概率解释下的成员身份过滤：通过多边形边界应用于多项式系统
A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding
AI总结：
本文提出通过概率方法重新解释成员身份过滤，利用概率测度集解决多项式系统中的多边形边界问题。
摘要：
成员身份估计通常在集合值微积分的背景下提出，无需概率计算。本文展示通过使用概率测度集可等价地在概率设置中表述成员身份估计。成员身份估计的推断通过计算更新的概率测度集P的期望进行，类似于概率情况。特别地，证明通过求解特定的半无限线性规划问题进行推断，该问题是截断矩形问题的特例，仅已知零阶矩（即支撑）。通过求解上述半无限线性规划问题的对偶问题，证明如果测量和过程方程中的非线性为多项式，并且初始状态、过程和测量噪声的边界集由多项式不等式描述，则可通过求和平方多项式优化理论高效近似该半无限线性规划问题。然后推导出一种智能贪心程序，通过计算最小体积多边形来计算真实成员集的外近似，该多边形外界包含所有相对于P计算的均值。
英文摘要：
Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zero-th order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P.
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