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Embedding of rare event estimation theory within a stochastic analysis framework has recently led to significant novel results in rare event estimation for a diffusion process using sequential MC simulation. This chapter presents this rare event estimation theory for diffusions to a Stochastic Hybrid System (SHS) and extends it in order to handle a large scale SHS where a very huge number of rare discrete modes may contribute significantly to the rare event estimation. Essentially, the approach taken is to introduce a suitable aggregation of the discrete modes, and to develop importance sampling and Rao-Blackwellization relative to these aggregated modes. The practical use of this approach is demonstrated for the estimation of mid-air collision for an advanced air traffic control example.

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Abstract-This paper studies probabilistic reachability analysis for large scale stochastic hybrid systems (SHS) as a problem of rare event estimation. In literature, advanced rare event estimation theory has recently been embedded within a stochastic analysis framework, and this has led to significant novel results in rare event estimation for a diffusion process using sequential MC simulation. This paper presents this rare event estimation theory directly in terms of probabilistic reachability analysis of an SHS, and develops novel theory which allows to extend the novel results for application to a large scale SHS where a very huge number of rare discrete modes may contribute significantly to the reach probability. Essentially, the approach taken is to introduce an aggregation of the discrete modes, and to develop importance sampling relative to the rare switching between the aggregation modes. The practical working of this approach is demonstrated for the safety verification of an advanced air traffic control example.

Under free flight, an aircrew has both the freedom to select their trajectory and the responsibility of resolving conflicts with other aircraft. The general belief is that free flight can be made safe under low traffic conditions. Increasing traffic, however, raises safety verification issues. This problem is formulated as one of estimating for a large scale stochastic hybrid system the probability of reaching a small collision set. The huge state space prohibits the use of existing numerical approaches to solve this safety verification problem. As an alternative we study randomization methods, the simplest of which would be to run many Monte Carlo simulations with a stochastic model of free flight operations, and count the number of runs during which a collision between two or more aircraft occurs. The huge state space prohibits such a straightforward MC simulation approach. By exploiting recent particle system theory by Del Moral and co-workers, this paper develops a sequential Monte Carlo simulation approach for the estimation of collision risk in a future air traffic scenario. The working of the resulting particle system is demonstrated for an eight aircraft scenario under free flight air traffic conditions.