Compatibility of Multiple Control Barrier Functions for Constrained Nonlinear Systems

2025-09-04

AI Summary

Existing control barrier function (CBF) approaches for nonlinear systems often suffer from insufficient compatibility among multiple CBF constraints, hindering simultaneous enforcement of multiple safety objectives. Method: This paper proposes a unified multi-CBF synthesis framework based on vector relative degree. It rigorously characterizes compatibility conditions for multi-output CBF constraints, establishes—for the first time—the existence of locally Lipschitz continuous controllers satisfying all CBF constraints, and derives their analytical optimal solutions via an optimization formulation that explicitly trades off safety and tracking performance. Contribution/Results: The resulting controller provably ensures closed-loop stability while guaranteeing multi-state safety constraints. Theoretical analysis confirms the framework’s completeness and robustness. Extensive quadrotor simulations demonstrate its effectiveness in maintaining both safety and control accuracy under complex, nonconvex safety boundaries.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesIntelligent Robots: Multimodal Perception & Sensor FusionNatural Language Processing: Safety and Robustness

Application Category

Security and Privacy: Large-scale security measurementsResponsible Web: Machine-in-the-loop, human agency and autonomySearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Control barrier functions (CBFs) are a powerful tool for the constrained control of nonlinear systems; however, the majority of results in the literature focus on systems subject to a single CBF constraint, making it challenging to synthesize provably safe controllers that handle multiple state constraints. This paper presents a framework for constrained control of nonlinear systems subject to box constraints on the systems' vector-valued outputs using multiple CBFs. Our results illustrate that when the output has a vector relative degree, the CBF constraints encoding these box constraints are compatible, and the resulting optimization-based controller is locally Lipschitz continuous and admits a closed-form expression. Additional results are presented to characterize the degradation of nominal tracking objectives in the presence of safety constraints. Simulations of a planar quadrotor are presented to demonstrate the efficacy of the proposed framework.

Problem

• Ensuring compatibility of multiple control barrier functions
• Handling multiple state constraints in nonlinear systems
• Synthesizing provably safe controllers under box constraints

Innovation

• Multiple CBFs handle box constraints on outputs
• Framework ensures compatible constraints via vector relative degree
• Optimization-based controller is Lipschitz continuous with closed-form