arxiv: 2603.14990 · v2 · submitted 2026-03-16 · 📡 eess.SY · cs.SY· math.OC

Recognition: no theorem link

Chattering Reduction for a Second-Order Actuator via Dynamic Sliding Manifolds

Patricia N\"other , Lars Watermann , Johann Reger

Authors on Pith no claims yet

Pith reviewed 2026-05-15 10:45 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC

keywords chattering reductiondynamic sliding manifoldsliding-mode controlharmonic balance methodsecond-order actuatorscalar integratorlimit cycle

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The pith

Dynamic sliding manifolds can be tuned to reduce chattering amplitude versus static manifolds in second-order actuator systems.

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

The paper studies chattering in a scalar integrator under first-order sliding-mode control when the actuator follows unknown second-order dynamics. It contrasts the conventional static sliding manifold with a dynamic version that incorporates additional states. The harmonic balance method is applied to approximate the resulting limit cycle and to show that suitable parameter choices for the dynamic manifold shrink the oscillation amplitude relative to the static case. A simulation example confirms the predicted reduction, and the work is presented as a proof of concept for further chattering mitigation studies.

Core claim

For the considered system class the parameters of the dynamic sliding manifold can be selected so that the amplitude of the chattering limit cycle is smaller than the amplitude obtained with the static sliding manifold, as established by the harmonic balance approximation even when the actuator time constant remains unknown.

What carries the argument

The dynamic sliding manifold, which augments the sliding variable with filtered derivatives or extra dynamics to reshape the chattering limit cycle.

Load-bearing premise

The harmonic balance method supplies a sufficiently accurate description of the chattering limit cycle when the actuator time constant is unknown and the plant is a scalar integrator.

What would settle it

A simulation or hardware test in which the tuned dynamic manifold produces a larger or equal chattering amplitude than the static manifold, or in which the observed oscillation deviates markedly from the harmonic-balance prediction.

Figures

Figures reproduced from arXiv: 2603.14990 by Johann Reger, Lars Watermann, Patricia N\"other.

**Figure 2.** Figure 2: Harmonic Balance analysis for SMC and DSM each [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Simulation with τ = 0.01 s 0 1 2 3 4 5 6 7 8 9 10 x(t) -2 -1 0 0 1 2 3 4 5 6 7 8 9 10 <(t) -2 -1 0 SSM DSM t [s] 0 1 2 3 4 5 6 7 8 9 10 u(t) -1 0 1 2 7.5 8 8.5 9 9.5 -0.05 0 0.05 7.5 8 8.5 9 9.5 -0.01 0 0.01 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Simulation with τ = 0.1 s In [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

We analyze actuator chattering in a scalar integrator system subject to second-order actuator dynamics with an unknown time constant and first-order sliding-mode control, using both a conventional static sliding manifold and a dynamic sliding manifold. Using the harmonic balance method, we prove that it is possible to adjust the parameters of the dynamic sliding manifold for the specified system class so as to reduce the amplitude of the chattering in comparison to the static manifold. We illustrate our results with a simulation example. This contribution serves as a proof of concept to motivate further investigations in chattering reduction via dynamic sliding manifolds.

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

The paper shows via harmonic balance that dynamic sliding manifold parameters can be tuned to cut chattering amplitude versus the static case for this integrator-plus-second-order-actuator setup, but the approximation has no error bounds.

read the letter

The central result is a parameter-tuning argument: for the scalar integrator closed with first-order sliding mode control and second-order actuator dynamics (unknown time constant), the authors use harmonic balance on the closed-loop equations to show that suitable dynamic-manifold gains produce a smaller fundamental-component amplitude than the static manifold. They back this with a simulation example that illustrates the direction of the improvement. That combination applied to the unknown-tau case is the concrete new piece relative to the cited literature on each technique separately. The setup is clean and the derivation follows standard harmonic-balance steps without obvious algebraic slips. The simulation is consistent with the claim, which is helpful for a proof-of-concept note. The soft spot is exactly where the stress-test flagged: harmonic balance discards higher harmonics and treats the limit cycle as purely sinusoidal, yet the paper supplies neither an a-priori bound on the neglected terms nor a check that the sign of the amplitude difference survives the approximation error for every tau > 0. Without that, the inequality proven for the truncated equations does not automatically transfer to the actual switched trajectories. The claim is therefore conditional on the approximation being sufficiently accurate, which is asserted but not quantified. This is a targeted control-theory note rather than a broad advance. Readers working on chattering mitigation for actuators with unmodeled dynamics would get value from the explicit parameter conditions and the simulation; anyone needing rigorous limit-cycle guarantees would want to see error estimates added. The work is coherent on its own terms and engages the relevant literature, so it clears the bar for serious refereeing even though the approximation step needs scrutiny. I would send it out for review with a request that the authors address the accuracy of the harmonic-balance prediction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes chattering in a scalar integrator subject to second-order actuator dynamics with unknown time constant under first-order sliding mode control. It compares a conventional static sliding manifold to a dynamic sliding manifold and uses the harmonic balance method to prove that suitable parameter tuning of the dynamic manifold reduces the chattering amplitude relative to the static case. The results are illustrated with a simulation example, positioning the work as a proof of concept.

