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Safe Control of Semi-Explicit Differential-Algebraic Systems: Control Barrier Functions with SOS Verification

Mohamad H. Kazma, Hongchao Zhang, Abdallah A. Albustami, Ben Wooding, Meiyi Ma, Taylor T. Johnson, Ahmad F. Taha

arXiv:2610.09634v1eess.SYcs.SYmath.OC

TL;DR

Algebraic constraints in DAE systems impose compatibility requirements that standard CBF safety filters may ignore, potentially making them infeasible. This paper builds DAE-aware CBF conditions using projected dynamics and SOS verification for correctness and feasibility. In power-system case studies, the DAE-aware filter maintains safety while the compared DAE-unaware and droop-only cases violate it.

  • Problem

    Standard CBF conditions may demand inputs that violate DAE algebraic constraints, because they do not account for compatibility requirements on the constraint manifold.

    Source: §I · p. 1

  • Method

    The framework uses projected vector fields to derive DAE-aware CBF conditions and SOS verification for correctness and feasibility.

    Source: §I · p. 2, §III · p. 3

  • Results

    The DAE-aware filter maintains b ≥ 0, while the DAE-unaware CBF reaches minimum b = −0.23 and droop-only reaches minimum b = −1.22.

    Source: A. Nonlinear Power System DAE Model · p. 7

  • Takeaways & Limitations

    Power-system case studies show that accounting for algebraic coupling is essential for reliable safety guarantees.

    Source: §VI · p. 8

Abstract

from arXiv · show

Differential-algebraic equation (DAE) systems arise in power networks, multibody mechanics, and chemical processes, where algebraic constraints couple dynamic and algebraic states. Standard control barrier function (CBF) methods, designed for ordinary differential equations, fail for DAE systems because the algebraic constraints impose compatibility requirements that may render safety filters infeasible. This paper develops a safety-critical control framework for semi-explicit DAE systems using CBFs that consider compatibility with the algebraic constraints, termed DAE-aware CBFs. We construct projected vector fields on the constraint manifold to account for the hidden dynamics, while ensuring compatibility conditions that keep the algebraic constraints satisfied. We then present sum of squares (SOS) verification conditions that certify both correctness and feasibility of the resulting DAE-aware CBF. The framework is validated on power system case studies.

I. INTRODUCTION

DAE algebraic constraints restrict trajectories to a manifold, so standard CBF filters may be infeasible when they ignore compatibility requirements. The paper develops DAE-aware safety and verification methods for regular semi-explicit systems, focusing on power-system models.

I. INTRODUCTION

  • Standard CBF conditions can demand inputs that violate algebraic constraints, making safety filters infeasible for DAE systems.The constraints restrict allowable trajectories to a manifold and impose compatibility requirements that ordinary CBF conditions do not account for.Source: §I · p. 1
  • The paper targets a gap in realtime safety filtering: prior DAE verification methods certify safety offline, while existing control methods do not address strict set invariance with algebraic compatibility.The paper also distinguishes its framework from prior DAE CBF work that does not explicitly derive manifold invariance or verify candidate CBFs.Source: §I · p. 1
  • Projected vector fields encode the coupling between dynamic and algebraic states, enabling CBF conditions that enforce both safe-set invariance and manifold compatibility.The conditions extend to high-order CBFs for barrier functions of arbitrary relative degree.Source: §I · p. 2
  • SOS conditions certify CBF correctness and feasibility, with both checks reducible to semidefinite programs.Correctness means the barrier set lies within the safe set on the manifold; feasibility means a compatible input exists that satisfies safety and algebraic constraints.Source: §I · p. 2

A. System Model

  • The framework focuses on regular semi-explicit DAE power-system models, where algebraic power-flow constraints couple with generator dynamics.Regularity requires a square, nonsingular algebraic Jacobian; the paper identifies this as a standard condition for index-1 power-system DAEs.Source: §I · p. 2, A. System Model · p. 2

B. Safety and Control Barrier Functions

For ODEs, CBFs define barrier sets and provide conditions for positive invariance, commonly implemented through a minimally modifying safety filter. For DAEs, the filter must additionally preserve algebraic compatibility and have a safe compatible input available throughout the relevant set.

