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REVIEW 5 major objections 6 minor 51 references

Multivariable Stochastic Newton-Based Extremum Seeking with Delays

T0 review · 5 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper shows that a stochastic Newton-based extremum-seeking controller with predictor feedback can stabilize a multi-input system around an unknown optimum even when each input channel has a different, arbitrarily long delay.

desk verdict Genuine gap in the literature addressed, but the proof of Theorem 1 has a load-bearing gap in §3.7 that a referee should push on. read the letter →

arxiv 2502.00861 v1 pith:W7WY3YW2 submitted 2025-02-02 math.OC cs.SYeess.SY

Paulo Cesar Souza Silva , Paulo Cesar Pellanda , Tiago Roux Oliveira This is my paper · ORCID
classification math.OCcs.SYeess.SY MSC 93C2393C4093E15
keywords extremumseekingcontrolNewton-basedstochasticperturbationsinputdelayspredictorfeedbackbacksteppingtransportPDEinfinite-dimensionalaveraging
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 tries to establish that a multivariable stochastic Newton-based extremum-seeking controller can solve real-time optimization problems in which each control input reaches the plant through a known, constant, and possibly very long delay. The proposed law combines predictor feedback with an online estimate of the inverse of the unknown Hessian, obtained from stochastic sinusoidal probing. If Theorem 1 is correct, the closed-loop error is exponentially stable in mean square, converges in probability to a neighborhood of the unknown maximizer, and the convergence rate is chosen by the designer rather than dictated by the unknown curvature. This matters because extremum seeking is normally not robust to long delays, and existing stochastic or Newton-based schemes either ignore delays, require uniform delays, or handle only single-input maps.

What carries the argument

The mechanism that carries the argument is the transport-PDE representation of the delay, $u_i(x,t)=U_i(t+x-D_i)$ for $x\in[0,D_i]$, together with the predictor feedback $U_i(t)=\frac{c_i}{s+c_i}[-k_i(z_i(t)+\int_0^{D_i}u_i(\sigma,t)\,d\sigma)]$. The measured signal $z(t)=\Gamma(t)G(t)$ is built from the stochastic demodulation of the output, and $\Gamma(t)$ solves the matrix Riccati equation (21) so that its average is $H^{-1}$. A backstepping transformation $w_i=u_i+k_i(\tilde\vartheta_i+\int_0^x u_i\,d\sigma)$ maps the linearized average system to a target system whose Lyapunov-Krasovskii functional decays exponentially once the filter gains $c_i$ exceed computable thresholds $c_i^*$. The stochastic averaging theorem is then invoked to transfer exponential stability from the averaged infinite-dimensional system to the original randomly perturbed system, yielding the $O(1/\omega)$ mean-square residue.

What would settle it

Simulate the two-input example with $D_1=50$, $D_2=100$ at a moderate frequency such as $\omega=2$ and with an initial Hessian-inverse error deliberately large, then measure the mean-square norm in (34) over a long horizon; if it fails to stay at order $1/\omega$ and instead grows or oscillates persistently, the discarded quadratic term $\tilde\Gamma H\tilde\theta$ is not negligible. A second, more direct check is to evaluate the finite-period average $\frac{1}{\Pi}\int_0^\Pi N(\sigma)y\,d\sigma$ with $\Pi=2\pi/\omega$ for the non-periodic signal $\eta$ from (14): it will differ from $H$, revealing the bias that the ergodic-limit interpretation of (17) must carry.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 1. For a locally quadratic map $Q(\theta)=y^*+\frac{1}{2}(\theta-\theta^*)^T H(\theta-\theta^*)$ with distinct input delays $D_1\le\cdots\le D_n$, there exist a threshold $c^*>0$ and, for each $c_i\ge c^*$, a frequency $\omega^*(c_i)$ such that for all $\omega\ge\omega^*$ the closed-loop system has a unique exponentially stable solution. The mean-square full-state norm obeys $E(|\tilde\Gamma(t)|^2+\sum_{i=1}^n[\tilde\theta_i(t-D_i)]^2+[U_i(t)]^2+\int_{t-D}^t[U_i(\sigma)]^2\,d\sigma)^{1/2}\le O(1/\omega)$, and the estimates satisfy $\theta(t)\to\theta^*$ and $y(t)\to y^*$ in probability with ultimate errors of order $O(|a|+1/\omega)$ and $O(|a|^2+1/\omega^2)$. The point is that the exponential decay rate is the user-assigned diagonal gain $K$, independent of the unknown Hessian $H$, while the delay-compensation step handles arbitrarily long and channel-specific delays.

