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

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.

Figures (8)

• Figure 1: Conceptual illustration of hidden infeasibility in standard CBF safety filters when applied to DAE systems. The algebraic constraint manifold $\mathcal{M} = \{x : \phi(x) = 0\}$ imposes an additional compatibility requirement on the control input that is not accounted for by standard CBF conditions, which may render the safety filter infeasible even when the state lies within the safe set $\mathcal{C}$.
• Figure 2: Geometric illustration of Problem \ref{['problem:dae_cbf']}. The state space contains the constraint manifold $\mathcal{M}$, the safe set $\mathcal{C}$, and the candidate barrier set $\mathcal{D}$. The goal is to design a barrier function $b(x)$ and a safe control policy $\mu$ such that $\mathcal{D} \cap \mathcal{M} \subseteq \mathcal{C} \cap \mathcal{M}$ is positively invariant under the closed-loop DAE dynamics \ref{['eq:dyn']}, ensuring that all system trajectories initialized in $\mathcal{D} \cap \mathcal{M}$ evolve on the constraint manifold and remain within the safe set for all $t \ge 0$.
• Figure 3: Simulation results for Example \ref{['exmpl:DAE_unaware_example_1']} using a DAE-unaware CBF. The figure shows (a) the CBF-QP trajectory on the constraint manifold $\phi(x)=0$, (b) the evolution of the barrier function $b(x)$ (with $b(x_0)=0.130$ and a minimum $b(x)=-0.1059$), and (c) the evolution of the safe set function $h(x)$ (a minimum $h(x)=-0.1059$). The CBF-QP becomes infeasible when the CBF inequality conflicts with the algebraic equality constraints, leading to violation of the safety condition $b(x)\ge 0$.
• Figure 4: Roadmap for the derivation of DAE-aware CBF conditions for index-1 systems. The algebraic constraints $\phi(x) = 0$ are differentiated to expose the compatibility condition and the constraint Jacobians $J_d(x)$ and $J_a(x)$. Under the invertibility assumption on $J_a(x)$, the algebraic states are eliminated via the implicit function theorem, and the DAE \ref{['eq:dyn']} is projected onto the constraint manifold $\mathcal{M}$. The CBF condition is then lifted back to the full state space, producing DAE-aware CBF conditions that jointly enforce manifold compatibility and positive invariance of $\mathcal{D} \cap \mathcal{M}$.
• Figure 5: Roadmap for deriving DAE-aware CBF conditions for higher-index systems ($\nu \geq 2$). The algebraic constraint $\phi(x)=0$ is differentiated successively up to order $\nu$, eliminating time derivatives via \ref{['eq:dyn_diff']} at each step, until the control input appears explicitly in the extended constraint system. Full row rank of $J_{ext}^{(\nu)}(x)$ then yields a projected control-affine representation on $\mathcal{M}$, from which the DAE-aware CBF conditions follow analogously to the index-1 case.

Theorems & Definitions (27)

• Definition 1: Differentiation index
• Definition 2: Regularity
• Definition 3: Safety
• Theorem 1: ames2019control
• Definition 4: Valid CBF
• Example 1: Motivating Example
• Definition 5: Lie derivative khalil2002nonlinear
• Definition 6: Relative degree Isidori2013
• Theorem 2: High-Order CBF (HOCBF) xiao2021high
• Theorem 3: Nagumo's Theorem blanchini2008set