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Optimization-Free Constrained Control with Guaranteed Recursive Feasibility: A CBF-Based Reference Governor Approach

Satoshi Nakano, Emanuele Garone, Gennaro Notomista

Abstract

This letter presents a constrained control framework that integrates Explicit Reference Governors (ERG) with Control Barrier Functions (CBF) to ensure recursive feasibility without online optimization. We formulate the reference update as a virtual control input for an augmented system, governed by a smooth barrier function constructed from the softmin aggregation of Dynamic Safety Margins (DSMs). Unlike standard CBF formulations, the proposed method guarantees the feasibility of safety constraints by design, exploiting the forward invariance properties of the underlying Lyapunov level sets. This allows for the derivation of an explicit, closed-form reference update law that strictly enforces safety while minimizing deviation from a nominal reference trajectory. Theoretical results confirm asymptotic convergence, and numerical simulations demonstrate that the proposed method achieves performance comparable to traditional ERG frameworks.

Optimization-Free Constrained Control with Guaranteed Recursive Feasibility: A CBF-Based Reference Governor Approach

Abstract

This letter presents a constrained control framework that integrates Explicit Reference Governors (ERG) with Control Barrier Functions (CBF) to ensure recursive feasibility without online optimization. We formulate the reference update as a virtual control input for an augmented system, governed by a smooth barrier function constructed from the softmin aggregation of Dynamic Safety Margins (DSMs). Unlike standard CBF formulations, the proposed method guarantees the feasibility of safety constraints by design, exploiting the forward invariance properties of the underlying Lyapunov level sets. This allows for the derivation of an explicit, closed-form reference update law that strictly enforces safety while minimizing deviation from a nominal reference trajectory. Theoretical results confirm asymptotic convergence, and numerical simulations demonstrate that the proposed method achieves performance comparable to traditional ERG frameworks.

Paper Structure

This paper contains 14 sections, 5 theorems, 31 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION
  3. PRELIMINARIES
  4. Control Barrier Functions and Softmin Aggregation
  5. Dynamic Safety Margins
  6. REFERENCE GOVERNOR DESIGN
  7. Steady-State Admissibility
  8. Gradient-Based Reference Update
  9. Safety and Convergence Analysis
  10. Discussion and Comparison
  11. Comparison with CBF and ERG frameworks
  12. Parameter selection and implementation
  13. SIMULATION RESULTS
  14. CONCLUSION

Key Result

Proposition 1

For any $(x,g)$ such that $H(x,g) \ge 0$, the choice $\rho = 0$ satisfies the CBF constraint eq:H_linear_constraint, i.e., $b_H(x,g) \ge 0$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure E1: Simulation results for the 2-DOF planar manipulator navigating around a static circular obstacle using the proposed ERG-CBF framework.

Theorems & Definitions (12)

  • Definition 1: Reference-dependent Lyapunov function 11006078freire2025a
  • Definition 2: Dynamic Safety Margin for constraint $i$
  • Definition 3: Steady-state admissible reference
  • Proposition 1: Feasibility of the trivial update
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 1: Safety and convergence to stationary points
  • proof
  • ...and 2 more