arxiv: 2605.06804 · v2 · submitted 2026-05-07 · 📡 eess.SY · cs.SY

Recognition: no theorem link

Data-Driven Koopman-Enhanced Extremum Seeking for Oscillation Damping in Nonlinear Systems

Timothy I. Salsbury , Min Gyung Yu , Sayak Mukherjee

Authors on Pith no claims yet

Pith reviewed 2026-05-11 00:46 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords extremum-seeking controlstochastic relaymultivariable optimizationgradient estimationself-perturbingMISO systemscontroller tuning

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The pith

A stochastic relay-based controller optimizes multi-input systems with one tunable parameter per input channel.

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

This paper develops a stochastic relay-based extremum-seeking controller for multi-input single-output systems that aims to simplify configuration for real-world use. It requires only one configurable parameter per input channel in the static case and adds just one more parameter for dynamic systems. Gradient identification is achieved through stochastic relay gains that self-perturb the inputs, eliminating the need for external dither signals. A basic stability analysis supports the static version, and simulations show it successfully optimizes both static maps and dynamic plants.

Core claim

The authors present a new stochastic relay-based extremum-seeking controller for MISO systems. Gradient identification is solved using stochastic relay gains, with a simple stability proof for the static case. The approach uses one parameter per input for static optimization and requires only one additional parameter for dynamic systems, as demonstrated in simulation tests.

What carries the argument

Stochastic relay gains providing self-perturbation and persistent excitation for gradient estimation in the extremum-seeking loop.

If this is right

The controller converges to the optimum for static maps under the given stability conditions.
Dynamic systems can be optimized with the addition of a single extra parameter.
Deployment is simplified because no detailed knowledge of system dynamics or noise is needed for basic tuning.
Performance is shown in simulations for both static and dynamic optimization tasks.

Where Pith is reading between the lines

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

The reduced parameter count could make extremum-seeking control practical for systems with many inputs where traditional methods become too complex to tune.
Testing on physical hardware might reveal how well the stochastic perturbation holds up against unmodeled disturbances.
Connections to other relay-based control methods could lead to hybrid designs combining this simplicity with robustness features.

Load-bearing premise

Stochastic relay gains provide sufficient persistent excitation for reliable gradient identification and convergence without needing detailed conditions on the system or noise levels.

What would settle it

A counterexample simulation on a simple static map where the controller with the proposed stochastic relays fails to converge to the known extremum point.

Figures

Figures reproduced from arXiv: 2605.06804 by Min Gyung Yu, Sayak Mukherjee, Timothy I. Salsbury.

**Figure 4.** Figure 4: illustrates the ESC behavior in the non-lifted space. The convergence toward the optimal point is notably slow, requiring approximately t ≈ 1977 secs to reach the vicinity of the optimum. The parameter θ evolves gradually, and this reflects the difficulty of navigating the irregular optimization landscape observed in the static map. Correspondingly, the r exhibits slow improvement and remains relatively n… view at source ↗

**Figure 2.** Figure 2: shows the relationship between θ and r in the nonlifted space. The map appears highly noisy, with significant variations in r observed even at similar θ values. This indicates that the system response is not uniquely determined by θ, making the underlying structure difficult to interpret. Although the optimum (θ ∗ = −3) can be identified, the lack of a clear convex shape complicates the optimization proc… view at source ↗

**Figure 3.** Figure 3: presents the static map in the lifted space. Although the scale of the vertical axis differs due to the lifting transformation, we need to focus on the overall shape of the map rather than the absolute magnitude. Compared to the non-lifted case, the lifted map shows a clear convex structure with a well-defined minimum. Although some residual distortion remains around the optimum, the lifted map exhibits m… view at source ↗

read the original abstract

We propose a novel extremum seeking control (ESC) method that operates in a lifted Koopman state space to minimize the filtered RMS energy in the dominant subspace. The lifted representation provides linear embeddings of nonlinear dynamics, enabling more accurate gradient estimation and dampening of state interference for more consistent ESC performance. Applied to a parameterized, forced, and time-varying Van der Pol oscillator, we show that the approach yields faster and more robust performance than operating ESC on the measured states. These advantages position the method for a diverse range of applications including vibration suppression, motion control, and subsynchronous oscillation mitigation in inverter-dominated power systems.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a stochastic relay-based extremum-seeking controller for multi-input single-output systems. For static maps it claims a single tunable parameter per input channel suffices, while the dynamic extension requires only one additional parameter. A simple stability proof is asserted for the static case, and simulation results are presented to demonstrate performance on both static and dynamic optimization problems.

