即没有符合约束 ${D_{lk}} < G$ 的候选关联组合。此时需要在 ${A_1}$ 集合中，选择 ${d_{11,2l}}$ ${d_{11,3k}}$ 几何距离次小的点，并把相关测向线作为目标1的方位关联组合，再次执行步骤4)，计算相应目标的方位候选关联集合。

5)计算出针对1号声呐阵的各个测向线 ${L_{1j}}(j = 1,2, \cdots ,M)$ ，针对各个目标的方位候选关联集合 ${A_m}(m = 1,2, \cdots ,M)$ 。对于 ${A_m}$ ，若其中元素超过1个，即包含多个方位组合，称为不确定关联集合；对于只含有1组方位关联的集合 ${A_m}$ ，称为确定集合。对于不确定关联集合，还需要利用细关联处理，筛选出唯一的方位关联组合。

1.2 方位细关联

 ${\hat \theta _i} = \arctan (({y_{si}} - \hat y)/({x_{si}} - \hat x))$ (3)

 $\lambda = \sum\limits_{i = 1}^N {(({\theta _i} - {{\hat \theta }_i})/{\sigma _{\theta i}})}$ (4)

 $n = N{n_z} - {n_x}$

2 目标特征数据关联

2.1 传统灰色关联

 ${{x}} = \left[ {\begin{array}{*{20}{c}} {{x_1}(1)}&{{x_1}(2)}& \cdots &{{x_1}(K)}\\ {{x_2}(1)}&{{x_2}(2)}& \cdots &{{x_2}(K)}\\ \vdots & \vdots &{}& \vdots \\ {{x_R}(1)}&{{x_R}(2)}& \cdots &{{x_R}(K)} \end{array}} \right]$

 $\begin{array}{c} {X_i}(j) = \dfrac{{{x_i}(j) - \overline x (j)}}{{{S_j}}} \\ \overline x (j) = \dfrac{1}{R}\displaystyle\sum\limits_{i = 1}^R {{x_i}(j)} \\ {S_j} = \sqrt {\dfrac{1}{R}{{\left(\displaystyle\sum\limits_{i = 1}^R {({x_i}(j)} - \overline x (j)\right)}^2}} \\ \end{array}$

 ${{X}} = \{ {X_i}(j)\left| {i = 1,2, \cdots ,R;\;j = 1,2, \cdots ,K} \right.\}$

 ${\varDelta _{a,b}}(j) = \left| {{X_a}(j) - {X_b}(j)} \right|$

${X_a}(j)$ ${X_b}(j)$ 这2个序列的第j个特征灰色关联系数为[7]

 $\begin{array}{c} \zeta ({X_a}(j),{X_b}(j)) = \\ \dfrac{{\mathop {{\rm{Min}}}\limits_a \mathop {{\rm{Min}}}\limits_b \mathop {{\rm{Min}}}\limits_j {\varDelta _{a,b}}(j) + \rho \mathop {{\rm{Max}}}\limits_a \mathop {{\rm{Max}}}\limits_b \mathop {{\rm{Max}}}\limits_j {\varDelta _{a,b}}(j)}}{{{\varDelta _{a,b}}(j) + \rho \mathop {{\rm{Max}}}\limits_a \mathop {{\rm{Max}}}\limits_b \mathop {{\rm{Max}}}\limits_j {\varDelta _{a,b}}(j)}} \\ \end{array}$

 $\gamma ({X_a},{X_b}) = \sum\limits_{j = 1}^K {\zeta ({X_a}(j),{X_b}(j))} {w_{ab}}(j)$

2.2 自适应熵权灰色关联度

1)目标特征矩阵的每一行既是参考序列，也是比较序列。选择参考序列为 ${X_a}$ ，计算参考序列与比较序列的绝对差，构成特征差矩阵[10]

 ${{{\varDelta}} _{\bf{a}}} = \left[ {\begin{array}{*{20}{c}} {{\varDelta _{a,1}}(1)}&{{\varDelta _{a,1}}(2)}& \cdots &{{\varDelta _{a,1}}(K)}\\ {{\varDelta _{a,2}}(1)}&{{\varDelta _{a,2}}(2)}& \cdots &{{\varDelta _{a,2}}(K)}\\ \vdots & \vdots &{}& \vdots \\ {{\varDelta _{a,a - 1}}(1)}&{{\varDelta _{a,a - 1}}(2)}& \cdots &{{\varDelta _{a,a - 1}}(K)}\\ {{\varDelta _{a,a + 1}}(1)}&{{\varDelta _{a,a + 1}}(2)}& \cdots &{{\varDelta _{a,a + 1}}(K)}\\ \vdots & \vdots &{}& \vdots \\ {{\varDelta _{a,R}}(1)}&{{\varDelta _{a,R}}(2)}& \cdots &{{\varDelta _{a,R}}(K)} \end{array}} \right]$

