ABSTRACT

It is known that the Kalman filter has both stochastic and deterministic interpretations, whereby the deterministic interpretation relates the prediction of the filter to the response of the plant driven by the minimising least squares disturbances acting thereon. Whilst the deterministic interpretation is known, the contribution of this note is to provide an alternative, simple and self-contained proof of these properties in the discrete case. The presentation allows an efficient derivation of the key deterministic properties, i.e. that the residuals computed by the Kalman filter (both forwards and backwards) are identical to the least squares disturbances. Results with variations are given for both zero and non-zero initial conditions. Finally, a numerical example is given to illustrate the deterministic properties.

摘要

已知卡尔曼滤波器具有随机和确定性两种解释，其中确定性解释将滤波器的预测与由作用于其上的最小二乘扰动驱动的装置的响应相关联。虽然确定性解释是已知的，但本笔记的贡献是在离散情况下提供这些属性的替代、简单和独立的证明。该演示允许有效推导关键的确定性属性，即由卡尔曼滤波器计算的残差（向前和向后）与最小二乘扰动相同。给出了零和非零初始条件的变化结果。最后，给出了一个数值例子来说明确定性特性。