自动化学报  2018, Vol. 44 Issue (6): 1107-1114   PDF    
带未知通信干扰和丢包补偿的多传感器网络化不确定系统的分布式融合滤波
祁波, 孙书利     
黑龙江大学电子工程学院 哈尔滨 150080
摘要: 研究了带有未知通信干扰、观测丢失和乘性噪声不确定性的多传感器网络化系统的状态估计问题.通过白色乘性噪声描述系统状态和观测中的随机不确定性,采用一组服从Bernoulli分布的随机变量描述网络传输过程中存在的观测丢失现象,且数据传输中存在未知的网络通信干扰.当发生丢包时,以当前丢失观测的预报值进行补偿.对每个单传感器子系统,应用线性无偏最小方差估计准则设计了不依赖于未知通信干扰的最优线性滤波器.推导了任两个局部滤波误差之间的互协方差阵.进而,应用矩阵加权融合估计算法给出了分布式融合状态滤波器.仿真例子验证了算法的有效性.
关键词: 未知通信干扰     丢包补偿     乘性噪声     分布式融合滤波     多传感器网络化系统    
Distributed Fusion Filtering for Multi-sensor Networked Uncertain Systems With Unknown Communication Disturbances and Compensations of Packet Dropouts
QI Bo, SUN Shu-Li     
School of Electronics Engineering, Heilongjiang University, Harbin 150080
Manuscript received : September 13, 2016, accepted: February 6, 2017.
Foundation Item: Supported by National Natural Science Foundation of China (61174139, 61573132), Outstanding Youth Fund in Heilongjiang Province (JC201412), and Postgraduate Innovation Project of Heilongjiang Province (YJSCX2016-068HLJU)
Author brief: QI Bo  Master student at the School of Electronic Engineering, Heilongjiang University. His main research interest is state estimation
Corresponding author. SUN Shu-Li   Professor at the School of Electronic Engineering, Heilongjiang University. His research interest covers the networked systems filtering and multi-sensor information fusion. Corresponding author of this paper
Recommended by Associate Editor GAO Hui-Jun
Abstract: This paper is concerned with the state estimation problem for multi-sensor networked systems with unknown communication disturbances, measurement losses and multiplicative noise uncertainties. The random uncertainties of the state and measurements of systems are described by white multiplicative noises. The phenomena of measurement losses during data transmissions through networks are described by a group of Bernoulli distributed random variables. Unknown communication disturbances exist in data transmissions. The predictors of the lost measurements are used as the compensation in the presence of packet losses. By applying the linear unbiased minimum variance estimation criterion, an optimal linear filter independent of unknown communication disturbances is designed for every single sensor subsystem. Filtering error cross-covariance matrices between any two local filters are derived. Further, a distributed fusion state filter is presented by using the matrix-weighted fusion estimation algorithm. A simulation example is given to verify the effectiveness of the proposed algorithms.
Key words: Unknown communication disturbance     packet loss compensation     multiplicative noise     distributed fusion filter     multi-sensor networked system    

近年来, 网络化控制系统的广泛应用使得有关网络化系统的控制与估计问题成为广大学者研究的热点[1-4].与传统的点对点控制系统相比较, 网络化控制系统具有信息交互速度快、控制范围广等优点.然而, 网络化系统也面临着在数据网络传输过程中的数据包丢失、随机滞后和未知干扰输入等问题.这些随机不确定性因素极大地影响了系统的性能, 甚至破坏系统稳定性.因此对带有未知干扰、丢失观测和乘性噪声不确定性的网络化系统进行滤波器的设计具有重要的实际意义.

目前, 针对网络化控制系统中涉及的丢包、滞后、未知输入、乘性噪声不确定性问题已有许多研究[5-15], 但综合考虑这些问题的研究文献还鲜见.文献[5-6]研究了带未知输入系统的观测器设计问题.文献[7]给出了线性离散随机系统未知输入和状态的统一形式滤波器.文献[8-10]研究了带有传输滞后、丢包或乘性噪声网络化系统的最优滤波问题.文献[11]对具有多数据包丢失线性离散随机系统设计了故障检测滤波器.然而, 文献[8-11]没有考虑多传感器融合估计问题.考虑到多传感器系统, 文献[12-15]对带有丢包和滞后的网络化多传感器系统研究了融合估计问题.然而, 文献[8-15]在数据包丢失时, 均采用前一时刻的观测近似代替丢失观测, 是一种简单的补偿.文献[16]对带有未知观测干扰和观测丢失的随机不确定多传感器系统给出了线性无偏最小方差最优融合预报器.然而, 对丢失观测没有补偿.文献[17]采用丢失观测的预报器作为补偿设计了稳态滤波器.采用相同的补偿方法, 文献[18]对带有未知通信干扰和丢包多传感器系统设计了融合预报器.由于使用了当前时刻之前的所有观测信息, 所以带预报补偿的估计比没有补偿和利用前一时刻观测补偿的估计具有更高精度.

