REVIEW 3 major objections 7 minor
A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers
T0 review · 3 major / 7 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read This paper argues that a prior experimental comparison finding sliding-mode observers practically inadmissible is an artifact of gain selection and pole placement, and that with standard perturbation-bound tuning a super-twisting observer o
desk verdict A sharp, checkable re-simulation that exposes the original comparison as artifact; the hardware 'STA wins' claim needs re-anchoring before it can stand. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The error equation e¨ + kv e˙ + kp e = ε, together with the nominal ISE integral, is the core object: it shows that when observer error is zero, the metric is fixed by the controller gains alone. The standard super-twisting gain prescription kp=1.5√L, ki=1.1L, which encodes a single physically meaningful bound L>sup|δ̇|, is the mechanism that makes the SM observer tunable; the invariant ratio α2/√α1=1.430 diagnoses whether published gains follow the prescription.
What would settle it
If a hardware run at the I&I-preferred ζ=5 (or with an explicitly computed L from measured perturbation) shows the STA losing ISE or ISC, the central hardware claim would be overturned.
Extended reading notes
Core claim
The reported 4.3x ISE advantage of the I&I observer is the nominal response of the closed-loop error dynamics with kp=1600, kv=1100 (poles at -1.455 and -1098.5), which a perfect observer would also produce; the super-twisting scheme sits exactly on this nominal trajectory and is invariant to its own gain L over a fifty-fold range. The I&I advantage reverses when the loop pole crosses the observer's induced pole at kv≈170, and the I&I results change threefold under step-size refinement while the SM results change by 0.2%. On hardware with a coarser encoder and a signum super-twisting observer tuned at the perturbation bound L=40, the STA wins all three indices.
Load-bearing premise
The hardware conclusion assumes that L=40 is the true Levant-Moreno perturbation bound for the real motor and that the ζ=1 operating point is a neutral comparison point; neither is demonstrated, and the plant identification shows a 34% model mismatch and 10.6% Coulomb asymmetry.
Editorial extensions
If this is right
- If the paper's central claim is correct, the conclusion that sliding-mode observers are practically inadmissible is unsupported and should not guide engineer choices.
- Comparisons between observers must verify gain-insensitivity and step-size convergence before attributing differences to the algorithms.
- The L-based prescription provides a concrete, physically motivated tuning rule for super-twisting observers, and the ratio test (α2/√α1 = 1.430) offers a quick check of whether published SM gains are within an admissible family.
- In steady state (t≥2s) all observers are indistinguishable at the tested operating point, so the entire comparison is a transient phenomenon; transient shape is set by pole placement and initial conditions, not by the observation algorithm.
- On hardware with actuator bandwidth effects, switching energy above mechanical bandwidth is filtered and does not produce audible chattering, so 'chattering' and 'switching' should be distinguished.
Reading between the lines
- A natural extension: the same 'gain-blind metric' critique could be applied to other adaptive-observer comparisons where the performance index is dominated by the nominal closed-loop response rather than the estimator.
- The step-size dependence of the I&I observer suggests that its continuous-time advantages may not carry over to discrete-time implementations without specific discretization care; this could be tested by running I&I with a higher-order integration or at smaller steps on the same hardware.
- The 34% model mismatch in the hardware identification suggests that the I&I observer's adaptation is compensating for unmodeled dynamics; the STA's disturbance-rejection approach may be more robust to model error, which could be tested with a deliberate model perturbation.
- A practical engineering rule emerges: if switching content lies above the mechanical bandwidth, it is electrically present but mechanically inert; this could be formalized as a hardware-dependent acceptance criterion for sliding-mode control.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper re-examines the experimental comparison in Cervantes-Pérez et al. (arXiv:2601.12545) between an immersion-and-invariance (I&I) adaptive controller/observer and a super-twisting sliding-mode (SM) observer. The authors re-simulate the published plant, reference trajectory, and controller gains using explicit Euler integration at Δt=1 ms and 0.1 ms, and they also implement both observers on a separate 500 Hz DC-motor platform. They report three main findings: (i) the published controller gains kp=1600, kv=1100 place the tracking-error poles at -1.455 and -1098.5 (ζ=13.7), so the nominal ISE is 0.0306 and the super-twisting observer's ISE is invariant to its own gain over a fifty-fold range under exact initialization; (ii) the I&I margin is monotone in kv and reverses near kv≈170; (iii) refining the sampling step tenfold changes the I&I figures by roughly a factor of three while the SM figures move by about 0.2%. The paper additionally argues that the published SM gains violate the Levant–Moreno tuning prescription by 22–105×, and it reports a hardware experiment in which a signum-based STA observer tuned at L=40 beats I&I on ISE, ITAE, and ISC at ζ=1, ωn=20, without audible chattering.
