Verification and Forward Invariance of Control Barrier Functions for Differential-Algebraic Systems

2026-03-13

AI Summary

This work proposes the first control barrier function (CBF) framework tailored for differential-algebraic equation (DAE) systems, addressing the challenge that algebraic constraints pose to existing CBF methods in simultaneously ensuring safety and constraint consistency. By integrating the differential-algebraic structure through projected vector fields, the approach guarantees forward invariance of safe sets while strictly satisfying algebraic constraints, even for high-index DAEs. The method establishes a DAE-aware CBF theory, providing necessary and sufficient conditions for geometric correctness and feasibility. Verification for polynomial systems is enabled via sum-of-squares (SOS) optimization, while falsification of non-polynomial or neural network candidate functions is supported through SMT solvers. Experimental validation on wind turbines and flexible-link robotic arms demonstrates the framework’s effectiveness in concurrently enforcing safety and algebraic constraints.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Differential-algebraic equations (DAEs) arise in power networks, chemical processes, and multibody systems, where algebraic constraints encode physical conservation laws. The safety of such systems is critical, yet safe control is challenging because algebraic constraints restrict allowable state trajectories. Control barrier functions (CBFs) provide computationally efficient safety filters for ordinary differential equation (ODE) systems. However, existing CBF methods are not directly applicable to DAEs due to potential conflicts between the CBF condition and the constraint manifold. This paper introduces DAE-aware CBFs that incorporate the differential-algebraic structure through projected vector fields. We derive conditions that ensure forward invariance of safe sets while preserving algebraic constraints and extend the framework to higher-index DAEs. A systematic verification framework is developed, establishing necessary and sufficient conditions for geometric correctness and feasibility of DAE-aware CBFs. For polynomial systems, sum-of-squares certificates are provided, while for nonpolynomial and neural network candidates, satisfiability modulo theories are used for falsification. The approach is validated on wind turbine and flexible-link manipulator systems.

Problem

• Control Barrier Functions
• Differential-Algebraic Equations
• Forward Invariance
• Algebraic Constraints
• Safety Verification

Innovation

• Control Barrier Functions
• Differential-Algebraic Equations
• Forward Invariance
• Projected Vector Fields
• Sum-of-Squares