在不准确方差下带随机系数矩阵的卡尔曼滤波稳定性

L_2-stability of Discrete-time Kalman Filter with Random Coefficients under Incorrect Covariance

针对离散时间线性随机系统,研究了卡尔曼滤波的L2-稳定性问题.考虑具有这两个特点的系统:1)系数矩阵是随机的;2)过程噪声、量测噪声、初始估计误差的方差矩阵不准确.在系数矩阵有界、条件能观测、初始估计误差有界的假设下,得到了卡尔曼滤波的L2-稳定性.同时,建立了卡尔曼滤波和状态空间最小二乘的等价性,并在该等价性基础上得到状态空间最小二乘的状态估计误差L2-稳定性.最后,数值仿真说明了卡尔曼滤波的有效性.

This paper studies the L2-stability of Kalman filter for discrete-time linear stochastic systems. Two main features, i.e., random coefficient matrices and incorrect covariances of process noise, measurement noise and initial value, are emphasized. Under suitable conditions, including boundedness of coefficient matrices, conditional observability and boundedness of initial error and noises, L2-stability of Kalman filter is achieved. The equivalence between Kalman filter and state-space least squares algorithm is established. Based on this equivalence, L2-stability of state estimation error by state-space least squares is also obtained. A numerical example is given to demonstrate the effectiveness of Kalman filtering algorithm.