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The Space-Time Transform: Memory-Augmented Control Barrier Functions

Avinash Malik

arXiv:2609.00079v1eess.SY

TL;DR

The paper addresses safety-control limitations arising from zero-memory CBF filtering, including noise sensitivity and QP infeasibility. It introduces the Space-Time Transform as a proper temporal-convolution framework and reports equivalent safety with drastically reduced control variance in third-order-system simulations.

  • Problem

    Standard CBF variants act as zero-memory filters, making control sensitive to bounded measurement noise and high-frequency disturbances while higher-order formulations can produce QP-infeasible trajectories.

  • Method

    The Space-Time Transform replaces instantaneous geometric sequences with a proper predictive temporal convolution implemented through a Linear Time-Invariant dynamic extension.

  • Results

    The proposed framework achieves safety rates equivalent to state-of-the-art parameterized barrier methods while drastically reducing control variance; external pre-filtering achieves a 0.0% safety rate.

  • Takeaways & Limitations

    Proper temporal convolution preserves active control authority, maintains QP feasibility under actuator limits, and supports robust physical forward invariance for hardware-oriented safety control.

  • Takeaways & Limitations

    The feasibility analysis is framed around the structural zero-memory-kernel properties of current safety methods and their frequency-domain behavior.

Abstract

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Control Barrier Functions (CBFs), their High-Order variants (HOCBFs) and Exponential CBFs (ECBFs) are standard geometric tools for enforcing nonlinear safety constraints. CBFs, and their variants, offer an elegant geometric framework for nonlinear safety, yet mathematically, they reduce to continuous-time convolutions restricted by zero-memory kernels. In the presence of high-frequency measurement noise, these memoryless operators act as improper filters, leading to significant control chattering and the potential loss of active control authority due to Quadratic Program (QP) infeasibility. To address this structural limitation, this paper introduces a space-time transform that embeds dynamic temporal filtering directly into the safety constraint synthesis. By designing a proper spatio-temporal kernel, this approach inherently attenuates high-frequency noise while preserving affine control authority. Crucially, we prove the robust forward invariance of the designed STT-CBF. Monte Carlo simulations of a third-order system demonstrate that the proposed framework achieves a 100% safety rate while reducing control total variation by over 99% compared to conventional parameterized barrier methods, mitigating hardware hazards and enabling reliable deployment on physical robotic platforms.

I. INTRODUCTION

Standard CBF, HOCBF, and ECBF methods enforce safety through instantaneous geometric differentiation, but their zero-memory structure creates fundamental robustness and feasibility limits. The Space-Time Transform replaces this structure with proper temporal convolution while preserving control authority and forward-invariance guarantees.

  • Classical geometric safety frameworks: CBFs, HOCBFs, and ECBFs enforce safety by mapping spatial boundaries to control inputs through instantaneous or successive Lie differentiation.Standard CBFs address relative degree one, while HOCBFs and ECBFs expose control at higher derivatives.
  • Space-Time Transform: The Space-Time Transform replaces instantaneous geometric sequences with proper predictive temporal convolution and a linear time-invariant dynamic extension.The framework is designed to bound demanded control effort, maintain QP feasibility under hardware saturation, and preserve robust physical forward invariance.
  • Convolutional perspective: Standard geometric safety filters are continuous-time convolutions restricted to zero-memory, improper distributional kernels.This convolutional perspective is established for linear affine HOCBF and ECBF formulations.
  • Structural limitations: Memoryless spatial mappings demand unbounded high-frequency gain, causing disturbance-induced actuator saturation and QP infeasibility in higher relative-degree systems.The paper characterizes this as a structural limitation rather than a tuning issue.
  • Structural limitations: External pre-filtering introduces a trade-off among affine control authority, physical forward invariance, and dynamic boundary penetration under temporal phase lag.The stated failure modes include total loss of affine control authority, loss of physical forward invariance, or boundary penetration.

III. CONVOLUTIONAL REPRESENTATION OF SAFETY KERNELS

The paper deconstructs geometric safety constraints as temporal convolution operators. Standard CBF, HOCBF, and ECBF formulations correspond to improper kernels composed of Dirac-delta impulses and therefore have zero temporal memory.