由于模型误差、传感器老化、外部干扰和网络通信不完全可靠等问题, 网络控制系统中未知通信干扰、丢包和乘性噪声不确定性现象不可避免地存在.本文针对带未知通信干扰、观测丢失和状态与观测中均有乘性噪声不确定性的网络化多传感器系统, 采用文献[17]的方法以丢失观测的一步预报估值作为丢包补偿, 应用线性无偏最小方差估计准则[19], 设计了基于单传感器子系统的递推状态滤波器和基于多传感器系统的分布式融合滤波器.推导了任意两传感器子系统局部滤波器之间的滤波误差互协方差阵.最后, 应用矩阵加权融合估计算法给出了分布式融合滤波器.

1 问题阐述

考虑带未知通信干扰、观测丢失和乘性噪声不确定的多传感器离散随机系统(图 1):

$ \begin{equation} {\pmb x}(t + 1) = [{\Phi_0}(t) + \xi (t){\Phi_1}(t)]{\pmb x}(t) + \Gamma (t){\pmb w}(t) \end{equation} $ (1)
$ \begin{align} {\pmb y_i}(t) = &[{H_{0i}}(t) + {\lambda _i}(t){H_{1i}}(t)]{\pmb x}(t)+ {\pmb v_i}(t), \\ & i = 1, 2, \cdots , L \end{align} $ (2)
$ \begin{equation} {\pmb z_i}(t) = {\gamma _i}(t){\pmb y_i}(t) + {D_i}(t){\pmb \theta _i}(t), i = 1, 2, \cdots , L \end{equation} $ (3)
图 1 分布式融合估计框图 Figure 1 Block diagram of distributed fusion estimation

其中, ${\pmb x}(t) \in {{\bf R}^n}$是系统的状态向量, ${\pmb y_i}(t) \in {{\bf R}^{{m_i}}}, i = 1, 2, \cdots , L$为传感器端观测输出, 它将经由网络传输给局部处理器(局部滤波器), ${\pmb z_i}(t) \in {{\bf R}^{{m_i}}}$是局部滤波器端收到的观测, 系统噪声${\pmb w}(t) \in {{\bf R}^r}$和观测噪声${\pmb v_i}(t) \in {{\bf R}^{{m_i}}}$是零均值、方差分别为${Q_{\pmb w}}(t)$${Q_{{\pmb v}_i}}(t)$的不相关白噪声, ${\pmb \theta _i}(t) \in {{\bf R}^{{p_i}}}$为未知的通信干扰. $\xi (t)$${\lambda _i}(t), i = 1, 2, \cdots , L$是互不相关且均与其他变量不相关的零均值、方差分别为${Q_\xi }(t)$${Q_{{\lambda _i}}}(t)$的标量白噪声. $\{ {\gamma _i}(t)\} $是Bernoulli分布的随机变量序列, 其概率分布为${\rm{Prob}}\left\{ {{\gamma _i}(t) = 1} \right\} = {\alpha _i}$, ${\rm{Prob}}\left\{ {{\gamma _i}(t) = 0} \right\} = 1 - {\alpha _i}$, $0 < {\alpha _i} \le 1$, 且不相关于其他变量. ${\Phi _0}(t)$, ${\Phi _1}(t)$, $\Gamma (t)$, ${H_{0i}}(t)$, ${H_{1i}}(t)$, ${D_i}(t)$分别为适当维数的矩阵, 下标$i$表示第$i$个传感器, $L$表示传感器的个数.

模型(1) $\sim$ (3)描述了网络化系统中存在的未知通信干扰、乘性噪声不确定性和可能的观测丢失现象.当${\gamma _i}(t) =1$时, 观测数据没有丢失, 传感器观测经由网络按时到达局部滤波器端; 当${\gamma_i}(t) =0$时, 传感器观测数据丢失.为了改善局部滤波器估计精度, 我们采用丢失观测的预报值作为补偿, 此时, 用于局部滤波器设计的观测数据满足如下方程:

$ \begin{equation} {{\bar {\pmb z}}_i}(t) = {\gamma _i}(t){\pmb y_i}(t) + {D_i}(t){\pmb \theta _i}(t) + (1-{\gamma _i}(t)){{\hat {\pmb y}}_i}(t|t-1) \end{equation} $ (4)

其中, 丢失观测的预报器${{\hat {\pmb y}}_i}(t|t-1) = {H_{0i}}(t){{\hat {\pmb x}}_i}(t|t-1)$, ${{\hat {\pmb x}}_i}(t|t - 1) = {\Phi _0}(t-1){{\hat {\pmb x}}_i}(t-1)$为状态预报值, 式中${{\hat {\pmb x}}_i}(t-1)$$t-1$时刻状态的滤波估值.

假设1. 初始状态${\pmb x}(0)$${\pmb w}(t)$, ${\pmb v_i}(t)$均不相关, 且满足:

$ \begin{equation} {\rm {E\{ }}{\pmb x}(0){\rm{\} }} = {\pmb \mu _0}, {\rm{E\{ [}}{\pmb x}(0) - {\pmb \mu _0}]{[{\pmb x}(0) - {\pmb \mu _0}]^{\rm{T}}}{\rm{\} }} = {P_0} \end{equation} $ (5)

其中, ${\rm E}$为期望, ${\rm T}$为转置号.

