High-Order Control Barrier Functions: Safety can lead to instability

This paper provides a benchmark example where a standard control system is modified with state-of-the-art safety guarantees implemented via a High-Order Control Barrier Function as safety filter. We will demonstrate that this modification ensures safety but induces unstable behaviour. In particular, we show that this instability is induced by a persistent switching between the original closed-loop dynamics and the CBF dynamics. Stability conditions will be needed to ensure that safety does not damage the stability of the closed-loop system.