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History-Aware Control Barrier Functionals with Nonsingular Fading-Memory Kernels

Ege C. Altunkaya, Esra Demir, Murad Abu-Khalaf, İbrahim Özkol

arXiv:2610.31525v1eess.SYcs.SY

TL;DR

Safety protection may need to account for how the safety margin evolved, because a historical-average bound need not be respected by its current value. The paper constructs a history-aware barrier functional and an affine input constraint to enforce reserve and physical safety; it proves safety under stated conditions and reports ADMIRE tracking improvements during calibration.

  • Problem

    A prescribed history-dependent protection policy must be enforced even when different histories reach the same plant state, while a bound on a historical average need not ensure the current value is safe.

    Source: §I · p. 1

  • Method

    The paper constructs a history-aware CBFal from a safety margin, fading kernel, and reserve fraction, making auxiliary-barrier nonnegativity equivalent to reserve satisfaction and enforcing it through an affine input constraint.

    Source: §I · p. 2, §III · pp. 2–3

  • Results

    Under the stated conditions, the certificate implies forward invariance of the physical safe set; in ADMIRE, tracking improved on the calibration set and both constraints were preserved in additional fixed-gain maneuvers.

    Source: §III · p. 3, §V · p. 6

  • Takeaways & Limitations

    For a prescribed history family, uniform history-blind enforcement is characterized by the exact worst-history intersection of certified input sets.

    Source: §I · p. 2, §V · p. 6

    The reported flight-control evaluation is in ADMIRE; nonlinear flight models and bounded-input feasibility maintenance are left for future work.

    Source: §V · p. 6

Abstract

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Safety-critical protection often depends not only on the state and physical limits of the system, but on how that state evolved over time. Here, the safety margin measures the remaining allowance before a physical limit is reached. In this context, the objective is to enforce both physical safety and history-dependent retention of this margin for integer-order control-affine systems. To this end, changes in this margin are converted into memory through fading-memory kernels with finite nonnegative weights. A history-aware control barrier functional (HA-CBFal) is constructed explicitly, with nonnegativity equivalent to retaining a prescribed fraction of this margin's weighted historical value. This condition is enforced by an affine input constraint and proved to imply physical safety under stated kernel, initialization, and feasibility conditions, without finite-dimensional memory realization. In 36 calibration cases on the ADMIRE flight-control benchmark, mean per-case tracking-RMSE reductions of 27.2\% and 7.0\% are obtained relative to calibrated state-only CBF and moving-average control barrier functional (MA-CBFal), respectively, with both constraints preserved to numerical tolerance. In a shorter-recovery maneuver outside calibration with frozen gains, both constraints are preserved to tolerance by HA-CBFal and state-only CBF, but the retention requirement is violated despite MA-CBFal nonnegativity.

I. INTRODUCTION

Safety protection may need to account for how physical quantities evolved, not only the current state and physical limit. The introduction develops a history-dependent reserve certificate and motivates its use for assigning different permitted actions to histories that reach the same state.

I. INTRODUCTION

  • Protection can depend on prior evolution as well as current physical limits, including thermal, strain, and high-incidence flight contexts,,,.The introduction notes that motor restart may depend on thermal loading and cooling, viscoelastic stress retains strain history, and aerodynamic response can reflect recent pitching motion.Source: §I · p. 1
  • Different histories can reach the same state and safety margin yet receive different permitted actions under a history-aware policy.A history-blind set must lie within the intersection of the history-specific sets and can be more restrictive for one history.Source: Figure 1 · p. 1, §I · p. 1
  • The proposed HA-CBFal converts a prescribed fading-memory reserve into an affine input constraint, with auxiliary-barrier nonnegativity equivalent to reserve satisfaction.The reserve requires the current safety margin to retain a prescribed fraction of its weighted historical value.Source: §I · p. 2
  • Joint reserve and physical-safety preservation is proved through a positive Volterra resolvent under explicit kernel, initialization, and feasible-enforcement conditions.The construction applies across admissible kernels without a separate implication proof for each kernel choice.Source: §I · p. 2
  • The paper characterizes certified input sets at a fixed state through a scalar history shift and establishes their exact worst-history intersection.This identifies the greatest history-blind set satisfying the same certificate over a prescribed history family.Source: §I · p. 2
  • In ADMIRE, tracking benefits and the distinction between functional preservation and reserve satisfaction are demonstrated against CBF/CBFal comparators calibrated under both requirements.The evaluation uses Gaussian memory with no exact finite-dimensional LTI realization.Source: §I · p. 2

II. PRELIMINARIES

The preliminaries define the state-based safety certificate and the fading-kernel assumptions used by the paper. They establish the plant, safety margin, and kernel setting for the later functional construction.

