⚡ electrical engineering

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

This paper introduces DAE-aware Control Barrier Functions that incorporate projected vector fields to ensure forward invariance of safe sets while respecting algebraic constraints in differential-algebraic systems, supported by a systematic verification framework and validated on wind turbine and manipulator applications.

Original authors: Hongchao Zhang, Mohamad H. Kazma, Meiyi Ma, Taylor T. Johnson, Ahmad F. Taha

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hongchao Zhang, Mohamad H. Kazma, Meiyi Ma, Taylor T. Johnson, Ahmad F. Taha

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are driving a car, but this isn't a normal car. It's a magical car that must obey two sets of rules simultaneously:

The Physics Rules: How the engine, steering, and brakes work (the differential equations).
The Invisible Tracks: The car is magically glued to a specific, invisible wire track on the ground. It cannot leave this track, no matter how hard you try (the algebraic constraints).

Your goal is to drive this car safely, avoiding a cliff (the "safe set").

The Problem: The Old Way of Driving

In the past, engineers used a "Safety Filter" (called a Control Barrier Function or CBF) to keep cars safe. Think of this filter as a smart co-pilot.

You tell the co-pilot, "I want to turn left."
The co-pilot checks: "If you turn left, will you hit the cliff?"
If yes, the co-pilot gently steers you away from the cliff.

But here's the catch: This old co-pilot only looked at the cliff. It didn't know about the invisible wire track.

If the co-pilot told you to turn left to avoid the cliff, but that turn would pull you off the invisible wire track, the car would crash.
In the real world (like power grids or chemical plants), these "invisible tracks" are laws of physics (like conservation of energy). If you ignore them, the math breaks, the system fails, and safety is lost.

The paper calls this the "Hidden Infeasibility" problem. The safety filter says, "I can save you!" but the physics says, "No, you can't move that way!" The result? The car freezes or crashes.

The Solution: The "Track-Aware" Co-Pilot

This paper introduces a new kind of co-pilot called a DAE-Aware CBF.

Imagine this new co-pilot has a special pair of glasses.

It sees the cliff (Safety).
It sees the invisible wire track (Algebraic Constraints).

When you ask to turn left, this new co-pilot doesn't just ask, "Is the cliff safe?" It asks, "Is the cliff safe AND can I stay on the wire track while doing it?"

If turning left would pull you off the track, the co-pilot knows it can't do that. Instead, it calculates a new, slightly different turn that keeps you safe from the cliff and glued to the track. It essentially projects your movement onto the track before making a decision.

How They Built It (The "Magic" Tricks)

The authors had to solve two hard puzzles to make this co-pilot work:

1. The "Index" Puzzle (The Depth of the Track)
Some tracks are simple (Index-1). If you pull the car, it moves immediately.
Other tracks are complex (High-Index). If you pull the car, nothing happens for a second, then it jerks, then it moves. The "pull" takes a few steps to show up.

The Paper's Fix: They figured out how to count these "steps" (differentiation index). They built a system that looks ahead multiple steps to see how your steering will eventually affect the track. This allows them to handle even the most stubborn, complex tracks.

2. The "Proof" Puzzle (Verifying the Co-pilot)
Before letting this new co-pilot drive a real nuclear plant or a wind turbine, you need to be 100% sure it won't make a mistake.

For Simple Math: They used a method called Sum-of-Squares (SOS). Think of this as a super-precise calculator that proves, mathematically, that the co-pilot's logic is perfect for every possible scenario.
For Complex/Neural Networks: Sometimes the co-pilot is a "black box" (like a neural network). You can't just calculate the proof. Instead, they used a "falsification" method (SMT). This is like a hacker trying to break the co-pilot. They threw millions of crazy driving scenarios at it. If the hacker couldn't find a single crash, they declared the co-pilot safe.

Real-World Tests

They tested this new system on two real-world "magical cars":

A Wind Turbine: The blades must spin within a safe range, but the electricity flow (the track) must stay balanced. The old co-pilot tried to save the blades but broke the electricity balance. The new co-pilot saved both.
A Flexible Robot Arm: The arm must reach a target without hitting a wall, but its joints are physically linked (the track). The new co-pilot successfully moved the arm safely without breaking the physical links.

The Big Takeaway

This paper is like upgrading a GPS navigation system.

Old GPS: "Avoid the red zone!" (But it might drive you off a cliff).
New GPS: "Avoid the red zone, but remember you are driving on a tightrope. Here is the exact path that keeps you safe from the red zone and on the tightrope."

By understanding the hidden rules of the system (the algebraic constraints), this new method ensures that safety filters don't just look good on paper—they actually work in the messy, constrained reality of engineering.

1. Problem Statement

Differential-Algebraic Equations (DAEs) are fundamental for modeling complex engineering systems such as power networks, chemical processes, and multibody mechanical systems. These systems are governed by both differential equations (describing dynamic evolution) and algebraic equations (encoding physical conservation laws like mass or energy).

The Core Challenge:
Standard Control Barrier Functions (CBFs), which are widely used for online safe control in Ordinary Differential Equation (ODE) systems, fail when directly applied to DAEs.

Infeasibility: Standard CBFs assume the existence of a control input that satisfies the barrier condition. However, in DAEs, the algebraic constraints restrict the allowable state trajectories to a specific manifold ( $\mathcal{M}$ ). A control input that satisfies the safety condition might violate the algebraic constraints, rendering the safety filter (typically a Quadratic Program, QP) infeasible.
Safety Violations: When the QP becomes infeasible, the controller cannot generate a valid input, leading to potential safety violations where the system leaves the safe set.
Hidden Constraints: Higher-index DAEs involve "hidden" constraints that only appear after differentiating the algebraic equations multiple times. Standard CBFs do not account for the relationship between the differentiation index of the DAE and the relative degree of the barrier function.

