Compact Amplified Laser Power Stabilization Using Robust Active Disturbance Rejection Control with Sensor Noise Decoupling

Chenyao Zhang; Jingxuan Zhang; Rui Feng; Yanpei Shi; Yuze Guo; Zhuo Shi

REVIEW 4 major objections 6 minor 30 references

Compact Amplified Laser Power Stabilization Using Robust Active Disturbance Rejection Control with Sensor Noise Decoupling

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A two-loop active disturbance rejection controller stabilizes compact amplified-laser power with one photodetector, cutting 1-hour instability by over 85.7% and improving long-term Allan variance tenfold versus standard ADRC.

desk verdict The hardware results are believable, but the paper's main theoretical claim — sensor noise decoupling — isn't supported by its own L∞ bounds, which actually predict amplification. read the letter →

arxiv 2506.08404 v1 pith:FQMPZG3Z submitted 2025-06-10 eess.SY cs.SY

Yanpei Shi , Jingxuan Zhang , Zhuo Shi , Chenyao Zhang , Yuze Guo , Rui Feng This is my paper

classification eess.SYcs.SY

keywords activedisturbancerejectioncontrolextendedstateobserverlaserpowerstabilizationopticallypumpedmagnetometermagnetoencephalographysensornoisedecouplingAllanvariancedual-loop

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 argues that the long-standing trade-off in active disturbance rejection control, between fast disturbance tracking and sensitivity to sensor noise, can be broken by splitting the observer into two loops, and that this makes compact, single-photodetector stabilization of watt-level amplified lasers practical. The proposed dual-loop ADRC (DLADRC) uses an inner extended state observer to absorb model uncertainty across operating points and a cascade of outer-loop observers with increasing bandwidths to filter photodetector noise before it enters the control signal. The paper backs this with a quantitative stability analysis: explicit exponential-decay bounds on the observation and control errors, derived through Lyapunov functionals. Physical experiments on a 1 W to 2 W amplified laser show over 85.7% lower 1-hour power instability and a tenfold improvement in Allan variance at $10^{2}$ to $10^{3}$ second correlation times compared with standard ADRC. If correct, this removes a practical obstacle to large-scale optically pumped magnetometer arrays for magnetoencephalography.

What carries the argument

The load-bearing mechanism is the dual-loop ADRC topology: a supplemental inner-loop extended state observer (ESO) that estimates generalized disturbance from model mismatches, combined with an outer cascade of $p$ ESOs whose bandwidths form a geometric sequence $\omega_{oj} = \alpha^{j-1}\omega_{o1}$ to progressively filter sensor noise. The analysis machinery is a set of Lyapunov-functional estimates (Theorems 1 through 5) that convert the coupled error dynamics into explicit exponential bounds, for instance $|\tilde{z}_j(t)| \le c_9 \omega_{op}^{j-1}\{e^{-c_4\omega_{op}t}\|\tilde{z}_p(0)\| + \sum_{i=1}^{p-1}\|\tilde{z}_i(0)\| + \|\rho\|_\infty\}$, showing how steady-state observation and control errors scale with the noise bound $\|\rho\|_\infty$ and with the observer and control bandwidths.

What would settle it

Run the same DLADRC controller at a fixed power while injecting broadband drive-current ripple whose spectrum overlaps the photodetector noise band, and compare the 1-hour Allan variance with standard ADRC; if the tenfold long-term improvement vanishes or the instability reduction drops below 85.7%, the noise-decoupling premise is false.

Watch

Extended reading notes

Core claim

The central claim is that a dual-loop ADRC architecture can simultaneously reject unknown disturbances and decouple sensor noise in amplified laser power control, something a single extended state observer cannot do because raising its bandwidth to track disturbances also amplifies measurement noise. The paper establishes this by constructing an inner loop whose ESO compensates for plant-model parameter variations, and an outer cascade of $p$ ESOs whose bandwidths start low and increase geometrically, so the first stage low-pass filters the photodetector noise while later stages correct estimation residuals. On the theory side it derives explicit time-decay estimates for the observation error of every ESO and for the closed-loop tracking error, showing that steady-state bounds scale with the noise bound and with ratios of observer bandwidths. On the experimental side it reports that DLADRC lowers 1-hour power instability by more than 57% at every tested operating point from 1 W to 2 W compared with standard ADRC, and by over 85.7% once the cascade outer loop is used, with Allan variance improved by an order of magnitude at correlation times of $10^2$ to $10^3$ seconds.