A. Control Barrier Functions

  • For a control-affine plant, the physical safe set is defined by a differentiable margin, and the relative-degree-one CBF condition constrains the input.The stated pointwise condition yields h(t) ≥ e^-kt h(0) and forward invariance under the usual closed-loop existence assumptions.Source: A. Control Barrier Functions · p. 2

B. Fading-Memory Kernels

  • The fading-memory kernel is continuous, nonnegative, finite and positive at zero age, and tends to zero with age.Its shape and decay time are selected for the desired protection horizon, subject to the stated kernel assumption and input feasibility.Source: B. Fading-Memory Kernels · p. 2

C. Convolutional Memory

This section introduces convolutional memory generated by a fading kernel. It applies that memory to changes in the physical safety margin and names the resulting retained history.

C. Convolutional Memory

  • Convolution is defined for causal scalar signals, and the convolutional memory operator is introduced for absolutely continuous signals using the fading kernel.The supplied passage identifies the operator as generated by the kernel defined in Section II-B.Source: C. Convolutional Memory · p. 2
  • The memory weights past signal changes by the kernel, with the physical-barrier memory denoted qκ(t).The passage requires only the stated kernel conditions; fractional kernels are optional specializations.Source: C. Convolutional Memory · p. 2
  • The memory origin is set to zero, and the retained history is denoted Ht.The history is the safety margin restricted to the interval from 0 to t.Source: C. Convolutional Memory · p. 2

III. MAIN RESULT

The HA-CBFal converts a prescribed reserve relative to the weighted history of the safety margin into an input constraint. Under the stated kernel, initialization, and feasibility conditions, satisfying that constraint preserves both the auxiliary barrier and physical safety.

III. MAIN RESULT

  • An input-affine constraint with a history-dependent offset certifies physical safety without requiring a finite-dimensional realization of memory.For a prescribed h, kernel, and reserve fraction, the auxiliary barrier and policy terms can be updated analytically while retaining the affine input coefficient.Source: §III · pp. 2–4
  • Admissible kernels are locally absolutely continuous, nonincreasing, nonnegative, finite at age zero, and decay to zero; their weights are nonnegative and have integral κ0.The kernel shape and decay time set the protection horizon, subject to the kernel assumption and input feasibility.Source: §III · pp. 2–3
  • Under Assumption 1, h(0) ≥ 0, and the stated input inequality, ψκ(t) ≥ 0 and h(x(t)) ≥ 0, rendering the physical safe set forward invariant.The proof reconstructs the margin as nonnegative terms from ψκ, the kernel, the positive resolvent, and h(0).Source: §III · p. 3
  • The barrier’s nonnegativity is equivalent to retaining h ≥ θ h̄κ, a prescribed fraction of the normalized weighted historical safety margin.For any prescribed θ ∈ (0, 1), γ = κ0(1−θ)/θ; this fraction is also the Neumann-resolvent contraction factor.Source: §III · p. 4
  • At a fixed state, history changes the certified input set through a scalar offset; intersecting these sets yields the greatest history-blind set satisfying the same certificate for every prescribed history.The result orders the history-parametrized sets by their offsets; Λ suffices to determine the instantaneous set, but a closed scalar memory evolution is not asserted.Source: §III · p. 4

A. History-Aware Safety Filter

The history-aware barrier is enforced through a minimum-intervention program around a nominal input. Its affine input constraint makes the program convex, and a QP when the admissible-input set is polyhedral.

A. History-Aware Safety Filter

A. ADMIRE Model and Nominal Load-Factor Control

The evaluation uses an invariant symmetric ADMIRE subsystem with zero rudder and equal elevons. A fixed offline nominal controller is clipped to the actuator box before online safety filtering.

A. ADMIRE Model and Nominal Load-Factor Control

  • The symmetric ADMIRE subsystem uses zero rudder and equal elevons at Mach 0.22 and 3000 m.Source: A. ADMIRE Model and Nominal… · p. 4
  • The trim-deviation state is x = [∆α q]⊤ and the input is u = [δC δRE δLE]⊤, in radians and radians per second.Source: A. ADMIRE Model and Nominal… · p. 4
  • The nominal load-factor controller uses offline-fixed stabilising and trim-allocation gains, clips its command to the actuator box, and leaves only the safety-filter QP for online solution.The load-factor output and nominal control law are given explicitly in the passage.Source: A. ADMIRE Model and Nominal… · p. 4

B. High-Incidence Reserve Policy and Gaussian-Kernel Synthesis

The reserve policy supplements the angle-of-attack safety margin with retention of a prescribed fraction of its weighted historical value. The Gaussian-kernel design is enforced through a QP and compared with calibrated state-only and moving-average barriers.

