面向机动目标跟踪的多传感器长时调度策略

乔成林,段修生,单甘霖,徐公国

(陆军工程大学(石家庄校区) 电子与光学工程系,石家庄 050003)



摘要:

为解决杂波环境下机动目标跟踪以及系统辐射风险控制的问题,提出了一种面向机动目标跟踪的多传感器长时调度策略.该方法首先以交互式多模型和概率数据关联算法为基础,估计杂波环境下机动目标跟踪精度.然后以辐射度影响量化辐射代价、推导有限时域内辐射代价,以后验克拉美-罗下界衡量目标跟踪性能、预测机动目标有限时域内后验克拉美-罗下界.最后,引入传感器切换代价,考虑跟踪精度约束,建立基于代价函数和后验克拉美-罗下界的多传感器长时调度策略,并将该约束调度问题转化为决策树优化问题,采用阈值剪枝搜索技术求解最优策略.仿真结果表明:该方法验证了所提策略的有效性,与标准代价搜索相比,所提搜索算法能够以辐射风险略上升为代价,显著降低节点打开数、加快搜索空间；与随机调度、最近调度和贪婪调度相比,所提调度策略能够在满足跟踪任务需求下获得更低的辐射代价；与随机调度和贪婪调度相比,所提调度策略切换代价更低,有效克服了传感器频繁调度问题,更利于实际实现.

关键词:  传感器调度  辐射控制  切换代价  交互式多模型概率数据关联  后验克拉美-罗下界  决策树

DOI：10.11918/j.issn.0367-6234.201801029

分类号:TP212

文献标识码:A

基金项目:国防预研基金项目(012015012600A2203)



Non-myopic multi-sensor scheduling policy for maneuvering target tracking

QIAO Chenglin,DUAN Xiusheng,SHAN Ganlin,XU Gongguo

(Department of Electronic and Optical Engineering, Army Engineering University, Shijiazhuang 050003, China)

Abstract:

To control the system radiation risk and track the maneuvering target in clutter, a non-myopic multi-sensor scheduling policy for maneuvering target tracking is proposed. Firstly, the maneuvering target tracking accuracy in clutter was estimated by the interacting multiple model and probability data association (IMMPDA) algorithm. Secondly, the radiation cost was quantized by the emission level impact (ELI) and the posterior carmér-rao lower bound (PCRLB) was utilized to represent the target tracking performance. Then the radiation cost and the PCRLB over the future finite time horizon were predicted, respectively. Finally, considering the switching cost and the tracking accuracy constraint, a non-myopic multi-sensor scheduling policy with cost function and PCRLB was set up. The constrained scheduling problem was converted to a decision optimization problem which can be solved by a search algorithm with threshold pruning technique. Simulation results show the effectiveness of the proposed policy. Compared with the uniform cost search (UCS), the proposed search algorithm can reduce the number of nodes opened and improve the search speed with a slightly increased radiation cost. Compared with random scheduling, closest scheduling, and greedy scheduling, the proposed policy can obtain lower radiation cost while satisfying the target tracking requirement. Furthermore, the proposed policy also has the lowest switching cost, and the sensor scheduling frequency has been reduced effectively.

Key words:  sensor scheduling  radiation control  switching cost  interacting multiple model and probability data association  posterior carmér-rao lower bound  decision tree


乔成林, 段修生, 单甘霖, 徐公国. 面向机动目标跟踪的多传感器长时调度策略[J]. 哈尔滨工业大学学报, 2019, 51(4): 123-130. DOI: 10.11918/j.issn.0367-6234.201801029.
QIAO Chenglin, DUAN Xiusheng, SHAN Ganlin, XU Gongguo. Non-myopic multi-sensor scheduling policy for maneuvering target tracking[J]. Journal of Harbin Institute of Technology, 2019, 51(4): 123-130. DOI: 10.11918/j.issn.0367-6234.201801029.
基金项目 国防预研基金项目(012015012600A2203) 作者简介 乔成林(1990—)，男，博士研究生;
单甘霖(1962—)，男，教授，博士生导师 通信作者 单甘霖，shanganlin@163.com 文章历史 收稿日期: 2018-01-05



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面向机动目标跟踪的多传感器长时调度策略
乔成林, 段修生, 单甘霖, 徐公国    
陆军工程大学(石家庄校区) 电子与光学工程系，石家庄 050003

收稿日期: 2018-01-05
基金项目: 国防预研基金项目(012015012600A2203)
作者简介: 乔成林(1990—)，男，博士研究生;
单甘霖(1962—)，男，教授，博士生导师
通信作者: 单甘霖，shanganlin@163.com


摘要: 为解决杂波环境下机动目标跟踪以及系统辐射风险控制的问题，提出了一种面向机动目标跟踪的多传感器长时调度策略.该方法首先以交互式多模型和概率数据关联算法为基础，估计杂波环境下机动目标跟踪精度.然后以辐射度影响量化辐射代价、推导有限时域内辐射代价，以后验克拉美-罗下界衡量目标跟踪性能、预测机动目标有限时域内后验克拉美-罗下界.最后，引入传感器切换代价，考虑跟踪精度约束，建立基于代价函数和后验克拉美-罗下界的多传感器长时调度策略，并将该约束调度问题转化为决策树优化问题，采用阈值剪枝搜索技术求解最优策略.仿真结果表明：该方法验证了所提策略的有效性，与标准代价搜索相比，所提搜索算法能够以辐射风险略上升为代价，显著降低节点打开数、加快搜索空间；与随机调度、最近调度和贪婪调度相比，所提调度策略能够在满足跟踪任务需求下获得更低的辐射代价；与随机调度和贪婪调度相比，所提调度策略切换代价更低，有效克服了传感器频繁调度问题，更利于实际实现.
关键词: 传感器调度    辐射控制    切换代价    交互式多模型概率数据关联    后验克拉美-罗下界    决策树    
Non-myopic multi-sensor scheduling policy for maneuvering target tracking
QIAO Chenglin, DUAN Xiusheng, SHAN Ganlin, XU Gongguo    
Department of Electronic and Optical Engineering, Army Engineering University, Shijiazhuang 050003, China