B. Safety and Control Barrier Functions

  • A CBF defines a barrier set D within the safe set C, and an appropriate input condition guarantees positive invariance for ODE systems.In practice, a CBF-QP minimally modifies a nominal controller to meet the barrier condition.Source: B. Safety and Control Barrier Functions · p. 2
  • A valid CBF requires correctness, D ⊆ C, and feasibility, meaning an input satisfying the barrier condition exists at every state in D.Feasibility ensures the CBF-QP admits a solution at every state.Source: B. Safety and Control Barrier Functions · p. 2

C. Problem Formulation

  • For DAEs, safety must hold on D ∩ M, and direct ODE-style CBF conditions may violate algebraic constraints or lack a compatible safe input.The problem is to design a policy making D ∩ M positively invariant and verify correctness and feasibility on that manifold.Source: C. Problem Formulation · pp. 2–3
  • The proposed solution constructs projected vector fields for joint manifold compatibility and set invariance, then uses SOS methods to verify correctness and feasibility.The framework is developed for semi-explicit DAEs satisfying the regularity assumption.Source: C. Problem Formulation · p. 3

III. DAE-AWARE CBFS FOR SEMI-EXPLICIT DAES

The paper formulates DAE safety on the constraint manifold using projected dynamics and CBF conditions that enforce both compatibility and barrier invariance. It also provides an autonomous special case for systems without control inputs.

III. DAE-AWARE CBFS FOR SEMI-EXPLICIT DAES

  • Projected vector fields capture hidden algebraic constraints and support a CBF condition for safety on the constraint manifold.The section also develops a high-order extension for barrier functions with arbitrary relative degree.Source: §III · p. 3

A. Projected Vector Fields and DAE-Aware CBFs

  • For regular index-1 DAEs, the algebraic Jacobian determines the algebraic-state derivative, yielding projected fields tangent to the constraint manifold.The projected fields use the Jacobian pseudoinverse; for a square invertible algebraic Jacobian, it equals the inverse.Source: A. Projected Vector Fields and… · p. 3
  • Theorem 2 ensures D ∩ M is positively invariant when inputs satisfy both the compatibility condition and the CBF inequality along projected dynamics.The compatibility condition keeps trajectories on M, while the projected Lie derivatives account for algebraic-state responses.Source: A. Projected Vector Fields and… · p. 3
  • When no control input is available, the compatibility requirement reduces to a drift-field constraint, yielding an autonomous invariance condition.The paper notes this case applies to power-system DAEs under fixed generation dispatch.Source: A. Projected Vector Fields and… · p. 4

B. Relative Degree and High-Order CBFs

The projected dynamics can raise a barrier function’s effective relative degree because control propagates through the algebraic coupling. The paper extends its DAE-aware CBF conditions with a high-order hierarchy for such cases.

B. Relative Degree and High-Order CBFs

  • The effective relative degree depends on how control enters the projected dynamics, including through the algebraic-state block.The projected system may require multiple differentiations of the barrier before control appears.Source: B. Relative Degree and High-Order CBFs · p. 4
  • For an index-ν DAE, the algebraic constraint’s relative degree is d = ν + d′ − 1, where d′ is the first derivative order at which the differential dynamics couple to control through the constraint Jacobian.For index-1 systems, d = d′; when d′ > 1, control does not appear in the first d − 1 derivatives of the barrier along M.Source: B. Relative Degree and High-Order CBFs · p. 4
  • The DAE-aware HOCBF hierarchy enforces manifold compatibility separately from the high-order safety inequalities built from projected Lie derivatives.When d = 1, the hierarchy recovers the ordinary DAE-aware CBF inequality.Source: B. Relative Degree and High-Order CBFs · p. 4

C. DAE-Aware CBF-QP

The DAE-aware CBF-QP combines nominal-control tracking with algebraic compatibility and barrier constraints. A valid DAE-aware CBF must be correct on the constraint manifold and admit a compatible safe input throughout its barrier set.

