Safety-Aware Learning-Based Control of Systems with Uncertainty Dependent Constraints (extended version)

The problem of safely learning and controlling a dynamical system - i.e., of stabilizing an originally (partially) unknown system while ensuring that it does not leave a prescribed 'safe set' - has recently received tremendous attention in the controls community. Further complexities arise, however, when the structure of the safe set itself depends on the unknown part of the system's dynamics. In particular, a popular approach based on control Lyapunov functions (CLF), control barrier functions (CBF) and Gaussian processes (to build confidence set around the unknown term), which has proved successful in the known-safe set setting, becomes inefficient as-is, due to the introduction of higher-order terms to be estimated and bounded with high probability using only system state measurements. In this paper, we build on the recent literature on GPs and reproducing kernels to perform this latter task, and show how to correspondingly modify the CLF-CBF-based approach to obtain safety guarantees. Namely, we derive exponential CLF and second relative order exponential CBF constraints whose satisfaction guarantees stability and forward in-variance of the partially unknown safe set with high probability. To overcome the intractability of verification of these conditions on the continuous domain, we apply discretization of the state space and use Lipschitz continuity properties of dynamics to derive equivalent CLF and CBF certificates in discrete state space. Finally, we present an algorithm for the control design aim using the derived certificates.