Abstract: To control the system radiation risk and track the maneuvering target in clutter, a non-myopic multi-sensor scheduling policy for maneuvering target tracking is proposed. Firstly, the maneuvering target tracking accuracy in clutter was estimated by the interacting multiple model and probability data association (IMMPDA) algorithm. Secondly, the radiation cost was quantized by the emission level impact (ELI) and the posterior carmér-rao lower bound (PCRLB) was utilized to represent the target tracking performance. Then the radiation cost and the PCRLB over the future finite time horizon were predicted, respectively. Finally, considering the switching cost and the tracking accuracy constraint, a non-myopic multi-sensor scheduling policy with cost function and PCRLB was set up. The constrained scheduling problem was converted to a decision optimization problem which can be solved by a search algorithm with threshold pruning technique. Simulation results show the effectiveness of the proposed policy. Compared with the uniform cost search (UCS), the proposed search algorithm can reduce the number of nodes opened and improve the search speed with a slightly increased radiation cost. Compared with random scheduling, closest scheduling, and greedy scheduling, the proposed policy can obtain lower radiation cost while satisfying the target tracking requirement. Furthermore, the proposed policy also has the lowest switching cost, and the sensor scheduling frequency has been reduced effectively.
Keywords: sensor scheduling    radiation control    switching cost    interacting multiple model and probability data association    posterior carmér-rao lower bound    decision tree    
网络化战争中，传感器调度通过实时分配战场的传感器资源，使得系统整体性能达到最优，其核心在于能够依据先验知识预测目标的未来状态.近年来，后验克拉美-罗下界PCRLB(posterior carmér-rao lower bound)被应用于传感器调度中，它表征了目标状态估计的理论下界.文献[1-2]提出基于PCRLB的无线传感器管理调度方法.进一步，考虑机动目标，文献[3-4]提出适用于机动目标的传感器调度策略，并给出基于最大模型概率的PCRLB预测方法.

实际上，有源传感器(如雷达)在探测目标时需要持续的向外辐射电磁波，因此如何隐蔽自身、提高自身生存能力显得尤为重要.文献[5-6]研究采用有源传感器开机次数衡量己方传感器辐射，当目标跟踪误差不满足要求则调度有源传感器，否则，不调度.对于多有源传感器系统，文献[7]提出基于空间的辐射控制方法.实际上，不同时刻不同传感器的辐射风险是不同的.为此，文献[8-9]提出采用截获概率熵量化传感器辐射风险，然而计算截获概率需要事先知道敌方窗函数参数，实际中不易获取.借鉴截获概率思想，文献[10-11]提出了辐射度影响ELI(emission level impact)的量化方法，之后文献[12]研究了面向跟踪任务需求的主动传感器调度方法.由于该方法仅以一步预测进行决策且没有考虑切换代价，系统辐射风险依然较大且存在频繁调度的缺陷，不利于实际应用.

针对上述问题，考虑跟踪任务需求，提出一种面向机动目标跟踪的传感器长时调度策略.仿真实验表明，所提策略有效降低了系统辐射、避免了频繁调度问题.

1 机动目标跟踪算法 1.1 机动目标模型在己方三维监视空间中，考虑一个敌方机动目标.不失一般性，假设目标以近似匀速直线和近似匀转弯(包含左转弯和右转弯)运动.则目标动态方程为

$\mathit{\boldsymbol{X}}_{k + 1}^j = {\mathit{\boldsymbol{F}}^j}\mathit{\boldsymbol{X}}_k^j + \mathit{\boldsymbol{v}}_k^j.$ (1)

式中:${\mathit{\boldsymbol{X}}_k} = {[{x_k}, {\dot x_k}, {y_k}, {\dot y_k}, {z_k}, {\dot z_k}]^{\rm{T}}}$为k时刻目标位置信息和速度信息，上标T表示取逆操作；F^j(j = 1, …, r)为模型j对应的状态转移矩阵，其中r为模型个数；v_k^j为零均值白高斯过程噪声，假设其协方差矩阵已知，即

$E\left[ {v_k^j{{\left( {v{{_k^j}^\prime }} \right)}^{\rm{T}}}} \right] = \left[ {\begin{array}{*{20}{c}}{{\tau ^3}/3}&{{\tau ^2}/2}\\{{\tau ^2}/2}&\tau \end{array}} \right] \otimes {\mathit{\boldsymbol{I}}_{3 \times 3}}.$

式中:E、?分别为求期望和求Kronecker积；τ为采样间隔；I_3×3为3×3的单位矩阵.

假设系统中有M个有源传感器(如雷达)，则传感器m(m = 1, …, M)的观测方程为

$\mathit{\boldsymbol{Z}}_{{X_k}}^m = {h^m}\left( {{\mathit{\boldsymbol{X}}_k}} \right) + \mathit{\boldsymbol{w}}_k^m = {\left[ {r_k^m,\theta _k^m,\varphi _k^m} \right]^{\rm{T}}} + \mathit{\boldsymbol{w}}_k^m.$ (2)

式中：h^m为传感器m的非线性观测方程；[r_k^m, θ_k^m, φ_k^m]为对应的目标状态观测值，包含斜距离、方位角和俯仰角；w_k^m为零均值白高斯量测噪声，对应的协方差矩阵为R^m.

1.2 多传感器IMMPDA为有效跟踪杂波环境下的机动目标，采用交互多模型和概率数据关联IMMPDA(interacting multiple model and probability data association)算法^[13].同时，考虑到容积卡尔曼滤波CKF(cubature Kalman filter)在处理高维非线性系统的优越性^[14]，以CKF估计系统状态和协方差矩阵.算法具体步骤如下.