C. DAE-Aware CBF-QP

  • The DAE-aware CBF-QP minimally modifies a nominal input while jointly enforcing manifold compatibility and the barrier inequality.For relative degree greater than one, the barrier constraint is replaced by the higher-order condition; the compatibility constraint is unchanged.Source: C. DAE-Aware CBF-QP · p. 4
  • A valid DAE-aware CBF requires its barrier set on the manifold to lie within the safe set and a compatible input to exist at every such state.Correctness and feasibility are the two validity conditions.Source: C. DAE-Aware CBF-QP · pp. 4–5

IV. SOS-BASED VERIFICATION

The verification framework checks whether a candidate barrier set is safe on the constraint manifold and whether compatible safe inputs exist throughout it. It uses Positivstellensatz-based certificates for correctness and dual feasibility tests that distinguish interior states from boundary states.

IV. SOS-BASED VERIFICATION

  • Verification asks whether the candidate barrier set on the manifold lies within the safe set and admits a compatible input at every state.The problem separates correctness from input feasibility.Source: §IV · p. 5

A. Correctness Verification

  • Correctness reduces to certifying that the barrier function is nonnegative throughout the candidate set on the constraint manifold.The Positivstellensatz characterizes emptiness of semialgebraic sets, and its certificate is used to rule out states in the set where safety fails.Source: A. Correctness Verification · p. 5

B. Feasibility Verification

  • Input feasibility is tested through affine inequalities that encode compatibility, the CBF condition, and input bounds, then through a dual alternative.Opposing compatibility inequalities encode the equality constraint; the CBF row is included for boundary tangency.Source: B. Feasibility Verification · p. 5
  • Theorem 4 checks interior feasibility using compatibility and input bounds, and boundary feasibility with the additional CBF tangency condition.The interior-boundary separation follows the tangent-cone characterization of positive invariance; the conditions do not fix a class-K∞ function α.Source: B. Feasibility Verification · p. 5

C. Polynomial SOS Certificates

Polynomial DAE-aware CBFs can be verified through SOS certificates for correctness and feasibility, provided the barrier set on the manifold is compact. The approach can also accommodate nonpolynomial algebraic constraints by lifting them to a polynomial DAE on an extended manifold.

C. Polynomial SOS Certificates

  • Polynomial mappings, polynomial descriptions of the manifold and safe set, and compactness of the barrier set on the manifold support polynomial feasibility certificates.Under these assumptions, the relevant interior and boundary matrix entries are polynomial, allowing dual infeasibility to be certified using the Positivstellensatz.Source: C. Polynomial SOS Certificates · pp. 5–6
  • The interior and boundary dual infeasibility conditions can be certified with SOS polynomials and multipliers, reducing verification to an SOS feasibility program.The certificates use cones generated by barrier and dual-variable constraints, with monoid terms encoding strict inequalities.Source: C. Polynomial SOS Certificates · p. 6
  • The correctness and feasibility certificates together form a complete SOS-based verification framework, with both reducing to semidefinite programs.Nonpolynomial algebraic constraints, including trigonometric power-flow equations, can be handled by auxiliary-variable lifting to a polynomial DAE on an extended manifold.Source: C. Polynomial SOS Certificates · p. 6

V. CASE STUDIES

The case studies examine DAE-aware CBF safety and SOS validity checks in semi-explicit power system models. They also test an autonomous two-bus power DAE using a positive-invariance result.

V. CASE STUDIES

  • The case studies ask whether the DAE-aware CBF guarantees manifold compatibility and safe-set invariance, and whether SOS programs certify candidate-barrier validity.The validity checks use the SOS programs from the correctness and feasibility results.Source: §V · p. 6
  • The study applies the DAE-aware CBF-QP and SOS verification framework to a nonlinear network-preserving DAE model from power systems.The model is identified as a semi-explicit power system DAE.Source: §V · p. 6
  • A two-bus polynomial benchmark with u ≡ 0 is used to test whether Corollary 1 certifies positive invariance of D ∩ M.Source: §V · p. 6

A. Nonlinear Power System DAE Model

The case study applies DAE-aware barrier control to a two-bus generator-load model with algebraic power-flow constraints and load-voltage safety limits. The filter preserves the barrier set in simulation, while the DAE-unaware and droop-only controllers do not; SOS checks address correctness and feasibility.