2. Methodology

The authors propose a novel framework called DAE-aware CBFs that explicitly incorporates the differential-algebraic structure into the safety synthesis and verification process.

A. Theoretical Framework: Projected Vector Fields

The core innovation is the construction of projected vector fields that evolve strictly on the constraint manifold $\mathcal{M}$ .

Compatibility Conditions: By differentiating the algebraic constraints $\phi(x)=0$ , the authors derive compatibility conditions that link the control input $u$ to the algebraic state derivatives.
Projected Dynamics: Using the Moore-Penrose pseudoinverse of the constraint Jacobian, they eliminate the algebraic state derivatives. This results in a control-affine representation on the manifold:
$\dot{x} = \hat{f}(x) + \hat{g}(x)u$
where $\hat{f}$ and $\hat{g}$ are the projected drift and input vector fields. These fields are guaranteed to be tangent to the manifold, ensuring that any trajectory generated by them remains consistent with the algebraic constraints.
Handling Differentiation Index ( $\nu$ ):
- Index-1 DAEs: The control input appears after one differentiation of the constraint. The projected fields are constructed directly.
- Higher-Index DAEs ( $\nu \ge 2$ ): The authors establish a relationship between the differentiation index $\nu$ and the relative degree of the barrier function. They derive High-Order CBF (HOCBF) conditions that account for the cascade of hidden constraints revealed by successive differentiations.
Safety Conditions: The safety condition is split into two parts:
- Manifold Invariance: A constraint ensuring the control input satisfies the compatibility conditions (keeping the system on $\mathcal{M}$ ).
- Set Invariance: A standard CBF inequality applied to the projected dynamics to ensure the system stays within the safe set $D \cap \mathcal{M}$ .

B. Verification Framework

The paper develops a systematic method to verify if a candidate function $b(x)$ is a valid DAE-aware CBF. This involves checking two properties:

Geometric Correctness: Ensuring the candidate barrier set $D$ is a subset of the true safe set $C$ on the manifold ( $D \cap \mathcal{M} \subseteq C \cap \mathcal{M}$ ).
Feasibility: Ensuring that for every state in $D \cap \mathcal{M}$ , there exists a control input that satisfies both the manifold compatibility and the safety barrier condition.

Verification Techniques:

Polynomial Systems: Uses Sum-of-Squares (SOS) programming and Positivstellensatz certificates to formally prove correctness and feasibility.
Non-Polynomial/Neural Networks: Uses Satisfiability Modulo Theories (SMT) solvers (e.g., Z3) to search for counterexamples (falsification). If no counterexample is found within a time limit, the system is considered safe.

3. Key Contributions

DAE-Aware CBF Formulation: Introduced a class of CBFs that explicitly account for algebraic constraints via projected vector fields, resolving the infeasibility issues of standard CBFs in DAE systems.
Higher-Index Extension: Generalized the framework to handle higher-index DAEs by formalizing the relationship between the differentiation index and the barrier function's relative degree, leading to necessary high-order CBF conditions.
Formal Verification: Developed a rigorous verification framework providing necessary and sufficient conditions for geometric correctness and feasibility. This includes SOS certificates for polynomial systems and SMT-based falsification for general systems.
Empirical Validation: Validated the approach on two distinct benchmarks:
- A Wind Turbine system (Index-1 DAE).
- A Flexible-Link Manipulator (Index-2 DAE).

4. Results

The paper presents simulation results comparing the proposed DAE-aware CBFs against standard (DAE-unaware) CBFs:

Wind Turbine (Index-1):
- Standard CBF: The QP became infeasible during simulation, causing the system trajectory to violate the safety constraints.
- DAE-Aware CBF: The controller successfully maintained the trajectory on the constraint manifold. Both the barrier function and the safety metric remained positive throughout the simulation, and the QP remained feasible.
Flexible-Link Manipulator (Index-2):
- The system was subjected to aggressive initial conditions. The DAE-aware HOCBF controller successfully modulated joint torques to keep the end-effector below a safety height limit (1.8m) while strictly respecting the algebraic position constraint.
- Verification: The candidate barrier functions were formally verified. For the polynomial case, SOS verification completed in under 10 seconds. For a neural network candidate, SMT-based falsification was used, confirming safety within the given time limits.

5. Significance

This work bridges a critical gap between control theory and differential-algebraic modeling.

Safety Guarantees: It provides the first computationally efficient method to guarantee safety for DAE systems without converting them to ODEs (which can be numerically unstable or lose physical structure).
Handling Complexity: By addressing higher-index DAEs, the framework applies to a broader class of physical systems (e.g., rigid body dynamics with constraints) that were previously difficult to control safely using CBFs.
Formal Assurance: The integration of formal verification (SOS/SMT) ensures that the safety guarantees are not just empirical but mathematically certified, addressing the "black box" nature of many modern control approaches.

Limitations & Future Work:
The authors note that SOS and SMT verification methods face scalability challenges for very high-dimensional systems (e.g., large power grids). Additionally, the current framework assumes exact model knowledge and requires a pre-defined candidate barrier function; future work aims to address synthesis (automatically generating the CBF) and robustness to uncertainty.