Load-bearing premise

The proof assumes that the fast random probing can be averaged out over one nominal period and that the product of the two estimation errors is small enough to be ignored; if either fails, the exponential-stability conclusion does not follow.

Editorial extensions

If this is right

  • Input delays that are known, constant, distinct, and arbitrarily large can be compensated, so extremum seeking remains stable where the uncompensated loop diverges.
  • The exponential convergence rate is set by $K$ and does not depend on the unknown Hessian $H$, so the user can prescribe how fast the estimates approach the extremum.
  • For multi-input, single-output static maps with cross-coupling, the estimates of the maximizer and of the Hessian inverse converge in probability to neighborhoods whose sizes shrink as the dither amplitudes $|a|$ and $1/\omega$ shrink.
  • Newton-based stochastic extremum seeking retains its faster transient behavior relative to gradient-based stochastic extremum seeking even when predictor feedback is used to compensate equal delays.

Reading between the lines

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

  • Beyond the paper, the same predictor-plus-backstepping construction should also cover simultaneous input and output delays by lumping them into one effective delay; the authors note the extension but do not prove a separate theorem for it.
  • Beyond the paper, a reader might test finite-horizon behavior at moderate $\omega$: since $\eta(t)=\omega\pi(1+\sin(W_\omega t))$ is not periodic, the finite-period average in (17) is only an idealization, and the true ergodic average will leave a time-dependent bias that the $O(1/\omega)$ bound must absorb.
  • Beyond the paper, the same machinery could be probed on the source-seeking problem with unequal Hessian cross terms, where the discarded bilinear term $\tilde\Gamma H\tilde\theta$ is larger; the paper does not demonstrate how large the cross-coupling can be before the linearization fails.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper proposes a multivariable Newton-based stochastic extremum-seeking controller for static quadratic maps with known distinct input delays. The control law combines sinusoidal stochastic perturbations with a dynamic Riccati filter that estimates the inverse Hessian, and predictor feedback implemented through low-pass filters; the input delays are represented as transport PDEs, and stability is analyzed by backstepping and infinite-dimensional averaging. Theorem 1 claims exponential stability of the closed-loop system in a mean-square norm with residual O(1/omega), together with practical convergence in probability to O(|a|+1/omega) for theta(t) and O(|a|^2+1/omega^2) for y(t). Numerical simulations for a two-input quadratic map with delays D1=50 and D2=100 are provided.

Significance. If Theorem 1 is correct, the paper closes a natural gap in the extremum-seeking literature: it extends Newton-based stochastic ESC to multivariable systems with distinct input delays, preserving a user-assignable convergence rate that is independent of the unknown Hessian. This is practically valuable for applications with long, known, channel-dependent actuation delays and cross-coupled channels. The backstepping and Lyapunov analysis of the linearized averaged system in Sections 3.4-3.6 follows a standard pattern and appears sound, and the simulations are consistent with the qualitative claims. The paper does not fit parameters to data; the convergence-rate claim is derived rather than calibrated, and there is no evident circularity. However, the proof's decisive stochastic averaging step in Section 3.7 is asserted rather than verified, so the theoretical contribution is currently not established at the level claimed by the theorem.

