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REVIEW 3 major objections 5 minor 30 references

Event-Triggered Newton-Based Extremum Seeking Control

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read An event-triggered Newton-based extremum seeking law drives a scalar static map to the extremum at a user-assigned exponential rate independent of the unknown Hessian, and provably avoids Zeno behavior.

desk verdict Interesting combination of Newton ES and event-triggered control, but the averaging proof has a time-scaling error that undermines Theorem 1 as written. read the letter →

arxiv 2502.00930 v1 pith:N5RBRITZ submitted 2025-02-02 math.OC cs.SYeess.SY

Victor Hugo Pereira Rodrigues , Tiago Roux Oliveira , Miroslav Krstic , Paulo Tabuada This is my paper · ORCID
classification math.OCcs.SYeess.SY MSC 93C5793D0534C2993C40
keywords extremumseekingcontrolevent-triggeredNewton-basedoptimizationRiccatiequationfilteraveragingtheorydiscontinuoussystemsscalarstaticmapsexponentialstability
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 combining static event-triggered control with Newton-based extremum seeking makes the convergence rate of an unknown-map optimizer user-assignable and independent of the unknown Hessian. The proposed loop uses a Riccati differential equation to estimate the inverse Hessian online, and an event trigger that updates the control only when the zero-order-held actuation error becomes a fixed fraction of the gradient estimate. If the main theorem is right, a networked controller can reach a small neighborhood of the optimum at a chosen exponential rate while issuing fewer control updates than the gradient-based event-triggered alternative, and without infinitely fast triggering. Numerical simulations for a quadratic map show the Newton version settling near the optimum after about 100 seconds with 43 updates, versus the gradient version's 47 updates over 500 seconds.

What carries the argument

The load-bearing mechanism is the Riccati filter $\dot{\Gamma} = \omega_r\Gamma - \omega_r\hat{H}\Gamma^2$, which yields an online estimate $\Gamma$ of $1/H^*$ even when the raw Hessian estimate $\hat{H}$ passes through zero, and multiplies the gradient estimate so that the unknown curvature cancels from the linearized error dynamics. The event-triggered component is the condition $\Xi(\hat{G},e)=\sigma|\hat{G}|-\beta|e|<0$ (with $\sigma\in(0,1)$, $\beta>0$) that fires a zero-order-held control update; between updates it keeps $|e|\le(\sigma/\beta)|\hat{G}|$, which preserves a fixed fraction of the ideal Lyapunov decay. The proof's engine is the Lyapunov function $V=\hat{G}_{av}^2$ on the averaged system (42)-(44), and an averaging theorem for discontinuous systems transfers the exponential decay back to the original fast-oscillating closed loop. The averaged event trigger operates on $\hat{G}_{av}$ and $e_{av}$, so both the stability and the inter-event-time arguments are carried out in averaged coordinates.

What would settle it

Simulate the original (non-averaged) event-triggered Newton loop for two maps with $H^*=1$ and $H^*=100$ under identical $K$, $a$, $\sigma$, and $\omega$; if the measured exponential rate of $|\theta(t)-\theta^*|$ differs by more than the $O(a+1/\omega)$ residue, the claimed user-assignable and Hessian-independent rate does not survive outside the averaged model.

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Extended reading notes

Core claim

The central claim is Theorem 1. For sufficiently large probing frequency $\omega$ and sufficiently small initial errors, the averaged closed-loop system (42)-(44) under the average trigger (48) is locally exponentially stable, and the original variables satisfy $$|\$\theta$(t)-\$\theta$^*| \le $e^{{-(1-\sigma)a^2K t/2}}$|\$\theta$(0)-\$\theta$^*| + O\left(a+\frac{1}{\omega}\right),$$ $$|y(t)-Q^*| \le $e^{{-(1-\sigma)a^2K t/2}}$|y(0)-Q^*| + O\left($a^{2}$+\frac{1}{\$omega^{2}$}\right),$$ and $$\left|\Gamma(t)-\frac{1}{H^*}\right| \le $e^{{-\omega_r t}}$\left|\Gamma(0)-\frac{1}{H^*}\right| + O\left(\frac{1}{\omega}\right).$$ The convergence exponent $(1-\sigma)a^2K/2$ contains no $H^*$, so the user can choose the speed through the gain $K$, the probing amplitude $a$, and the trigger parameter $\sigma$. The paper also derives a positive lower bound on inter-event times, so the event-triggered controller avoids Zeno behavior.

Load-bearing premise

The proof's load-bearing premise is that the discontinuous, fast-oscillating event-triggered loop is well approximated by its average system; the paper does not verify the averaging theorem's hypotheses for the zero-order-held error and event times, and the time scaling in Section 4 has an internal inconsistency between $\bar{t}=\omega t$ and the $\omega\bar{t}$ arguments, so the transfer from the averaged to the original system is the fragile link.