Step 1 ?初始化，计算混合概率和混合初始状态.

根据先验信息，假设k时刻模型j的模型概率为μ_k^j，对应的状态估计和协方差矩阵分别为${\mathit{\boldsymbol{\hat X}}_k}^j$和P_k^j.则运用全期望公式，由文献[4]计算混合概率μ_k^ij、混合初始状态$\mathit{\boldsymbol{\hat X}}_k^{0j}$和混合初始协方差矩阵P_k^0j.

Step 2 ?杂波环境下，模型条件滤波.

1) 状态预测和观测预测.

结合目标动态方程及各模型目标状态，计算其状态预测及对应的协方差矩阵，即:

$\mathit{\boldsymbol{\hat X}}_{k + 1\left| k \right.}^j = {\mathit{\boldsymbol{F}}^j}\mathit{\boldsymbol{\hat X}}_k^{0j},$

$\mathit{\boldsymbol{P}}_{k + 1\left| k \right.}^j = {\mathit{\boldsymbol{F}}^j}\mathit{\boldsymbol{P}}_k^{0j}{\left( {\mathit{\boldsymbol{F}}_k^j} \right)^{\rm{T}}}.$

进一步，依据传感器m的观测方程，根据文献[14]，计算容积点χ_k+1|k^{l, j}(l = 1, …, 2L)，其中L为状态维数.进一步，根据文献[14]，计算k+1时刻模型j的预测观测$\mathit{\boldsymbol{\dot Z}}_{{X_{k + 1\left| k \right.}}}^{m, j}$及相应的新息协方差S_k+1^{m, j}.

2) 观测确认.

判断观测$\mathit{\boldsymbol{Z}}_{{X_{k + 1}}}^m$是否满足下式:

$\begin{array}{l}\left( {\mathit{\boldsymbol{Z}}_{{X_{k + 1}}}^m - \mathit{\boldsymbol{\hat Z}}_{{X_{k + 1}}\left| k \right.}^m} \right){\left( {\mathit{\boldsymbol{S}}_{k + 1}^l} \right)^{ - 1}}{\left( {\mathit{\boldsymbol{Z}}_{{X_{k + 1}}}^m - \mathit{\boldsymbol{\hat Z}}_{{X_{k + 1}}\left| k \right.}^m} \right)^{\rm{T}}} < {g^2},\\\;\;\;\;\;l = \mathop {\max }\limits_j \left| {\mathit{\boldsymbol{S}}_{k + 1}^{m,j}} \right|.\end{array}$ (3)

式中:$\mathit{\boldsymbol{\dot Z}}_{{X_{k + 1\left| k \right.}}}^m$为预测观测值，可由文献[6]计算得到；l为求最大有效区域对应的模型；g为跟踪波门参数；|·|为求行列式操作.

如果观测满足式(3)，则将该观测作为候选回波；否则，舍弃该观测.

3) 根据候选回波(假设共有n_k+1^m个)，对模型j进行状态估计为

$\mathit{\boldsymbol{\hat X}}_{k + 1}^{m,j} = \mathit{\boldsymbol{\hat X}}_{k + 1\left| k \right.}^j + \mathit{\boldsymbol{K}}_{k + 1}^j\nu _{{X_{k + 1}}}^{m,j}.$

式中:K_k+1^j为滤波增益；ν_Xk+1^{m, j}为组合新息，具体为

$\nu _{{X_{k + 1}}}^{m,j} = \sum\limits_{i = 1}^{n_{k + 1}^m} {\beta _{k + 1}^{m,i}\left( {\mathit{\boldsymbol{Z}}_{{X_{k + 1}}}^{m,i} - \mathit{\boldsymbol{\hat Z}}_{{X_{k + 1}}\left| k \right.}^{m,j}} \right)} ,$

式中β_k+1^{m, i}为量测Z_Xk+1^{m, i}来自于目标的概率，对于非参数模型，应用贝叶斯公式，有

$\beta _{k + 1}^{m,i} = \left\{ \begin{array}{l}b_{k + 1}^m{\left[ {b_{k + 1}^m + \sum\limits_{l = 1}^{n_{k + 1}^m} {\beta _{k + 1}^{m,l}} } \right]^{ - 1}},\;\;\;\;i = 0;\\e_{k + 1}^{m,i}{\left[ {b_{k + 1}^m + \sum\limits_{l = 1}^{n_{k + 1}^m} {e_{k + 1}^{m,l}} } \right]^{ - 1}},\;\;\;\;\;其他.\end{array} \right.$

式中b_k+1^m、e_k+1^{m, i}(i = 1, …, n_k+1^m)可由文献[13]计算得到.

Step 3 ?更新模型概率.

依据n_k+1^m个新息的联合概率密度函数，结合文献[13]，计算似然函数为

$\Lambda _{k + 1}^{m,j} = \left[ {b_{k + 1}^m + \sum\limits_{i = 1}^{n_{k + 1}^m} {e_{k + 1}^{m,l}} } \right]\frac{{{P_D}{P_G}}}{{n_{k + 1}^m}}{\left( {{c_{{n_z}}}{g^{{n_z}}}{{\left| {\mathit{\boldsymbol{S}}_{k + 1}^{m,j}} \right|}^{1/2}}} \right)^{ - n_{k + 1}^m + 1}},$

则模型概率更新为

$\mu _{k + 1}^j = \frac{1}{c}\Lambda _{k + 1}^{m,j}\mathit{\boldsymbol{\pi }}_j^{\rm{T}}{\mathit{\boldsymbol{\mu }}_\mathit{k}}.$

式中：π_j为列向量，表示其他模型转移到模型j的概率；μ_k为模型概率；c为归一化因子.

Step 4 ?滤波综合.

