Barrier certificate explained

A barrier certificate^[1] or barrier function is used to prove that a given region is forward invariant for a given ordinary differential equation or hybrid dynamical system.^[2] That is, a barrier function can be used to show that if a solution starts in a given set, then it cannot leave that set.

Showing that a set is forward invariant is an aspect of safety, which is the property where a system is guaranteed to avoid obstacles specified as an unsafe set.

Barrier certificates play the analogical role for safety to the role of Lyapunov functions for stability. For every ordinary differential equation that robustly fulfills a safety property of a certain type there is a corresponding barrier certificate.^[3]

History

The first result in the field of barrier certificates was the Nagumo theorem by Mitio Nagumo in 1942.^[4] The term "barrier certificate" was introduced later based on similar concept in convex optimization called barrier functions.

Barrier certificates were generalized to hybrid systems in 2004 by Stephen Prajna and Ali Jadbabaie.

Variants

There are several different types of barrier functions. One distinguishing factor is the behavior of the barrier function at the boundary of the forward invariant set

S . A barrier function that goes to zero as the input approaches the boundary of S is called a zeroing barrier function.^[5] A barrier function that goes to infinity as the inputs approach the boundary of S are called reciprocal barrier functions.^[5] Here, "reciprocal" refers to the fact that a reciprocal barrier functions can be defined as the multiplicative inverse of a zeroing barrier function.

Notes and References

1. Prajna, Stephen, and Ali Jadbabaie. "Safety verification of hybrid systems using barrier certificates." International Workshop on Hybrid Systems: Computation and Control. Springer, Berlin, Heidelberg, 2004.
2. ((Maghenem, M.)), ((Sanfelice, R. G.)) . Automatica . Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions . 124 . 109328 . February 2021 . 00051098 . 10.1016/j.automatica.2020.109328. 1908.03980 .
3. Stefan Ratschan: "Converse Theorems for Safety and Barrier Certificates".IEEE Trans. on Automatic Control, Volume 63, Issue 8, 2018
4. . English translation in 2406.18614 . Menner . Marcel . Lavretsky . Eugene . Translation of Nagumo's Foundational Work on Barrier Functions: On the Location of Integral Curves of Ordinary Differential Equations.
5. ((Ames, A. D.)), ((Xu, X.)), ((Grizzle, J. W.)), ((Tabuada, P.)) . IEEE Transactions on Automatic Control . Control Barrier Function Based Quadratic Programs for Safety Critical Systems . 62 . 8 . 3861–3876 . August 2017 . 1558-2523 . 10.1109/TAC.2016.2638961. free . 1609.06408 .