假设2  ${\rm{rank}}(D_i(t))=p_i$, $m_i>p_i$, $i = 1, 2, \cdots, L$. ${\rm rank}( * )$表示矩阵$*$的秩.

问题是基于补偿后的观测$({{\bar {\pmb z}}_i}(t), {{\bar {\pmb z}}_i}(t - 1), \cdots, {{\bar {\pmb z}}_i}(1)), i = 1, 2, \cdots, L$, 利用线性无偏最小方差估计准则[19]设计局部滤波器, 进而基于局部估计和按矩阵加权融合估计算法[20], 设计分布式融合递推状态滤波器.

2 分布式融合滤波器

分布式融合滤波由于具有并行结构, 使其具有容错性好、可靠性高且易于故障诊断等优点.我们首先, 给出基于单传感器的线性无偏最小方差估计; 然后, 推导任两个局部估计误差间的互协方差阵; 最后, 应用按矩阵加权融合算法[20]给出分布式融合滤波器.

2.1 局部单传感器的滤波器

对系统(1) $\sim$ (4), 我们设计具有如下Kalman形式的局部递推状态滤波器

$ \begin{equation} {{\hat {\pmb x}}_i}(t + 1) = {F_i}(t){{\hat {\pmb x}}_i}(t) + {L_i}(t + 1){{\bar {\pmb z}}_i}(t + 1) \end{equation} $ (6)

其中增益矩阵${F_i}(t)$${L_i}(t+1)$由如下定理1计算.

定理1.   在假设1和2下, 多传感器系统(1) $\sim$ (4)中局部单传感器子系统的递推状态滤波器(6)的增益阵${F_i}(t)$${L_i}(t+1)$可计算如下:

$ \begin{equation} {F_i}(t) = {\Phi _0}(t) - {L_i}(t + 1){H_{0i}}(t + 1){\Phi _0}(t) \end{equation} $ (7)
$ \begin{equation} {L_i}(t + 1) = [G_i^{\rm{T}}(t + 1) - {\Lambda _i}(t + 1)D_i^{\rm{T}}(t + 1)]C_i^{ - 1}(t + 1) \end{equation} $ (8)

其中

$ \begin{align} {G_i}&(t + 1)= {\alpha _i}{H_{0i}}(t + 1)[{\Phi _0}(t){P_i}(t)\Phi _0^{\rm{T}}(t) + \\ & {Q_\xi }(t){\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t) + \Gamma (t){Q _{\pmb w}}(t){\Gamma ^{\rm{T}}}(t)] \end{align} $ (9)
$ \begin{align} {\Lambda _i}&(t + 1)= G_i^{\rm{T}}(t + 1)C_i^{ - 1}(t + 1){D_i}(t + 1)\times \\ &{[D_i^{\rm{T}}(t + 1)C_i^{ - 1}(t + 1){D_i}(t + 1)]^{ - 1}} \end{align} $ (10)
$ \begin{align} {C_i}&(t + 1) = {\alpha _i}\{ {Q_{{\lambda _i}}}(t + 1){H_{1i}}(t + 1)X(t + 1)\times \\ & H_{1i}^{\rm{T}}(t + 1)+{H_{0i}}(t + 1)[{\Phi _0}(t){P_i}(t)\Phi _0^{\rm{T}}(t) + \\ & {Q_\xi }(t){\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t)+\Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t)]\times \\ & H_{0i}^{\rm{T}}(t + 1) + {Q_{{\pmb v}_i}}(t + 1)\} \end{align} $ (11)

状态二阶矩$X(t) = {\rm{E}}[{\pmb x}(t){{\pmb x}^{\rm{T}}}(t)]$计算如下:

$ \begin{align} X&(t + 1) = {\Phi _0}(t)X(t)\Phi _0^{\rm{T}}(t) + {Q_\xi }(t)\times \\ & {\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t) + \Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t) \end{align} $ (12)

初值$X(0) = {P_0} + {\pmb \mu _0} {{\pmb \mu}^{\rm{T}} _0}$.状态滤波误差方差计算为

$ \begin{align} {P_i}&(t + 1)= {\Phi _0}(t){P_i}(t)\Phi _0^{\rm{T}}(t) + {Q_\xi }(t){\Phi _1}(t)X(t)\times \\ & \Phi _1^{\rm{T}}(t) + \Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t) + {L_i}(t + 1){C_i}(t + 1)\times \\ & L_i^{\rm{T}}(t + 1)-{L_i}(t + 1){G_i}(t + 1) - \\ & G_i^{\rm{T}}(t + 1)L_i^{\rm{T}}(t + 1) \end{align} $ (13)

初值${{\hat {\pmb x}}_i}(0) = {\pmb \mu _0}$${P_i}(0) = {P_0}$.