Significance. The paper has substantial strengths. Equation (12) is the correct modal ISE for real negative poles, and Eq. (15) is a parameter-free necessary condition for Levant–Moreno tuning that is used to expose an order-of-magnitude deviation in the published gains. Table 3's step-size comparison is a useful convergence diagnostic, and Table 6's panel structure helpfully separates pole-placement effects from observer effects. If the hardware claim were fully supported, the paper would be an important correction to the published comparison. However, the central hardware conclusion rests on two unverified assumptions: the admissibility of L=40 as a true perturbation bound, and the neutrality of ζ=1 as an operating point. These are load-bearing for the headline claim that the STA 'wins all three indices,' and they must be established before that conclusion is accepted.
major comments (3)
- [§9.2–9.3, Eq. (9), Table 7] Equation (9) requires L>sup|δ̇|, and the hardware experiment sets L=40. Section 9.2 reports a 34% model mismatch and 10.6% Coulomb asymmetry but does not give a computation, an experimental estimate, or an upper bound for sup|δ̇| on the real motor. Without an admissible bound, the STA gains in §9.3 are not shown to lie within the Levant–Moreno prescription; if the true bound exceeds 40, the comparison is no longer 'within the prescribed tuning' and the headline that the STA outperforms I&I is unsupported. Please provide the bound calculation from the identified model plus an explicit uncertainty model, or an experimental estimate, and report sensitivity of Table 7 to L.
- [§9.3, Table 6B] Table 6B indicates that the two schemes have disjoint optimal pole placements: I&I's best is ζ=5, STA's best is ζ=1. Section 9.3 nevertheless fixes ζ=1, ωn=20 for the hardware comparison. A single operating point chosen from one scheme's preferred region is not a neutral or fair comparison. The conclusion 'STA wins all three indices' is therefore conditional on that choice. Please either justify ζ=1 as neutral by an independent criterion, sweep ζ over both schemes' preferred regions, or report results at a common nonpreferred point. Note also that the ISC margin in Table 7 (0.74 vs 0.77) is within the reported standard deviations and should not be treated as a decisive win.
- [§3, Table 1] The invariance of the STA metric to L in Table 1 is derived under the assumption that θ̄=θ and x̂(0)=x(0), which makes the observer error identically zero and ε≡0. The paper states that this reproduces [1]'s setup, but no evidence from [1] is cited for those initial conditions. If [1] initialized the STA differently, Table 1 does not describe [1]'s runs, and the statement that the comparison is 'not measuring that method' is not established. Please document the initialization used in [1] or provide a sensitivity analysis with nonzero initial observer error.
minor comments (7)
- [Abstract and §1] The abstract uses 'We got three findings' and §1 contains 'It must be mention' and 'We take no position on the broader argument...' — these are informal and should be revised for a journal submission.
- [§2.4] The notation θ̄ appears in Eq. (8) but is not defined until §9. Define it in the set-up section, and clarify how θ̄ differs from the adaptive θ̂ used in the I&I scheme.
- [Table 4] The columns 'implied L' and 'required k_p^{sta}' are not defined in the text. State how they are computed from (α1, α2), including the inversion of the Levant–Moreno formulas.
- [Figure 2] The cumulative-energy text says '95% at 82 Hz' and '95% at 1 Hz,' while the band table reports 87.5% below 5 Hz for STA. Reconcile these numbers or clarify the time window over which each statistic is computed.
- [Title and references] The title contains a typo: 'Immersion-and-Ivariance' should be 'Immersion-and-Invariance.' Also check the reference numbering in §2.4 and §10: 'Proposition 1 of [1]' and 'Proposition 1 of [2]' may refer to different publications.
- [§9.4, Figure 1] The STA ISE in the Figure 1 caption (0.00096) differs from the Table 7 mean (0.00118). Clarify whether this is a single representative run, and label the figure accordingly.
