Probabilistic Optimal Estimation and Filtering under Uncertainty

Details Probabilistic Optimal Estimation and Filtering under Uncertainty Probabilistic Optimal Estimation and Filtering under Uncertainty

2012-03-07

arxiv.org/abs/1203.1429v1

Computer Science

Systems and Control

Scientific paper

The classical approach to system identification is based on statistical assumptions about the measurement error, and provides estimates that have stochastic nature. Worst-case identification, on the other hand, only assumes the knowledge of deterministic error bounds, and provides guaranteed estimates, thus being in principle better suited for its use in control design. However, a main limitation of such deterministic bounds lies on the fact that they often turn out to be overly conservative, thus leading to estimates of limited use. In this paper, we propose a rapproachement between these two paradigms, stochastic and worst-case, and propose a novel probabilistic framework for system identification that combines elements from information-based complexity with recent developments in the theory of randomized algorithms. The main idea in this line of research is to "discard" sets of measure at most $\epsilon$, where $\epsilon$ is a probabilistic accuracy, from the set of deterministic estimates. Therefore, we are decreasing the so-called worst-case radius of information at the expense of a given probabilistic "risk." In this setting, we compute a trade-off curve, called \textit{violation fuction}, which shows how the radius of information decreases as a function of the accuracy. To this end, we construct randomized and deterministic algorithms which provide approximations of this function. The obtained results are based upon specific properties regarding the intersection of convex sets.

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