Probabilistic reach-Avoid problems in nondeterministic systems with time-Varying targets and obstacles

Abstract

The probabilistic reachability problem, which involves the computation of probabilistic reachable sets, is studied for nondeterministic systems. In the existing works, the system evolution initialized from the probabilistic reachable set is required to reach the target set with a certain probability in a given time horizon. In this paper, the definition of probabilistic reachable sets is refined by taking into account time-varying target set and obstacle. In the context of this definition, the evolution of the system is required not only to reach the target set but also to avoid obstacle. We address two distinct interpretations of probabilistic reachability problem via dynamic planning. In the first case, the control policy is given. In the second case, the control policy is a parameter to be optimized. A numerical method is proposed to compute probabilistic reachable sets. First, a scalar function in the state space is constructed by backward recursion and grid interpolation, and then the probabilistic reachable set is represented as a nonzero upper level set of this scalar function. In addition, based on the constructed scalar function, the optimal control policy can be designed. Two examples are provided at the end of this article. The first example consists of a one-dimensional problem where the analytical solution is compared with the results of the proposed method in order to analyze the computational accuracy. The second example, which contains a three-dimensional problem, is used to demonstrate the effectiveness of the proposed method.