A Preliminary Study on A Physical Model Oriented Learning Algorithm with Application to UAVs

Minghui Zheng; Xiao Liang; Zhu Chen

REVIEW 5 major objections 4 minor 27 references

A Preliminary Study on A Physical Model Oriented Learning Algorithm with Application to UAVs

T0 review · 5 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proposes a physical-model-based learning algorithm that lets a target UAV improve its trajectory tracking by learning from other UAVs' flight errors in near real time, even when the UAVs differ in dynamics and reference path…

desk verdict A plausible ILC extension that stumbles on an unstated filter-sum condition in the main derivation; worth a referee but needs a fix. read the letter →

arxiv 1908.03478 v1 pith:DUI3URKC submitted 2019-08-09 eess.SY cs.SY

Minghui Zheng , Zhu Chen , Xiao Liang This is my paper · ORCID

classification eess.SYcs.SY

keywords physicalmodelorientedlearningiterativecontrolUAVtrajectorytrackingfeedforwardsignalheterogeneousUAVsfromothernearreal-time

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 proposes a physical-model-based learning algorithm that lets a target UAV improve its trajectory tracking by learning from other UAVs' flight errors in near real time. The algorithm is motivated by iterative learning control but replaces repeated trials on the same task with an analytical mapping: scaling matrices convert reference-path differences and transfer-function matrices convert dynamic differences, turning a training UAV's error into a feedforward learning signal. The central claim is that this signal drives the target's tracking error to a weighted combination of the training UAVs' errors, with convergence guaranteed when a frequency-domain norm condition holds. The authors verify the claim in numerical studies where the target UAV tracks an aggressive trajectory while learning from another UAV flying one sampling step ahead. If the claim is right, it would give UAVs a way to share control experience across different airframes and missions without offline training.

What carries the argument

The load-bearing object is the unified learning signal of Eq. (21), $s_k=\Lambda_k^{-1}M_k^{-1}(\alpha_k\{s_k\}+L_k\{h_k\})$. Here $\Lambda_k$ is a diagonal scaling matrix that maps the target's reference trajectory onto the training UAV's reference, $M_k$ is a transfer-function matrix representing the dynamic mapping from the training UAV to the target UAV, $\alpha_k$ is a robust filter, and $L_k$ is the learning filter. The learning filter is chosen so that $\alpha_k I + F L_k \approx 0$, i.e., $L_k\approx \alpha_k F^{-1}$; this identity is what converts another UAV's tracking error into an approximately cancelling correction for the target. The convergence proof rests on the same style of condition used in robust iterative learning control: the weighted composite filter must have infinity norm below $1/N$ for $N$ training UAVs. Inserting the correction in the feedforward loop keeps stability of the target's tracking system unchanged.

What would settle it

Simulate the same tracking problem with two training UAVs and choose weights $\alpha_1=0.3$, $\alpha_2=0.4$ so that $\sum\alpha_k=0.7\neq1$. If the paper's convergence claim is exact, the target error should still collapse to the weighted combination; the derivation predicts a leftover $0.3\,T\{r_d\}$ term instead. Comparing the simulated error floor with this predicted residual directly settles whether the dropped term matters.

Watch

Extended reading notes

Core claim

The paper's central claim is that the target UAV's tracking error $h$ can be made to follow a weighted combination of training UAVs' errors $h_k$, rather than being driven by its own unlearned error. The unified learning algorithm uses $s_k=\Lambda_k^{-1}M_k^{-1}(\alpha_k\{s_k\}+L_k\{h_k\})$, where $\Lambda_k$ scales the target's reference path to the training UAV's scenario and $M_k$ maps the training UAV's open-loop dynamics to the target's. The paper derives $h\approx \sum_k \Lambda_k^{-1}M_k^{-1}(\alpha_k I+FL_k)\{h_k\}$ and designs the learning filters $L_k\approx \alpha_k F^{-1}$ so that each term $\alpha_k I+FL_k$ is small. Convergence is stated as an infinity-norm bound on the composite filters ($\|[\cdots]\|_\infty<1/N$ in the multi-UAV case), and because the learning signal enters only the feedforward path, it does not alter closed-loop stability. The numerical section shows the target tracking a sharp-turn reference with small overshoot while learning from a training UAV with different dynamics and a different reference.

Load-bearing premise

The derivation assumes that the target UAV's own tracking error, $T\{r_d\}$, cancels out of the final formula; that cancellation requires the learning weights to add to one, or that error to be zero, and the paper never states this requirement.