计算状态估计值和协方差矩阵为：

${{\mathit{\boldsymbol{\hat X}}}_{k + 1}} = \sum\limits_{j = 1}^r {\mu _{k + 1}^j\mathit{\boldsymbol{\hat X}}_{k + 1}^{m,j}} ,$

${\mathit{\boldsymbol{P}}_{k + 1}} = \sum\limits_{j = 1}^r {\mu _{k + 1}^j\left[ \begin{array}{l}\mathit{\boldsymbol{P}}_{k + 1}^{m,j} + \left( {\mathit{\boldsymbol{\hat X}}_{k + 1}^{m,j} - {{\mathit{\boldsymbol{\hat X}}}_{k + 1}}} \right) \times \\{\left( {\mathit{\boldsymbol{\hat X}}_{k + 1}^{m,j} - {{\mathit{\boldsymbol{\hat X}}}_{k + 1}}} \right)^{\rm{T}}}\end{array} \right]} ,$

式中, P_k+1^{m, j}为$\mathit{\boldsymbol{\hat X}}_{k + 1}^{m, j}$对应的协方差矩阵.

2 基于代价函数和PCRLB的传感器长时调度策略 2.1 辐射代价以ELI量化传感器辐射，建立部分可观马尔可夫决策过程模型^[15]，其模型元素可由数组〈 a, E, T, Z_E, O, b 〉表征，具体描述如下.

1) 调度动作a.

定义k时刻所有传感器的调度动作空间为a_k = [a_k¹, …, a_k^M]^Τ，其中a_k^m = 1为k+1时刻选择传感器m跟踪目标，否则，a_k^m = 0.考虑到不同平台多传感器配准的困难性，假定每个时刻只有一个传感器跟踪目标，则存在

$\sum\limits_{m = 1}^M {a_k^m} = 1.$

2) 状态空间E及状态转移T.

${\mathit{\boldsymbol{E}}_k} = {\left[ {E_k^1, \cdots ,E_k^m} \right]^{\rm{T}}}.$

式中E_k^m为截止k时刻传感器m被敌方累积截获的辐射量^[10]，将其量化，记为{1, …, N_s}，其中N_s为最大辐射量对应的量化值.

依据ELI理论，当a_k^m = 1，即调度传感器m时，定义ELI状态转移矩阵为T_k^m = T^m = (t_{i, j})_Ns×Ns.其中，元素t_{i, j} = p(E_k+1^m = j|E_k^m = i, a_k^m)为k时刻在调度动作a_k^m下传感器m从状态i转移到状态j的条件概率.否则，T_k^m为N_s维单位矩阵I_Ns×Ns.

3) 观测值Z_E及观测矩阵O.

定义k时刻ELI状态观测值为Z_Ek，即

${\mathit{\boldsymbol{Z}}_{{E_{\rm{k}}}}} = {\left[ {Z_{{E_{\rm{k}}}}^1, \cdots ,Z_{{E_{\rm{k}}}}^m} \right]^{\rm{T}}},$

式中，Z^m_Ek(m = 1, …, M)为调度传感器m时的瞬间威胁度观测值，将其量化，记为{1, …, M_s}，其中M_s为最大瞬间威胁度观测值对应量化值.

同理，当a_k-1^m = 1，即调度传感器m时，定义不同ELI状态观测值的观测矩阵为O_k^m(l) = O^m(l) = (q_{i, j, l}^m)_Ns×Ns(l = 1, …, M_s).其中，q_{i, j, l}^m = p(Z_Ek^m = l|E_k－1^m = i, E_k^m = j, a_k－1^m)表示在调度动作a_k－1^m下ELI前后状态分别为i和j时，瞬间威胁度观测值为l的条件概率.否则，O_k^m(l)为M_s维单位矩阵I_Ms×Ms.

4) 信念状态b.

为了有效描述ELI状态的后验分布，引入信念状态b_k = [(b_k¹)^Τ, …, (b_k^M)^Τ]^Τ. b_k是依据初始状态、所有历史观测值及所有调度动作η_k = {E_{0, pE0}, Z_E1, …, Z_Ek, a₀, …, a_k－1}，利用贝叶斯准则对ELI状态的后验估计，即b_k = p(E_k|η_k).

因此，对于H时域内任意传感器调度序列A_k+H = [a_k, …, a_k+H－1]，其辐射代价为

$\Re \left( {{\mathit{\boldsymbol{A}}_{k + H}}} \right) = \sum\limits_{h = 1}^H {\sum\limits_{m = 1}^M {a_{k + h - 1}^m{\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{\tilde b}}_{k + h\left| k \right.}^m} } .$

式中: V为N_s×1维列向量，对应ELI状态量化值；$\mathit{\boldsymbol{\tilde b}}_{k + h{\rm{|}}k}^m$为已知k时刻ELI信念状态b_k^m，k+h时刻ELI信念状态估计.

则信念状态估计为

$\begin{array}{l}\mathit{\boldsymbol{\tilde b}}_{k + 1\left| k \right.}^m\left( {E_k^{m + 1}} \right) = \\p\left( {E_k^{m + 1}\left| {{\mathit{\boldsymbol{a}}_k},\mathit{\boldsymbol{b}}_k^m} \right.} \right) = \\\sum\limits_{Z_{{E_{k + 1}}}^m = 1}^{{M_{\rm{s}}}} {p\left( {E_k^{m + 1}\left| {{\mathit{\boldsymbol{a}}_k},\mathit{\boldsymbol{b}}_k^m,\mathit{\boldsymbol{Z}}_{{E_{k + 1}}}^m} \right.} \right)} \times \\p\left( {Z_{{E_{k + 1}}}^m\left| {{\mathit{\boldsymbol{a}}_k},\mathit{\boldsymbol{b}}_k^m} \right.} \right),\end{array}$ (4)

式中，Z_Ek+1^m为k+1时刻瞬间威胁度观测值.

