On the Correspondence Between Output Limiting Backstepping and Higher-Order Control Barrier Functions

Output Limiting Backstepping (OLB) and Higher-Order Control Barrier Functions (HOCBFs) both protect constrained system outputs through recursive constructions. This paper investigates their mathematical relationship and identifies conditions under which they yield identical limiting control inputs. The OLB formulation considered here requires a strict-feedback structure, whereas HOCBFs apply to a broader class of systems. For systems admitting both methods, the analysis shows how the HOCBF recursion corresponds to OLB phase-plane boundaries and tracking errors. It further establishes how to choose the safety function and design gains so that both methods yield identical limiting control inputs for relative degrees two and three. Aircraft angle-of-attack protection simulations confirm the correspondence and show that neglecting actuator dynamics can cause oscillations or instability under aggressive protection settings. The resulting correspondence provides a physical interpretation of HOCBF design parameters through OLB phase-plane limits and can guide HOCBF design for systems to which both methods apply.