We address the problem of safe adaptive control for a class of nonlinear systems with dynamical uncertainties, while satisfying control barrier function (CBF) type safety constraints with user-defined risk tolerances at all times. We develop a model reference adaptive control framework that provably guarantees safety in two stages. In the first stage, we design a safe reference model to generate reference trajectories that satisfy CBF-based safety conditions. However, asymptotically tracking a safe reference trajectory does not automatically guarantee safety at every time step. Therefore, in the second stage, we formulate a chance-constrained optimization problem for the nonlinear system with dynamical uncertainties to track the reference model, while provably guaranteeing CBF-based safety constraint satisfaction at each time step up to a user-defined risk bound. We then provide a risk-tunable sampling-based scenario design approach to tune parameterized controllers that solve this optimization problem. In addition, for the special case of linear dynamics, we provide conditions on the uncertainty samples for the existence of controller parameters that can guarantee safe tracking. We illustrate the performance of our framework on a quadcopter navigation problem with obstacle avoidance constraints.
Sharma et al. (Thu,) studied this question.