证明.由式(6), 多传感器系统(1) $\sim$ (4)的基于第$i$个传感器子系统的局部滤波误差方程为

$ \begin{align} {{\tilde {\pmb x}}_i}&(t + 1) = {\pmb x}(t + 1) - {{\hat {\pmb x}}_i}(t + 1)= \\ & \{ [{\Phi _0}(t) + \xi (t){\Phi _1}(t)] - {F_i}(t) - {L_i}(t + 1) \times \\ & {H_{0i}}(t + 1){\Phi _0}(t) - {\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1) \times \\ & \xi (t){\Phi _1}(t) - {\gamma _i}(t + 1){\lambda _i}(t + 1){L_i}(t + 1) \times \\ & {H_{1i}}(t + 1)[{\Phi _0}(t) + \xi (t){\Phi _1}(t)] \} {\pmb x}(t)+ \\ & [{F_i}(t) + {L_i}(t + 1){H_{0i}}(t + 1){\Phi _0}(t) - {\gamma _i}(t+ 1)\times \\ & {L _i}(t + 1){H _{0i}}(t + 1){\Phi _0}(t)]{{\tilde {\pmb x}}_i}(t) +{\Gamma }(t){\pmb w}(t) - \\ & {\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1)\Gamma (t){\pmb w}(t) - \\ & {\gamma _i}(t + 1){\lambda _i}(t + 1){L_i}(t + 1){H_{1i}}(t + 1)\Gamma (t){\pmb w}(t) - \\ & {\gamma _i}(t + 1){L_i}(t + 1){{\pmb v}_i}(t + 1) - \\ & {L_i}(t + 1){D_i}(t + 1){{\pmb \theta} _i}(t + 1) \end{align} $ (14)

对任意的未知输入${\pmb \theta _i}(t)$, 为了使状态估计满足无偏性, 即满足${\rm{E[}}{\tilde {\pmb x}}(t)] = 0$, 由(14)可得:

$ \begin{equation} {\Phi _0}(t) - {F_i}(t) - {L_i}(t + 1){H_{0i}}(t + 1){\Phi _0}(t)=0 \end{equation} $ (15)
$ \begin{equation} {L_i}(t + 1){D_i}(t + 1) = 0 \end{equation} $ (16)

则由式(15)引出式(7)成立.因此, 式(14)可化简为

$ \begin{align} {{\tilde {\pmb x}}_i}&(t + 1) = \{ \xi (t){\Phi _1}(t) - {\gamma _i}(t + 1){L_i}(t + 1)\times \\ & {H_{0i}}(t + 1)\xi (t){\Phi _1}(t) - {\gamma _i}(t + 1){\lambda _i}(t + 1)\times \\ & {L_i}(t + 1){H_{1i}}(t + 1)[{\Phi _0}(t) + \xi (t){\Phi _1}(t)]\} {\pmb x}(t)+ \\ & [{\Phi _0}(t) -{\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1){\Phi _0}(t)] \times \\ & {{\tilde {\pmb x}}_i}(t) +[{I_n} -{\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1) - %\times \\ & {\gamma _i}(t + 1) {\lambda _i}(t + 1){L_i}(t + 1){H_{1i}}(t + 1)] \Gamma (t){\pmb w}(t) - \\ & {\gamma _i}(t + 1){L_i}(t + 1){{\pmb v}_i}(t + 1) \end{align} $ (17)
$ \begin{align} {P_i}&(t + 1) = {\rm{E}}[{{\tilde {\pmb x}}_i}(t + 1){{\tilde {\pmb x}}_i}^{\rm{T}}(t + 1)] = \\ & {\rm{E}} \{ \{ \xi (t){\Phi _1}(t) - {\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1)\xi (t){\Phi _1}(t) - {\gamma _i}(t + 1){\lambda _i}(t + 1){L_i}(t + 1){H_{1i}}(t + 1) \times \\ & [{\Phi _0}(t) + \xi (t){\Phi _1}(t)] \} {\pmb x}(t){{\pmb x}^{\rm{T}}}(t)\{ \xi (t){\Phi _1}(t) - {\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1)\xi (t){\Phi _1}(t) - {\gamma _i}(t + 1){\lambda _i}(t + 1) \times \\ & {L_i}(t + 1){H_{1i}}(t + 1)[{\Phi _0}(t) + \xi (t){\Phi _1}(t)]{\} ^{\rm{T}}}\} + {\rm{E}} \{ [{\Phi _0}(t) - {\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1){\Phi _0}(t)]{{\tilde {\pmb x}}_i}(t){{{\tilde {\pmb x}}_i}^{\rm{T}}}(t) \times \\ & {[{\Phi _0}(t) - {\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1){\Phi _0}(t)]}^{\rm{T}} \} + {\rm{E}} \{ [{I_n} - {\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1) - {\gamma _i}(t + 1) \times \\ & {\lambda _i}(t + 1){L_i}(t + 1){H_{1i}}(t + 1)]\Gamma (t){\pmb w}(t){{\pmb w}^{\rm{T}}}(t){\Gamma ^{\rm{T}}}(t) [{I_n} - {\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1) - {\gamma _i}(t + 1) \times \\ & {\lambda _i}(t + 1){L_i}(t + 1){H_{1i}}(t + 1)]^{\rm{T}} \} + {\rm{E}}\{ \gamma _i^{\rm{2}}(t + 1){L_i}(t + 1){{\pmb v}_i}(t + 1){{\pmb v}_i}^{\rm{T}}(t + 1)L_i^{\rm{T}}(t + 1) \} \end{align} $ (18)
$ \begin{align} {P_i}&(t + 1) = {Q_\xi }(t){\Phi _1}(t)X(t){\Phi _1 ^{\rm{T}}}(t) + {\alpha _i}{Q_\xi }(t){L_i}(t + 1){H_{0i}}(t + 1){\Phi _1}(t)X(t)\Phi _1 ^{\rm{T}}(t)H_{0i} ^{\rm{T}}(t + 1)L_i ^{\rm{T}}(t + 1) - \\ & {\alpha _i}{Q_\xi }(t){L_i}(t + 1){H_{0i}}(t + 1){\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t) - {\alpha _i}{Q_\xi }(t){\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t)H_{0i}^{\rm{T}}(t + 1)L_i^{\rm{T}}(t + 1) + {\alpha _i} \times \\ & {Q _{\lambda _i}}(t + 1){L_i}(t + 1){H_{1i}}(t + 1){\Phi _0}(t)X(t)\Phi _0 ^{\rm{T}}(t)H_{1i} ^{\rm{T}}(t + 1) L_i^{\rm{T}}(t + 1) + {\alpha _i}{Q_{{\lambda _i}}}(t + 1){Q_\xi }(t) {L_i}(t + 1) \times \\ & {H_{1i}}(t + 1){\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t)H_{1i}^{\rm{T}}(t + 1)L_i ^{\rm{T}}(t + 1) + {\Phi _0}(t){P_i}(t)\Phi _0 ^{\rm{T}}(t) - {\alpha _i}\Phi _0(t){P_i}(t) \Phi _0 ^{\rm{T}}(t)H_{0i}^{\rm{T}}(t + 1) \times \\ & L_i^{\rm{T}}(t + 1) - {\alpha _i}{L_i}(t + 1){H_{0i}}(t + 1){\Phi _0}(t){P_i}(t)\Phi _0 ^{\rm{T}}(t) + {\alpha _i}{L_i}(t + 1){H_{0i}}(t + 1){\Phi _0}(t){P_i}(t)\Phi _0^{\rm{T}}(t) \times \\ & H_{0i}^{\rm{T}}(t + 1)L_i^{\rm{T}}(t + 1) + \Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t)- {\alpha _i}\Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t)H_{0i}^{\rm{T}}(t + 1)L_i^{\rm{T}}(t + 1) - {\alpha _i}{L_i}(t + 1) \times \\ & {H_{0i}}(t + 1)\Gamma (t){Q_{\pmb w}}(t) {\Gamma ^{\rm{T}}}(t) + {\alpha _i}{L_i}(t + 1){H_{0i}}(t + 1)\Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t)H_{0i}^{\rm{T}}(t + 1)L_i^{\rm{T}}(t + 1) +{\alpha_i}{L_i}(t + 1) \times \\ & {Q_{{{\pmb v}_i}}}(t + 1)L_i^{\rm{T}}(t + 1)+ {\alpha _i}{Q_{{\lambda _i}}}(t + 1){L_i}(t + 1){H_{1i}}(t + 1)\Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t)H_{1i}^{\rm{T}}(t + 1)L_i^{\rm{T}}(t + 1) \end{align} $ (19)

