When are safety filters safe? On minimum phase conditions of control barrier functions

Abstract: In emerging control applications involving multiple and complex tasks, safety filters are gaining prominence as a modular approach to enforcing safety constraints. Among various methods, control barrier functions (CBFs) are widely used for designing safety filters due to their simplicity, imposing a single linear constraint on the control input at each state. In this work, we focus on the internal dynamics of systems governed by CBF-constrained control laws. Our key observation is that, although CBFs guarantee safety by enforcing state constraints, they can inadvertently be "unsafe" by causing the internal state to diverge. We investigate the conditions under which the full system state, including the internal state, can remain bounded under a CBF-based safety filter. Drawing inspiration from the input-output linearization literature, where boundedness is ensured by minimum phase conditions, we propose a new set of CBF minimum phase conditions tailored to the structure imposed by the CBF constraint. A critical distinction from the original minimum phase conditions is that the internal dynamics in our setting is driven by a nonnegative virtual control input, which reflects the enforcement of the safety constraint. We include a range of numerical examples, including single-input, multi-input, linear, and nonlinear systems, validating both our analysis and the necessity of the proposed CBF minimum phase conditions.