﻿ 被动多传感器改进交互多模型粒子滤波算法
 舰船科学技术  2017, Vol. 39 Issue (3): 145-149 PDF

1. 中国人民解放军92038部队，山东  青岛 266108;
2. 北海舰队参谋部信息保障处，山东  青岛 266071

Improved interacting multiple model particle filter for passive multi-sensor
LIU Wei1, SONG Wei2, HUANG Zun-quan1
1. No.92038 Unit of PLA, Qingdao 266108, China;
2. North China Sea Fleet Information Security Department, Qingdao 266071, China
Abstract: The proposed method in this paper is the binding of improved particle filter which called uniform resampling particle filter and interacting multiple model algorithm. The proposed method is called interacting multiple model uniform resampling particle filter, and is applied to target tracking for passive multi-sensor. The resampling process is improved in uniform resampling particle filter compared with standard particle filter. While ensuring solving the degeneration problem of particle, the variety of particle and the filtering properties are improved. Uniform resampling particle filter is applied in interacting multiple model to improving the tracking accuracy of maneuvering target for passive multi-sensor. By comparing the proposed method and interacting multiple model particle filter, better tracking performance of proposed method is presented.
Key words: uniform resampling     passive multi-sensor     maneuvering target     interacting multiple model     particle filter
0 引 言

1 被动跟踪模型 1.1 离散状态方程

 ${ X}\left( {k + 1} \right) = { F}X(k) + { G}v(k)\text{，}$ (1)

1） CV 模型

 $\begin{array}{l}{ F} = \left[ {\begin{array}{*{20}{l}}{1\quad 0\quad T\quad 0}\\{0\quad 1\quad 0\quad T}\\{0\quad 0\quad 1\quad 0}\\{0\quad 0\quad 0\quad 1}\end{array}} \right]\text{，}\\{ G} = \left[ {\begin{array}{*{20}{l}}{{T^2}/2\quad 0\quad T\quad 0}\\{0{T^2}/2\quad 0\quad T}\end{array}} \right]\text{。}\end{array}$

2） CT 模型

 $\begin{array}{l}{ F} = \left[ {\begin{array}{*{20}{c}}1 & 0 & {\displaystyle\frac{{\sin \omega T}}{\omega }} & {\displaystyle\frac{{\cos \omega T}}{\omega }}\\[8pt]0 & 1 & {\displaystyle\frac{{1 - \cos \omega T}}{\omega }} & {\displaystyle\frac{{\sin \omega T}}{\omega }}\\[8pt]0 & 0 & {\cos \omega T} & { - \sin \omega T}\\[8pt]0 & 0 & {\sin \omega T} & {\cos \omega T}\end{array}} \right]\text{，}\\[45pt]{ G} = \left[ \begin{array}{l}{T^2}/2\quad 0\quad \quad 0\\0\quad \quad {T^2}/2\quad 0\end{array} \right]\text{。}\end{array}$

1.2 离散观测方程

 ${ Y}\left( {k + 1} \right) = {h_k}\left( {{x_k}} \right) + {\omega _k}\text{，}$ (2)

 ${h_k}\left( {{x_k}} \right) = {\left[ {atg\frac{{x - {x_1}}}{{y - {y_1}}}, \cdots ,atg\frac{{x - {x_i}}}{{y - {y_i}}}, \cdots } \right]^{\rm T}}\text{。}$ (3)

2 AUPF算法 2.1 标准的粒子滤波算法 [ 7]

 $p\left( {{x_k}/{z_k}} \right) \approx \sum\limits_{i = 1}^N {{\omega ^i}_k \cdot \delta \left( {{x_k} - {x^i}_k} \right)} \text{，}$ (4)

 ${\omega ^i}_k \propto {\omega ^i}_{k - 1}\frac{{p\left( {{z_k}/{x^i}_k} \right) \cdot p\left( {{x^i}_k/{x^i}_{k - 1}} \right)}}{{q\left( {{x^i}_k/{x^i}_{k - 1},{z_k}} \right)}}\text{，}$ (5)

 $q\left( {{x^i}_k/{x^i}_{k - 1},{z_k}} \right) = p\left( {{x^i}_k/{x^i}_{k - 1}} \right)\text{，}$ (6)

 ${\omega ^i}_k \propto {\omega ^i}_{k - 1} \cdot p\left( {{z_k}/{x^i}_k} \right)\text{。}$ (7)

1）初始化。由先验概率 $p\left( {{x_0}} \right)$ 产生粒子群 $\left\{ {{x^i}_0} \right\}_{i = 1}^N$