根据滤波误差方程(17), 有滤波误差方差阵为(见式(18) (见下页)), 经计算可得(见式(19) (见下页)).合并整理化简得:

$ \begin{align} {P_i}&(t + 1)= {\Phi _0}(t){P_i}(t)\Phi _0^{\rm{T}}(t) + {Q_\xi }(t){\Phi _1}(t)X(t)\times \\ & \Phi _1^{\rm{T}}(t) + \Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t) + {L_i}(t + 1){C_i}(t + 1)\times \\ & L_i^{\rm{T}}(t + 1)-{L_i}(t + 1){G_i}(t + 1) - \\ & G_i^{\rm{T}}(t + 1)L_i^{\rm{T}}(t + 1) \end{align} $ (20)

即式(13)成立, 其中${C_i}(t+1)$${G_i}(t+1)$分别由式(11)和式(9)定义.

应用线性无偏最小方差估计准则[19], 并由约束条件式(16)可引出如下辅助方程:

$ \begin{align} {J_i}(t + 1) = &{\rm {tr}} \{ {P_i}(t + 1) \} + 2{\rm {tr}}\{ {\Lambda _i ^{\rm{T}}}(t + 1) \times \\ &{L_i}(t + 1){D_i}(t + 1) \} \end{align} $ (21)

为了极小化性能指标${J_i}(t + 1)$, 令$\frac{{\partial {J_i}(t + 1)}}{{\partial {L_i}(t + 1)}} = 0$, 应用矩阵迹的求导公式[21]有:

$ \begin{equation} {L_i}(t + 1){C_i}(t + 1) + {\Lambda _i}(t + 1)D_i^{\rm{T}}(t + 1) = G_i^{\rm{T}}(t + 1) \end{equation} $ (22)

将式(22)和约束条件(16)联立得矩阵方程组

$ \begin{equation} \left \{ \begin{array}{lll} {L_i}(t + 1){C_i}(t + 1) + \\\qquad\qquad {\Lambda _i}(t + 1)D_i^{\rm{T}}(t + 1) = G_i^{\rm{T}}(t + 1) \\ {L_i}(t + 1){D_i}(t + 1) = 0 \\ \end{array} \right. \end{equation} $ (23)