式(4)中$p(E_k^{m + 1}\left| {{\mathit{\boldsymbol{a}}_k}, {\mathit{\boldsymbol{b}}_k}^m, \mathit{\boldsymbol{Z}}_{{E_{k + 1}}}^m} \right.)$表示已知Z_Ek+1^m时，k+1时刻ELI状态的概率分布.则由文献[10]可得

$p\left( {E_k^{m + 1}\left| {{\mathit{\boldsymbol{a}}_k},\mathit{\boldsymbol{b}}_k^m,\mathit{\boldsymbol{Z}}_{{E_{k + 1}}}^m} \right.} \right) = \frac{{\sum\limits_{i = 1}^{{N_{\rm{s}}}} {t_{i,E_k^{m + 1}}^mq_{i,E_k^{m + 1},Z_{{E_{k + 1}}}^m}^mb_k^m\left( i \right)} }}{{\sum\limits_{i = 1}^{{N_{\rm{s}}}} {\sum\limits_{k = 1}^{{N_{\rm{s}}}} {t_{i,w}^mq_{i,w,Z_{{E_{k + 1}}}^m}^mb_k^m\left( i \right)} } }}.$ (5)

进一步，$p(Z_{{E_{k + 1}}}^m\left| {{\mathit{\boldsymbol{a}}_k}, {\mathit{\boldsymbol{b}}_k}^m} \right.)$表示已知b_k^m时，瞬间威胁度观测值的概率分布.则依据贝叶斯准则，可得

$\begin{array}{l}p\left( {Z_{{E_{k + 1}}}^m\left| {{\mathit{\boldsymbol{a}}_k},\mathit{\boldsymbol{b}}_k^m} \right.} \right) = \\\sum\limits_{E_k^{m + 1}} {p\left( {{Z_{{E_{k + 1}}}},E_{k + 1}^m\left| {{\mathit{\boldsymbol{a}}_k},\mathit{\boldsymbol{b}}_k^m} \right.} \right)} = \\\sum\limits_{E_k^{m + 1}} {p\left( {E_{k + 1}^m\left| {{\mathit{\boldsymbol{a}}_k},\mathit{\boldsymbol{b}}_k^m} \right.} \right)p\left( {{Z_{{E_{k + 1}}}}\left| {E_{k + 1}^m,{\mathit{\boldsymbol{a}}_k},\mathit{\boldsymbol{b}}_k^m} \right.} \right)} = \\\sum\limits_{E_k^{m + 1}} {\sum\limits_{E_k^m} {\left\{ \begin{array}{l}p\left( {E_{k + 1}^m\left| {E_k^m,{\mathit{\boldsymbol{a}}_k}} \right.} \right)\\\;\;\;\;\;\;\;p\left( {Z_{{E_{k + 1}}}^m\left| {E_{k + 1}^m,E_k^m,{\mathit{\boldsymbol{a}}_k}} \right.} \right)b_k^m\left( {E_k^m} \right)\end{array} \right\}} } = \\\sum\limits_{i = 1}^{{N_{\rm{s}}}} {\sum\limits_{w = 1}^{{N_{\rm{s}}}} {t_{i,w}^mq_{i,w,Z_{{E_{k + 1}}}^m}^m\mathit{\boldsymbol{b}}_k^m\left( i \right)} } .\end{array}$ (6)

将式(5)、(6)代入式(4)，并写为矩阵形式，可得

$\begin{array}{l}\mathit{\boldsymbol{\tilde b}}_{k + 1\left| k \right.}^m = \sum\limits_{Z_{{E_{k + 1}}}^m} {\left\{ {{{\left[ {{\mathit{\boldsymbol{O}}^m}\left( {Z_{{E_{k + 1}}}^m} \right)} \right]}^{\rm{T}}} \odot {{\left( {{\mathit{\boldsymbol{T}}^m}} \right)}^{\rm{T}}}\mathit{\boldsymbol{b}}_k^m} \right\}} = \\\;\;\;\;\;\;\;\;\;\;{\left\{ {\left[ {\sum\limits_{Z_{{E_{k + 1}}}^m = 1}^{{M_{\rm{s}}}} {{\mathit{\boldsymbol{O}}^m}\left( {Z_{{E_{k + 1}}}^m} \right)} } \right] \odot {\mathit{\boldsymbol{T}}^m}} \right\}^{\rm{T}}}\mathit{\boldsymbol{b}}_k^m = {\left( {{T^m}} \right)^{\rm{T}}}\mathit{\boldsymbol{b}}_k^m,\end{array}$

式中，⊙为Hadamard积运算.

进一步，可得

$\tilde b_{k + h\left| k \right.}^m = {\left[ {{{\left( {{\mathit{\boldsymbol{T}}^m}} \right)}^{\sum\limits_{c = k}^{k + h - 1} {a_c^m} }}} \right]^{\rm{T}}}b_k^m,$

因此，H时域内辐射代价为

$\begin{array}{l}\Re \left( {{\mathit{\boldsymbol{A}}_{k + H}}} \right) = \sum\limits_{h = 1}^H {\sum\limits_{m = 1}^M {a_{k + h - 1}^mE\left( {\mathit{\boldsymbol{\tilde b}}_{k + h\left| k \right.}^m} \right)} } = \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{h = 1}^H {\sum\limits_{m = 1}^M {a_{k + h - 1}^m{\mathit{\boldsymbol{V}}^{\rm{T}}}{{\left[ {{{\left( {{\mathit{\boldsymbol{T}}^m}} \right)}^{\sum\limits_{c = k}^{k + h - 1} {a_c^m} }}} \right]}^{\rm{T}}}\mathit{\boldsymbol{b}}_k^m} } .\end{array}$ (7)

2.2 基于PCRLB的跟踪精度预测传感器调度的核心在于能够依据先验知识预测目标的未来状态.理想条件下，文献[9,11]依据目标运动方程和量测方程预测了匀速运动目标未来一步^[9]和多步^[11]的协方差矩阵，并给出了基于协方差控制的传感器调度方法.显然，该方法不适用于本文场景.考虑PCRLB能够根据当前先验信息预测未来时刻目标状态估计的下界，为此给出一种基于PCRLB的跟踪精度预测方法.