  • Convolutional representation: Spatial safety constraints can be deconstructed into continuous-time convolution operators mapping spatial trajectory signals.The instantaneous spatial boundary dynamics are represented along the state trajectory before applying the convolutional formulation.
  • Standard CBF kernel: Standard CBFs use the memoryless kernel g_CBF(τ) = δ(τ) to map spatial boundary geometry to safety constraints.The Dirac delta evaluates the signal instantaneously at the current time.
  • High-order formulations: For linear class-K functions, HOCBF and ECBF constraints share the same n-th-order affine constraint structure.The equivalence is stated for systems where the control input enters at order n.
  • Distributional kernel theorem: Every linear affine HOCBF or ECBF formulation of relative degree n ≥1 admits a unique continuous-time convolution representation with an improper distributional kernel.The representation follows by identifying ordinary derivatives with convolutions involving distributional derivatives of the Dirac delta.
  • Distributional kernel theorem: The effective HOCBF and ECBF kernel consists entirely of Dirac-delta impulses at τ = 0 and therefore has zero temporal memory.Sequential distributional derivatives synthesize the high-order kernel.

IV. STRUCTURAL LIMITATIONS OF INSTANTANEOUS GEOMETRY

Instantaneous geometric safety operators lack proper filtering and are sensitive to high-frequency disturbances. The paper formalizes actuator saturation and QP infeasibility as consequences of improper kernels, including for high-order CBF formulations and external pre-filtering.

  • Memorylessness of Standard CBFs: The standard CBF kernel lacks internal state or integral action, making commanded control highly sensitive to bounded measurement noise and high-frequency disturbances.Its zero-memory operation provides no proper filtering mechanism between boundary trajectories and control inputs.
  • Actuator saturation: An improper safety kernel can require control effort outside any finite actuator bound under a valid bounded disturbance with arbitrarily high-frequency energy.The result applies to affine safety constraints under strict input saturation limits.
  • Actuator saturation: Because an improper kernel has unbounded frequency response, bounded disturbances can make the required control exceed the admissible set and render the safety constraint infeasible.The paper derives this from sup_ω≥0 |G(jω)| = ∞ and the corresponding control-feasibility bound.
  • Higher-order and exponential CBFs: For relative degree n ≥3, classical HOCBF and ECBF operators exhibit gain divergence O(ω^n), making them susceptible to disturbance-induced actuator saturation.This is presented as a corollary of the general improper-kernel saturation result.
  • Scope of the limitation: The actuator-saturation conclusion extends to general nonlinear class-K functions because local linearization retains an improper polynomial kernel of degree n.Nonlinear intermediate decay rates alter local boundary curvature but do not eliminate high-frequency gain divergence.
  • External pre-filtering: External upstream filtering can remove the control input from the active safety constraint when the safety operator evaluates only n derivatives of the filtered boundary.In that case, L_gC(t) = 0 and the QP becomes independent of u(t).

2) Loss of Physical Forward Invariance (Surrogate Mismatch):

External temporal filtering can preserve a surrogate barrier set while failing to preserve the physical safe set. Phase lag and insufficient control authority explain how trajectories can penetrate the true boundary.

  • Surrogate mismatch: Forward invariance is preserved exclusively for the surrogate set ˜C, which generally does not guarantee invariance of the physical set C.The surrogate set is defined by ˜h(t) ≥ 0, whereas the physical set is defined by h(x) ≥ 0.
  • Boundary penetration: Uncompensated causal memory introduces phase lag that permits physical boundary penetration while the surrogate filter still registers safety.This mismatch motivates a proper spatio-temporal kernel that preserves control authority while addressing the physical set.
  • Loss of control authority: A proper low-pass filter can remove direct dependence of the n-th order safety constraint on u(t), causing LgC(t) = 0 and total loss of control authority.The filtered output’s successive derivatives truncate below the n-th spatial derivative where the control input enters.
  • Surrogate mismatch: Restoring control authority by differentiating the augmented state raises the relative degree from n to n + m.The resulting (n + m)-th order CBF concerns the augmented surrogate dynamics.
  • Boundary penetration: At physical boundary contact, causal filtering can keep ˜h(t) positive because the signal accumulates historical interior states.The controller may therefore withhold maximal boundary-deflecting effort while the physical state enters h(x(t)) < 0.

V. THE SPACE-TIME TRANSFORM

The Space-Time Transform replaces instantaneous spatial differentiation with a proper causal spatio-temporal convolution. Its bounded kernel embeds temporal filtering into the safety constraint while retaining finite-gain control and forward invariance of the augmented state.

