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Safe Control Synthesis Using Environmentally Robust Control Barrier Functions

Vahid Hamdipoor, Nader Meskin, Christos G. Cassandras

TL;DR

This work addresses safe control for dynamical systems operating in dynamically changing environments by introducing Environmentally Robust Control Barrier Functions (ER-CBFs) that account for worst-case environmental uncertainty. It demonstrates how ER-CBF constraints can be formulated as a SOCP, and further shows that using the nominal safe input as a reference leads to a QP-based robust controller with minimal modification, enabling a closed-form or near-closed-form solution. The approach is validated on an adaptive cruise control example, where nominal CBFs alone fail safety under uncertainty, but ER-CBF-SOCP and ER-CBF-QP maintain forward invariance of the safe set with real-time solvability. The results indicate that ER-CBF-QP offers a faster, more feasible alternative with slightly more conservative behavior, making robust safety practical for autonomous and mixed-traffic scenarios.

Abstract

In this paper, we study a safe control design for dynamical systems in the presence of uncertainty in a dynamical environment. The worst-case error approach is considered to formulate robust Control Barrier Functions (CBFs) in an optimization-based control synthesis framework. It is first shown that environmentally robust CBF formulations result in second-order cone programs (SOCPs). Then, a novel scheme is presented to formulate robust CBFs which takes the nominally safe control as its desired control input in optimization-based control design and then tries to minimally modify it whenever the robust CBF constraint is violated. This proposed scheme leads to quadratic programs (QPs) which can be easily solved. Finally, the effectiveness of the proposed approach is demonstrated on an adaptive cruise control example.

Safe Control Synthesis Using Environmentally Robust Control Barrier Functions

TL;DR

This work addresses safe control for dynamical systems operating in dynamically changing environments by introducing Environmentally Robust Control Barrier Functions (ER-CBFs) that account for worst-case environmental uncertainty. It demonstrates how ER-CBF constraints can be formulated as a SOCP, and further shows that using the nominal safe input as a reference leads to a QP-based robust controller with minimal modification, enabling a closed-form or near-closed-form solution. The approach is validated on an adaptive cruise control example, where nominal CBFs alone fail safety under uncertainty, but ER-CBF-SOCP and ER-CBF-QP maintain forward invariance of the safe set with real-time solvability. The results indicate that ER-CBF-QP offers a faster, more feasible alternative with slightly more conservative behavior, making robust safety practical for autonomous and mixed-traffic scenarios.

Abstract

In this paper, we study a safe control design for dynamical systems in the presence of uncertainty in a dynamical environment. The worst-case error approach is considered to formulate robust Control Barrier Functions (CBFs) in an optimization-based control synthesis framework. It is first shown that environmentally robust CBF formulations result in second-order cone programs (SOCPs). Then, a novel scheme is presented to formulate robust CBFs which takes the nominally safe control as its desired control input in optimization-based control design and then tries to minimally modify it whenever the robust CBF constraint is violated. This proposed scheme leads to quadratic programs (QPs) which can be easily solved. Finally, the effectiveness of the proposed approach is demonstrated on an adaptive cruise control example.
Paper Structure (7 sections, 6 theorems, 44 equations, 5 figures, 1 table)

This paper contains 7 sections, 6 theorems, 44 equations, 5 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Problem Statement
  4. Results
  5. Adaptive cruise control example
  6. Simulation Results
  7. conclusion

Key Result

Theorem 1

Given a CBF $h(x,t)$ as in Definition (def:CBF) with the associated set $\mathcal{C}$, any Lipschitz continuous controller $u(t)$ that satisfies (eq:CBF), $\forall t \geq 0$ renders $\mathcal{C}$ forward invariant for system (eq:system).

Figures (5)

  • Figure 1: Adaptive cruise control example for mixed traffic where ego vehicle is AV following the lead vehicle which is an HDV.
  • Figure 2: CBF and its uncertainty bound for keeping safe distance in cruise control using nominal CBF (top), robust CBF with SOCP (middle), and robust CBF with QP (bottom).
  • Figure 3: Velocity and distance of ego and lead vehicles in the adaptive cruise control along with the control input for ego vehicle using nominal CBF, robust CBF with SOCP, and robust CBF with QP.
  • Figure 4: CBF for keeping safe distance with lead vehicle and CLF for achieving desired distance with lead vehicle in cruise control using nominal CBF, robust CBF with SOCP, and robust CBF with QP.
  • Figure 5: Obtaining robust control input via closed form solution of robust QP.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1: ames2019control
  • Definition 5
  • Theorem 2
  • proof
  • Lemma 1
  • Theorem 3
  • ...and 7 more