Adaptive Safety with a RISE-Based Disturbance Observer

Ensuring the safety (formally, forward invariance) of control systems using control barrier functions typically requires being overly conservative when the dynamics of the control system are uncertain. In this work, we develop a disturbance observer based on the robust integral of the sign of the error (RISE) control paradigm that exponentially identifies the unknown dynamics when certain gain conditions are satisfied. The estimate of the dynamics is used in an optimization-based control law that ensures safety while expanding the operating region of the dynamical system as the dynamic model is identified. We provide conditions for when the control law is locally Lipschitz continuous. The RISE-based disturbance observer can provide safety guarantees to a secondary model of the dynamics of unknown accuracy. A simulation example is provided to demonstrate the performance of the disturbance observer and to illustrate the benefits of combining the observer with a secondary model given by a pretrained deep neural network.