写为分块矩阵形式

$ \begin{align} &\left[ {\begin{array}{*{20}{c}} {{C_i}(t + 1)}&{{D_i}(t + 1)} \\ {D_i^{\rm{T}}(t + 1)}&0 \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {L_i^{\rm{T}}(t + 1)} \\ {\Lambda _i^{\rm{T}}(t + 1)} \\ \end{array}} \right] = \\ &\qquad\left[ {\begin{array}{*{20}{c}} {{G_i}(t + 1)} \\ 0 \\ \end{array}} \right] \end{align} $ (24)

由假设2可知方程式(24)的系数矩阵的逆存在[21], 解分块矩阵方程(24)得:

$ \begin{align} {\Lambda _i}&(t + 1) = G_i^{\rm{T}}(t + 1)C_i^{ - 1}(t + 1){D_i}(t + 1)\times \\ & {[D_i^{\rm{T}}(t + 1)C_i^{ - 1}(t + 1){D_i}(t + 1)]^{ - 1}} \end{align} $ (25)
$ \begin{equation} {L_i}(t + 1) = [G_i^{\rm{T}}(t + 1) - {\Lambda _i}(t + 1)D_i^{\rm{T}}(t + 1)]C_i^{ - 1}(t + 1) \end{equation} $ (26)

即(10)与(8)成立.

注1. 由定理1可知, 由于通信干扰是未知的, 为了避免干扰对滤波器的影响, 我们设计了不依赖于未知干扰的滤波器式(6), 使其满足无偏性和滤波误差方差的迹最小.为了保证此类滤波器的存在性, 即矩阵方程式(24)的解存在, 要求假设2成立.

2.2 局部估计误差互协方差阵的计算

定理2.   在假设1和2下, 多传感器系统(1) $\sim$ (4)的第$i$个和第$j$个传感器子系统间的滤波误差互协方差阵可计算如式(27) $(i, j = 1, 2, \cdots , L)$, 初值${P_{ij}}(0) ={P_0}$.

$ \begin{align} {P_{ij}}&(t + 1) = {Q_\xi }(t){\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t) + {\alpha _i}{\alpha _j}{L_i}(t + 1){H_{0i}}(t + 1){Q_\xi }(t){\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t)H_{0j}^{\rm{T}}(t + 1)L_j^{\rm{T}}(t + 1)- \\ & {\alpha _i}{Q_\xi }(t){L_i}(t + 1){H_{0i}}(t + 1){\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t) - {\alpha _j}{Q_\xi }(t){\Phi _1}(t)X(t)\Phi _1^{\rm{T}}(t)H_{0j}^{\rm{T}}(t + 1) L_j^{\rm{T}}(t + 1)+ {\Phi _0}(t) \times \\ & {P_{ij}}(t)\Phi _0^{\rm{T}}(t) - {\alpha _i}{L_i}(t + 1){H_{0i}}(t + 1) {\Phi _0}(t){P_{ij}}(t)\Phi _0^{\rm{T}}(t) - {\alpha _j}{\Phi _0}(t){P_{ij}}(t) \Phi _0^{\rm{T}}(t)H_{0j}^{\rm{T}}(t + 1)L_j^{\rm{T}}(t + 1)+ \\ & {\alpha _i}{\alpha _j}{L_i}(t + 1){H_{0i}}(t + 1) {\Phi _0}(t){P_{ij}}(t)\Phi _0^{\rm{T}}(t)H_{0j}^{\rm{T}}(t + 1)L_j^{\rm{T}}(t + 1) + \Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t) + {\alpha _i}{\alpha _j}{L_i}(t + 1) \times \\ & {H_{0i}}(t + 1)\Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t)H_{0j}^{\rm{T}}(t + 1) L_j^{\rm{T}}(t + 1) - {\alpha _i}{L_i}(t + 1){H_{0i}}(t + 1)\Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t) - \\ & {\alpha _j}\Gamma (t){Q_{\pmb w}}(t){\Gamma ^{\rm{T}}}(t) H_{0j}^{\rm{T}}(t + 1) L_j^{\rm{T}}(t + 1) \end{align} $ (27)
$ \begin{align} {P_{ij}}&(t + 1) = {\rm{E}} \{ \{ \xi (t){\Phi _1}(t) - {\gamma _i}(t + 1){L_i}(t + 1){H_{0i}}(t + 1)\xi (t){\Phi _1}(t) - {\gamma _i}(t + 1){\lambda _i}(t + 1){L_i}(t + 1) \times \\ & {H_{1i}}(t + 1)[{\Phi _0}(t) + \xi (t){\Phi _1}(t)] \} {\pmb x}(t) { {\pmb x}^{\rm{T}}}(t) \{ \xi (t){\Phi _1}(t) - {\gamma _j}(t + 1){L_j}(t + 1){H_{0j}}(t + 1)\xi (t){\Phi _1}(t) - \\ & {\gamma _j}(t + 1){\lambda _j}(t + 1){L_j}(t + 1){H_{1j}}(t + 1)[{\Phi _0}(t) + \xi (t){\Phi _1}(t)]{\} ^{\rm{T}}}\}+ {\rm{E}}\{ [{\Phi _0}(t) - {L_i}(t + 1){\gamma _i}(t + 1)\times \\ & {H_{0i}}(t + 1){\Phi _0}(t)]{{\tilde {\pmb x}}_i}(t){{\tilde {\pmb x}}_j}^{\rm{T}}(t)[{\Phi _0}(t) - {L_j}(t + 1){\gamma _j}(t + 1){H_{0j}}(t + 1){\Phi _0}(t){]^{\rm{T}}}\} + {\rm {E}} \{ [{I_n} - {\gamma _i}(t + 1) \times \\ & {L_i}(t + 1){H_{0i}}(t + 1) - {\gamma _i}(t + 1){\lambda _i}(t + 1) {L_i}(t + 1){H_{1i}}(t + 1)]\Gamma (t){\pmb w}(t){{\pmb w} ^{\rm{T}}}(t){\Gamma ^{\rm{T}}}(t) \times \\ & {[{I_n} - {\gamma _j}(t + 1){L_j}(t + 1){H_{0j}}(t + 1) - {\gamma _j}(t + 1){\lambda _j}(t + 1){L_j}(t + 1){H_{1j}}(t + 1)]^{\rm{T}} }\} \end{align} $ (28)

