论文标题
重新考虑数学框架和设置会员过滤的最佳性
Rethinking the Mathematical Framework and Optimality of Set-Membership Filtering
论文作者
论文摘要
在存在未知统计量的有限噪声的情况下，已经对Set-Membership滤波器（SMF）进行了广泛的研究以进行状态估计。自1960年代首次推出以来，对SMF的研究将基于设定的描述用作其数学框架。被忽略的一个重要问题是SMF的最佳性。在这项工作中，我们使用不确定变量的概念提出了一个新的数学框架，用于SMF。我们首先建立了不确定变量的两个基本属性，即总范围定律（总概率定律的非传统版本）和等效的贝叶斯规则。这使我们能够以既定的最优性提出一个一般的SMFing框架。此外，我们在非探索马尔可夫条件下获得了最佳SMF，该条件从根本上等同于贝叶斯过滤器。请注意，文献中的经典SMF仅等于我们在非策略马尔可夫条件下获得的最佳SMF。当违反这种情况时，我们表明经典SMF不是最佳的，并且仅在最佳估计上给出外部界限。
Set-Membership Filter (SMF) has been extensively studied for state estimation in the presence of bounded noises with unknown statistics. Since it was first introduced in the 1960s, the studies on SMF have used the set-based description as its mathematical framework. One important issue that has been overlooked is the optimality of SMF. In this work, we put forward a new mathematical framework for SMF using concepts of uncertain variables. We first establish two basic properties of uncertain variables, namely, the law of total range (a non-stochastic version of the law of total probability) and the equivalent Bayes' rule. This enables us to put forward a general SMFing framework with established optimality. Furthermore, we obtain the optimal SMF under a non-stochastic Markov condition, which is shown to be fundamentally equivalent to the Bayes filter. Note that the classical SMF in the literature is only equivalent to the optimal SMF we obtained under the non-stochastic Markov condition. When this condition is violated, we show that the classical SMF is not optimal and it only gives an outer bound on the optimal estimation.
