Safety-Aware Learning-Based Control of Systems with Uncertainty Dependent Constraints

In this paper, we tackle the problem of safely stabilizing an originally (partially) unknown system while ensuring that it does not leave a prescribed ’safe set’ whose structure itself depends on the unknown part of the system’s dynamics. For this aim, we apply 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. However, with the mentioned safety set structure, we witness 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 Gaussian Processes (GPs) and reproducing kernels to address the challenge and show how to modify the CLF-CBF-based approach correspondingly to obtain safety guarantees. 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 discuss the strategy for the control design aim using the derived certificates.