Safety assessment for stochastic linear systems using enclosing hulls of probability density functions

This contribution addresses the problem of computing the probability that a linear system with uncertain inputs reaches an unsafe region in the state space. The probability of reaching unsafe states is bounded by an upper limit, by intersecting enclosing hulls of the probability density function for time intervals with the specified unsafe regions. Disturbances are modeled as white Gaussian noise, whose mean is modeled as uncertain within specified sets. This allows to consider white non-Gaussian noise in an over-approximative manner. The presented approach is efficient in the sense that high-dimensional systems can be handled.

I. Introduction

In recent years, the growing complexity of technical systems together with an increasing demand for the compliance with system requirements has driven the development of algorithmic verification methods. A focus was to investigate techniques for verifying safety properties of systems with deterministic dynamics. In many applications, however, a pure yes/no-decision about safety is unsatisfactory, because a strict safety requirement for a system with partly uncertain dynamics may severely limit the system's operability or availability; examples are scenarios in air traffic [1] or road traffic [2]. This paper hence investigates stochastic verification techniques that lead to a probability for the safe execution of linear systems. The verification is performed algorithmically by computing enclosing hulls of probability density functions, and determining the probability of intersecting with unsafe sets.