Control Barrier Functions for Complete and Incomplete Information Stochastic Systems

Abstract

Real-time controllers must satisfy strict safety requirements. Recently, Control Barrier Functions (CBFs) have been proposed that guarantee safety by ensuring that a suitably-defined barrier function remains bounded for all time. The CBF method, however, has only been developed for deterministic systems and systems with worst-case disturbances and uncertainties. In this paper, we develop a CBF framework for safety of stochastic systems. We consider complete information systems, in which the controller has access to the exact system state, as well as incomplete information systems where the state must be reconstructed from noisy measurements. In the complete information case, we formulate a notion of barrier functions that leads to sufficient conditions for safety with probability 1. In the incomplete information case, we formulate barrier functions that take an estimate from an extended Kalman filter as input, and derive bounds on the probability of safety as a function of the asymptotic error in the filter. We show that, in both cases, the sufficient conditions for safety can be mapped to linear constraints on the control input at each time, enabling the development of tractable optimization-based controllers that guarantee safety, performance, and stability. Our approach is evaluated via simulation study on an adaptive cruise control case study.