Power-based control of output oscillations with online estimation of biased harmonics

Denis Efimov; Michael Ruderman

REVIEW 3 major objections 5 minor 29 references

Power-based control of output oscillations with online estimation of biased harmonics

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A control that fires two impulses per oscillation period, with frequency, amplitude, and bias estimated online, stabilizes the otherwise diverging load position of a fifth-order actuator.

desk verdict A genuine plug-in demonstration: power-based control plus online biased-harmonic estimation stabilizes a real fifth-order actuator, but the phase-synchronization robustness and single-run evidence keep it incremental. read the letter →

arxiv 2504.16445 v1 pith:452XPX5X submitted 2025-04-23 eess.SY cs.SY

Michael Ruderman , Denis Efimov This is my paper · ORCID

classification eess.SYcs.SY

keywords power-basedcontroloscillationscompensationfrequencyestimationbiasedharmonicsfinite-timeparameterdiscretefeedback

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a discrete power-based controller, which applies only two constant impulses per oscillation period, can stabilize an otherwise unstable oscillatory output when the frequency, amplitude, and bias of that output are estimated online. The authors extend the earlier power-based control scheme with a finite-time biased-harmonic estimator and an improved analytic expression for the impulse gain. They demonstrate experimentally that the combined scheme keeps the load position bounded on a fifth-order actuator with a free hanging load under gravity and measurement noise, even though the base PI feedback alone drives the same output to diverge. The significance is practical: the control loop needs only two updates per oscillation period, drastically reducing communication effort in non-collocated sensing-actuation configurations.

What carries the argument

The central mechanism is the discrete power-based compensator: at each detected extremum of the detrended output $y(t) - \hat{Y}_0$, a rectangular impulse $u' = k\hat{\omega}^2\hat{A} \operatorname{sign}(y(t^*) - \hat{Y}_0)$ is applied with $k = \sqrt{3}/(2\pi)$, scaled by the gain $K$, and delayed by $T$ to align it with the internal double-integrator input of the plant. The supporting estimator uses the regression identity $y(t-3\tau) - y(t-2\tau) + y(t-\tau) - y(t) = 2\cos(\omega\tau)(y(t-2\tau)-y(t-\tau))$, which turns frequency estimation into a scalar linear regression whose finite-time gradient update (10) produces $\hat{\omega} = \tau^{-1}\arccos(\hat{\theta}_0)$. A second regression in the basis $[1, \sin \hat{\omega} t, \cos \hat{\omega} t]$ then recovers the bias $\hat{Y}_0$ and amplitude $\hat{A}$, giving the controller all the harmonic parameters it needs.

What would settle it

Set up the same fifth-order actuator, deliberately misidentify the feed-forward model by a phase error of $\pm 30^\circ$ at the oscillation frequency, and run the power-based controller with the nominal gain $K = 2.4$; if the load oscillation amplitude grows rather than shrinks, or if no $K$ in $1 < K < 4.24$ restores boundedness, the stabilization claim is contingent on an exact model phase.

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Extended reading notes

Core claim

The paper establishes that the discrete power-based controller (Eqs. (2)–(6)), augmented by the online estimator in Eqs. (10)–(11), stabilizes the load position of the fifth-order actuator that an unstable PI loop alone makes diverge. The control commutates twice per oscillation period at the extrema of the detrended output, applying a rectangular impulse whose magnitude depends on the estimated frequency $\hat{\omega}$ and amplitude $\hat{A}$, and whose timing is synchronized through a delay $T$ computed from the feed-forward sub-dynamics $\tilde{G}(s)$. With the power-based control switched on at $t = 2.5$ s, the measured load position in Fig. 6 remains bounded around the reference, while the online estimates of $\hat{\omega}$, $\hat{A}$, and $\hat{Y}_0$ stay convergent. The paper also provides an improved analytic calculation of the impulse weighting factor $K$, bounding it as $1 < K < |\tilde{G}(j\omega)|^{-1}$ and selecting $K = 2.4$ in the experiment.

Load-bearing premise

The load-bearing premise is that the feed-forward model $\tilde{G} \approx G(s)s^2$ predicts the phase lag accurately enough that the fixed delay $T$ from Eq. (5) places the two impulses per cycle at the part of the oscillation where they remove energy rather than add it; if the model phase is wrong, the same impulses can amplify the oscillation.