- [§9.4] The phrase 'Click Here to Watch!' is informal and should be replaced with a proper link or data-availability statement.
Circularity Check
No significant circularity; core quantitative claims are anchored externally.
full rationale
The paper's central derivation is self-contained against external benchmarks. The STA tuning rule is Levant's 1998 prescription (k_p=1.5√L, k_i=1.1L) with Moreno-Osorio Lyapunov footing, cited as [5,6] and not derived from the paper's own results. The nominal ISE=0.0306 is computed from [1]'s published gains k_p=1600, k_v=1100 using the error dynamics (10), with no fitted constants. Table 1's L-invariance is explicitly explained as a consequence of the stated zero-error initialization (¯θ=θ, ˆx(0)=x(0)), so it is a transparent mathematical reduction rather than a hidden circular prediction. The hardware comparison uses L=40 asserted as the 'computed perturbation bound' without showing the computation (Section 9.3), and the experimental operating point (ζ=1) matches STA's best regime while I&I's best is ζ=5 (Table 6B); however, these are unvalidated empirical assumptions or potential biases, not circular reductions in the sense of a result being equivalent to its inputs by definition. There is no load-bearing self-citation: references [2]–[6] are external prior work with disjoint authorship from the present paper. Accordingly, the circularity score is 0.
Assumptions & free parameters
free parameters (3)
- L (hardware super-twisting gain bound) =
40
- nu (boundary-layer width) =
5
- Hardware loop poles =
omega_n=20, zeta=1 (kp=400, kv=40)
assumptions (6)
- domain assumption Explicit Euler with a fixed step is the honest integrator for relay-type right-hand sides (sign and tanh(330*x)); a higher-order RK would degrade on the switching surface.
- domain assumption The Levant-Moreno gain prescription k_p=1.5*sqrt(L), k_i=1.1*L with L>sup|delta-dot| is the correct/standard STA tuning.
- ad hoc to paper The re-simulation reproduces [1]'s setup exactly, including observer initialization x-hat(0)=x(0) and theta-bar=theta, so the STA error is an exact invariant and epsilon=0.
- domain assumption The hardware plant model (16) with parameters (17) adequately represents the motor despite the disclosed 34% Phase-B discrepancy and 10.6% Coulomb asymmetry.
- domain assumption Winding inductance and the 131:1 gearbox filter the 500 Hz switching content before it reaches the shaft, so chattering is electrical-only and inaudible.
- standard math Standard modal analysis: the integral of the square of a sum of real exponentials (Eq. 12) is the exact ISE for the linear error dynamics.
Cite this review
Pith. "Pith review of A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers." pith.science (2026). https://pith.science/paper/VNQVCEHS
@misc{pith2026260802000,
author = {Pith},
title = {Pith review of: A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers},
year = {2026},
howpublished = {\url{https://pith.science/paper/VNQVCEHS}},
note = {Machine review of arXiv:2608.02000}
}
abstract
Cervantes-P\'erez et al. (arXiv:2601.12545) claim, on experimental grounds, that high-gain injection is practically inadmissible, taking the SM observer of Davila, Fridman & Poznyak as a prototypical example. We re-simulate their plant, trajectory and gains with explicit Euler integration at \Delta t = 1 ms and 0.1 ms, and implement both observers on independent hardware. We got three findings: i) the controller gains k_p=1600, k_v=1100 place the tracking-error poles at the "pathological" locations -1.455 and -1098.5 (\zeta = 13.7); the super-twisting scheme lands on the resulting nominal response and is invariant to its own observer gain over a fifty-fold range; ii) the I&I margin is monotone in k_v and reverses near k_v \approx 170, where the loop pole crosses the induced observer pole at -9.0 rad/s; iii) refining the sampling step tenfold moves every I&I figure by a factor of three, while the SM figures move by 0.2%. On hardware, a signum-based super-twisting observer tuned at the Levant-Moreno perturbation bound outperforms I&I on ISE, ITAE and ISC simultaneously, without audible chattering, on a $60 motor with a 0.043^\circ encoder at 500 Hz. The gains of Cervantes-Perez et al. exceed the same prescription by two orders of magnitude; the chattering the authors report is the designed consequence of that excess.
Figures
Reviewed August 4, 2026 · model on record in the stance chip above.
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