Load-bearing premise

The load-bearing premise is that photodetector noise sits at higher frequencies than external disturbances and model variations; if the two spectra overlap, the cascade's first observer cannot filter the sensor noise without also rejecting real disturbances, and the reported gains would not generalize.

Editorial extensions

If this is right

A compact amplified laser with a single photodetector can hold 1-hour power instability below 0.05% at 1.5 W, removing the need for the electro-optic or acousto-optic modulator chains used in conventional active stabilization.
With three cascade levels the closed-loop performance becomes nearly insensitive to the first-level observer bandwidth, so the controller can be commissioned without delicate bandwidth tuning.
The exponential-decay estimates predict that convergence speed is governed mainly by the control bandwidth $\omega_c$; step experiments confirm this by matching decay coefficients across different set-point changes.
The inner loop absorbs model variation across the 1 W to 2 W range, so one fixed tuning holds at all tested power levels, whereas standard ADRC develops persistent oscillations away from its design point.
The quantitative observation-error framework extends to any ADRC-based system with spectrally separated disturbances and sensor noise, as the paper's note to practitioners states.

Reading between the lines

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

If external disturbances ever occupy the same frequency band as photodetector noise, the first cascade ESO cannot low-pass one without also rejecting the other; the paper's central premise, stated in its experiment design, would fail and the reported advantage over standard ADRC should shrink.
The same cascade-ESO arrangement could be tried on other single-sensor precision systems, such as frequency locking of lasers or interferometric displacement sensing, whenever the sensor noise is broadband and the target signal is low-frequency.
A direct power-spectral-density comparison before and after the first cascade stage would separate the noise-filtering contribution from the disturbance-rejection contribution; the paper reports time-domain and Allan-variance data but not that spectral decomposition.
The bounds require only boundedness of the disturbance derivative and noise, so the controller should port to lower sampling rates if the geometric bandwidth spacing $\alpha$ is scaled down to respect the Nyquist limit.

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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

Quick read of arXiv:2506.08404. The useful thing here is the hardware validation: the proposed DLADRC — an inner ESO plus a p-level cascade of outer ESOs with geometrically increasing bandwidths — does appear to stabilize a compact amplified laser better than standard ADRC in their setup. E2 shows 57-67% one-hour instability reduction across the full 1-2 W range, and E3 shows another factor of several at 1.5 W. The step-response decay rates in E1 matching ω_c scaling is a nice check. If you work on OPM pump lasers, this is worth a look.

The soft spots are real, though. The headline 85.7% number comes from E3, at 1.5 W only; the abstract generalizes it to the full range. No error bars or repeated trials, so we're trusting single traces. More substantively, the paper's central theoretical claim — sensor noise decoupling — is not actually established by the theorems. The stress-test note is right: Theorem 5's proof has a sign slip in (38) (should be −u_out), and the resulting L∞ bound (36) scales as ω_op^n ||ρ||∞, which is the opposite of decoupling. The spectral-separation assumption in Section IV-A is only qualitative; it never enters the estimates. So the analysis proves boundedness, not noise attenuation. The mechanism might still be real — the cascade's first low-bandwidth ESO plausibly does low-pass filter the photodetector noise — but a serious referee should ask for a frequency-domain analysis (transfer functions from ρ to the error) or an experimental noise-spectrum measurement.

The sign error is minor in itself, but it reflects a general carelessness in the theorem statements. That said, the architecture is novel enough and the experiments are credible enough that this deserves peer review, not a desk reject. The authors should be asked to either provide the frequency-domain support or reframe the contribution as an empirical one. I'd bring it to reading group as a case study in the gap between L∞ observer bounds and frequency-domain noise behavior, but I wouldn't cite the theory as it stands.