Editorial extensions

If this is right

The same controller should stabilize marginally damped oscillations ($\sigma = 0$) as well as slowly diverging ones ($\sigma > 0$), provided the output channel can be approximated as a double integrator in series with a low-pass plant.
Only two control updates per oscillation period are required, so the sensor-to-actuator communication rate can be extremely low compared with conventional sampled feedback.
The estimator converges within a few periods under persistence of excitation, so the power-based control can be switched on after divergence has already begun, as the experiment does at $t = 2.5$ s.
The analytic gain bounds $1 < K < |\tilde{G}(j\omega)|^{-1}$ give a concrete tuning interval; the experiment selects $K = 2.4$, comfortably inside the computed bound of $4.24$.
If the phase model $\tilde{G} \approx G(s)s^2$ is accurate, the same two-impulse strategy transfers to other non-collocated oscillatory systems, such as flexible structures or drill strings, without requiring full state feedback.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the delay $T$ is the only model-dependent quantity, the scheme could be made adaptive by replacing the fixed feed-forward model with an online-estimated phase lag, directly addressing the acknowledged model uncertainty in Section 2.
The regression identity used here is specific to a single sinusoid plus bias; an analogous construction with more delay taps should yield finite-time estimates for multi-harmonic signals, which would matter for applications like torsional drill-string vibrations with several dominant harmonics.
A natural testable extension is to start the estimator and the controller simultaneously from rest and measure how many oscillation periods are needed before the amplitude stops growing, which would quantify the transient cost of online estimation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real integration of two known components, and the experimental stabilization of a PI-destabilized load (Fig. 6) is a genuine result. It is not a new theory paper, and the advertised 'finite-time' label covers only the frequency estimator. The main soft spot is the phase synchronization delay T, which is load-bearing and has no robustness margin.

The paper combines Ruderman's discrete power-based control with the Ahmed/Efimov biased-harmonic estimator. That combination is new, and it is evaluated on a real fifth-order actuator with gravity and noise. The experiment shows a diverging load position being brought to reference once the power-based loop is switched on. That is worth having, especially because the control only needs output extrema and three parameter estimates, which is exactly the low-communication setting the authors claim. The PE analysis in Section 4 that guides the choice of tau is also reasonable and useful.

The weaknesses are in proportion to how incremental the contribution is. First, the phase delay T in Eq. (5) is computed from an approximate feed-forward model G_tilde approx G(s)s^2, with the paper itself admitting that feedback propagation and disturbances at the input of the double integrator are not taken into account. Since an error in T can flip damping into excitation, the lack of a phase-error margin or sensitivity analysis is a genuine gap. Second, the finite-time claim is overstated: Eq. (10) gives finite-time frequency estimation, but the amplitude and bias estimates come from the standard gradient descent (11), which is only asymptotic. The abstract and conclusions say 'finite-time estimation of the biased harmonics,' which will mislead a reader. Third, the experimental support is one successful run with one manually tuned gain K=2.4. No repeated trials, no error bars, no quantitative settling or noise metrics. For a claim that the controller 'stabilizes' the system, that is thin. Fourth, the 'improved analytic calculation of the impulse weighting factor' is not actually derived; we see a bound in (7) and a reference to Ruderman 2024b for the optimal k. If the improvement is in the bound, it should be shown step by step.

The citation pattern is heavy on the authors' own prior work, but that is legitimate: the building blocks are theirs, and they say so. I do not see a circularity problem; the estimation algorithms are generic and the experimental success is an external check.

Who should read this: people working on non-collocated oscillation compensation, drill strings, flexible structures. It deserves a serious referee, but with the expectation of major revisions: the phase robustness needs a quantitative treatment, the finite-time language needs correcting, and the experiments need more runs and a sensitivity study.

Recommendation: send to peer review, but flag these issues. It is not a desk reject, and it is not a breakthrough.

Referee Report

3 major / 5 minor

Summary. The paper combines a recently proposed discrete power-based oscillation compensator (Ruderman 2024b) with online biased-harmonic estimation following Ahmed et al. (2022) and Wang et al. (2020b). The control law (2)-(6) applies a sign-definite impulse at output extrema, with an analytically computed gain factor and a delay T that synchronizes the impulse with the input of a double integrator in the plant. The amplitude, bias, and frequency estimates are obtained from linear regressions (8)-(11), with a finite-time option for the frequency estimate. The combined scheme is demonstrated experimentally on a fifth-order voice-coil actuator with a free-hanging load under an unstable PI feedback, where the power-based control is switched on at t=2.5 s and the output is stabilized (Fig. 6).