对于任意时刻的目标状态估计，存在：

${\mathit{\boldsymbol{P}}_k} = {\rm{E}}\left( {{{\mathit{\boldsymbol{\hat X}}}_k} - {\mathit{\boldsymbol{X}}_k}} \right){\left( {{{\mathit{\boldsymbol{\hat X}}}_k} - {\mathit{\boldsymbol{X}}_k}} \right)^{\rm{T}}} \ge \mathit{\boldsymbol{J}}_k^{ - 1}.$

式中：${\mathit{\boldsymbol{J}}_k} = {\rm{E}}\{-\Delta _{{X_k}}^{{X_k}}{\rm{log}}\; p({\mathit{\boldsymbol{X}}_k}, {\mathit{\boldsymbol{Z}}_k})\} $为Fisher信息矩阵；Δ_Xk^X_k为求二阶偏导；J_k^－1为克拉美-罗下界.

目标状态转移先验概率密度函数^[4]为

$\begin{array}{l}p\left( {{\mathit{\boldsymbol{X}}_{k + 1}}\left| {{\mathit{\boldsymbol{X}}_k}} \right.} \right) = \frac{1}{{\sqrt {{{\left( {2{\rm{ \mathit{ π} }}} \right)}^6}\left| {{\mathit{\boldsymbol{Q}}^{{j_{k + 1}}}}} \right|} }} \times \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\exp \left\{ { - \frac{1}{2}\left( {{\mathit{\boldsymbol{X}}_{k + 1}} - {\mathit{\boldsymbol{F}}^{{j_{k + 1}}}}{\mathit{\boldsymbol{X}}_k}} \right){{\left( {{\mathit{\boldsymbol{Q}}^{{j_{k + 1}}}}} \right)}^{ - 1}} \cdot } \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {{{\left( {{\mathit{\boldsymbol{X}}_{k + 1}} - {\mathit{\boldsymbol{F}}^{{j_{k + 1}}}}{\mathit{\boldsymbol{X}}_k}} \right)}^{\rm{T}}}} \right\}.\end{array}$ (8)

进一步，结合式(8)和式(3)，Fisher信息矩阵递推如下

${\mathit{\boldsymbol{J}}_{k + 1}} = \mathit{\boldsymbol{D}}_k^{22} - \mathit{\boldsymbol{D}}_k^{21}{\left( {{\mathit{\boldsymbol{J}}_k} + \mathit{\boldsymbol{D}}_k^{11}} \right)^{ - 1}}\mathit{\boldsymbol{D}}_k^{12} + \sum\limits_{m = 1}^M {a_k^m\mathit{\boldsymbol{J}}_{k + 1}^m} ,$ (9)

其中：

$\mathit{\boldsymbol{D}}_k^{11} = {\left( {{\mathit{\boldsymbol{F}}^{{j_{k + 1}}}}} \right)^{\rm{T}}}{\left( {{\mathit{\boldsymbol{Q}}^j}} \right)^{ - 1}}{\mathit{\boldsymbol{F}}^{{j_{k + 1}}}},\mathit{\boldsymbol{D}}_k^{12} = - {\left( {{\mathit{\boldsymbol{F}}^{{j_{k + 1}}}}} \right)^{\rm{T}}}{\left( {{\mathit{\boldsymbol{Q}}^j}} \right)^{ - 1}},$

$\mathit{\boldsymbol{D}}_k^{21} = {\left( {\mathit{\boldsymbol{D}}_k^{12}} \right)^{{\rm{T}}}},\mathit{\boldsymbol{D}}_k^{22} = {\left( {{\mathit{\boldsymbol{Q}}^{{j_{k + 1}}}}} \right)^{ - 1}},$

$\mathit{\boldsymbol{J}}_{k + 1}^m = {\left( {\mathit{\boldsymbol{H}}_{k + 1}^m} \right)^{\rm{T}}}{\left( {{\mathit{\boldsymbol{R}}^m}} \right)^{ - 1}}\mathit{\boldsymbol{H}}_{k + 1}^m.$

式中：J_k+1^m为从传感器m获得的信息增益；H为雅可比矩阵；j_k+1为k+1时刻目标的运动模型，在k时刻是未知的.

为了计算J_k+1，需要获知k+1时刻目标运动模型.依据当前先验信息，文献[3-4]提出以当前最大模型概率对应的模型作为未来时刻的运动模型，即

${{\tilde j}_{k + 1}} = \arg \mathop {\max }\limits_{j \in \left\{ {1, \cdots ,r} \right\}} \mu _k^j.$

因此，算法具体流程见图 1.

Fig. 1
图 1 基于PCRLB的跟踪精度预测流程 Fig. 1 Flow of tracking accuracy prediction by PCRLB


2.3 传感器长时调度策略考虑传感器辐射代价和切换代价，则基于代价函数和PCRLB的传感器长时调度问题可描述为:

$\begin{array}{l}\mathit{\boldsymbol{A}}_{k + H}^{{\rm{opt}}} = \arg \mathop {\max }\limits_{{A_k} + H} \left[ \begin{array}{l}\sum\limits_{h = 1}^H {\sum\limits_{m = 1}^M {a_{k + h - 1}^m{\mathit{\boldsymbol{V}}^{\rm{T}}}{{\left[ {{{\left( {{T^m}} \right)}^{\sum\limits_{c = k}^{k + h - 1} {a_c^m} }}} \right]}^{\rm{T}}}\mathit{\boldsymbol{b}}_k^m} } + \\\sum\limits_{h = 1}^H {\sum\limits_{m = 1}^M {{c^m}{{\left( {a_{k + h - 1}^m - a_{k + h - 2}^m} \right)}^2}} } \end{array} \right] = \\\;\;\;\;\;\;\;\;\;\;\arg \mathop {\max }\limits_{{A_k} + H} \left[ {\sum\limits_{h = 1}^H {{\mathit{\Gamma }_{k + h}}} + \sum\limits_{h = 1}^H {{\Upsilon _{k + h}}} } \right],\\\;\;\;\;\;\;\;\;\;\;{\rm{s}}.{\rm{t}}\;\;\;\left\{ \begin{array}{l}\sum\limits_{m = 1}^M {a_{k + h - 1}^m} = 1,\\{{\tilde \rho }_{k + h}} \le {\rho _{\rm{d}}}.\;\;\;h = 1, \cdots ,H\end{array} \right.\end{array}$ (10)

