  This paper addresses learning safe output feedback control laws from partial
observations of expert demonstrations. We assume that a model of the system
dynamics and a state estimator are available along with corresponding error
bounds, e.g., estimated from data in practice. We first propose robust output
control barrier functions (ROCBFs) as a means to guarantee safety, as defined
through controlled forward invariance of a safe set. We then formulate an
optimization problem to learn ROCBFs from expert demonstrations that exhibit
safe system behavior, e.g., data collected from a human operator or an expert
controller. When the parametrization of the ROCBF is linear, then we show that,
under mild assumptions, the optimization problem is convex. Along with the
optimization problem, we provide verifiable conditions in terms of the density
of the data, smoothness of the system model and state estimator, and the size
of the error bounds that guarantee validity of the obtained ROCBF. Towards
obtaining a practical control algorithm, we propose an algorithmic
implementation of our theoretical framework that accounts for assumptions made
in our framework in practice. We validate our algorithm in the autonomous
driving simulator CARLA and demonstrate how to learn safe control laws from
simulated RGB camera images.