证明. 将式(17)代入${P_{ij}}(t+1) = {\rm{E[}}{{\tilde {\pmb x}}_i}(t+1){{\tilde {\pmb x}}_j}^{\rm{T}}(t+1){\rm{]}}$中, 由${{\tilde {\pmb x}}_i}(t)$${\pmb v_j}(t)$, ${{\tilde {\pmb x}}_i}(t)$${\pmb w}(t)$, ${\pmb v_i}(t)$${\pmb v_j}(t)$, $i \ne j$, 均不相关, 可得(见式(28))又由${\lambda _i}(t+1)$${\lambda _j}(t+1)$不相关, 展开计算式(28)可得式(27).

2.3 按矩阵加权融合滤波器

基于定理1的局部滤波器和定理2的任两个局部滤波误差互协方差阵, 应用在线性最小方差意义下的按矩阵加权融合估计算法[20]有如下分布式融合状态滤波器:

$ \begin{equation} {{\hat {\pmb x}}_o}(t) = \sum\limits_{i = 1}^L {{{\bar A}_i}} (t){{\hat {\pmb x}}_i}(t) \end{equation} $ (29)

最优加权矩阵${\bar A_i}(t)$, $i = 1, 2, \cdots , L$, 计算如下:

$ \begin{equation} [{\bar A_1}(t), {\bar A_2}(t), \cdots, {\bar A_L}(t)] = {[{e^{\rm{T}}}{\Sigma ^{ - 1}}(t)e]^{ - 1}}{e^{\rm{T}}}{\Sigma ^{ - 1}}(t) \end{equation} $ (30)

其中, $e = {[{I_n}, {I_n}, \cdots , {I_n}]^{\rm{T}}}$$nL \times n$的矩阵, 矩阵$\Sigma (t)$为第$(i, j)$块元素为${P_{ij}}(t)$$nL \times nL$矩阵.分布式融合估计误差方差阵计算为

$ \begin{equation} {P_o}(t) = {[{e^{\rm{T}}}{\Sigma ^{ - 1}}(t)e]^{ - 1}} \end{equation} $ (31)

并且有关系

$ \begin{equation} {P_o}(t) \le {P_i}(t), \quad i = 1, 2, \cdots , L \end{equation} $ (32)

注2. 在图 1中, 我们假设从各局部滤波器到融合中心的通信是完美的, 即无数据丢失.如果有数据丢失, 只要存在局部滤波器到达融合中心, 就可应用上面的融合算法获得融合估计.若某时刻局部估计都丢失了, 则可用上一时刻的融合估计进行预报.

3 仿真实例

考虑如下跟踪系统:

$ \begin{align} {\pmb x}&(t + 1) = \left( \left[ {\begin{array}{*{20}{c}} {0.95}&{{T}} \\ 0&{0.95} \\ \end{array}} \right] + \xi (t) \times \right. \\ & \left. \left[ {\begin{array}{*{20}{c}} {0.05}&{\rm{0}} \\ 0&{0.05} \\ \end{array}} \right] \right) {\pmb x}(t) + \left[ {\begin{array}{*{20}{c}} \frac{T^2}{2} \\ {{T}} \\ \end{array}} \right]{\pmb w}(t) \end{align} $ (33)
$ \begin{equation} {\pmb y_i}(t) = ({H_{0i}} + {\lambda _i}(t){H_{1i}}){\pmb x}(t) + {\pmb v_i}(t), \quad i = 1, 2, 3 \end{equation} $ (34)
$ \begin{equation} {\pmb z_i}(t) = {\gamma _i}(t){\pmb y_i}(t) + {D_i}{\pmb \theta _i}(t), \quad i = 1, 2, 3 \end{equation} $ (35)

