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High-order Barrier Functions: Robustness, Safety and Performance-Critical Control

Xiao Tan, Wenceslao Shaw Cortez, Dimos V. Dimarogonas

arXiv:2104.00101v2eess.SY

TL;DR

The paper addresses how to enforce safety for nonlinear systems with higher-relative-degree constraints while retaining robustness and performance-critical control. It proposes high-order zeroing barrier functions and controlled constructions that establish invariance, asymptotic stability, flexible relative-degree handling, locally Lipschitz control, and nominal-control regions. A rigid-body attitude case study demonstrates safe trajectories and nominal-control coincidence in the designated region.

  • Problem

    Nonlinear control requires safety guarantees for higher-relative-degree constraints while balancing computational cost, robustness, and performance objectives.

  • Method

    The paper generalizes zeroing barrier functions to high-order derivatives and develops controlled, singularity-free constructions with performance-critical regions.

  • Results

    The formulation guarantees forward invariance and asymptotic stability, does not require forward completeness for invariance, permits varying relative degree, yields locally Lipschitz control, and preserves bounded nominal control in designated regions.

  • Takeaways & Limitations

    The approach supports robust safety certification while allowing designers to specify where an existing bounded nominal controller must remain unmodified.

Abstract

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In this paper, we propose a notion of high-order (zeroing) barrier functions that generalizes the concept of zeroing barrier functions and guarantees set forward invariance by checking their higher order derivatives. The proposed formulation guarantees asymptotic stability of the forward invariant set, which is highly favorable for robustness with respect to model perturbations. No forward completeness assumption is needed in our setting in contrast to existing high order barrier function methods. For the case of controlled dynamical systems, we relax the requirement of uniform relative degree and propose a singularity-free control scheme that yields a locally Lipschitz control signal and guarantees safety. Furthermore, the proposed formulation accounts for "performance-critical" control: it guarantees that a subset of the forward invariant set will admit any existing, bounded control law, while still ensuring forward invariance of the set. Finally, a non-trivial case study with rigid-body attitude dynamics and interconnected cell regions as the safe region is investigated.

I. INTRODUCTION

The paper motivates high-order barrier functions as a way to combine safety guarantees with performance-oriented control for nonlinear systems. It addresses higher-relative-degree constraints while targeting robustness, flexible control design, and reduced computational burden relative to MPC.

  • Motivation: MPC enforces safety constraints across a finite horizon but incurs heavy online computation, whereas barrier functions provide modular forward-invariance certificates.Barrier functions can be designed alongside a performance-optimizing controller.
  • Limitations of existing methods: Reciprocal barrier functions can require large control signals near safety boundaries, while barrier certificates impose conditions that are too strong and restrictive.These limitations motivate alternative barrier formulations.
  • Related work: Zeroing barrier functions are defined inside and outside the safe set and provide robustness through asymptotic stability and input-to-state stability properties.Their controlled versions originally addressed relative-degree-one constraints.
  • Need for high-order methods: Higher-order derivative conditions are relevant because many practical constraints have higher relative degrees, including position constraints in mechanical systems.They also provide an alternative way to find barrier functions.
  • Contributions: The paper introduces high-order barrier functions using extended class-K functions and develops controlled formulations without requiring forward completeness or uniform relative degree.The controlled construction yields a locally Lipschitz signal and supports pre-defined performance-critical regions.

II. HIGH-ORDER BARRIER FUNCTIONS

The paper formulates high-order barrier functions for nonlinear systems by recursively transforming a differentiable constraint with extended class-K functions. The resulting construction is designed to generalize zeroing barrier functions and remain robust to perturbations.

  • High-order barrier function formulation: The proposed high-order barrier function definition generalizes zeroing barrier functions and is presented as more general than previous constructions.The formulation is also described as robust to perturbations.
  • System setting: The underlying nonlinear system assumes locally Lipschitz dynamics, with forward invariance defined over each solution’s maximal time interval of existence.No forward completeness assumption is included in this setup.
  • Set construction: A differentiable constraint h defines the superlevel set C_h = {x : h(x) ≥ 0} and its stricter subset C_h,δ = {x : h(x) ≥ δ}.These sets provide the basic safe-set notation.
  • Class-K functions: High-order barrier functions recursively use sufficiently smooth extended class-K functions, which are continuous, strictly increasing, and zero at zero.The paper takes these functions to be defined on the entire real line.

