The last few decades have witnessed much development in filtering of systems with Gaussian noises and arbitrary unknown inputs. Nonetheless, there are still some important design questions that warrant thorough discussions. Especially, the existing literature has shown that for unbiased and minimum variance estimation of the state and the unknown input, the initial guess of the state has to be unbiased. This clearly raises the question of whether and under what conditions one can design an unbiased and minimum variance filter, without making such a stringent assumption. The above-mentioned question will be investigated systematically in this paper, i.e., design of the filter is sought to be independent of a priori information about the initial conditions. In particular, for both cases with and without direct feedthrough, we establish necessary and sufficient conditions for unbiased and minimum variance estimation of the state/unknown input, independently of a priori initial conditions, respectively. When the former conditions do not hold, we carry out a thorough analysis of all possible scenarios. For each scenario, we present detailed discussions regarding whether and what can be achieved in terms of unbiased estimation, independently of a priori initial conditions. Extensions to the case with time-delays, conceptually like Kalman smoothing where future measurements are allowed in estimation, will also be presented, amongst others.

在过去的几十年中，在对具有高斯噪声和任意未知输入的系统进行滤波方面取得了长足发展。尽管如此，仍有一些重要的设计问题需要进行深入的讨论。特别是，现有文献表明，对于状态和未知输入的无偏差和最小方差估计，必须对状态的初始猜测无偏差。显然，这提出了一个问题，即在不做如此严格的假设的情况下，是否可以以及在什么条件下可以设计无偏和最小方差滤波器。本文将系统地研究上述问题，即，滤波器的设计应与先验条件无关有关初始条件的信息。特别是，对于有或没有直接馈通的两种情况，我们分别建立独立于先验初始条件的状态和未知输入的无偏和最小方差估计的必要条件和充分条件。如果以前的条件不成立，我们将对所有可能的情况进行彻底的分析。对于每种情况，我们都会进行关于是否可以实现无偏估计以及与先验初始条件无关的无条件估计的详细讨论。除其他外，还将介绍对延时进行扩展的情况，例如在概念上类似于卡尔曼平滑法，其中允许将来进行测量。