取采样周期${{T}} = 1$, 观测阵${H_{01}} = \left[ {\begin{array}{*{20}{c}} {0.5}&{0.6} \\ 0&{0.9} \\ \end{array}} \right]$, ${H_{02}} = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&{2.2} \\ \end{array}} \right]$, ${H_{03}} = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 1&{0.7} \\ \end{array}} \right]$, ${H_{11}} = \left[ {\begin{array}{*{20}{c}} {0.01}&{0.09} \\ {0.05}&{0.15} \\ \end{array}} \right]$, ${H_{12}} = \left[ {\begin{array}{*{20}{c}} {0.1}&{0.15} \\ {0.08}&{0.1} \\ \end{array}} \right]$, ${H_{13}} = \left[ {\begin{array}{*{20}{c}} {0.2}&{0.1} \\ 0&{0.1} \\ \end{array}} \right]$, 其中不相关白噪声${\pmb w}(t)$${\pmb v_i}(t)$的方差分别为${Q _{\pmb w}} =2$, ${Q_{{\pmb v}_1}} = {I_2}$, ${Q_{{\pmb v}_2}} = 1.2{I_2}$, ${Q_{{\pmb v}_3}} = 0.8{I_2}$, ${I_2}$为2维单位阵; 互不相关且与其他噪声均不相关的白噪声$\xi (t)$${\lambda _i}(t)$的方差分别为${Q_\xi } = 0.8$, ${Q_{{\lambda _1}}} = 0.5$, ${Q_{{\lambda _2}}} = 0.7$, ${Q_{{\lambda _3}}} = 0.6$.取干扰系数阵${D_1} = {[\begin{array}{*{20}{c}} 1&{0.8} \\ \end{array}]^{\rm{T}}}$, ~${D_2} = {[\begin{array}{*{20}{c}} {1.2}&{0.9} \\ \end{array}]^{\rm{T}}}$, ~${D_3} = {[\begin{array}{*{20}{c}} {0.3}&1 \\ \end{array}]^{\rm{T}}}$, 通道干扰${\pmb \theta _1}(t) = 1$, ${\pmb \theta _2}(t) = t/2$, ${\pmb \theta _3}(t) = \sin t$, 初值${\hat {\pmb x}}(0) = {[0, 0]^{\rm{T}}}$, ${P_0} = 0.1{I_2}$.取200个采样数据, 且不同传感器的接收率分别为${\alpha _1} = 0.5$, ${\alpha _2} = 0.8$, ${\alpha _3} = 0.4$.求多传感器分布式按矩阵加权融合递推状态滤波器${{\hat {\pmb x}}_o}(t)$.

图 2是分布式按矩阵加权融合状态滤波器的跟踪图, 由图 2可以看出本文所设计的分布式融合状态滤波器具有良好的跟踪特性.图 3是各单传感器局部滤波器与分布式融合滤波器的估计误差方差比较图, 表明了分布式融合滤波器的估计误差方差小于各局部滤波器的估计误差方差.这验证了融合估计比单传感器估计精度高, 达到了融合的目的.

图 2 分布式融合状态滤波器跟踪图 Figure 2 Tracking performance of distributed fusion state filter
图 3 局部与分布式融合状态滤波器估计误差方差比较图 Figure 3 Comparison of estimation error variances of local and distributed fusion state filters

图 4图 5分别给出了带补偿与无补偿的第3个局部单传感器子系统滤波器和分布式融合滤波器经过100次Monte-Carlo试验的MSE (Mean square error)比较, 从图中可以看出, 带有补偿的滤波精度比无补偿的滤波精度高.这验证了采用丢包补偿方法可以改善滤波器估计精度.

图 4 带补偿与无补偿的第3传感器子系统滤波器的MSE比较 Figure 4 MSE comparison of the 3rd sensor subsystem filters with compensation and no compensation
图 5 带补偿与无补偿的分布式融合滤波器的MSE比较 Figure 5 MSE comparison of distributed fusion filters with compensation and no compensation

图 6给出了第3个局部单传感器子系统采用本文预报补偿的滤波算法和文献[10]采用以前收到的最新数据补偿的算法进行MSE比较图, 因为采用预报补偿用到了以前收到的所有观测数据, 且文献[10]未考虑未知干扰, 所以本文的滤波精度高于文献[10]的滤波精度.

图 6 第3传感器子系统的文献[10]和本文的算法的MSE比较 Figure 6 MSE comparison of algorithms of [10] and ours for the 3rd sensor subsystem
4 结论

针对带有未知通信干扰、丢失观测和乘性噪声不确定性的多传感器网络化系统, 考虑从不同传感器到局部滤波器的数据传输中具有不同丢失率情形, 当观测丢失时采用当前丢失观测的一步预报作为补偿.在线性无偏最小方差意义下, 提出了不依靠未知通信干扰的最优局部子系统状态滤波器.推导了任意两传感器子系统间的估计误差互协方差阵, 应用矩阵加权融合估计算法给出了分布式融合状态滤波器.下一步将开展系统噪声与观测噪声相关情形下状态滤波器的设计, 以及未知通信干扰的估计问题.

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