Significance. If the claims hold, the paper offers a low-communication plug-in option for damping oscillatory outputs using only amplitude, frequency, and bias estimates, and it improves the impulse weighting by analytic calculation. The experimental demonstration on a real fifth-order system with noise and an unstable base loop is a valuable independent check of the estimation algorithms. The finite-time frequency estimator and the persistence-of-excitation analysis in Section 4 are useful, and the authors are transparent about the approximation in the synchronization model.

major comments (3)

[Section 2, Eq. (5)] The phase-synchronization delay T is computed from arg G̃(j2ω̂), where G̃ ≈ G(s)s² is explicitly approximative and ignores disturbances and feedback propagation at the double-integrator input. Since the sign-definite impulse (6) damps only if it arrives at the correct phase and can add energy otherwise, the central stability claim rests on an unquantified phase-model assumption. The manuscript should provide a phase-error bound or a sensitivity/stability analysis, for example a sweep over T or a phase-margin computation, and connect it to the experimental run with K=2.4.
[Section 5.3, Fig. 6] The assertion that the power-based control stabilizes the otherwise unstable load position is supported by a single experimental run with one manually selected gain K=2.4 and no repeated trials, error bars, or quantitative performance metrics. Please provide multiple runs, statistics such as mean and variance of convergence time and residual amplitude, and ideally a robustness test varying K within the allowed range (7).
[Abstract and Section 3] The abstract and conclusions describe 'finite-time estimation of the biased harmonics,' but finite-time convergence is claimed and cited only for the frequency estimator (10). The bias and amplitude estimates (11) are standard gradient descent with exponential or asymptotic convergence under persistence of excitation, not finite-time. Please restrict the finite-time claim to the frequency estimate or provide a finite-time estimator for the full parameter vector.

minor comments (5)

[Section 1, paragraph 4] The phrase 'convergence prosperities' should be 'convergence properties.'
[Eq. (3)] The definition of Â(t*) uses sign(y(t*) − Ŷ0) multiplied by Â, but Â is introduced as a positive amplitude; please clarify that this expression denotes a signed amplitude estimate.
[References] The reference 'Vediakova et al. (2020)' is incomplete, ending with 'report'; please provide the full bibliographic entry.
[Fig. 1] Several labels in Fig. 1 appear corrupted or garbled (for example '1s/g16', 'u/g99'); please provide a clean version with all blocks and signals properly labeled.
[Section 5.3, Eq. (7)] The upper gain bound is quoted as |G̃(jω)|^{-1}=4.24, but Eq. (7) uses ω while the online estimate ω̂ varies; please clarify which frequency is used in the experimental calculation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the control and estimation blocks are imported from prior work, but the central claim rests on an independent experimental stabilization that is not forced by construction.

full rationale

The paper combines the discrete power-based controller of Ruderman (2024b) with the biased-harmonic estimator of Ahmed et al. (2022) and Wang et al. (2020b), and evaluates the combination on a fifth-order actuator. The control law (2), the optimal gain (4), and the time-delay synchronization (5) are stated as results from Ruderman (2024b); the frequency estimator (10) and amplitude/bias regression (11) are likewise based on cited prior work. These are self-citations, but they are not circular in the load-bearing sense: each cited component is a parameter-free mathematical construction with explicit assumptions, and the paper does not redefine the output in terms of the control or fit a parameter to the target result. The experimental claim in Fig. 6 is an external check: the load position is stabilized by augmenting an unstable PI loop, and the gain K = 2.4 is selected within the derived bounds (7), not estimated from the measured stabilized response. The phase model G̃ in (17) is identified from the same setup, but using an identified plant model to design a controller is standard practice and does not make the observed stability an identity or a tautology. The acknowledged uncertainty in the approximate G̃ ≈ G(s)s² and in the synchronization delay T is a robustness concern, not a circularity. Therefore no step in the derivation chain reduces, by construction or by self-citation, to its own inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central scheme depends on a single-harmonic signal model, a double-integrator plant structure, an approximate phase model for impulse synchronization, an assumed frequency band for choosing τ, and four manually tuned parameters (K, τ, γ1, γ2). No new physical entities are introduced.

free parameters (4)

K = 2.4
Gain amplification factor in Eq. (6), manually tuned between bounds (7) for the experiment.
tau = 0.075 s
Delay in the regression (8), chosen from the assumed frequency interval as recommended in Section 4.
gamma1 = 1.5e5
Adaptation gain for the finite-time frequency estimator (10), tuned for the experiment.
gamma2 = 1e6
Adaptation gain for the amplitude/bias estimator (11), tuned for the experiment.