Referee Report

4 major / 6 minor

Summary. This paper proposes a dual-loop active disturbance rejection control (DLADRC) architecture for stabilizing the output power of an amplified laser (AL) used in optically pumped magnetometer (OPM) arrays. The inner loop uses a supplemental ESO to handle model uncertainties, while the outer loop uses a cascade of ESOs intended to decouple photodetector noise. The authors derive explicit exponential-decay bounds for ESO observation errors and control errors (Theorems 1–5) and validate the approach with three hardware experiments (E1–E3) comparing DLADRC with standard ADRC. The headline claims are an 85.7% reduction in 1-hour power instability and a tenfold improvement in Allan variance at long correlation times. The experimental results show qualitative improvements, and E1's decay-rate ratio matches the theoretical dependence on the controller bandwidth ω_c.

Significance. If the claims are fully substantiated, the proposed compact single-photodetector controller could be a practical improvement for OPM-array laser power stabilization. The paper's strengths include a transparent hardware description, a layered theoretical framework, and direct experimental comparison with standard ADRC. E1's exponential decay rates and their consistency with the ω_c-dependence in Eq. (35) provide a valuable quantitative check; E2 demonstrates robustness across the 1–2 W operating range. However, the advertised 'sensor noise decoupling' mechanism is not proven by the L∞ bounds in Theorems 4–5, and the headline 85.7% figure is measured at a single operating point without repeated trials. The paper's value to the OPM community is potentially high, but the analytical support for the key mechanism and the statistical support for the headline quantitative claim need significant revision.

major comments (4)

[Section III-C, Eq. (34) and Eq. (38)] There is a sign and index error in the derivation of the control-error dynamics. From the plant (4), dot z_n = -u_out, so with the control law (9), dot ǫ_n = -u_out = -ω_c^n(ǫ_1+ρ) - Σ_{q=1}^{n-1} K_out^{q+1} ǫ_q + K_out \tilde{z}_p. The printed Eq. (38) omits the minus sign, and Eq. (34) has +ω_c^n ρ and the summation over ǫ_{q+1} instead of ǫ_q. Although the absolute-value bounds in (36) are unaffected because they use |ρ|, the proof as written does not correctly represent the closed-loop system and must be corrected.
[Theorem 5 and Eq. (36)] The bound ls∞|ǫ_q| ≤ c10 ω_op^n ω_c^{q-1} ||ρ||∞ grows with the p-th-level ESO bandwidth ω_op to the n-th power. With n=3 and the experimental values ω_op=250–1000, this predicts a noise amplification factor of 10^7–10^9, which is the opposite of 'sensor noise decoupling.' Theorems 2–5 use only L∞ bounds on ρ and never encode the spectral-separation assumption stated in Section IV-A. Therefore the claimed low-pass filtering behavior of the cascade is not a consequence of the theorems. To support the headline mechanism, the authors need a frequency-domain or stochastic analysis, or a revised bound showing attenuation as ω_o1 decreases or as p increases.
[Section III-A and E3] The qualitative statement that the first-level ESO 'acts as a low-pass filter' is not reflected in Theorem 4, which gives ls∞|\tilde{z}_p^j| ≤ c8 ω_op^{j-1}||ρ||∞, independent of the first-level bandwidth ω_o1 and growing with the highest bandwidth. The experimental E3 indeed shows improved Allan variance, but only at 1.5 W, with no error bars or repeated trials. The conclusion that the 85.7% reduction holds 'throughout the 1 W–2 W range' is not supported: E2 (Table III) reports 57–67% improvements at other operating points with p=1, and E3's 85.7% is a single point. Please clarify the scope of the headline claim and add statistical replicates or error bars.
[Theorem 5 derivation, Eq. (35)] The bound ||K_out|| ≤ ω_c^n used in the proof is loose: from Eq. (10), K_out's nonzero entries scale as ω_c^{n-1} at most. Recomputing the constants with the correct scaling would make the dependence on ω_c and ω_op transparent and would likely change the form of (36). As written, the inflated bound obscures the actual noise-dependence of the control error.