式中:c^m为传感器m的切换代价(也称为启动/停止代价)^[16]；Γ、Υ分别为一步辐射代价和切换代价；ρ_d为任务需求对应的跟踪精度；

对于式(10)，共有M^H种传感器组合，且随着时域长度呈指数爆炸增长.为了求解最优传感器调度序列，将该问题建立为约束条件下决策树寻优问题.依据决策树原理，决策树深度和分支因子分别对应于调度序列长度H和传感器数目M，图 2为H = 3，M = 4的决策树.考虑到式(10)中的约束，不同深度的分支数是不同的.对于满足约束的分支，其权重即为式(10)中的代价函数.因此，求解最优传感器调度序列即为搜索具有最优代价函数的节点路径.

Fig. 2
图 2 决策树(H = 3，M = 4) Fig. 2 Decision tree


引入阈值剪支技术，给出基于阈值ε剪支的标准代价搜索UCS(uniform cost search)算法求解最优传感器调度序列，其步骤见图 3.

Fig. 3
图 3 基于阈值剪支的UCS算法 Fig. 3 UCS algorithm based on threshold pruning


2.4 长时调度策略流程长时调度策略流程如图 4所示，具体步骤如下.

Fig. 4
图 4 长时调度策略流程图 Fig. 4 Flow chart of non-myopic scheduling policy


Step 1 ?初始化，获得k时刻目标状态估计(${\mathit{\boldsymbol{\hat X}}_k}$, P_k)及传感器辐射信念状态b_k.

Step 2 ?建立H时长决策树，由图 1预测每个节点的跟踪精度，式(10)预测传感器辐射代价和切换代价，并依据图 3搜索最优调度序列.

Step 3 ?从最优调度序列中选择第1个传感器，用于下一个时刻观测目标.

Step 4? k = k+1，若任务没有结束，则依据多传感器IMMPDA算法更新目标状态、结合威胁度观测值更新传感器辐射状态；否则，转到Step 5.

Step 5?任务结束.

2.5 复杂度分析本文长时调度策略的复杂度主要包括：目标状态及传感器辐射状态的更新、最优调度序列的计算.其中，传感器辐射状态更新只是简单的矩阵运算^[12]，而目标状态更新为滤波过程，其复杂度可参考文献[17].与其他调度策略相比，长时调度策略的复杂度主要体现在最优调度序列的计算.

由式(7)和图 1可知，每个节点的复杂度是相同的，则计算最优调度序列的复杂度取决于节点打开数.不失一般性，假设每个节点的复杂度为单位1，则对于给定的M和H，理论上需打开M^H个节点，即其复杂度为O(M^H).实际中，UCS以代价为顺序进行搜索，其复杂度要小于O(M^H).进一步，引入阈值剪支技术，UCS依据阈值ε可以逐步的缩紧下界，从而降低算法复杂度.结合仿真实验，选择合适的阈值ε可以有效的减少节点打开数，并保证搜索正确率.以H = 3、ρ_d = 70 m为例，UCS(即ε = 0)节点打开率为68%，而本文方法取ε = 0.3，在保证传感器正确率不低于98%时，其节点打开率仅为31%.

3 实验及结果分析 3.1 仿真参数设置考虑M = 4个传感器在杂波环境下跟踪单个机动目标.目标的初始位置和速度分别为(15, 4, 5)km和(－280, －260, 0)m/s.采样间隔为τ = 1 s，仿真时长为100τ.不失一般性，假设目标运动模型为r = 3，即匀速直线运动、匀速左转弯和匀速右转弯.进一步，在26τ~50τ目标以角速度ω = 5°向右转弯，在51τ~74τ以角速度ω = 5°向左转弯，其余时间做直线运动.其中，模型切换概率为π_ij = 0.025(i≠j).假定杂波服从泊松分布，虚假量测密度为3×10^－9/(m·mrad²)，检测为1，波门参数为4，门概率为0.999 7.

传感器位置分别为(0, －5)、(－5, 0)、(5, 0)、(0, 5) km，其斜距离标准差分别为200、100、100、10 m，方位角标准差分别为0.010、0.005、0.005、0.001 rad，对应的俯仰角标准差与方位角一致.将传感器ELI状态量化为{1, 2, 3}(其中：1、2、3分别为低辐射、中辐射、高辐射)；瞬间辐射的观测值量化为{1, 2, 3}(其中：1、2、3分别为小增量、中增量、高增量)^[12].不失一般性，假定传感器1更易处于低ELI状态、传感器4更易处于高ELI状态.则ELI状态转移矩阵设为：

${\mathit{\boldsymbol{T}}^1} = \left[ {\begin{array}{*{20}{c}}{0.8}&{0.1}&{0.1}\\{0.5}&{0.3}&{0.2}\\{0.4}&{0.3}&{0.3}\end{array}} \right],{\mathit{\boldsymbol{T}}^2} = \left[ {\begin{array}{*{20}{c}}{0.6}&{0.2}&{0.2}\\{0.2}&{0.5}&{0.3}\\{0.1}&{0.3}&{0.6}\end{array}} \right],$

${\mathit{\boldsymbol{T}}^3} = \left[ {\begin{array}{*{20}{c}}{0.6}&{0.2}&{0.2}\\{0.2}&{0.5}&{0.3}\\{0.1}&{0.3}&{0.6}\end{array}} \right],{\mathit{\boldsymbol{T}}^4} = \left[ {\begin{array}{*{20}{c}}{0.5}&{0.3}&{0.2}\\{0.2}&{0.5}&{0.3}\\{0.1}&{0.2}&{0.7}\end{array}} \right].$

此外，取切换代价为0.25，所有仿真结果均为200次蒙特卡洛仿真取平均.