assumptions (5)

domain assumption The measured output is a single biased sinusoid plus bounded noise, with quasi-constant parameters (Eq. 1).
Stated in Section 2, Eq. (1); the estimation and control algorithms both assume this signal form.
domain assumption The plant transfer G(s) contains a double integrator in series at the output channel and a dominant lightly damped or unstable pole pair.
Section 2, first paragraph; the power-based control law is designed for this plant structure.
ad hoc to paper The feed-forward sub-dynamics are approximated as G̃ ≈ G(s)s² so the phase-synchronization delay T in Eq. (5) is valid.
Section 2, paragraph before Eq. (6); the paper acknowledges this is approximate but does not quantify the error.
domain assumption The oscillation frequency lies in a known interval [ω, ω̄] used to choose τ.
Section 4 derives the recommended τ range and the experiment fixes τ=0.075 s; if ω leaves the band, excitation is lost.
standard math Convergence theorems for gradient descent under persistence of excitation apply to the noisy experimental setting.
Section 4 invokes Sastry and Bodson (1989), Wang et al. (2020b), and assumes a reasonable signal-to-noise ratio.

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Cite this review

Pith. "Pith review of Power-based control of output oscillations with online estimation of biased harmonics." pith.science (2026). https://pith.science/paper/452XPX5X

@misc{pith2026250416445,
  author       = {Pith},
  title        = {Pith review of: Power-based control of output oscillations with online estimation of biased harmonics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/452XPX5X}},
  note         = {Machine review of arXiv:2504.16445}
}

read the original abstract

The recently introduced discrete power-based control (Ruderman (2024b)) reduces largely the communication efforts in the control loop when compensating for the marginally damped or even slowly diverging output oscillations. The control commutates twice per oscillations period (at the amplitude peaks) and uses the measured harmonic output only. The power-based control scheme requires the knowledge of the instantaneous frequency, amplitude, and bias parameters of the harmonic signal. This paper extends the power-based control by the finite-time estimation of the biased harmonics (Ahmed et al. (2022)). Also an improved analytic calculation of the impulse weighting factor is provided. The power-based oscillations control with online estimation of the harmonic parameters is evaluated experimentally on the fifth-order actuator system with a free hanging load under gravity and measurement noise.

Figures

Figures reproduced from arXiv: 2504.16445 by the authors.

**Figure 1.** Discrete power-based control u ′ subject to the gain shaping and time-delay synchronization. It is assumed that the single measured output of the process has the form y(t) = Y0 + A sin(ωt + φ) + v(t), t ≥ 0, (1) where y(t), v(t) ∈ R are the measured signal with the respective bounded noise. A > 0 and ω > 0 are the amplitude and the frequency of oscillations, correspondingly, Y0 ∈ R is the bias and φ ∈ [0, 2π) is th… view at source ↗

**Figure 2.** Simulated response of the stabilized oscillating [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Experimental oscillatory setup: laboratory view [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Unstable load position response of PI-control. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 6.** Figure 6: Stabilized load position response of PI-control [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 5.** Figure 5: Online estimation of harmonic parameters at unsta [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 7.** Figure 7: Online estimation of harmonic parameters at stable [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

and van de Wouw, N

Aarsnes, U.J.F. and van de Wouw, N. (2018). Dynamics of a distributed drill string system: Characteristic parameters and stability maps. Journal of Sound and Vibration, 417, 376--412

work page 2018

[2] [2]

Ahmed, H., Ushirobira, R., and Efimov, D. (2022). On biased harmonic signal estimation: Application to electric power grid monitoring. IEEE Transactions on Control Systems Technology, 30(6), 2743--2750

work page 2022

[3] [3]

Aranovskiy, S., Bobtsov, A., Ortega, R., and Pyrkin, A. (2017). Performance enhancement of parameter estimators via dynamic regressor extension and mixing. IEEE Tran. on Automatic Control, 62(7), 3546--3550

work page 2017

[4] [4]

Astolfi, A., Karagiannis, D., and Ortega, R. (2008). Nonlinear and Adaptive Control with Applications. Springer

work page 2008

[5] [5]

Chowdhary, G., Yucelen, T., M\"uhlegg, M., and Johnson, E.N. (2012). Concurrent learning adaptive control of linear systems with exponentially convergent bounds. International Journal of Adaptive Control and Signal Processing, 27(4), 280--301

work page 2012

[6] [6]

Efimov, D., Barabanov, N., and Ortega, R. (2019). Robust stability under relaxed persistent excitation conditions. International Journal of Adaptive Control and Signal Processing, 33(12), 1885--1900

work page 2019

[7] [7]

and Fradkov, A

Efimov, D. and Fradkov, A. (2015). Design of impulsive adaptive observers for improvement of persistency of excitation. International Journal of Adaptive Control and Signal Processing, 29(6), 765--782

work page 2015

[8] [8]

and Polyakov, A

Efimov, D. and Polyakov, A. (2021). Finite-time stability tools for control and estimation. Foundations and Trends in Systems and Control, 9(2-3), 171--364

work page 2021