BLF families for arbitrary class- comparison rates

Develop a Barrier Lyapunov Function family beyond the logarithmic and power forms that recovers an arbitrary class- comparison rate in the vanishing-clearance-margin limit.

Background

The paper studies a Barrier Lyapunov Function safety filter for continuous-time transformer dynamics. Its logarithmic and power BLFs yield a particular limiting comparison rate as the prescribed clearance margin tends to zero, recovering an exponential second-order control-barrier-function condition.

The authors explicitly note that obtaining an arbitrary class- comparison rate would require a more general family of BLFs than the two forms analyzed. This remains unresolved in the paper.