﻿ 一种基于随机理论的寻北滤波噪声影响分析方法
 舰船科学技术  2016, Vol. 38 Issue (6): 132-136 PDF

Stochastic theory approach to analyzing noise influences on north-finding filtering
ZHU Hai, YU Hua-peng, CAI Peng
Navy Submarine Academy, Qingdao 266199, China
Abstract: The north-finding Kalman filter is studied from the aspect of the propagation mechanism of filtering noises. Explicit analytic expressions of the state error covariance matrix with different noise statistical characteristics are derived based on the stochastic controllability and stochastic observablity theory, and the proposed analytic method is detailed demonstrated on the single-axis rotary inertial navigation system (INS). The experimental results show that, the proposed analytic method is effective and applicable to analyzing the north-finding filter, and the analytic expressions can provide with us more information about the system performance.
Key words: stochastic theory     noise characteristics     north-finding filtering     rotary
0 引 言

1 基于随机理论的解析分析方法

1.1 误差协方差矩阵的解析式

 ${\bf{\dot {P}}} = {FP} + {P}{{F}^{\text{T}}} + {GQ}{{G}^{\rm T}} - {P}{{H}^{\rm T}}{{R}^{ - 1}}{HP}$ (1)

 $\begin{array}{l} {{\bf{P}}_{SC}}\left( {k{t_s}} \right) = \\ [10pt] \displaystyle\sum\limits_{i = 0}^{k - 1} {{{\varPhi }}\left( {k{t_s},i{t_s}} \right){G}\left( {i{t_s}} \right){Q}\left( {i{t_s}} \right){{G}^{\text{T}}}\left( {i{t_s}} \right){{{\varPhi }}^{\rm T}}\left( {k{t_s},i{t_s}} \right)} \text{。} \end{array}$ (2)

 $\begin{array}{l} {{P}_{SO}}\left( {k{t_s}} \right) = \\ [6pt] \displaystyle\sum\limits_{i = 0}^k {{{{\varPhi }}^{\rm T}}\!\!\left( {i{t_s},k{t_s}} \right){{H}^{\text{T}}}\!\!\left( {i{t_s}} \right){{R}^{ - 1}}\left( {i{t_s}} \right){H}\left( {i{t_s}} \right){{\varPhi }}\left( {i{t_s},k{t_s}} \right)} \text{。} \end{array}$ (3)

1.2 寻北 Kalman 滤波模型的建立

 ${\bf{X}} = {\left[{\begin{array}{*{20}{c}} {{\phi _N}} \!\!\! & \!\!\! {{\phi _E}} \!\!\! & \!\!\! {{\phi _D}} \!\!\! & \!\!\! {\delta {v_N}} \!\!\! & \!\!\! {\delta {v_E}} \! & \! {\varepsilon _{gx}^b} \!\!\! & \!\!\! {\varepsilon _{gy}^b} \!\!\! & \!\!\! {\varepsilon _{gz}^b} \!\!\! & \!\!\! {\nabla _{ax}^b} \! & \! {\nabla _{ay}^b} \end{array}} \right]^{\text{T}}} \text{。}$ (4)

 ${\bf{\dot {X}}}\left( t \right) = {F}\left( t \right){X}\left( t \right) + {G}\left( t \right){u}\left( t \right),{u}\left( t \right) \sim N\left( {0,{Q}} \right)$ (5)

 ${Q} = \left[{\begin{array}{*{20}{c}} {n_{grw}^2{I_{3 \times 3}}} & {{0_{3 \times 2}}} & {{0_{3 \times 3}}} & {{0_{3 \times 2}}}\\ [6pt] {{0_{2 \times 3}}} & {n_a^2{I_{2 \times 2}}} & {{0_{2 \times 3}}} & {{0_{2 \times 2}}}\\ [6pt] {{0_{3 \times 3}}} & {{0_{3 \times 2}}} & {n_{gbrw}^2{I_{3 \times 3}}} & {{0_{3 \times 2}}}\\ [6pt] {{0_{2 \times 3}}} & {{0_{2 \times 2}}} & {{0_{2 \times 3}}} & {n_{abrw}^2{I_{2 \times 2}}} \end{array}} \right] \text{。}$ (6)

 ${z}\left( t \right) = \left[{\begin{array}{*{20}{c}} {\delta {v_N}}\\ [6pt] {\delta {v_E}} \end{array}} \right] = {H}\left( t \right){X}\left( t \right){\bf{ + \upsilon }}\left( t \right), {\bf{\upsilon }}\left( t \right) \sim N\left( {0,{R}} \right) \text{。}$ (7)

 ${R} = \left[{\begin{array}{*{20}{c}} {r_v^2} & 0\\ [10pt] 0 & {r_v^2} \end{array}} \right] \text{。}$ (8)
1.3 实现需考虑的问题

