Preventing Unconstrained CBF Safety Filters Caused by Invalid Relative Degree Assumptions

Abstract: Control barrier function (CBF)-based safety filters are used to certify and modify potentially unsafe control inputs to a system such as those provided by a reinforcement learning agent or a non-expert user. In this context, safety is defined as the satisfaction of state constraints. Originally designed for continuous-time systems, CBF safety filters typically assume that the system's relative degree is well-defined and is constant across the domain; however, this assumption is restrictive and rarely verified -- even linear system dynamics with a quadratic CBF candidate may not satisfy this assumption. In real-world applications, continuous-time CBF safety filters are implemented in discrete time, exacerbating issues related to violating the condition on the relative degree. These violations can lead to the safety filter being unconstrained (any control input may be certified) for a finite time interval and result in chattering issues and constraint violations. We propose an alternative formulation to address these challenges. Specifically, we present a theoretically sound method that employs multiple CBFs to generate bounded control inputs at each state within the safe set, thereby preventing incorrect certification of arbitrary control inputs. Using this approach, we derive conditions on the maximum sampling time to ensure safety in discrete-time implementations. We demonstrate the effectiveness of our proposed method through simulations and real-world quadrotor experiments, successfully preventing chattering and constraint violations. Finally, we discuss the implications of violating the relative degree condition on CBF synthesis and learning-based CBF methods.