Editorial extensions

If this is right

A fleet of UAVs with different airframes and different reference paths can pool their flight errors, so a new mission can be learned one sampling step before it begins instead of after many offline trials.
Because the learning signal goes into the feedforward path, closed-loop stability is untouched; the only design check is the norm bound on the composite filters.
The learning filter design has an explicit target, $L_k\approx \alpha_k F^{-1}$, so the algorithm is implementable whenever a model of the position loop's transfer function is available.
The algorithm inherits iterative learning control's ability to reject repetitive tracking errors, but relaxes the requirement that the same task be repeated by the same system.

Reading between the lines

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

The derivation between lines two and three of Eq. (11) drops the target's own baseline-error term $T\{r_d\}$; the final expression is exact only when $\sum\alpha_k=1$ or $T\{r_d\}=0$. The numerical example sets $\alpha_1=1$, which sidesteps the issue, but the general convergence claim depends on an unstated condition.
The approximation $\|G\|\gg 1$, used to replace each training UAV's dynamics by the target's, becomes doubtful at frequencies where the position-control loop gain is not large; a Bode-magnitude comparison would show where the learning transfer guarantee should be restricted.
A natural strengthening is to require the robust filters to satisfy $\sum_k\alpha_k=1$ (or to keep the residual term as an explicit disturbance), which would close the cancellation gap without changing the algorithm's structure.

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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

The paper's real contribution is modest but real: it takes the authors' earlier model-based learning framework for identical UAVs and adds scaling matrices Λ_k and dynamics mappings M_k so a target UAV can learn from training UAVs with different reference trajectories and different dynamics. That is a natural ILC extension, not a breakthrough, but it is clearly spelled out and the unification in Algorithm 1 is useful.

The physical-model motivation is sensible, the LTI assumption for the position loop is standard, and the convergence conditions are at least framed in H-infinity terms, citing their prior ILC design. The simulations show qualitative improvement, and the paper is honest about being preliminary.

The load-bearing issue is Eq. (11). Moving from the second to the third line, the term T{rd} − Σ Λ_k^{-1} α_k T{Λ_k rd} is dropped. Since T is diagonal LTI and Λ_k are diagonal constants, this equals (1 − Σ α_k) T{rd}. The final expression h = Σ Λ_k^{-1}(F L_k + α_k I){h_k} only holds if Σ α_k = 1 or T{rd} = 0. That condition is never stated, and conditions (12), (20), (23) don't include it. In the simulations there is a single training UAV with α_1 = 1, so the issue is invisible. If Σ α_k ≠ 1, the unlearned target error term survives and the convergence guarantee collapses. This is not a minor typo; it is the central derivation.

The second issue is Eq. (19): the notation is confused (M_k^{-1} pulled inside a sum over k in a way that doesn't parse), and it relies on F_k ≈ F via ‖G‖ ≫ 1, which is a frequency-dependent approximation that can fail. The numerical studies are purely qualitative: no error metrics, no baselines, no comparison to flying without learning. The convergence analysis is also cited from prior work rather than proved here, which is acceptable for a preliminary study but means the reader has to trust that the conditions carry over.

Who is this for? People working on ILC transfer or multi-UAV learning. It deserves a serious referee because the idea is sound enough to fix and the flaw is specific and addressable. I'd want the authors to state the Σ α_k = 1 condition (or redesign the law), clean up Eq. (19), and add quantitative comparisons.

Send it to review, but expect major revision.

Referee Report

5 major / 4 minor

Summary. The paper proposes a feedforward learning algorithm for UAV trajectory tracking in which the learning signal is generated from the tracking errors and control signals of other ('training') UAVs. The claimed contribution is that, by explicitly using physical-model transformations (reference scaling matrices Lambda_k and dynamics-mapping matrices M_k), a target UAV can learn in near real time from training UAVs with different dynamics and different reference trajectories. Sections 3, 4, and 5 present formulations for heterogeneous scenarios, heterogeneous agents, and their unification, each followed by a numerical simulation; convergence is stated in terms of H-infinity norm conditions. The paper concludes that the algorithm demonstrates great learning efficiency and reliability.

