Analytical Construction of CBF-Based Safety Filters for Simultaneous State and Input Constraints (Extended Version)

TL;DR

This paper presents an analytic safety filter for chained integrator systems that guarantees simultaneous state and input constraints, extending previous maximal invariant set methods with a recursive, easy-to-implement approach.

Contribution

The paper introduces a recursive, analytic safety filter for chained integrator systems that handles multiple simultaneous constraints, offering a practical alternative to maximal invariant set methods.

Findings

Provides an explicit safety filter for single-input, single-output chains.

Extends the safety filter to multi-input, multi-output systems with multiple constraints.

Offers a trade-off between conservatism and simplicity in safety guarantees.

Abstract

We revisit the problem explored in [1] of guaranteeing satisfaction of multiple simultaneous state constraints applied to a single-input, single-output plant consisting of a chain of n integrators subject to input limitations. For this problem setting, we derive an analytic, easy-to-implement safety filter which respects input limitations and ensures forward-invariance of all state constraints simultaneously. Additionally, we provide a straightforward extension to the multi-input, multi-output chained integrator setting, and provide an analytic safety filter guaranteeing satisfaction of arbitrarily many simultaneous hyperplane constraints on the output vector. Whereas the approach in [1] obtains maximal invariant sets, our approach trades off some degree of conservatism in exchange for a recursive safety filter which is analytic for any arbitrary n >= 1.