A. Nonlinear Power System DAE Model

  • The model couples second-order generator dynamics to algebraic load-bus power-flow constraints, with generator angle and speed dynamic and load-bus angle and voltage algebraic states.The safety specification bounds load voltage between Vmin and Vmax using h(x) = (Vℓ−Vmin)(Vmax−Vℓ).Source: A. Nonlinear Power System DAE Model · p. 6, Figure 3 · p. 6
  • The DAE-aware filter kept b(x) positive (minimum 3.5 × 10^-2) with max |u| = 0.061; the DAE-unaware and droop-only cases reached -0.23 and -1.22, respectively.The DAE-unaware filter also used max |u| = 5.26; the phase portrait shows only the DAE-aware trajectory remaining inside D.Source: A. Nonlinear Power System DAE Model · p. 7
  • Polynomial lifting replaces trigonometric power-flow terms with variables constrained to remain consistent with a physical angle difference.The lifted variables and constraints yield a polynomial DAE on the lifted constraint manifold.Source: A. Nonlinear Power System DAE Model · p. 7
  • The SOS correctness certificate establishes that b(x) ≥ 0 implies h(x) ≥ 0, and MOSEK found a feasible certificate in 0.02 s.The certificate uses no algebraic constraint multipliers.Source: A. Nonlinear Power System DAE Model · p. 7
  • Feasibility was supported by min |L̂_g b| = 0.34 across 282 sampled points near ∂D ∩ M, allowing a compatible control to satisfy the barrier condition.This verifies feasibility for the nonlinear power DAE case study.Source: A. Nonlinear Power System DAE Model · p. 7

B. Verification of Voltage Constraints

The voltage-constraint case study uses a lifted polynomial DAE and a candidate barrier for bus-voltage bounds. Its verification supports positive invariance and correctness, with regularity and the projected barrier derivative checked on the operating set.

B. Verification of Voltage Constraints

  • The polynomial DAE uses algebraic variables for bus-2 voltage and the sine and cosine of its angle, with active and reactive power-flow expressions in lifted coordinates.The model and parameter values follow the two-bus system in.Source: B. Verification of Voltage Constraints · pp. 7–8
  • The bus-2 safety specification bounds voltage using h(x) = (z1−Vmin)(Vmax−z1), and the barrier set is D = {x : b(x) ≥ 0}.The voltage limits are Vmin = 0.9 and Vmax = 1.1.Source: B. Verification of Voltage Constraints · p. 8
  • For 2,000 sampled points in D ∩ M, min |det Ja| = 500.9, confirming regularity; since ∇xab = 0, the projected Lie derivative reduces to (2β3/τ)x2.The skew-symmetric oscillator block of A cancels the x1-x2 cross terms.Source: B. Verification of Voltage Constraints · p. 8
  • The case study reports min h(x) = 4.2 × 10^-3 on D ∩ M and certifies positive invariance of D ∩ M.The barrier condition holds because L̂_f b ≥ 0 ≥ −α(b) on D.Source: B. Verification of Voltage Constraints · p. 8

VI. CONCLUSION

The paper presents DAE-aware CBFs for regular semi-explicit DAEs, using projected dynamics to combine safety invariance with algebraic consistency. SOS conditions verify candidate-barrier correctness and feasibility, and power-system case studies show the importance of accounting for algebraic coupling.

VI. CONCLUSION

  • Projected dynamics on the constraint manifold yield safety conditions that enforce set invariance together with algebraic consistency.Source: §VI · p. 8
  • SOS verification conditions address both correctness and feasibility of candidate barrier functions.Source: §VI · p. 8
  • Power-system case studies show that accounting for algebraic coupling is essential for reliable safety guarantees.Source: §VI · p. 8
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