A. High-order barrier functions

The high-order barrier construction recursively imposes conditions on transformed constraint functions to certify forward invariance. Its use of extended class-K functions broadens the formulation beyond linear or ordinary class-K choices.

  • Definition: Given a sufficiently differentiable constraint h and extended class-K functions α_1 through α_r, the paper defines a recursive series of high-order barrier functions.The construction is based on successive transformed constraints ψ_k.
  • Forward invariance: If the intersection C = ⋂_{k=1}^r C_{ψ_{k−1}} lies in the definition domain, the proposed condition guarantees that C is forward invariant.The proof establishes that the dynamics remain in the tangent cone on the boundary.
  • Proof mechanism: The proof uses active constraints on the boundary to show f(x) belongs to the tangent cone T_C(x), then applies Brezis’s theorem for locally Lipschitz dynamics.This avoids relying on a forward-completeness assumption for invariance.
  • Generality: Unlike earlier formulations, the paper allows arbitrary extended class-K functions instead of restricting α_k to linear functions or ordinary class-K functions.Extended class-K functions remain well-defined when ψ_{k−1}(x) < 0, including outside the safe set.

B. Asymptotic stability of the set C

The proposed high-order barrier formulation establishes asymptotic stability of the forward invariant set C under the paper’s stated conditions. The proof uses comparison arguments and a cascade of auxiliary systems to establish attraction and uniform stability.

  • A comparison proposition bounds the high-order state variables using the auxiliary system’s solution.The comparison relies on a quasimonotone nondecreasing vector field and a generalized comparison lemma.
  • Under the compactness condition on C, the high-order barrier formulation makes C asymptotically stable.
  • The proof establishes asymptotic stability by showing global asymptotic stability of an auxiliary cascade system.The cascade is analyzed inductively, beginning with its final subsystem and proceeding toward the full auxiliary system.
  • The comparison argument proves that C is attractive and uniformly stable, completing the asymptotic-stability result.Attraction follows from convergence of the auxiliary system, while uniform stability is established inductively across subsystems.
  • The result extends relative-degree-one zeroing-barrier stability to high-order barrier functions and supports robustness to vanishing or sufficiently small perturbations.For sufficiently small nonvanishing perturbations, the paper states that a new asymptotically stable set containing C can be established.

III. HIGH-ORDER CONTROL BARRIER FUNCTIONS

High-order control barrier functions extend zeroing control barrier functions to higher-order constraints while relaxing uniform relative-degree requirements. Locally Lipschitz controllers satisfying the resulting condition preserve forward invariance, and under additional assumptions can yield asymptotic stability.

  • The formulation requires only least relative degree, which is weaker than the uniform relative-degree condition used in earlier approaches.Uniform relative degree additionally requires the highest-order input derivative term to be nonzero throughout the domain.
  • High-order control barrier functions use higher-order constraint derivatives and differentiable extended class K functions to define safety conditions.The construction uses recursively defined ψ_k terms over an open domain containing the relevant constraint sets.
  • When r = 1, the high-order control barrier function reduces to the zeroing control barrier function.
  • Any locally Lipschitz controller whose values satisfy the HOCBF admissible-control set renders C forward invariant.
  • If C is compact and the closed-loop system is forward complete, a controller satisfying the HOCBF condition makes C asymptotically stable.The paper notes that states starting outside the safe set asymptotically reach C under these additional assumptions.
  • The paper also gives a disturbance-robustified HOCBF condition when the disturbance enters with matching least relative degree and has a known bound.Under that condition, the set C remains forward invariant for the perturbed system.

IV. SINGULARITY-FREE, PERFORMANCE-CRITICAL HOCBFS

The paper identifies HOCBF construction as nontrivial when the highest-order input derivative term vanishes at some points. It contrasts this singular case with the uniformly nonzero condition, where feasibility and local Lipschitz continuity are guaranteed.

  • HOCBF construction is not straightforward when the highest-order input derivative term vanishes at some points in the domain.
  • Under uniform relative degree and U = R^m, the safety quadratic program is feasible throughout the domain and produces a locally Lipschitz controller.