minor comments (6)

[Theorem 1 proof, Eq. (15)] The step 'dividing both sides of (15) by m(P) exp{...} N(t)' is not fully justified: one needs a comparison lemma, and the case N(t)=0 must be handled separately. Please rewrite this step for rigor.
[Theorem 4 proof] In the definition of c8, the index i in ω_{oi} is not defined; it should presumably be p. Please fix this typo.
[Section II-B, notation] The symbols c, d, and b are overloaded: c is used both as a vector and as constants c1, c2, ...; d is used both as the disturbance and as the vector d in Eq. (3). Consider renaming the vectors (e.g., e_1, e_n) to avoid confusion.
[Section IV-B, E1] Please state explicitly which variable (e.g., ǫ_1 or the output error) was exponentially fitted, the number of data points, and the fitting procedure. Also clarify that the 'decay rate' refers to the exponent in (35).
[Abstract and Conclusion] The abstract says 'tenfold decrease in Allan variance for correlation times 10^2 s–10^3 s', while the conclusion says '10^1 s–10^3 s'. Please unify the stated timescales; the data in Figs. 7 and 8 appear to cover 10^-1 s to 10^3 s.
[Conclusion] The phrase 'spanning 10^1 s–10^3 s' in the conclusion should be checked against the actual Allan-variance curves; some curves in Fig. 7(b) show long-term behavior up to 10^3 s, but the abstract's 10^2–10^3 s may be more consistent with the reported factor-of-ten improvement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability theorems are derived from Lyapunov arguments without fitted constants, the E1 exponential fit corroborates rather than defines the theoretical predictions, and the headline performance numbers are direct hardware measurements.

full rationale

Walking the claimed derivation chain: the plant model (1)-(2) is built from circuit parameters, the DLADRC law (5)-(9) is an explicit controller parameterized by bandwidths, and Theorems 1-5 are proved from Lyapunov inequalities (11)-(15) without invoking any fitted constants or the experimental data. The bounds (28)-(36) are consequences of the model and observer/controller definitions, not restatements of those definitions. Experiment E1 fits exponentials to measured step responses and compares the fitted decay rates with the omega_c dependence predicted by (35); the fit is used only as corroboration and is not used to define or tune the theoretical bound. Experiments E2 and E3 are direct hardware comparisons against SADRC, so the claimed 85.7% reduction in 1-hour instability and the tenfold Allan variance improvement are measured outputs, not encoded inputs. I found no self-citation chain carrying a load-bearing premise, no parameter fitted to a target and then renamed as a prediction, and no definition that presupposes the result. The reviewer-flagged sign inconsistency in Eq. (34)/(38) and the resulting omega_op^n scaling in (36) is a possible correctness defect in the theoretical support for the noise-decoupling claim, but it is not circular reasoning: the bound is derived rather than assumed, and the hardware result stands independently of that bound.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard ADRC plant model and on three domain assumptions: bounded smooth disturbances, a constant-gain third-order laser model, and spectral separation of sensor noise from disturbances. The only fitted quantities are controller bandwidths and the E1 exponential fit coefficients used for validation. No new physical entities are introduced.

free parameters (2)

Controller bandwidth choices (omega_c, omega_in, omega_o1, alpha, p) = Controller bandwidths: chosen from Table II per experiment, e.g., omega_c=150/300, omega_in=800, omega_o1=150…
Standard ADRC tuning knobs selected by the experimenter; they affect the comparison but are not fitted to reproduce the stability metrics.
E1 exponential decay fit coefficients = 0.335, 0.336 for omega_c=150; 0.617, 0.616 for omega_c=300.
Fitted to step-response data to claim consistency with the theoretical exponential decay; the theory's constants c4 are not computed, so the comparison is qualitative.