major comments (5)
  1. [Sec. 3.7, Eq. (71)] The decomposition d/dt u^epsilon = G(u^epsilon_t) + epsilon F(t,u^epsilon_t,eta,epsilon) is not derived from the exact system (69). In (69), the terms U_i(t-D_i), -c_i k_i z_i(t) - c_i k_i integral_{t-D_i}^t U_i(tau) dtau, and the Riccati term carry no factor epsilon=1/omega, while (71) places them in epsilon F. Unless an additional rescaling of U_i, Gamma-tilde, or the dynamics is performed and stated, the hypotheses of the averaging theorem [30] are not satisfied by the system as written.
  2. [Sec. 3.7, after Eq. (71)] The assertion F(tau,0,eta,epsilon)=0 is false. At u^epsilon=0 one has Gamma=H^{-1} and theta_D=theta*+S(eta_D), so y(t) is the nonzero stochastic signal (1/2)S(eta_D)^T H S(eta_D), and z_i(t) is nonzero; hence the U_i-component of F contains -c_i k_i z_i(t) at u^epsilon=0. The Riccati component is also nonzero at Gamma-tilde=0 because the instantaneous Hessian estimate hat H(t) is not equal to its average H. The origin is therefore not an equilibrium of the exact stochastic system, so the zero-equilibrium condition needed for the cited averaging theorem must be replaced by a practical-stability formulation.
  3. [Secs. 3.3-3.7, Eqs. (25)-(26), (49)] The averaged system whose exponential stability is proved is the linearized system: Eq. (25) drops the bilinear term Gamma-tilde^av H theta-tilde^av in passing to (26), and Eq. (49) drops the quadratic Riccati term. The stochastic averaging theorem [30] applies to the exact averaged system associated with (69), which contains both nonlinear terms. The manuscript does not show that the exact averaged system is locally exponentially stable with an explicit region of attraction; a local Lyapunov argument for the full nonlinear averaged system is missing.
  4. [Sec. 2.1 and Sec. 3.7, Eqs. (14)-(17)] The Markov-parameter and averaging hypotheses of [30] are not verified. The signal eta_i(t)=omega pi(1+sin(W^i_{omega t})) is not by itself a Markov process, since the SDE in (15) is driven by W^i_tau and cannot be closed in eta_i alone; the circular Brownian motion or the pair (eta_i,W^i) should be used as the Markov parameter. Moreover, Eq. (17) states the average property as a finite-period average over Pi=2pi/omega even though eta is non-periodic; this must be formulated as an ergodic average with a convergence estimate in omega.
  5. [Sec. 3.8, Eqs. (72)-(74)] The stopping-time argument is not mathematically meaningful as written: tau^Delta(epsilon)_epsilon := inf{forall t>=0 : |u^epsilon(t)| > M|u^epsilon(0)|e^{-lambda t}+Delta} takes the infimum of a proposition rather than a random time, and Eq. (73) uses a similar ill-posed expression. Since this section is the only bridge from the stability inequality (34) to the practical convergence claims (35)-(36), it should be replaced by a standard stochastic practical-stability or stopping-time argument with a correctly defined stopping time.
minor comments (6)
  1. [Abstract] The abstract paragraph is duplicated verbatim at the beginning of the paper; one copy should be removed.
  2. [Eq. (57)] The first term of V_i(t) should read (1/2)[theta-tilde^av_i(t)]^2; the square is missing, although Eq. (58) uses the squared quantity.
  3. [Eq. (62)] The notation (KH)^2 and e^{KH(D_i-sigma)} is undefined and appears to be a leftover from a different derivation; these objects should be removed or defined consistently with the scalar arguments used in the backstepping transformation.
  4. [Eq. (44)] In Eq. (44) the boundary condition writes -k_i z(t) with vector z(t), while the left-hand side is the scalar u_i(D_i,t); it should read -k_i z_i(t).
  5. [Sec. 4.3] The organization of Section 4.3 is confusing: Figures 5 and 6 are labeled as no-delay simulations even though the section heading promises distinct input delays; the captions or the text should be corrected to make clear which simulation is being reported.
  6. [Sec. 4] The simulation parameters state c=20, but the controller in (33) requires a vector of filter constants c_i; the values used for each channel should be stated explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the claimed distinct-delay stochastic Newton ESC result is not equivalent to its inputs by construction; the main weakness is an unverified application of an external stochastic averaging theorem, which is a correctness gap rather than a circular reduction.

