Probabilistic Reachability Analysis for Large Scale Stochastic Hybrid Systems

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

Sequential Monte Carlo simulation for the estimation of small reachability probabilities for stochastic hybrid systems

The problem of estimating the probability that a system reaches a given set within some time horizon is considered. Standard Monte Carlo methods for reachability probability estimation do not require specific assumptions on the system under consideration. However, they are computationally demanding when the probability to be estimated is small. An Interacting Particle System (IPS) approach has been developed for the estimation of small reachability probabilities for diffusion processes. IPS has then been extended so as to estimate small reachability probabilities for a certain family of stochastic hybrid processes, namely switching diffusions. This contribution further improves the hybrid IPS method by adopting an importance sampling approach that uses the interaction equations characterizing stochastic hybrid systems

Sequential Monte Carlo simulation of collision risk in free flight air traffic

Within HYBRIDGE a novel approach in speeding up Monte Carlo simulation of rare events has been developed. In the current report this method is extended for application to simulating collisions with a stochastic dynamical model of an air traffic operational concept. Subsequently this extended Monte Carlo simulation approach is applied to a simulation model of an advanced free flight operational concept; i.e. one in which aircraft are responsible for self separation with each other. The Monte Carlo simulation results obtained for this advanced concept show that the novel method works well, and that it allows studying rare events that stayed invisible in previous Monte Carlo simulations of advanced air traffic operational concepts

A Particle System for Safety Verification of Free Flight in Air Traffic

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

Estimating Rare Event Probabilities in Large Scale Stochastic Hybrid Systems by Sequential Monte Carlo Simulation

We study the problem of estimating small reachability probabilities for large scale stochastic hybrid processes through Sequential Monte Carlo (SMC) simulation. Recently, [Cerou et al., 2002, 2005] developed an SMC approach for diffusion processes, and referred to the resulting SMC algorithm as an Interacting Particle System (IPS). In [Krystul&Blom, 2004, 2005] it was shown that this IPS approach works very well for a diffusion example, but has its limits when applied to a switching diffusion with large differences in discrete state (mode) probabilities or with rare mode switching. In order to cope with these problems, in [Krystul&Blom, 2004, 2005, 2006] the IPS approach has been extended to Hybrid IPS (HIPS) versions. Unfortunately, these HIPS versions may need impractically many particles when the space of the discrete state component is very large. Such situation typically occurs when the stochastic process considered is highly distributed and incorporates many local discrete valued switching processes. Then the vector of local discrete valued components has a state space the size of which is exponentially large. The aim of the current work is formulate the estimation of extremely small rare event probabilities in stochastic hybrid systems with a large state space for the discrete valued process component into one of a hierarchical estimation process, and to use this for the derivation of a Hierarchical HIPS version. The effectiveness of the approach is illustrated for evaluating the risk of collision between two aircraft in a scenario of the future

A Particle System for Safety Verification of Free Flight in Air Traffic

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

Authors H.A.P. Blom NLR

WP9: Risk assessment for a distributed control syste