3.2 结果分析图 5为不同精度和决策时长下，系统累积总代价.由图 5可知，随着跟踪精度ρ_d降低，其跟踪需求逐渐升高，为了满足该需求，系统会频繁调度性能好、辐射风险高的传感器(如传感器4)，因此其累积辐射总代价也逐渐升高.另一方面，对于给定跟踪精度ρ_d约束，随着决策时长增大，系统累积总代价总体下降，即能够获得更好地调度序列使得辐射代价和切换代价降低.进一步，以H = 3为例，图 6为不同精度下，目标均方根误差(root mean square error, RMSE).由图 6可知，本文滤波方法能够较好的估计杂波环境下机动目标状态.在不同跟踪精度ρ_d约束下，本文方法能够自适应的调度传感器满足精度约束.同时，由于目标机动，在模型切换过程中，目标RMSE可能会高于精度约束，符合实际情形.因此，不失一般性，考虑到计算复杂度，之后的实验中取H = 3、ρ_d = 70 m.

Fig. 5
图 5 不同精度和决策时长下，累积总代价 Fig. 5 Cumulative total cost of different accuracies and decision steps


图 7为不同阈值ε下，传感器选择正确率.由图 7可知，随着阈值ε增大，传感器选择正确率逐渐下降，即系统有时不能获得最优的调度序列.图 8为对应的平均节点打开数，随着阈值ε增大，平均节点打开数逐渐减少，即加快了算法搜索速度.进一步，图 9为对应的累积总代价，随着阈值ε增大，其累积总代价逐渐提高，即系统的辐射代价和切换代价更大.综合上述因素，取阈值ε = 0.3较为适宜.图 10为ε = 0和ε = 0.3时，节点打开百分比对比.由图 10可知，在整个调度过程中，取阈值ε = 0.3能够显著地降低节点打开百分比，提高调度效率.

Fig. 6
图 6 不同精度下，目标均方根误差 Fig. 6 RMSE of different accuracies


Fig. 7
图 7 不同阈值ε下，传感器选择正确率 Fig. 7 Accuracy rate versus threshold ε


Fig. 8
图 8 不同阈值ε下，平均节点打开数 Fig. 8 Average number of nodes opened versus threshold ε


Fig. 9
图 9 不同阈值ε下，累积总代价 Fig. 9 Cumulative total cost versus threshold ε


Fig. 10
图 10 节点打开百分比对比 Fig. 10 Comparison of the percentage of nodes opened


为了验证本文方法的有效性，选取随机调度^[15]、最近调度^[16]和贪婪调度(或短时调度)^[12]3种调度策略进行对比.表 1为不同策略下，调度效果对比.图 11为不同调度策略下，目标RMSE.图 12为本文方法下，目标航迹、航迹估计及其传感器选择.为了显示清楚，将其投影到x－y平面，其中黑色实线为目标真实航迹、红色实点为航迹估计.由表 1可知，随机调度和最近调度由于不考虑传感器辐射风险和切换代价，其累积总代价较大.同时，这两种调度策略也不考虑跟踪精度ρ_d约束，其RMSE往往不能满足要求，由图 11可知，初始阶段随机调度(10 s内)和最近调度(15 s)的RMSE均高于ρ_d.相比而言，结合图 6，贪婪调度和本文方法能够依据ρ_d变化自适应的调度传感器以满足精度约束(由于目标机动，在模型切换过程中，可能不满足精度约束，符合实际情形).此外，贪婪调度和本文方法考虑传感器辐射风险和切换代价，其对应的累积总代价能够显著下降.进一步，由于本文方法以多步时长进行决策，相比于贪婪调度，其能够获得更优的调度序列.结合图 12和表 1，本文方法的累积切换次数更少，更易实现.同时，观察表 1，不同调度策略的累积辐射代价和累积ELI值近似相等，这也验证了目标优化函数的合理性.

表 1
表 1 不同策略下，调度效果对比 Tab. 1 Comparison of scheduling performance of different policies 不同调试策略 累积总代价 累积辐射代价 累积ELI值 累积切换代价 累积切换次数

随机调度 231.43 194.39 194.25 37.04 74.07

最近调度 207.08 205.08 205.00 2.00 4.00

贪婪调度 197.25 179.05 178.82 18.21 36.41

本文方法 190.83 174.99 174.73 15.84 31.67



表 1 不同策略下，调度效果对比 Tab. 1 Comparison of scheduling performance of different policies


Fig. 11
图 11 不同策略下，目标均方根误差 Fig. 11 RMSE of different scheduling policies


Fig. 12
图 12 本文方法下，目标航迹、航迹估计及其传感器选择 Fig. 12 Target trajectory, estimated trajectory, and sensor selection of the proposed method


4 结论1) 以ELI量化传感器辐射风险，设计辐射代价函数，该函数能够有效衡量传感器真实ELI状态；以PCRLB预测目标跟踪精度，该方法能够有效控制跟踪误差、满足跟踪任务需求.

2) 提出基于阈值ε剪支的UCS快速搜索算法，与UCS搜索算法相比，该算法能够有效降低节点打开数、提高搜索实时性.

3) 与已有调度策略相比，本文方法能够在满足跟踪任务需求下，获得更优的调度序列，系统辐射代价更低.

4) 引入传感器切换代价，有效权衡了系统辐射代价和切换次数，避免了频繁调度问题.


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