面向无线传感器网络节点定位的自适应卡尔曼滤波算法收敛条件分析

Convergence of Locating Nodes' Positions in Wireless Sensor Networks in a Complex Environment

分析了新息序列是有色噪声时自适应卡尔曼滤波算法(Adaptive Kal man Filter,AKF)的滤波效果,在范数意义下,证明了k时刻AKF算法中估计误差协方差矩阵和k时刻最优KF算法中估计误差协方差矩阵间距离与新息序列相关性成正比。利用上述结论,证明了所有AKF算法中估计误差协方差矩阵必逐渐远离1时刻最优KF算法中估计误差协方差矩阵。总结上述结论,发现AKF算法收敛条件可描述成以下几个等价命题:1)AKF算法中估计误差协方差矩阵与1时刻最优KF算法中估计误差协方差矩阵差有极限;2)k时刻AKF算法中估计误差协方差矩阵和k时刻最优KF算法中估计误差方差矩阵间距离极限是0;3)AKF算法渐进收敛于k时刻最优KF算法;4)AKF算法中新息序列渐进收敛于白噪声序列;5)k时刻AKF算法中滤波增益矩阵与k时刻最优KF算法中滤波增益矩阵间距离极限是0。上述理论为最终解决复杂环境下无线传感器网络节点定位问题奠定了基础。

The filtering effects are analyzed when the innovations＇ series in an adaptive Kalman filter（AKF） is colored noise. Taken the matrix norm as a tool, it is proved that at time k, the distance between the covariance matrix in the AKF and the covariance matrix in the kth optimal KF is in proportion to the innovations＇ correlation. Then, it is derived that covariance matrices in all AKFs are gradually away from the covariance matrix in the 1th optimal KF based on the above-mentioned conclusions. At last, it is summarized that convergent conditions of AKFs are： 1 The difference between the covariance matrix in an AKF and the covariance matrix in the 1th optimal KF has limit value; 2 At time k, the limit value of the distance between the covariance matrix in an AKF and the covariance matrix in the kth optimal KF is zero; 3 The AKF is gradually convergent to the kth optimal KF; 4 The innovations＇ series is gradually convergent to white noise; 5 At time k,the limit value of the distance between the gain matrix in an AKF and the gain matrix in the kth optimal KF is zero. These theories found groundworks for perfectly locating nodes＇ positions in wireless sensor networks.