1）在 IMU 器件的零偏可观情况下，一般寻北 Kalman 滤波模型均将零偏状态进行扩展，则不可避免地增加滤波模型的维数，意味着本文所研究的滤波模型是一个高阶线性系统。当寻北过程中 IMU 位置保持固定时，系统矩阵 Ft）是常值；然而，当寻北过程中改变 IMU 位置以改善系统滤波状态的可观测度时，系统矩阵是时变的，导致基于随机理论进行误差协方差矩阵分析时，计算量非常大；

2）基于随机理论的误差协方差矩阵分析方法在 SINS 对准中的成功应用成果仅有文献 [16-17]（为得到解析表达式使用了简化的三维或四维模型，降低了解析式的准确度）。

2 实验结果及分析

2.1 系统噪声影响分析

 \begin{aligned} P_{\left( {3,3} \right)}^0\left( t \right) = & n_{grw}^2\left( {t + \frac{1}{3}{\Delta _1}^2{t^3} + \frac{1}{{80}}{\Delta _2}^2{t^5}} \right)+ \\ & n_{gbrw}^2\left( {\frac{1}{3}{t^3} + \displaystyle\frac{1}{{20}}{\Delta _1}^2{t^5} + \frac{1}{{1008}}{\Delta _2}^2{t^7}} \right) \text{，} \end{aligned} (9)
 $P_{\left( {3,3} \right)}^1\left( t \right) \!=\! n_{grw}^2\left( {t + \frac{1}{3}{\Delta _1}{t^3} \!+\! \frac{1}{{80}}{\Delta _2}^2{t^5}} \right)\\ \!+\! \frac{{n_{gbrw}^2}}{{480\varOmega _C^7}}\left( \begin{array}{l} 160\varOmega _C^7{t^3} \!+\! 80\omega _{ie}^2\varOmega _C^3\left( {6 \!+\! \varOmega _C^2{t^2}} \right)t \!+\! 6{\Delta _2}^2\varOmega _C^{}\left( {40 \!+\! \varOmega _C^4{t^4}} \right)t\\ \!+\! \omega _{ie}^2\left( \begin{array}{l} 80\varOmega _C^{}\cos \left( {2L} \right)t\left( \begin{array}{l} 6\varOmega _C^2 \!+\! \varOmega _C^4{t^4}\\ \!-\! 12{\Delta _1}^2\cos \left( {\varOmega _C^{}t} \right) \end{array} \right)\\ \!+\! 160{\cos ^2}L\sin \left( {\varOmega _C^{}t} \right)\left( \begin{array}{l} \varOmega _C^2{\Delta _3}\left( {12 \!+\! \varOmega _C^2{t^2}} \right)t\\ \!+\! 6\cos \left( {\varOmega _C^{}t} \right)\left( \begin{array}{l} \varOmega _C^3 \!+\! \omega _{ie}^2\varOmega _C^{}\\ \!+\! 3{\Delta _3}\varOmega _C^2 \end{array} \right)t\\ \!+\! 3\left( {\varOmega _C^{} \!+\! {\Delta _3}} \right)\left( \begin{array}{l} \!-\! 4\varOmega _C^{} \!-\! 6{\Delta _3}\\ \!+\! \varOmega _C^2{t^2}{\Delta _3} \end{array} \right) \end{array} \right) \end{array} \right) \end{array} \right) \text{。}$ (10)

1）旋转调制对陀螺角随机游走噪声对方位角误差的影响没有抑制效果；

2）旋转调制对角速率随机游走噪声对方位角误差的影响有抑制效果。

 图 1 旋转调制对陀螺角速率随机游走噪声对方位角误差作用分量的相对抑制比率 Fig. 1 Attenuation ratio of gyro angular rate random walk noise influence on azimuth error under different turntable rate（> 0）compared to zero turntable rate

2.2 观测噪声影响分析

 $\sqrt {P_{\left( {3,3} \right)}^{}\left( t \right)} = {\rm{0}}{\rm{.01448427}}{r_\nu } \text{。}$

 图 2 寻北结束时方位角误差标准差与观测噪声标准差之间的关系曲线 Fig. 2 Standard deviation curve between azimuth error and measurement noise after north-filtering

3 结 语

Kalman 滤波在捷联惯性导航系统的初始对准和标定等各个环节广泛应用。为了能够更全面地深入研究 SINS 和系统运行环境中噪声信号特性对滤波器运行的作用机理，得到关于滤波模型各状态的全面信息，本文利用随机可控制性和随机可观测性得到滤波噪声在寻北 Kalman 滤波器影响作用的解析表达式，利用单轴旋转式寻北系统对所提出的基于随机理论的解析分析方法进行了计算验证。根据详细验证结果可见，基于随机理论的解析分析方法适用于寻北系统，分析结论与已公开研究结论一致，并且利用该方法还能够得到更全面的系统性能分析结果，具有较高的理论参考价值。