B. High-Incidence Reserve Policy and Gaussian-Kernel Synthesis

  • The angle-of-attack margin hα measures allowance to 26°; physical safety requires hα ≥ 0, while the policy retains fraction θα of its weighted historical value.Source: B. High-Incidence Reserve Policy and… · p. 4
  • The Gaussian kernel uses κ0 = 2, τG = 3.0 s, and θα = 0.40, and its kernel excludes an exact finite-dimensional LTI realization.Memory is evaluated directly from retained history.Source: B. High-Incidence Reserve Policy and… · p. 4
  • Figure 2 plots command, incidence, load-factor response, policy residual, retained memory, weighted historical margin, and control commands; physical states coincide at QP activation while memory differs.The second pulse is magnified in the reserve inset, and gθ < 0 regions are shaded.Source: Figure 2 · p. 5
  • With ḣα = −Aαx−Bαu, the implemented QP uses QG,α = (κ0 + γ)(ġθ + βgθ) subject to the actuator box.Source: B. High-Incidence Reserve Policy and… · p. 5
  • The calibrated PP-CBF gain is 0.2393 s−1, while MA-CBFal uses η = 0.55, T = 3 s, and βMA = 0.2777 s−1; HA-CBFal β = 3.8341 s−1 is fixed before comparator calibration.Source: B. High-Incidence Reserve Policy and… · p. 5

C. History-Prepared Twin-Square Evaluation

In the shorter-recovery evaluation, HA-CBFal and PP-CBF preserve both requirements to tolerance, while MA-CBFal preserves physical safety but violates the reserve requirement. The evaluation also compares input sets at a common checkpoint across retained histories.

C. History-Prepared Twin-Square Evaluation

  • Three physically safe, unfiltered preparations begin at t0 = −9 s with qG(t0) = 0, and their activation endpoints are matched in incidence and pitch rate while full memory is retained.The evaluation commands two 2.6-g, 2-s squares, separated by a 1-s release for margin recovery.Source: C. History-Prepared Twin-Square… · p. 5
  • Both requirements are preserved to tolerance by PP-CBF and HA-CBFal, with lower HA-CBFal tracking RMSE; MA-CBFal violates the reserve requirement for all three preparations despite Φ ≥ 0.For the neutral preparation, MA-CBFal's reserve minimum is −0.0149 rad; the 1-s release was outside calibration and gains were not retuned.Source: C. History-Prepared Twin-Square… · p. 5
  • At the common checkpoint, HA-CBFal input-set volumes are 0.07763, 0.50500, and 0.86112 versus 0.07197 for PP-CBF, with nested sets and the adverse set as their intersection.Figure 3 compares instantaneous incidence-barrier input polytopes at fixed state, current command, and actuator box, varying retained history.Source: C. History-Prepared Twin-Square… · pp. 5–6, Figure 3 · p. 6

D. Calibration and Maneuver Tests

HA-CBFal had lower tracking RMSE than both comparators in the original calibration cases, and preserved both constraints in additional tests with frozen gains. MA-CBFal’s reserve violations despite functional nonnegativity distinguish calibration success from online reserve enforcement.

D. Calibration and Maneuver Tests

  • Calibration used 36 original cases and an expanded 48-case set; 27 additional profiles under three histories were tested outside both sets with parameters frozen.On each calibration set, PP-CBF and MA-CBFal mean RMSE was minimized subject to hα, gθ ≥0 and QP feasibility throughout each trajectory, while the HA-CBFal gain was retained.Source: D. Calibration and Maneuver Tests · p. 6
  • 27.2% versus PP-CBF and 7.0% versus MA-CBFal were the mean per-case RMSE reductions with HA-CBFal; all filters met both constraints on the original calibration set.With 48-case calibration, MA-CBFal violated the reserve requirement in 17 of 27 tests despite hα, Φ ≥0, while HA-CBFal met both constraints to tolerance in all 27.Source: D. Calibration and Maneuver Tests · p. 6
  • Gaussian-memory and uniform-history policies each met both constraints to tolerance in all 27 maneuvers after policy terms were recomputed analytically.A common θc ≃0.2567 was selected from preparation data for admissible initialization, replacing 0.4; β was retained.Source: D. Calibration and Maneuver Tests · p. 6
  • MA-CBFal reserve satisfaction was checked during calibration, but only its functional inequality was enforced online, so new reserve violations require further tuning or redesign.HA-CBFal gains did not require retuning in these tests; joint preservation was certified under Theorem 1.1’s hypotheses.Source: D. Calibration and Maneuver Tests · p. 6

V. CONCLUSION

The paper presents a conditional certificate for jointly preserving prescribed reserve requirements and physical safety, then reports ADMIRE evaluations and identifies remaining scope boundaries.

V. CONCLUSION

  • An explicit construction jointly certifies prescribed reserve requirements and physical safety for integer-order plants under stated conditions, without finite-dimensional memory realization.The same implication proof applies across admissible kernels.Source: §V · p. 6
  • In ADMIRE, tracking improved on the calibration set, and both constraints were preserved in additional maneuvers with fixed HA-CBFal gains.MA-CBFal reserve violations occurred despite functional nonnegativity, distinguishing calibration success from certified reserve enforcement.Source: §V · p. 6
  • Uniform history-blind enforcement is characterized by the exact worst-history intersection.This describes the common admissible input set required across histories.Source: §V · p. 6
  • Nonlinear flight models and bounded-input feasibility maintenance are left for future work.Source: §V · p. 6
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