A. Singularity-free HOCBF design

The paper addresses singularities in high-order control barrier functions by allowing varying relative degree and transforming the constraint so the control condition remains feasible. The resulting controller is locally Lipschitz and renders the constructed safe set forward invariant.

  • Motivation: Uniform relative degree is unnecessary for an HOCBF candidate, addressing singularities that arise even for double-integrator systems with circular constraints.In the double-integrator example, singular points satisfy L_gL_f b(x)=0 on the set p=0, which lies inside the circular safe region.
  • Singularity-free construction: If singular points remain strictly inside the safe region, a transformed barrier function makes the quadratic-program constraint feasible everywhere.The construction uses a smooth rth-order truncating function χ with χ(0)=0, χ(τ)=1 for τ≥1, and positive derivative for τ<1.
  • Guarantees: For unconstrained inputs U=R^m, the transformed function h is an HOCBF, and the associated intersection of superlevel sets is forward invariant.The proof handles points where L_gψ_{r−1}=0 separately and shows a feasible control exists elsewhere because the condition imposes a linear constraint on u.
  • Controller properties: The quadratic-program controller is locally Lipschitz continuous and renders the constructed set forward invariant.The controller agrees with the nominal controller on the singularity-containing region, and continuity is established at the boundary between the two regions.

B. Performance-Critical HOCBF

The performance-critical formulation preserves a nominal controller on a prescribed interior region while maintaining safety outside it. This requires the performance-critical region to lie strictly inside the safety region.

  • Motivation: Performance-critical control specifies in advance where the nominal controller must remain unmodified, supporting learning-based control and high-precision motion tasks.The paper distinguishes the safety region from the forward-invariant safe set and defines both through smooth superlevel-set functions.
  • Construction: When the performance-critical region is strictly inside the safety region, the transformed barrier construction recovers the nominal control there while preserving safety.The interior separation ensures the barrier constraint is trivially satisfied on the performance-critical set.
  • Guarantees: The resulting HOCBF controller renders the constructed set forward invariant and equals the nominal controller for states in C_s.The theorem states these properties together with the HOCBF property of h.
  • Special case: The uniform-relative-degree case is presented as a corollary of the performance-critical construction.The corollary assumes b has uniform relative degree r in an open set containing its superlevel set.

V. AN APPLICATION TO RIGID-BODY ATTITUDE DYNAMICS

The proposed HOCBF framework is applied to rigid-body attitude stabilization over interconnected cell regions. Simulations show safe trajectories under additive disturbances and nominal-control recovery in the performance-critical region.

  • Application setup: The attitude application uses a simple nominal stabilizing controller and the HOCBF framework to obtain a modular, safe, stabilizing control design.The paper contrasts this with prior work that used a more complicated nominal controller.
  • Safe-region construction: The safe region is the connected union of attitude cells centered at sampled orientations R_i, with the barrier function measuring trajectory margin to that region.Each cell is defined by a smooth orientation-distance constraint, and b(x) is constructed from transformed cell functions with a margin parameter δ.
  • Safety experiment: The barrier-enabled trajectory remains within the safe region despite an additive control signal that can drive the nominal trajectory outside it.Figure 1 compares barrier-enabled and unprotected trajectories, with the yellow region denoting safety.
  • Performance-critical behavior: When b(t) ≥ ξ = 0.6, the actual control coincides with the nominal control, validating the performance-critical property in the attitude simulation.The additive signal is applied during t∈[20,25] and is treated as part of the nominal input to the quadratic program.

VI. CONCLUSION

The paper formulates high-order zeroing barrier functions and controlled counterparts that extend zeroing barrier functions while providing stability, robustness, singularity handling, and performance-critical control. A rigid-body attitude case study demonstrates the framework.

  • Contributions: The formulation removes the need for forward completeness and proves asymptotic stability of the intersection of superlevel sets associated with the high-order barrier function.This stability property is used to generalize robustness results associated with standard zeroing barrier functions.
  • Contributions: For controlled systems, the method permits varying relative degree, handles singular states, and produces a locally Lipschitz safe controller.The construction also supports a priori performance-critical regions where the nominal control is retained.
  • Validation: The proposed formulation is implemented in a non-trivial rigid-body attitude dynamics case study.
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