  • Kernel construction: The STT resolves unbounded high-frequency gain by replacing the improper spatial kernel with a proper spatio-temporal kernel.The construction combines the improper spatial operator with a causal temporal filter through convolution.
  • Kernel construction: When m ≥ n + p, the Space-Time Kernel contains no Dirac delta distributions or derivatives and becomes a bounded distributed function over time.Here p and m are the numerator and denominator degrees of the temporal filter.
  • Transform operator: The STT maps a spatial boundary trajectory to a feasible control constraint by integrating the spatial history against the Space-Time Kernel.The operator evaluates the causal trajectory history and embeds filter dynamics directly in the kernel.
  • Guarantees and unification: The STT bounds demanded control effort, prevents QP crashes associated with zero-memory HOCBFs and ECBFs, and maintains exact forward invariance of the augmented state.Classical CBF, HOCBF, and ECBF formulations arise as the infinite-bandwidth, memoryless limit of the STT.
  • Transform operator: By convolution associativity, the STT can be expressed as temporal filtering of spatial Lie-derivative mappings evaluated over the causal history.The spatial mapping is applied before the outer temporal convolution, yielding the equivalent Lie-derivative formulation.
  • Design restriction: The transform requires a linear extended decay function αL, although the underlying spatial barrier h(x) may remain nonlinear and non-convex.The restriction follows from requiring an LTI finite-gain kernel in the Laplace domain.

A. Operational Calculus of the Space-Time Transform

The STT is presented as a mathematical operator with linearity, causal representation, and invertibility properties. These properties support a kernel-based calculus for spatio-temporal safety filters.

  • Linearity: The STT is a linear functional over extended boundary trajectories, so superposition follows from linear Lie differentiation and temporal convolution.For boundaries h1(x) and h2(x), scalar combinations are mapped linearly.
  • Causal representation: Valid proper affine safety constraints with bounded control authority admit a unique causal impulse-response kernel in L1(R≥0).The STT operator class is characterized by proper filters with m ≥ n.
  • Invertibility: The STT has a physical inverse mapping that recovers the required continuous spatial geometry from a designated temporal survival horizon.This inversion is defined through the temporal settling-time map.

VI. SYNTHESIS OF SPATIAL DECAY VIA TEMPORAL HORIZON INVERSION

The paper links spatial barrier decay to physical time-to-impact so that temporal filter bandwidth and spatial coefficients are synthesized consistently. Under the LTI restriction, this produces a unique linear decay mapping.

  • Motivation: Standard CBF practice can heuristically choose spatial decay parameters without synchronizing them to actuator temporal limits.The paper identifies this desynchronization as a source of phase lag and boundary overshoot.
  • Survival horizon: The survival horizon τ(x) measures the maximum allowable time for a trajectory with margin h(x) > 0 to reach a boundary threshold ϵ.It is obtained by integrating the spatial decay dynamics over the margin-to-boundary interval.
  • Survival horizon: The survival horizon collapses the state into a scalar time-to-impact variable that can parameterize filter bandwidths, gain schedules, and supervisory switches.This provides a low-dimensional scheduling coordinate across the n-dimensional state space.
  • Horizon inversion: Under the LTI restriction, inverse horizon mapping uniquely yields the linear decay law α(h) = λhh.The coefficient λh converts spatial margin into the required temporal horizon in rad/s.
  • Horizon inversion: The actual survival time is bounded below by the synthesized horizon, τactual(x) ≥ τ(x), including the most aggressive allowable approach to the boundary.The bound ensures the temporal filter retains sufficient bandwidth for state transients.
  • Hardware synchronization: The plant bandwidth bounds the filter bandwidth, whose dominant pole determines the spatial decay rate and coefficients.For an n-th order system, the coefficients mirror the characteristic polynomial of the temporal filter.

VII. ROBUST FORWARD INVARIANCE VIA THE SPACE-TIME TRANSFORM

The STT framework combines proper temporal filtering with direct control authority, then proves robust forward invariance under explicit filter, initialization, and regularity assumptions.

  • Assumptions: The forward-invariance result requires six explicit mathematical and structural assumptions concerning the temporal filter and its synchronization with barrier relative degree.These include EMA-cascade structure, positive real poles, m ≥ n, pole synchronization, architectural separation, and consistent initialization.
  • Architectural basis: STT replaces external pre-filtering with a proper causal convolution while preserving instantaneous control authority through architectural separation.The filter acts on the unactuated drift certificate, while g(x)u(t) bypasses the convolution.
  • Filter properties: EMA-cascade filters preserve positivity and monotonicity, enabling bounded perturbation analysis of the filtered drift certificate.Non-negative impulse responses preserve inequalities under convolution.
  • Discrepancy bound: Under consistent initialization and bounded drift-certificate derivative, the filter discrepancy remains uniformly bounded, with ultimate magnitude inversely proportional to bandwidth.For cascades, the bound depends on κ(F) and the slowest pole; non-ideal initialization adds a decaying transient.
  • Robust invariance: Continuous enforcement of the STT constraint yields robust forward invariance of an ISS-relaxed safe set with an explicit trade-off between noise attenuation and spatial deformation.The perturbation bound is ϵ = κ(F)L/pmin, and the safe set is CISS = {x : h(x) ≥ −δ(ϵ)}.

