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Safety-Critical Control via Recurrent Tracking Functions

Jixian Liu, Enrique Mallada

TL;DR

This work tackles safety-critical control for high-order nonlinear systems where constructing valid CBFs is intractable. It introduces Recurrent Tracking Functions (RTFs) that replace strict Lyapunov decay with finite-time recurrence, enabling transient tracking deviations while preserving safety. By augmenting reduced-order-model CBFs with RTFs, it constructs Recurrent CBFs (RCBFs) with $h_V(z,\dot e) = -V(z,\dot e) + \alpha_e h(z)$ and a recurrent safe set $S_V$ that guarantees FoM safety when $\beta > \alpha$; this holds even under disturbances via an ISS-based robustness margin. The approach is validated through a 2D double-integrator case study showing safe FoM behavior and exponential tracking, illustrating practical viability for layered RoM-FoM control and suggesting data-driven avenues for constructing RTFs. Overall, the recurrence-based framework provides a scalable, theory-backed pathway to safety certification in complex safety-critical systems.

Abstract

This paper addresses the challenge of synthesizing safety-critical controllers for high-order nonlinear systems, where constructing valid Control Barrier Functions (CBFs) remains computationally intractable. Leveraging layered control, we design CBFs in reduced-order models (RoMs) while regulating full-order models' (FoMs) dynamics at the same time. Traditional Lyapunov tracking functions are required to decrease monotonically, but systematic synthesis methods for such functions exist only for fully-actuated systems. To overcome this limitation, we introduce Recurrent Tracking Functions (RTFs), which replace the monotonic decay requirement with a weaker finite-time recurrence condition. This relaxation permits transient deviations of tracking errors while ensuring safety. By augmenting CBFs for RoMs with RTFs, we construct recurrent CBFs (RCBFs) whose zero-superlevel set is control $τ$-recurrent, and guarantee safety for all initial states in such a set when RTFs are satisfied. We establish theoretical safety guarantees and validate the approach through numerical experiments, demonstrating RTFs' effectiveness and the safety of FoMs.

Safety-Critical Control via Recurrent Tracking Functions

TL;DR

This work tackles safety-critical control for high-order nonlinear systems where constructing valid CBFs is intractable. It introduces Recurrent Tracking Functions (RTFs) that replace strict Lyapunov decay with finite-time recurrence, enabling transient tracking deviations while preserving safety. By augmenting reduced-order-model CBFs with RTFs, it constructs Recurrent CBFs (RCBFs) with and a recurrent safe set that guarantees FoM safety when ; this holds even under disturbances via an ISS-based robustness margin. The approach is validated through a 2D double-integrator case study showing safe FoM behavior and exponential tracking, illustrating practical viability for layered RoM-FoM control and suggesting data-driven avenues for constructing RTFs. Overall, the recurrence-based framework provides a scalable, theory-backed pathway to safety certification in complex safety-critical systems.

Abstract

This paper addresses the challenge of synthesizing safety-critical controllers for high-order nonlinear systems, where constructing valid Control Barrier Functions (CBFs) remains computationally intractable. Leveraging layered control, we design CBFs in reduced-order models (RoMs) while regulating full-order models' (FoMs) dynamics at the same time. Traditional Lyapunov tracking functions are required to decrease monotonically, but systematic synthesis methods for such functions exist only for fully-actuated systems. To overcome this limitation, we introduce Recurrent Tracking Functions (RTFs), which replace the monotonic decay requirement with a weaker finite-time recurrence condition. This relaxation permits transient deviations of tracking errors while ensuring safety. By augmenting CBFs for RoMs with RTFs, we construct recurrent CBFs (RCBFs) whose zero-superlevel set is control -recurrent, and guarantee safety for all initial states in such a set when RTFs are satisfied. We establish theoretical safety guarantees and validate the approach through numerical experiments, demonstrating RTFs' effectiveness and the safety of FoMs.

Paper Structure

This paper contains 15 sections, 5 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Layered Control System
  4. Problem Statement
  5. Control Barrier Function for RoM
  6. Recurrent Lyapunov and Control Barrier Functions
  7. Layered Safety-Critical Control via Recurrence
  8. Recurrent Tracking Function
  9. Safety Assessment
  10. Effect of Disturbances
  11. Case Study
  12. Conclusion and Future Work
  13. Acknowledgments
  14. Proof of Theorem \ref{['theo:RCBF_set']}
  15. Proof of Theorem \ref{['theo:safe_assess']}

Key Result

Theorem 1

As a direct consequence of Definition def:CBF, any Lipschitz-continuous controller $k(z)$ that satisfies renders the set $h_{\ge 0}:=\{z \mid h(z) \ge 0\}$ forward invariant. In particular, $h_{\ge0}$ is control invariant.

Figures (2)

  • Figure 1: (a) $V, h, S_V$ and a safe trajectory's projection when $n =1$. (b) Time evolution of V. (c) Time evolution of h.
  • Figure 2: (a) 2D path with circular obstacles. (b) Barrier value $h$ vs. time for the three $\alpha$. (c) Speeds for $\alpha=0.5$ of $\|\dot z_d\|$, $\|\dot z_s\|$, and $\|\dot z\|$. (d) Tracking-error speed $\|\dot e\|$ for $\alpha=0.5$.

Theorems & Definitions (26)

  • Definition 1: Full-Order Model
  • Definition 2: Reduced-order model
  • Definition 3: Safe state of the RoM
  • Definition 4: Safe state of the FoM
  • Definition 5: Extended Class-$\mathcal{K}$ Function
  • Definition 6: Control Barrier Function acenst2019ecc
  • Theorem 1: acenst2019ecc
  • Definition 7: Exponential Stability
  • Definition 8: Reachable Tube
  • Definition 9: Containment Times
  • ...and 16 more