full rationale

The central claim of the paper is a new controller construction for multivariable stochastic Newton-based extremum seeking with distinct input delays. The result is not obtained from fitted data: no parameter is calibrated from measured outputs, and the convergence bounds in Theorem 1, (34)-(36), are expressed in terms of O(1/omega) and O(|a|), not from fitted constants. The averaging identities (17), (20), and (24) are standard stochastic-ESC results cited from [6,18], and their use is not a renamed version of the target theorem. Self-citations such as [11,17,24] are present but are not load-bearing in a circular sense: [24] is the earlier gradient-based conference version that the paper explicitly extends, and [11,17] are cited to motivate the low-pass-filtered predictor form, not to supply the exponential-stability conclusion. The serious weakness is in Section 3.7, where the paper writes the exact system as (71), asserts F(tau,0,eta,epsilon)=0, assumes eta is a homogeneous exponentially ergodic Markov process, and invokes the external averaging theorem [30] to conclude (34). However, the stability analysis in Sections 3.3-3.6 establishes exponential stability only for the linearized averaged system, where (25) is replaced by (26) and (49) linearizes the Riccati equation. The theorem [30] would require exponential stability of the exact averaged system and verified hypotheses on F and the Markov parameter; at u^epsilon=0, z_i(t) is a nonzero stochastic signal and U_i contains -c_i k_i z_i(t), so F(tau,0,eta,epsilon)=0 is not obviously satisfied. These are non-circular correctness gaps: the conclusion is not equivalent to an input by construction, and it is not a fitted parameter renamed as a prediction. The derivation is substantially self-contained and benchmarked against simulations, so the circularity score is low.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. Its load-bearing assumptions are the locally quadratic map, known constant delays, the ergodicity and averaging properties of the stochastic signals, and two approximations in the proof: the linearization dropping ~Gamma H ~theta and the unverified applicability of the stochastic averaging theorem. The controller gains are design degrees of freedom, not fitted values.

free parameters (1)
  • Design gains k_i, c_i, omega_r, a_i = Simulation: k=0.005, c=20, omega_r=0.007, a1=a2=0.22; theorem requires only k_i>0, c_i>c*_i (64), omega_r>0, a_i nonzero
    These are user-chosen controller parameters, not fitted to data; they set convergence rate K and residual size O(|a|), so the central performance claims depend on them, but no calibration against measurements is involved.
assumptions (5)
  • domain assumption Unknown static map is locally quadratic, y=y*+0.5(theta-theta*)^T H(theta-theta*) with H=H^T<0 (eq. 4).
    This is the standard local model for extremum seeking; it makes the Hessian constant and enables the average property (17).
  • domain assumption Input delays are known, constant, distinct and ordered, 0<=D1<=...<=Dn (eq. 3).
    Predictor feedback (33) uses exact delay values; the paper only claims robustness to small delay errors.
  • domain assumption The stochastic perturbation eta_i(t)=omega*pi*(1+sin(W_omega t)) is a homogeneous ergodic Markov process with symmetric density and fast oscillation, and the demodulation signals satisfy the average property (17).
    Needed for the stochastic averaging theorem; as written, the finite-period average in (17) is not justified for the non-periodic eta.
  • ad hoc to paper The stochastic averaging theorem of Katafygiotis and Tsarkov [30] applies to (71) and yields exponential p-stability with p=2 for the original random system.
    Invoked in Section 3.7 without checking Lipschitz continuity, exponential ergodicity, or the O(epsilon) scaling of F.
  • ad hoc to paper The bilinear term ~Gamma_av H ~theta_av is negligible relative to ~theta_av in (25)-(26), i.e., the linearization around Gamma=H^{-1} is valid for the stability analysis.
    This is the step that turns z_av into ~theta_av and yields the predictor (33); no quantitative bound is supplied.

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Cite this review

Pith. "Pith review of Multivariable Stochastic Newton-Based Extremum Seeking with Delays." pith.science (2026). https://pith.science/paper/W7WY3YW2

@misc{pith2026250200861,
  author       = {Pith},
  title        = {Pith review of: Multivariable Stochastic Newton-Based Extremum Seeking with Delays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7WY3YW2}},
  note         = {Machine review of arXiv:2502.00861}
}
read the original abstract

This paper presents a Newton-based stochastic extremum-seeking control method for real-time optimization in multi-input systems with distinct input delays. It combines predictor-based feedback and Hessian inverse estimation via stochastic perturbations to enable delay compensation with user-defined convergence rates. The method ensures exponential stability and convergence near the unknown extremum, even under long delays. It extends to multi-input, single-output systems with cross-coupled channels. Stability is analyzed using backstepping and infinite-dimensional averaging. Numerical simulations demonstrate its effectiveness in handling time-delayed channels, showcasing both the challenges and benefits of real-time optimization in distributed parameter settings.

Discussion (0). Continue with ORCID to comment.

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Pith tools

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