最优滤波器的稳定逼近方案
Stable approximation schemes for optimal filters
AI总结：
本文通过一般截断方案构造与给定可能不稳定滤波器接近的稳定近似滤波器，并引入自然拓扑证明稳定滤波器在最优滤波器集合中稠密。
摘要：
稳定滤波器具有渐近“遗忘”初始扰动的性质。由于这一性质，可以构造此类滤波器的逼近，使其误差在时间上保持较小，换言之，这些逼近在时间变量上一致收敛。由于一致逼近从实际角度看是理想的，寻找滤波器稳定性的判据已成为许多论文的主题。本文旨在构造与给定的（可能不稳定的）滤波器保持接近的近似滤波器。此类滤波器通过一般的截断方案获得，并且在某些约束条件下是稳定的。该构造使我们能够刻画最优滤波器集合的拓扑性质。特别地，我们在该集合上引入一个自然拓扑，在该拓扑下稳定滤波器的子集是稠密的。
英文摘要：
A stable filter has the property that it asymptotically `forgets' initial perturbations. As a result of this property, it is possible to construct approximations of such filters whose errors remain small in time, in other words approximations that are uniformly convergent in the time variable. As uniform approximations are ideal from a practical perspective, finding criteria for filter stability has been the subject of many papers. In this paper we seek to construct approximate filters that stay close to a given (possibly) unstable filter. Such filters are obtained through a general truncation scheme and, under certain constraints, are stable. The construction enables us to give a characterisation of the topological properties of the set of optimal filters. In particular, we introduce a natural topology on this set, under which the subset of stable filters is dense.
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