Reachability Analysis and its Application to the Safety Assessment of Autonomous Cars

This thesis is about the safety verification of dynamical systems using reachability analysis. Novel solutions have been developed for three major problem classes: Classical reachability analysis, stochastic reachability analysis, and their application to the safety assessment of autonomous cars. Classical reachability analysis aims to compute the exact or over-approximative set of states that can be reached by a system for a given set of initial states, inputs, and parameters. If the reachable set does not intersect any set of unsafe states, the safety of the system is guaranteed – this cannot be achieved by simulation techniques since only single solutions out of infinitely many can be checked. Specialized algorithms have been developed for linear systems, nonlinear systems, and hybrid systems. The concept of reachability analysis is extended to stochastic reachability analysis which measures the probability of reaching an unsafe set. One pursued approach computes overapproximative enclosures of stochastic reachable sets for linear systems. Another developed approach generates a Markov chain which approximately reproduces arbitrary dynamics, allowing the computation of the stochastic reachable set on the simplified dynamics. Finally, stochastic reachable sets are applied to the safety assessment of autonomous cars. Autonomous cars are driverless, i.e. they drive automated according to a planned trajectory which is computed based on detected roads, obstacles, and traffic participants. The safety assessment relies on a prediction of traffic scenes, where the stochastic reachable set of each relevant traffic participant is computed by the developed abstraction by Markov chains. This allows the crash probability of the autonomous car to be predicted for a planned trajectory.