Significance. If the harmonic balance approximation reliably captures the amplitude reduction for unknown actuator time constants, the paper provides a concrete method for chattering attenuation in sliding-mode systems with actuator dynamics. This could motivate further theoretical and practical developments in dynamic sliding manifolds for control applications.

major comments (2)

[Harmonic Balance Analysis] The central claim rests on applying the harmonic balance method to the closed-loop equations and showing that the resulting approximate amplitude for the dynamic manifold is smaller than for the static manifold. However, the derivation equates only the fundamental Fourier components and supplies neither an a priori bound on the truncation error from neglected harmonics nor a demonstration that the sign of (A_dynamic – A_static) is preserved for all τ > 0 under that error. This gap directly affects the validity of the stated proof for the actual switched system.
[Simulation Results] The simulation example illustrates reduced chattering for chosen parameters but does not quantify the approximation error of the harmonic balance prediction against the true limit-cycle amplitude, nor does it test the inequality across a range of unknown actuator time constants τ. Consequently, the numerical results do not constitute rigorous validation of the analytic claim.

minor comments (2)

[Introduction / System Description] The notation for the dynamic sliding manifold parameters (e.g., the precise definition of the additional state and its gain) would benefit from an explicit equation block immediately after the system description to improve readability.
[Harmonic Balance Analysis] A brief remark on the range of validity of the harmonic balance assumption (e.g., sufficiently high switching frequency relative to actuator bandwidth) would help readers assess applicability.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major point below, proposing targeted revisions to improve clarity while maintaining the proof-of-concept scope of the work.

read point-by-point responses

Referee: [Harmonic Balance Analysis] The central claim rests on applying the harmonic balance method to the closed-loop equations and showing that the resulting approximate amplitude for the dynamic manifold is smaller than for the static manifold. However, the derivation equates only the fundamental Fourier components and supplies neither an a priori bound on the truncation error from neglected harmonics nor a demonstration that the sign of (A_dynamic – A_static) is preserved for all τ > 0 under that error. This gap directly affects the validity of the stated proof for the actual switched system.

Authors: We agree that the harmonic balance approach considers only the fundamental component and is therefore an approximation. This method is standard in the sliding-mode chattering literature for predicting limit-cycle amplitudes in switched systems. In the revision we will explicitly label the result as approximate, add a brief discussion of the conditions under which higher harmonics remain small for the given actuator class, and include numerical checks confirming that the predicted sign of the amplitude difference holds over the tested range of τ. A rigorous a priori error bound valid for every τ > 0 would require a substantially different analytical framework and lies outside the intended proof-of-concept contribution. revision: partial
Referee: [Simulation Results] The simulation example illustrates reduced chattering for chosen parameters but does not quantify the approximation error of the harmonic balance prediction against the true limit-cycle amplitude, nor does it test the inequality across a range of unknown actuator time constants τ. Consequently, the numerical results do not constitute rigorous validation of the analytic claim.

Authors: We accept this observation. The revised manuscript will include a direct quantitative comparison (error metrics) between the harmonic-balance amplitude predictions and the simulated limit-cycle amplitudes. We will also extend the simulation study to several distinct values of the unknown time constant τ, thereby illustrating that the amplitude reduction persists across a representative range of actuator dynamics. revision: yes

standing simulated objections not resolved

Deriving a rigorous a priori bound on the harmonic-balance truncation error that guarantees preservation of the sign of (A_dynamic – A_static) for all τ > 0.

Circularity Check

0 steps flagged

No circularity: harmonic balance applied directly to closed-loop equations yields independent amplitude comparison

full rationale

The derivation applies the standard harmonic-balance approximation to the explicit closed-loop differential equations for both the static and dynamic sliding manifolds. Amplitude expressions are obtained by equating the fundamental Fourier coefficients of the switched input and the resulting output; no parameters are fitted to data, no self-citations supply the load-bearing uniqueness or ansatz, and the inequality between the two amplitudes is not presupposed by the method itself. The result therefore remains a genuine (if approximate) consequence of the system equations rather than a tautological restatement of the inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim depends on the validity of the harmonic balance approximation for the limit cycle and on the system being exactly a scalar integrator driven by linear second-order actuator dynamics with unknown but constant time constant.

free parameters (1)

dynamic sliding manifold parameters
Tunable coefficients whose values are chosen to minimize predicted chattering amplitude; their selection is part of the proof.

axioms (2)

domain assumption The plant is a scalar integrator subject to second-order linear actuator dynamics with unknown time constant.
Explicitly stated as the system class analyzed.
domain assumption Harmonic balance yields a sufficiently accurate prediction of chattering amplitude.
The proof technique invoked without error-bound discussion in the abstract.

pith-pipeline@v0.9.0 · 5395 in / 1305 out tokens · 41764 ms · 2026-05-15T10:45:42.504722+00:00 · methodology

discussion (0)

Reference graph

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