Abstract It is known that Kalman–Bucy filter is stable with respect to initial conditions under the assumptions of uniform complete controllability and uniform complete observability. In this paper, we prove the stability of Kalman-Bucy filter for the case of noise free dynamical system, i.e., for deterministic processes. The earlier stability results cannot be applied for this case, as the system is not controllable. We further show that the optimal linear filter for a general class of non-Gaussian initial conditions is asymptotically proximal to Kalman–Bucy filter. It is also shown that the filter corresponding to non-zero system noise in the limit of small system noise approaches the filter corresponding to zero system noise in the case of Gaussian initial conditions.

中文翻译：

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确定性过程线性滤波器的渐近特性

摘要 已知Kalman-Bucy滤波器在均匀完全可控和均匀完全可观测的假设下相对于初始条件是稳定的。在本文中，我们证明了Kalman-Bucy 滤波器在无噪声动力系统情况下的稳定性，即确定性过程。由于系统不可控，早期的稳定性结果不适用于这种情况。我们进一步表明，对于一般类别的非高斯初始条件的最佳线性滤波器渐近地接近于 Kalman-Bucy 滤波器。还表明在小系统噪声极限下对应于非零系统噪声的滤波器在高斯初始条件下接近于对应于零系统噪声的滤波器。