2）所有粒子权值为 $\displaystyle\frac{1}{N}$

3）预测。在 k 时刻，根据状态转移方程预测下一时刻粒子的值 ${x^i}_k$ 和观测值 ${h_k}\left( {{x^i}_k} \right)$

4）更新。按式（7）更新粒子权值，并且归一化。

5）重采样。得到新的粒子集合 $\left\{ {{x^{{i^*}}}_{_k},i = 0,1,2, \cdots ,N} \right\}$ 。则可得 k 时刻状态的最小均方估计为： $\overline x \approx \sum\limits_{i = 1}^N {{\omega ^i}_k \cdot {x^i}_k}$

6）时刻 k = k + 1，转到第 2 步。

2.2 AUPF 粒子滤波算法 [ 4, 8]

1）对所有的 N 个粒子按权值进行排序，选出权值最大的 N s 个粒子，其中 ${N_s} = nN\left( {0.8 < n < 1} \right)$

2）计算粒子的概率累积和 $\left( {{a_j}} \right)_{i = 1}^{{N_s}}$ ，并假设 a 0 = 0；

3）随机采样第 $i\left( {i = 1,2, \cdots ,{N_s}} \right)$ 个服从 [0 1] 均匀分布的数 η i ，若 ${a_{j - 1}} < {\eta _i} < {a_j}$ ，则可以取得第 i 次随机采样的结果为 x j

5）将 2 次得到的粒子合并，得到将要进入下一次迭代的全部粒子 $\left\{ {{x^i}_k} \right\}_{i = 1}^N$ ，并重新分配权值。

3 IMM 均匀重采样粒子滤波 3.1 IMM 算法介绍

IMM 算法原理如 图 1 所示。IMM [ 9] 算法：混合概率计算、输入交互、模型滤波、模型概率更新及状态和协方差计算等步骤。

 图 1 IMM 的原理框图 Fig. 1 the principle diagram of IMM

3.2 IMM-AUPF 算法介绍

IMM-AUPF 算法具体实现步骤如下：

1）输入交互 [ 9]

 $\overline {{c_{{l_1}}}} = \sum\limits_{i = 1}^{{N_s}} {{p_{{l_2}{l_1}}} \cdot {u_{k - 1,{l_2}}}} \text{，}$ (8)
 $\overline {{c_{{l_1}}}} = \sum\limits_{i = 1}^{{N_s}} {{p_{{l_2}{l_1}}} \cdot {u_{k - 1,{l_2}}}} \text{，}$ (9)
 ${x^0}_{k - 1/k - 1,{l_1}} = \sum\limits_{{l_2} = 1}^{{N_s}} {{x_{k - 1/k - 1,{l_2}}} \cdot {u_{k - 1/k - 1,{l_2}/{l_1}}}} \text{，}$ (10)
 $\begin{array}{l}{P^0}_{k - 1/k - 1,{l_2}} = \sum\limits_{{l_2} = 1}^{{N_s}} {{u_{k - 1/k - 1,{l_2}/{l_1}}}}\times \\\quad \left\{ \begin{array}{l}{P_{k - 1/k - 1,{l_2}}} + \left[ {{x_{k - 1/k - 1,{l_2}}} - {x^0}_{k - 1/k - 1,{l_1}}} \right]\\{\left[ {{x_{k - 1/k - 1,{l_2}}} - {x^0}_{k - 1/k - 1,{l_1}}} \right]^T}\end{array} \right\}\text{。}\end{array}$ (11)

2）并行滤波

$N\left( {{x^0}_{k - 1/k - 1,{l_1}},{P^0}_{k - 1/k - 1,{l_1}}} \right)$ 中随机抽取样本点： ${x^{\left( i \right)}}_{k - 1/k - 1,{l_1}}\sim N\left( {{x^0}_{k - 1/k - 1,{l_1}},{P^0}_{k - 1/k - 1,{l_1}}} \right),i = 1, \cdots N;$ 根据状态转移方程进行粒子预测，得到新的粒子集 ${x^{\left( i \right)}}_{k - 1/k - 1,{l_1}}$ ，将其代入观测方程得到 ${z^{\left( i \right)}}_{k - 1/k - 1,{l_1}}$ ，计算残差：

 ${\delta ^{\left( i \right)}}_{k,{l_1}} = {z_k} - {z^{\left( i \right)}}_{k/k - 1,{l_1}}\text{。}$ (12)

3）模型概率更新

 $\begin{array}{l}{S_{k,{l_1}}} = R + \sum\limits_{i = 1}^N {{\omega ^{\left( i \right)}}_{k,{l_1}} \cdot } \times\\\quad \quad \quad \left( {{z^{\left( i \right)}}_{k/k - 1,{l_1}} - \overline {{z_{k,{l_1}}}} } \right){\left( {{z^{\left( i \right)}}_{k/k - 1,{l_1}} - \overline {{z_{k,{l_1}}}} } \right)^{\rm T}}\text{，}\end{array}$ (13)