Significance. The core idea is potentially useful: if the derivation were correct, it would turn ILC-style error updates into a one-shot, near-real-time transfer between different vehicles by using model knowledge to overcome heterogeneity. Algorithm 1 is concrete enough to implement, and the simulation plots in Figs. 3, 5, and 6 show plausible improvement over tracking without learning. However, the central error-propagation equations contain unstated cancellations and notational errors, and the numerical evidence is entirely qualitative. The contribution is incremental over the authors' prior ILC design [23], and the convergence conditions are not proved in this manuscript; the value of the paper is in the concept and preliminary demonstration rather than in a fully verified algorithm. No machine-checked proofs or reproducible code are provided, so the positive assessment rests on the analytic derivations, which currently require correction.

major comments (5)

[Section 3, Eq. (11)] The step from the fourth displayed line to the fifth displayed line drops the term T{r_d} - sum_{k in Omega} Lambda_k^{-1} alpha_k T{Lambda_k r_d}. Since T is a diagonal LTI system and Lambda_k is a diagonal constant matrix, T{Lambda_k r_d} = Lambda_k T{r_d}, so the dropped term equals (1 - sum_k alpha_k) T{r_d}. Eq. (11) is therefore valid only if sum_k alpha_k = 1 or T{r_d} = 0, and this condition is not stated anywhere in Section 3 and is not included in the convergence condition (12). If it is not enforced, the actual target error contains the additional unlearned term (1 - sum_k alpha_k) T{r_d}, which can be large even when the H-infinity condition (12) holds. The numerical verification in Section 3.2 uses a single training UAV with alpha_1 = 1, so the missing condition is satisfied trivially and the flaw is invisible. This must be fixed by either imposing sum_k alpha_k = 1 as an explicit design constraint or carrying the residual term through the convergence analysis.
[Section 4, Eq. (19)] The derivation of Eq. (19) is not valid as written. The substitution F{s_k} = h_k - T_k{r_d} requires F = F_k, while the text has only argued F_k approx F under ||G|| >> 1. In addition, the simplification of T{r_d} - sum M_k^{-1} alpha_k T_k{r_d} to zero requires both the approximation T_k approx T M_k and the same sum_k alpha_k = 1 condition from the previous comment; neither is stated. The displayed expression M^{-1}_k( sum_{k in Omega} alpha_k I + F L_k ){h_k} also has a free k on M^{-1}_k outside the sum, making the equation ill-formed. Please rewrite Eq. (19) with correct summation indices, state the approximations explicitly, and either impose sum_k alpha_k = 1 or include the residual terms in the convergence bound.
[Section 2, Eq. (6)] From Eq. (5), alpha_k I + F L_k approx 0, the implied learning filter is L_k approx -alpha_k F^{-1}, not L_k approx alpha_k F^{-1} as written in Eq. (6). The missing negative sign would change the sign of the feedforward correction and, if implemented as printed, would amplify rather than cancel the error. Please correct Eq. (6) and state which sign is actually used in the numerical simulations.
[Sections 3-5, Eqs. (12), (20), (23)] The convergence conditions are stated by reference to prior work [23] with the phrase 'designed such that the above condition is satisfied [23]', but no proof is given that the stated H-infinity condition bounds the target error in the heterogeneous multi-UAV setup. The conditions are derived from the error-propagation equations (11), (19), and (22), which currently contain the unstated cancellations described above. A self-contained proof, or a precise statement of how [23] applies after the added transformation matrices Lambda_k and M_k, is necessary to support the central convergence claim.
[Sections 3.2, 4.2, and 5] The numerical studies report only qualitative statements such as 'very small overshoot and oscillations' and show trajectory plots without quantitative error metrics. No tracking-error norms, overshoot percentages, or comparisons with a no-learning baseline are provided. Since the central claim is a quantitative error-reduction property, the paper should report quantitative performance measures for the target UAV with and without the proposed learning signal.

minor comments (4)

[Section 5, Eq. (21)] Eq. (21) uses the symbol s_k on both sides of the definition; the training UAV's stored learning signal and the contribution sent to the target UAV should be distinguished notationally.
[Algorithm 1] The input list 'F_1, F_2, ..., F_k, F' is ambiguous because k is used both as a running index and as the last index; use explicit maximum index N or a different letter.
[Throughout] There are typographical errors including 'scearios' in Section 2, 'feedforward look' in Section 4, and 'quad-robot' in Section 4; these should be corrected.
[Figure 4] The Bode plot figure would benefit from clearer labeling of which subplot corresponds to which input-output channel and from a legend identifying the training versus target UAV consistently across all subplots.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the learning signal derivation is a control-design identity with explicit norm conditions; the algebraic gap at Eq. (11) and the self-citations are correctness or citation issues, not reductions by construction.

