arxiv: 2604.03138 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.SY

Recognition: no theorem link

Logarithmic Barrier Functions for Practically Safe Extremum Seeking Control

Qixu Wang , Patrick McNamee , Zahra Nili Ahmadabadi

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:52 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords logarithmic barrier functionextremum seeking controlpractical safetyforward invarianceaveraging theorymodel-free optimizationsafe setdither signal

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The pith

Logarithmic barriers augment the cost in extremum seeking to keep trajectories inside a safe set and ensure interior convergence for fast dithers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper augments an unknown objective with a logarithmic barrier that adds a repulsive term near the safety boundary. Averaging theory then shows that sufficiently rapid dither signals keep the original closed-loop trajectories inside a positive safety margin, rendering the safe set forward invariant. Stability analysis establishes local practical convergence to the minimizer of the modified cost, which lies strictly inside the safe set. The method embeds safety directly into the optimization rather than applying an external filter, avoiding chattering. Sequential reduction of small parameters (dither amplitude, adaptation gain, barrier weight) yields the practical safety and convergence results.

Core claim

By augmenting the objective with a logarithmic barrier function, the extremum-seeking law produces a modified cost whose minimizer is interior to the safe set. Averaging arguments then prove that the original trajectories remain confined to a positive distance from the boundary whenever the dither frequency is high enough relative to the barrier and adaptation gains, establishing forward invariance of the safe set. Local practical asymptotic stability to this interior point follows from sequential tuning of the small parameters.

What carries the argument

Logarithmic barrier function that augments the objective and creates a repulsive potential penalizing proximity to the safety boundary.

If this is right

The safe set is forward invariant for sufficiently fast dithers.
Trajectories converge locally and practically to the minimizer of the barrier-augmented cost.
Safety is achieved without chattering induced by external safety filters.
The result holds for model-free extremum seeking under the stated averaging conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same barrier-augmented formulation could be combined with other periodic excitation signals beyond standard dithers.
Tuning guidelines for the barrier parameter might be derived from the explicit safety-margin bounds obtained via averaging.
The approach may extend to systems with slowly time-varying constraints if the barrier is updated on a slower timescale.
Practical hardware tests would need to verify that actuator limits do not interfere with the required dither speed.

Load-bearing premise

The dither signal must be sufficiently fast relative to the barrier and adaptation gains for averaging theory to guarantee that trajectories stay close to the averaged safe behavior.

What would settle it

Numerical simulation or experiment in which the dither frequency is progressively lowered until trajectories exit the safety margin or approach the boundary closer than the predicted margin.

Figures

Figures reproduced from arXiv: 2604.03138 by Patrick McNamee, Qixu Wang, Zahra Nili Ahmadabadi.

**Figure 4.** Figure 4: Simulation of a 2D point mass case with passage navigation. The [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 3.** Figure 3: Simulation of a 2D point mass case with island-shaped obstacle [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

This paper presents a methodology for Practically Safe Extremum Seeking (PSfES), designed to optimize unknown objective functions while strictly enforcing safety constraints via a Logarithmic Barrier Function (LBF). Unlike traditional safety-filtered approaches that may induce chattering, the proposed method augments the cost function with an LBF, creating a repulsive potential that penalizes proximity to the safety boundary. We employ averaging theory to analyze the closed-loop dynamics. A key contribution of this work is the rigorous proof of practical safety for the original system. We establish that the system trajectories remain confined within a safety margin, ensuring forward invariance of the safe set for a sufficiently fast dither signal. Furthermore, our stability analysis shows that the model-free ESC achieves local practical convergence to the modified minimizer strictly within the safe set, through the sequential tuning of small parameters. The theoretical results are validated through numerical simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Log barrier inside the ESC cost gives chattering-free safety but the averaging proof leaves the required dither speed unquantified.

read the letter

The main contribution is folding a logarithmic barrier straight into the extremum-seeking cost so the optimizer naturally repels from the constraint boundary. This avoids the chattering that often comes with separate safety filters or projections. They run standard ESC on the augmented cost and use averaging theory to argue that fast dither keeps the real trajectories inside a safety margin, making the safe set forward invariant while the system practically converges to the modified minimizer inside that set.

Referee Report

2 major / 2 minor

Summary. The paper proposes a practically safe extremum seeking control (PSfES) scheme that augments an unknown objective with a logarithmic barrier function to enforce safety constraints without chattering. Averaging theory is applied to the closed-loop dynamics to prove that, for a sufficiently fast dither signal, trajectories remain confined within a safety margin (forward invariance of the safe set) and that the system achieves local practical convergence to the modified minimizer strictly inside the safe set via sequential tuning of small parameters. Numerical simulations are provided to illustrate the results.