 ${v^{{l_1}}} = {z_k} - \overline {{z_{k,{l_1}}}}\text{，}$ (14)

 ${\Lambda _{k,{l_1}}}\sim N\left( {{v_{k,{l_1}}};0,{S_{k,{l_1}}}} \right)\text{，}$

 ${u_{k,{l_1}}} = \frac{1}{c}{\Lambda _{k,{l_1}}} \cdot \overline {{c_{{l_1}}}} \text{。}$ (15)

 $c = \sum\limits_{i = 1}^N {{\Lambda _{k,{l_1}}} \cdot \overline {{c_{{l_1}}}} } \text{。}$ (16)

4）重采样

$\left\{ {{x^i}_{k/k - 1,{l_1}},i = 1, \cdots ,{N_s}} \right\}$ 中根据重要性权值利用均匀重采样方法，重新采样得到 1 组新的粒子 $\left\{ {{x^j}_{k/k - 1,{l_1}},j = 1, \cdots ,{N_s}} \right\}$ ，并重新分配权值 ${\omega ^{\left( i \right)}}_{k,{l_1}} = \frac{1}{N}$

5）状态估计及协方差

 ${x_{k/k,{l_1}}} = \sum\limits_{j = 1}^N {{\omega ^{\left( j \right)}}_{k,{l_1}}{x^{\left( j \right)}}_{k/k,{l_1}}}\text{，}$ (17)
 $\begin{array}{l}{P_{k/k,{l_1}}} = \displaystyle\sum\limits_{j = 1}^N {{\omega ^{\left( j \right)}}_{k,{l_1}} \cdot } \times\\\quad \quad \quad \left[ {{x_{k/k,{l_1}}} -\displaystyle {x^{\left( j \right)}}_{k/k,{l_1}}} \right]{\left[ {{x_{k/k,{l_1}}} - {x^{\left( j \right)}}_{k/k,{l_1}}} \right]^{\rm T}}\text{。} \end{array}$ (18)

6）输出交互

 ${x_{k/k}} = \sum\limits_{{l_1} = 1}^{{N_s}} {{x_{k/k,{l_1}}}{u_{k,{l_1}}}}\text{。}$ (19)
4 仿真验证与比较 4.1 精度评价指标 [ 8]

 $RMSE = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {{{\left( {{x_k} - \overline {{x_k}} } \right)}^2}} } \text{。}$ (20)

J 次蒙特卡洛试验的 RMSE 均值:

 $\mu = \frac{1}{J}\sum\limits_{i = 1}^J {RMS{E_i}} \text{，}$ (21)

J 次蒙特卡洛试验的 RMSE 均方差:

 $\sigma = \sqrt {\frac{1}{J}\sum\limits_{i = 1}^J {{{\left( {RMS{E_i} - \mu } \right)}^2}} } \text{。}$ (22)
4.2 仿真分析

 ${ p} = \left[ \begin{array}{l}0.9\quad 0.05\quad 0.05\\0.2\quad 0.795 0.005\\0.2\quad 0.005 0.795\end{array} \right]\text{。}$

3 个模型的初始概率为 u 1（0）= 0.9， u 2（0）= 0.05 和 u 3（0）= 0.05。初始协方差矩阵设定为 ${ P}\left( 0 \right) = diag\left( {{{10}^{ - 6}} \times \left[ {400,400,100,100} \right]} \right)$ ，3 个模型的状态噪声方差为 100 m/s 2

 图 2 IMM-PF 跟踪结果 Fig. 2 The tracking result of IMM-PF

 图 3 IMM-AUPF 跟踪结果 Fig. 3 The tracking result of IMM-AUPF

 图 4 IMM-PF 各模型概率 Fig. 4 model probability of IMM-PF

 图 5 IMM-AUPF 各模型概率 Fig. 5 model probability of IMM-AUPF

 图 6 x 方向均方根误差对比 Fig. 6 RMSE error comparison of x direction

 图 7 y 方向均方根误差对比 Fig. 7 RMSE error comparison of y direction

5 结 语

AUPF 算法在重采样中引入均匀分布的思想，在保证粒子对真实后验概率密度逼近能力的情况下，使粒子具有比较广的分布范围，增加了粒子的多样性，改善了粒子的贫化问题，使其对状态的估计精度有了进一步的提升。机动目标的跟踪最有效的方法是交互式多模型算法，在 IMM 算法的滤波阶段采用 AUPF，即为本文算法 IMM-AUPF，将其应用于被动多传感器系统中，在不完全观测的情况下，实现对机动目标的有效跟踪。将本文方法与 IMM-PF 算法进行对比，仿真结果表明本文方法的状态估计精度优于 IMM-PF。

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