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

Peter A. Fisher, Anuradha M. Annaswamy

TL;DR

This work addresses guaranteeing forward-invariance of multiple state constraints for a single-input, $n$-th order integrator under input saturation by constructing analytic, recursive control barrier-function (CBF) based safety filters. The authors derive a family of square-root based CBFs that yield implementable safety filters for any $n\ge1$, complemented by a finite-parameter tuning algorithm that ensures feasibility and forward-invariance of a composite safe set. They extend the approach to a multi-input setting, discuss implementability conditions, and validate the method through simulations on a linearized quadrotor model navigating a non-convex environment. The results demonstrate that the analytic safety filters enforce safety without relying on expensive Hamilton–Jacobi or SOS computations, offering a scalable, practical tool for safety-critical control under input constraints.

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.

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

TL;DR

This work addresses guaranteeing forward-invariance of multiple state constraints for a single-input, -th order integrator under input saturation by constructing analytic, recursive control barrier-function (CBF) based safety filters. The authors derive a family of square-root based CBFs that yield implementable safety filters for any , complemented by a finite-parameter tuning algorithm that ensures feasibility and forward-invariance of a composite safe set. They extend the approach to a multi-input setting, discuss implementability conditions, and validate the method through simulations on a linearized quadrotor model navigating a non-convex environment. The results demonstrate that the analytic safety filters enforce safety without relying on expensive Hamilton–Jacobi or SOS computations, offering a scalable, practical tool for safety-critical control under input constraints.

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.
Paper Structure (28 sections, 7 theorems, 89 equations, 4 figures, 1 algorithm)

This paper contains 28 sections, 7 theorems, 89 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. A Useful Technical Lemma
  5. Problem Statements
  6. Solution to Problem \ref{['plm:simplified']}
  7. Intuition: Why is Square Root the Magic Function?
  8. Solution to Problem \ref{['plm:full']}
  9. Parameter Selection
  10. Main Result
  11. Extension to the Multi-Input Setting
  12. Extended Problem Statement
  13. Solution Setup for Problem \ref{['plm:extended']}
  14. Discussion of Implementability
  15. Simulation: Quadrotor Safety
  16. ...and 13 more sections

Key Result

Lemma 1

Consider a collection of sets $S_1, \dots, S_p \subset \mathbb{R}^n$ defined by smooth functions $h_1, \dots, h_p : \mathbb{R}^n \to \mathbb{R}$ as in eqn:S-eqn:S_interior, and define the safe set $\overline{S} = \bigcap_{\ell=1}^p S_\ell$. Suppose that, for some choice of $\mathbf{b}_k$ and $c_k$,

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (24)

  • Definition 1: ames2019CBF
  • Definition 2: ames2019CBF
  • Definition 3: agrawal2021ICCBFs
  • Definition 4: ames2019CBF
  • Definition 5: ames2019CBF
  • Definition 6
  • Definition 7
  • Remark 1
  • Lemma 1
  • proof
  • ...and 14 more