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Control Barrier Functions (CBFs) aim to ensure safety by constraining thecontrol input at each time step so that the system state remains within adesired safe region. This paper presents a framework for CBFs in stochasticsystems in the presence of Gaussian process and measurement noise. We firstconsider the case where the system state is known at each time step, andpresent reciprocal and zero CBF constructions that guarantee safety withprobability 1. We extend our results to high relative degree systems withlinear dynamics and affine safety constraints. We then develop CBFs forincomplete state information environments, in which the state must be estimatedusing sensors that are corrupted by Gaussian noise. We prove that our proposedCBF ensures safety with probability 1 when the state estimate is within a givenbound of the true state, which can be achieved using an Extended Kalman Filterwhen the system is linear or the process and measurement noise are sufficientlysmall. We propose control policies that combine these CBFs with ControlLyapunov Functions in order to jointly ensure safety and stochastic stability.Our results are validated via numerical study on a multi-agent collisionavoidance example.

Control Barrier Functions for Stochastic Systems