Input-to-State Safety with High-Order Control Barrier Functions

Input-to-state safety is a crucial property for evaluating the robustness of a safe set against unknown disturbances. To guarantee the input-to-state safety of the safe set, input-to-state safe control barrier functions (ISSf-CBFs) have been proposed and further extended to a high-order version, termed as input-to-state safe high-order control barrier functions (ISSf-HOCBFs) to handle safety constraints with arbitrary relative degree. However, existing results on ISSf-HOCBFs are limited to matched disturbances, which cannot affect the high-order control barrier function design until the control input is applied. In this paper, we extend the input-tostate safety guarantee from CBFs to HOCBFs under unmacthed disturbances, and propose a new type of ISSf-HOCBFs to handle safety constraints with arbitrary relative degree. A case study of an adaptive cruise control system is provided to demonstrate the effectiveness of the proposed methods.