assumptions (5)

domain assumption Assumption 1: all AL physical quantities, including disturbance d(t) and system states, are bounded and continuously differentiable.
Stated in Section II-A; it ensures the generalized disturbance derivative norm and sensor noise bound used in Theorems 1-5 are finite.
domain assumption The circuit-derived third-order plant model in Eqs (1)-(2), with constant optical gain phi_AL, accurately describes the amplified laser across the 1-2 W range.
Section II-A derives alpha0-alpha3 and beta from a high-frequency equivalent circuit and assumes instantaneous optical amplification; the control design and stability analysis depend on this model.
domain assumption Photodetector measurement noise rho occupies higher frequency bands than external disturbances and model parameter variations.
Invoked in Section IV-A to justify expanding the outer-loop observation range and the cascade noise-filtering design; if false, the noise decoupling benefit is lost.
domain assumption The disturbance-compensated inner loop behaves as a nominal integrator chain for the outer loop.
Section III-A.2 treats the compensated inner loop as nominal plant; the outer-loop error dynamics in Eq (4) rely on this separation.
standard math For the companion matrix Gamma defined in Theorem 2, there exists a positive definite P solving Gamma^T P + P Gamma = -I.
Used in Steps 1-2 of Theorem 2 and Theorem 3 via Lyapunov equation theory (Ref [30]); standard linear algebra, not an empirical input.

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Cite this review

Pith. "Pith review of Compact Amplified Laser Power Stabilization Using Robust Active Disturbance Rejection Control with Sensor Noise Decoupling." pith.science (2026). https://pith.science/paper/FQMPZG3Z

@misc{pith2026250608404,
  author       = {Pith},
  title        = {Pith review of: Compact Amplified Laser Power Stabilization Using Robust Active Disturbance Rejection Control with Sensor Noise Decoupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FQMPZG3Z}},
  note         = {Machine review of arXiv:2506.08404}
}

read the original abstract

Laser power instability, encompassing random jitter and slow drift, severely limits the performance of optically pumped magnetometers (OPMs) in detecting ultra-weak magnetic fields, especially in large-scale OPM arrays for magnetoencephalography. Although a unified amplified laser (AL) architecture improves integration, fluctuations in the pump beam progressively degrade performance across all channels, exacerbated by environmental disturbances and system uncertainties. To address this challenge, this paper presents a compact AL power stabilization approach based on an innovative dual-loop active disturbance rejection control (DLADRC) strategy, while integrating a comprehensive quantitative stability analysis through novel exponential decay estimates for extended state observers (ESOs) and control error dynamics. As validated through physical experimental results, the proposed method significantly improves AL's long-term stability with sensor noise decoupling, achieving an over 85.7% reduction in 1-hour power instability and a tenfold decrease in Allan variance for correlation times 10^2 s--10^3 s, compared to standard ADRC. Crucially, the strategy demonstrates robust effectiveness across diverse operating scenarios, enabling AL-based OPM systems to achieve their full potential in high-sensitivity biomagnetic field detection.

Figures

Figures reproduced from arXiv: 2506.08404 by the authors.

**Figure 3.** The proposed DLADRC strategy. where uin := uCTL is the control input, ˆbin represents a precise-enough estimate of the input gain β/α3 from (1), yin := uPWR donates the noise-free plant output, r := uREF is the set point, while the coefficients for ADRC application are given by A := 0 n×1 I n×n 0 0 1×n , b := [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** (a) Picture of the compact AL power stabilization [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Performance with PID strategy. (a) 1-hour AL power [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Results of E1. (a) ωc = 150. (b) ωc = 300. TABLE II: Bandwidth Configuration in Three Experiments Strategy Parameters ωc ωin ωo1 E1 DLADRC (p = 3) 150, 300 800 150 E2 SADRC 300 — 1000 DLADRC (p = 1) 300 800 1000 E3 DLADRC (p = 2) 300 800 300, 400, 500 DLADRC (p = 3) 30…

**Figure 7.** Figure 7: Results of E2. (a) Time domain. (b) Allan variance. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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Pith tools

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