full rationale

The paper's central derivation is an algebraic rearrangement of the plant relation h=T{rd}+F{s} with the learning signal s defined in Eq. (7). The target error is expressed as a weighted sum of the training UAV errors, and the convergence conditions (12), (20), and (23) are explicit norm bounds on the coefficient operators in that sum. This is a control-design derivation, not a prediction fitted to data: the filters α_k and L_k are chosen from the model F, not estimated from the h_k, and the numerical studies simulate the resulting closed loop rather than claiming an independent empirical forecast. The citations to the authors' prior ILC design [23] and model-based learning [26] motivate the filter design and the LTI assumption, but the convergence condition is stated in the paper itself and does not depend on a self-citation for its mathematical content. There is a genuine correctness gap in the derivation of Eq. (11): the term T{rd}−Σ_k Λ_k^{-1}α_k T{Λ_k rd} is dropped, which is valid only if Σα_k=1 (or T{rd}=0); this is an unstated algebraic assumption that makes the convergence guarantee incomplete, but it is not a circular step because the final expression is not assumed as an input—it is derived, albeit incorrectly. Likewise, the approximations in Section 4 (||G||≫1 leading to F_k≈F) are modeling assumptions, not circular inputs. No pattern of self-definition, fitted-input-as-prediction, or author-imported uniqueness applies, so the circularity score is 0.

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

The algorithm introduces no fitted parameters; all coefficients are specified by the user or derived from known models. The main assumptions are the LTI/decoupled model of the position loop, the availability of exact dynamics mappings, and the unstated cancellation of the target's own tracking error in the derivation of the learning dynamics.

assumptions (5)

domain assumption The position control loop of the UAV is a linear time-invariant system, with attitude dynamics ignored.
Stated in Section 2: 'the dynamics, T and F, are reasonably assumed as linear time-invariant (LTI) systems'.
domain assumption The position dynamics and scaling matrices are diagonal/decoupled.
Section 3.1: 'we assume that the position control loop of the UAV is decoupled, that is, T and F are diagonal-matrix systems, and Λ_k's are diagonal matrices.'
domain assumption The open-loop gain is large, so T ≈ G^{-1} and F_k ≈ F.
Section 4.1: 'Considering that the position controller guarantees transfer function approximately 1, we have ||G|| >> 1 and T = (1+G)^{-1} ≈ G^{-1}'.
domain assumption Exact dynamics mapping M_k such that G = M_k G_k is known.
Section 4.1: 'We assume that the mapping from G_k to G could be represented by a transfer function matrix M_k.' This requires accurate models.
ad hoc to paper Cancellation of the target's own error term, i.e., T{rd} - Σ α_k T{rd} = 0, is implicitly assumed.
In Eq (11), the terms T{rd} and the α_k T{Λ_k rd} terms are dropped to obtain h = Σ Λ^{-1}(FL_k + α_k I){h_k}. This requires Σ α_k = 1 or T{rd}=0, which is never stated.

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Cite this review

Pith. "Pith review of A Preliminary Study on A Physical Model Oriented Learning Algorithm with Application to UAVs." pith.science (2026). https://pith.science/paper/DUI3URKC

@misc{pith2026190803478,
  author       = {Pith},
  title        = {Pith review of: A Preliminary Study on A Physical Model Oriented Learning Algorithm with Application to UAVs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DUI3URKC}},
  note         = {Machine review of arXiv:1908.03478}
}

read the original abstract

This paper provides a preliminary study for an efficient learning algorithm by reasoning the error from first principle physics to generate learning signals in near real time. Motivated by iterative learning control (ILC), this learning algorithm is applied to the feedforward control loop of the unmanned aerial vehicles (UAVs), enabling the learning from errors made by other UAVs with different dynamics or flying in different scenarios. This learning framework improves the data utilization efficiency and learning reliability via analytically incorporating the physical model mapping, and enhances the flexibility of the model-based methodology with equipping it with the self-learning capability. Numerical studies are performed to validate the proposed learning algorithm.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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M. Zheng, C. Wang, L. Sun, and M. Tomizuka, “Design of arbitrary-order robust iterative learning control based on robust control theory,” Mechatronics, vol. 47, pp. 67–76, 2017

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work page 2015

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X. Liang, “Image-based post-disaster inspection of rein- forced concrete bridge systems using deep learning with bayesian optimization,” vol. 34, no. 5. Wiley Online Library, 2019, pp. 415–430

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work page 2006