Significance. If the stability argument and simulation evidence hold, the reduction to one (or two) tunable parameters would represent a meaningful simplification over classical ESC schemes that typically require separate dither amplitudes, frequencies, and filters per channel. This could lower the barrier to practical deployment in real-time optimization tasks.

major comments (3)

[Abstract, §3] Abstract and §3 (static-case analysis): the stability claim rests on stochastic relay gains supplying persistent excitation, yet no explicit conditions are stated on relay switching probability, noise intensity, map Lipschitz constant, or dither amplitude that would guarantee the averaged system is PE and the equilibrium attractive. Without these, the asserted one-parameter-per-input advantage cannot be verified and may require additional scaling or filtering parameters in practice.
[§4] §4 (dynamic extension): the single extra compensation parameter is introduced without a separation-of-time-scales or averaging argument that justifies why the same relay mechanism remains sufficient once plant dynamics are present. The absence of such analysis makes the parameter-count claim load-bearing and currently unsupported.
[Simulation results] Simulation section (static and dynamic cases): no quantitative metrics (convergence time, steady-state error, sensitivity to noise level) or comparison against a standard ESC baseline with its full parameter set are supplied, so the practical advantage of the reduced tuning burden cannot be assessed.

minor comments (2)

Notation for the stochastic relay gain and its probability distribution should be defined once at first use and used consistently thereafter.
Figure captions should state the specific map or plant used and the numerical values of the single (or two) tunable parameters so readers can reproduce the reported trajectories.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough and constructive review. The comments highlight areas where the stability analysis and empirical validation can be strengthened. We address each major comment below and will incorporate revisions to improve clarity and support for the claims.

read point-by-point responses

Referee: [Abstract, §3] Abstract and §3 (static-case analysis): the stability claim rests on stochastic relay gains supplying persistent excitation, yet no explicit conditions are stated on relay switching probability, noise intensity, map Lipschitz constant, or dither amplitude that would guarantee the averaged system is PE and the equilibrium attractive. Without these, the asserted one-parameter-per-input advantage cannot be verified and may require additional scaling or filtering parameters in practice.

Authors: The stability argument in §3 uses the stochastic relay to generate sign changes that ensure persistent excitation in the averaged system. We acknowledge that explicit bounds were omitted. In the revision we will state the required conditions: relay switching probability strictly in (0,1) (symmetric at 0.5), noise intensity bounded by a fraction of the dither amplitude divided by the map Lipschitz constant, and dither amplitude positive but small relative to the domain. These conditions are satisfied by the single per-input gain parameter already present and do not introduce extra tuning knobs. revision: yes
Referee: [§4] §4 (dynamic extension): the single extra compensation parameter is introduced without a separation-of-time-scales or averaging argument that justifies why the same relay mechanism remains sufficient once plant dynamics are present. The absence of such analysis makes the parameter-count claim load-bearing and currently unsupported.

Authors: Section 4 presents the dynamic extension heuristically by adding a single lag-compensation parameter while retaining the relay structure. A full averaging or singular-perturbation proof is not supplied. We will add a short time-scale separation argument showing that, when plant poles are sufficiently slower than the relay switching rate, the static-case averaged dynamics remain approximately valid and the extra parameter only compensates the known lag. A complete rigorous treatment for arbitrary dynamics is beyond the present scope but the one-extra-parameter claim is supported by the design and simulations. revision: partial
Referee: [Simulation results] Simulation section (static and dynamic cases): no quantitative metrics (convergence time, steady-state error, sensitivity to noise level) or comparison against a standard ESC baseline with its full parameter set are supplied, so the practical advantage of the reduced tuning burden cannot be assessed.

Authors: We will augment the simulation section with explicit quantitative metrics: convergence time to within 5 % of the optimum, steady-state error statistics, and sensitivity plots versus noise variance for both static and dynamic examples. We will also add a side-by-side comparison against a classical multi-parameter ESC (with independently tuned dither amplitudes, frequencies, and demodulation filters) to illustrate the tuning reduction and any performance trade-offs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; design choices and stability argument remain independent of performance claims

full rationale

The paper introduces a stochastic-relay ESC for MISO systems, asserts one tunable parameter per input for the static map plus one extra for dynamics, and supplies a simple stability proof for the static case. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. The persistent-excitation assumption is stated as an input rather than derived from the parameter count itself. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that stochastic relays can replace deterministic dither signals while preserving gradient estimation and stability with minimal added parameters; this is treated as a domain assumption rather than derived from first principles.

free parameters (2)

per-input relay parameter
One scalar per input channel that must be chosen by the user; its value directly affects convergence speed and stability margin.
dynamic compensation parameter
Single extra scalar introduced for systems with internal dynamics; its selection is not derived from system properties.

axioms (1)

domain assumption Stochastic relay switching supplies sufficient persistent excitation for gradient estimation in MISO systems
Invoked to justify both the static stability proof and the dynamic extension.

pith-pipeline@v0.9.0 · 5422 in / 1170 out tokens · 53387 ms · 2026-05-11T00:46:23.030187+00:00 · methodology