Probabilistic reachable sets of stochastic nonlinear systems with contextual uncertainties
CoRR(2024)
摘要
Validating and controlling safety-critical systems in uncertain environments
necessitates probabilistic reachable sets of future state evolutions. The
existing methods of computing probabilistic reachable sets normally assume that
the uncertainties are independent of the state. However, this assumption falls
short in many real-world applications, where uncertainties are state-dependent,
referred to as contextual uncertainties. This paper formulates the problem of
computing probabilistic reachable sets of stochastic nonlinear states with
contextual uncertainties by seeking minimum-volume polynomial sublevel sets
with contextual chance constraints. The formulated problem cannot be solved by
the existing sample-based approximation method since the existing methods do
not consider the conditional probability densities. To address this, we propose
a consistent sample approximation of the original problem by leveraging the
conditional density estimation and resampling. The obtained approximate problem
is a tractable optimization problem. Additionally, we prove the almost uniform
convergence of the proposed sample-based approximation, showing that it gives
the optimal solution almost consistently with the original ones. Through a
numerical example, we evaluate the effectiveness of the proposed method against
existing approaches, highlighting its capability to significantly reduce the
bias inherent in sample-based approximation without considering a conditional
probability density.
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