VIII. ENGINEERING DESIGN PROCEDURE AND REAL-TIME EXECUTION

The paper turns the STT theory into a six-step real-time design procedure that links hardware limits, spatial safety geometry, temporal filtering, state realization, and bounded QP execution.

  • Procedure overview: The six-step procedure replaces improper differentiation with a causal LTI extension and connects hardware characterization directly to discrete-time implementation.The workflow is designed to eliminate heuristic gain tuning.
  • Hardware and geometry: Hardware characterization identifies actuator bandwidth and strict saturation bounds before selecting the safety boundary and its relative degree.These quantities constrain subsequent filter and controller design.
  • Dynamic extension: The temporal filter is realized as a continuous LTI state-space system whose output evaluates the causal Space-Time integral.Instantaneous spatial Lie derivatives feed the filter dynamics at each time step.
  • Real-time execution: Zero-Order Hold discretization produces discrete filter matrices for sample-time state updates and exact bounded Quadratic Program execution.The QP uses the instantaneous control coefficient b(xk).

IX. EXPERIMENTAL VALIDATION

Experimental validation uses a noisy, saturated third-order triple-integrator approaching a hard boundary, evaluated over repeated discrete-time Monte Carlo runs.

  • Plant and scenario: The experiment simulates a third-order triple-integrator with symmetric jerk saturation of umax = 3.2 m/s3 and an aggressive approach toward h(x) = x ≥ 0.The initial state is (9.3 m, −6.0 m/s, 0.0 m/s2), with nominal command unom = −3.2 m/s3.
  • Noise conditions: State-feedback noise has standard deviations σx = 0.1 m and σv = 0.3 m/s, with a stochastic safety buffer ϵrobust = 0.05 m.The buffer uses kσ = 0.5.
  • Evaluation protocol: All architectures run at 200 Hz for 6.5 s across N = 50 Monte Carlo simulations using exact Zero-Order Hold discretization.Identical noise sequences and discrete-time execution support controlled comparison.

A. Evaluated Safety Filter Architectures

The benchmark compares six safety-filter configurations, with external pre-filtering losing control authority and becoming infeasible under the evaluated conditions.

  • Evaluated architectures: Six configurations span nominal HOCBF, nominal ECBF, parameterized barriers, external pre-filtering, and two STT filter topologies.The STT variants use third-order kernels with ωf = 14.0 rad/s.
  • External pre-filtering: External pre-filtering reduces the affine control coefficient ∂u to zero throughout the tested horizon.The safety inequality therefore contains no active control input.
  • External pre-filtering: 0.0% safety rate is reported for external pre-filtering, alongside TV(u) = 0.0 ± 0.0 and mean boundary penetration of −176.14 m.The QP remains permanently infeasible and falls back to u = unom = −3.2 m/s3.

C. Validation of Theorem 2: Chatter Suppression and Robotic Hardware Implications

Monte Carlo validation shows that STT Real Poles preserves safety and affine control authority while sharply suppressing control chattering, unlike conventional memoryless barriers and externally filtered variants.

  • Observed trajectories: The single-run trajectories show extreme commanded-jerk chattering for standard and PBF CBFs versus smooth STT control under position and velocity measurement noise.Figure 1 uses n = 3, σx = 0.1 m, and σv = 0.3 m/s; its fourth row reports the affine control coefficient.
  • Safety and control performance: STT Real Poles maintains full affine control authority with LgC(t) = 1.0 while filtering high-frequency measurement noise through a spatio-temporal kernel.This contrasts with external pre-filtering, which can produce complete control-authority loss, LgC(t) = 0.
  • Safety and control performance: TV(u) falls from 2178.3 ± 42.6 with PBF to 20.9 ± 1.6 with STT Real Poles, a reduction of over 99%.PBF preserves spatial safety and active control authority but does not attenuate temporal noise.
  • Filter topology and hardware implications: Real-pole STT yields 100.0% safety with hmin = 0.06 ± 0.02 m, whereas Bessel STT yields 92.0% safety and TV(u) = 46.7 ± 2.9.The Bessel filter’s higher initial group delay delays maximum braking and produces transient overshoot, causing boundary violations in 8% of runs.
  • Filter topology and hardware implications: The results connect STT’s proper temporal convolution to safety bounds that respect physical hardware limits, while the paper identifies broader functional analysis as future work.The stated future work concerns fully characterizing the operator space across broader classes of control systems.
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