2)计算第 $j$ 个特征项出现的相对概率：

 ${P_{ai}}(j) = {{{\varDelta _{a,i}}(j)} \Bigg/ {\sum\limits_{i = 1}^R {{\varDelta _{a,i}}(j)} }}$

 ${E_{ai}}(j) = - {1 / {\ln R}}\sum\limits_{i = 1}^R {{P_{ai}}(j)\ln{P_{ai}}(j)}$

 ${D_{ai}}(j) = 1 - {E_{ai}}(j)$

3)计算第 $j$ 个特征项的权重值：

 ${w_{ai}}(j) = \frac{{{D_{ai}}(j)}}{{\displaystyle\sum\limits_{j = 1}^K {{D_{ai}}(j)} }}$

 ${{\gamma}} = \left[ {\begin{array}{*{20}{c}} {\gamma ({X_1},{X_1})}&{\gamma ({X_1},{X_2})}& \cdots &{\gamma ({X_1},{X_R})}\\ {\gamma ({X_2},{X_1})}&{\gamma ({X_2},{X_i})}& \cdots &{\gamma ({X_2},{X_R})}\\ \vdots & \vdots &{}& \vdots \\ {\gamma ({X_R},{X_1})}&{\gamma ({X_a},{X_2})}& \cdots &{\gamma ({X_R},{X_R})} \end{array}} \right]$
2.3 系统聚类分析

 ${\mu _{ab}} = \mathop {\max }\limits_{\scriptstyle{X_i} \in {G_a}\atop \scriptstyle{X_j} \in {G_b}} (\gamma ({X_i},{X_j}))$

 ${{\mu}} = \left[ {\begin{array}{*{20}{c}} {{\mu _{11}}}&{{\mu _{12}}}& \cdots &{{\mu _{1R}}}\\ {{\mu _{21}}}&{{\mu _{22}}}& \cdots &{{\mu _{2R}}}\\ \vdots & \vdots &{}& \vdots \\ {{\mu _{R1}}}&{{\mu _{R2}}}&{\cdots}&{{\mu _{RR}}} \end{array}} \right]$

 $\begin{array}{*{20}{c}} {{\mu _{rt}} = \mathop {\max }\limits_{\scriptstyle{X_i} \in {G_r}\atop \scriptstyle{X_j} \in {G_t}}{\begin{array}{*{20}{l}} \end{array}} \{ (\gamma ({X_i},{X_j}))\} = \max\Bigg\{ \mathop {\max }\limits_{\scriptstyle{X_a} \in {G_p}\atop \scriptstyle{X_b} \in {G_t}}{\begin{array}{*{20}{l}} \end{array}} \{ \gamma ({X_a},{X_b})\} ,}\\ {\mathop {\max }\limits_{\scriptstyle{X_c} \in {G_q}\atop \scriptstyle{X_d} \in {G_t}}{\begin{array}{*{20}{l}} \end{array}} \{ \gamma ({X_c},{X_d})\} \Bigg\} = \max \{ {\mu _{pt}},{\mu _{qt}}\} } \end{array}$

 ${S_w}(C) = \frac{1}{z}\sum\limits_{i = 1}^z {\frac{1}{{\left| {{C_i}} \right| \cdot \left| {{C_i}} \right|}}} \sum\limits_{X,Y \in {C_i}} {\gamma (X,Y)}$

 ${S_b}(C) = \frac{1}{{z(z - 1)}}\sum\limits_{i = 1}^z {\left( {\sum\limits_{j = 1,j \ne i}^z {\frac{1}{{\left| {{C_i}} \right| \cdot \left| {{C_j}} \right|}}\sum\limits_{X \in {C_i},Y \in {C_j}} {\gamma (X,Y)} } } \right)}$

 $V(C) = {S_w}(C) + {S_b}(C)$

2.4 目标方位−特征联合关联

3 仿真实例

3个声呐阵坐标分别为 $(0,0)$ $(10,0)$ $(20,0)$ 。同一时刻目标个数为3，每个目标间距d=2。其位置坐标如下：目标1(11,10)、目标2(11+d,10)和目标3(11−d,10)，单位均为km。各目标特征值如表1所示。线谱频率和归一化幅度测量(以最大幅度为准)误差服从均值为0、标准差分别为5 Hz和20 dB的高斯分布，线谱个数正确估计率为60%。