Significance. If the central claims hold with explicit bounds, the work would offer a constructive, chattering-free alternative to safety-filtered ESC for model-free constrained optimization, with potential value in applications such as process control or robotics. The combination of logarithmic barriers with averaging analysis for practical safety is a clear technical contribution, and the emphasis on sequential parameter tuning is a positive step toward implementable guarantees.

major comments (2)

[stability analysis / averaging proof] In the stability analysis section (averaging-based proof of practical safety): the claim that trajectories remain within a safety margin for a 'sufficiently fast' dither signal is established only in the limit as dither frequency ω → ∞. No explicit lower bound on ω (in terms of barrier gain/width, adaptation gain, or plant Lipschitz constants) or O(1/ω) trajectory-error estimate is supplied, so it is impossible to verify that the safety margin is respected for any finite ω on a concrete plant.
[abstract and stability analysis] Abstract and § on sequential tuning: the statement that 'sequential tuning of small parameters' yields local practical convergence inside the safe set lacks accompanying error-bound calculations or explicit conditions relating the barrier parameter, adaptation gain, and dither frequency. Without these, the distance between the original and averaged trajectories cannot be guaranteed to stay below the safety margin.

minor comments (2)

[problem formulation] Notation for the logarithmic barrier function and its gradient should be introduced with an explicit equation number at first use to improve readability.
[numerical simulations] The simulation section would benefit from a table listing the exact parameter values (barrier width, dither amplitude/frequency, adaptation gains) used in each figure so that the 'sufficiently fast' regime can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below and indicate the revisions we plan to make.

read point-by-point responses

Referee: In the stability analysis section (averaging-based proof of practical safety): the claim that trajectories remain within a safety margin for a 'sufficiently fast' dither signal is established only in the limit as dither frequency ω → ∞. No explicit lower bound on ω (in terms of barrier gain/width, adaptation gain, or plant Lipschitz constants) or O(1/ω) trajectory-error estimate is supplied, so it is impossible to verify that the safety margin is respected for any finite ω on a concrete plant.

Authors: We agree that our proof relies on the standard averaging theory result that guarantees the existence of a sufficiently large ω without providing an explicit lower bound or an O(1/ω) error estimate in the main text. This is a common approach in ESC literature to maintain focus on the core ideas, but we acknowledge it limits immediate applicability for specific plants. In the revised version, we will add a new remark in the stability analysis section noting that explicit bounds can be obtained by following the constructive proofs in averaging theory references (e.g., by estimating the Lipschitz constants of the averaged system and the perturbation terms), and we will include a high-level outline of how the safety margin scales with 1/ω. However, deriving fully explicit expressions for general cases would require additional assumptions on the plant and significantly extend the paper length. revision: partial
Referee: Abstract and § on sequential tuning: the statement that 'sequential tuning of small parameters' yields local practical convergence inside the safe set lacks accompanying error-bound calculations or explicit conditions relating the barrier parameter, adaptation gain, and dither frequency. Without these, the distance between the original and averaged trajectories cannot be guaranteed to stay below the safety margin.

Authors: The sequential tuning procedure is intended to first fix the barrier function parameters to shape the modified objective, then select small adaptation gains and large dither frequency to ensure practical convergence. We recognize that the manuscript does not provide explicit error bounds or relations between these parameters. To address this, we will revise the abstract and the relevant section to include a more detailed description of the tuning order, along with a statement that the trajectory error between the original and averaged systems is bounded by a term that vanishes as the dither frequency increases, ensuring it remains within the safety margin for sufficiently large ω. We will also add a note that quantitative bounds depend on the specific system Lipschitz constants and can be computed on a case-by-case basis. revision: partial

Circularity Check

0 steps flagged

No circularity: standard averaging applied to augmented ESC dynamics

full rationale

The derivation applies classical averaging theory to the closed-loop system that includes the logarithmic barrier augmentation of the cost. Practical safety (forward invariance within a margin) and local practical convergence to the modified minimizer are obtained by sequential tuning of small parameters (dither frequency ω, barrier coefficient, adaptation gains). No step reduces a claimed result to a fitted quantity by construction, no self-definitional loop appears in the equations, and no load-bearing uniqueness theorem or ansatz is imported from prior self-citations. The central claims therefore remain independent of the inputs they are derived from.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard averaging theory for time-scale separation and on the assumption that the barrier parameters can be chosen small enough without destroying the extremum-seeking property. No new physical entities are introduced.

free parameters (2)

barrier gain and width parameters
Chosen small enough to keep the modified minimizer inside the safe set while still allowing convergence.
dither amplitude and frequency
Tuned sufficiently small in amplitude and large in frequency for averaging to hold and for practical safety.

axioms (2)

domain assumption Averaging theory applies to the closed-loop dynamics when the dither is fast
Invoked to separate the fast dither from the slow adaptation and to prove practical safety.
domain assumption The original plant is smooth and the objective has a local minimum inside the safe set
Standard assumption for local convergence results in extremum seeking.

pith-pipeline@v0.9.0 · 5456 in / 1604 out tokens · 38869 ms · 2026-05-13T18:52:16.619594+00:00 · methodology

discussion (0)

Reference graph

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