Editorial extensions

If this is right

  • The convergence rate can be dialed in by the user through $K$, $a$, and $\sigma$ without knowing or estimating the Hessian $H^*$, removing the conservative tuning that the gradient version requires.
  • For sufficiently large $\omega$, a positive minimum inter-event time exists, so the controller is compatible with networked actuation that needs a guaranteed spacing between updates.
  • The final neighborhood shrinks as $O(a+1/\omega)$ for the optimizer and $O(a^2+1/\omega^2)$ for the map output, so probing amplitude and frequency can be traded against steady-state accuracy.
  • In the paper's simulation, the event-triggered Newton controller reaches the optimum region within about 100 seconds with 43 updates, while the event-triggered gradient controller remains in transient over 500 seconds with 47 updates.

Reading between the lines

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

  • A self-triggered implementation is a natural extension: the dwell-time formula in (A.38) gives an explicit rule for scheduling the next update in advance from current data, which the paper does not pursue.
  • For multivariable static maps, the scalar Riccati filter would have to become a matrix-valued estimate of the inverse Hessian and the trigger would have to operate on a norm of the gradient error; the convergence-rate benefit would be larger in higher dimensions if the scalar argument carries over.
  • Because the Hessian-independence bound is proven for the averaged system, a direct check on the original system with two very different Hessian values would test whether the assignable rate survives outside the averaging idealization; this is an experimental question the paper leaves open.
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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

3 major / 5 minor

Summary. The paper proposes a static event-triggered Newton-based extremum seeking (SET-NewtonES) scheme for scalar static maps. The control law uses a Riccati-filter estimate of the inverse Hessian and a zero-order hold updated by a static event-triggering rule based on the gradient estimate. The main claim (Theorem 1) is local exponential practical stability of the average closed-loop system with a user-assignable convergence rate independent of the unknown Hessian, together with O(a+1/omega) closeness of the original variables to the averaged ones and avoidance of Zeno behavior. The proof combines a Lyapunov argument for the averaged system with an averaging theorem for discontinuous systems attributed to Plotnikov, and the paper includes a numerical comparison with an earlier event-triggered gradient method.

Significance. If the main theorem were fully established, the contribution would be a useful combination of two known ideas: Newton-based extremum seeking with a Riccati inverse-Hessian estimator and event-triggered actuation. The claimed rate independent of the Hessian is standard for continuous Newton ES but has not been demonstrated in the event-triggered setting, so the paper addresses a meaningful gap. The explicit exponential bounds and the attempt to prove a positive dwell time are valuable. However, the manuscript does not provide machine-checked proofs or reproducible code, and the significance is moderated by the fact that the central proof steps, especially the time-scaling and the transfer from averaged to actual event times, are not rigorously justified as written.

major comments (3)
  1. [Section 4, Eqs. (24)-(31) and (27)-(28)] The time-scaling transformation \bar t = \omega t is applied inconsistently. After this transformation, 2\omega t becomes 2\bar t, but equations (27)-(28) contain cos(2\omega\bar t), sin(\omega\bar t), and cos(3\omega\bar t). More importantly, the original right-hand sides (18)-(19) contain terms of order O(\omega), such as -3a^2\omega H^*/8 cos(3\omega t) and a^2\omega H^* sin(2\omega t)\tilde\theta. When written as dX/d\bar t = (1/\omega) F(\bar t,X,1/\omega), these terms become O(1) forcing terms, so the right-hand side is not of the standard small-parameter form with uniformly bounded F required by the cited averaging theory. Consequently, the invocation of Plotnikov's theorem and the O(1/\omega) estimates (A.16)-(A.18) are unsupported. The proof must either correct the scaling and then handle the remaining zero-mean O(1) terms with an appropriate averaging lemma, or explicitly verify the hypotheses of the theorem it cites.
  2. [Appendix B, Eqs. (A.26)-(A.38)] The Zeno analysis is carried out for the averaged signals \hat G_av and e_av and then transferred to the original system through (A.35). However, the actual gradient estimate (8) contains the persistent O(a) sinusoidal terms (aQ^*+3a^3H^*/8) sin(\omega t) - a^3H^*/8 sin(3\omega t), which do not vanish as \tilde\theta converges and are not captured by the average \hat G_av = a^2H^*/2 \tilde\theta_av. The actual trigger condition (20) therefore sees zero crossings of \hat G(t) even near the optimum, and the lower bound (A.38) computed from the average system does not control the original inter-event times. In fact, if an event occurs exactly at a zero of \hat G(t) and \beta>\sigma, then immediately after the event |e(t)|=|\hat G(t)|, so \sigma|\hat G(t)|-\beta|e(t)|=(\sigma-\beta)|\hat G(t)|<0, which contradicts the claimed positive dwell time. The proof needs to analyze the actual event times rather than the averaged event times.
  3. [Appendix A, Eq. (A.2)] The stability argument uses the implicit condition \beta>|H^*| when bounding -a^2/\omega H^*K\hat G e by a^2/\omega \beta K|\hat G||e| and then combining this with the trigger bound |e|\le \sigma/\beta|\hat G|. This condition on the trigger gain \beta is not stated among Assumptions (A1)-(A5) nor in Theorem 1. Since H^* is unknown, the theorem should either include this explicit sufficient condition on \beta, or prove stability under a weaker and verifiable condition. Without it, the proof of exponential decay in (A.3) does not follow from the stated assumptions.
minor comments (5)
  1. [Section 6] The simulation description states that the map parameters satisfy \theta^*=5, but later the text says the green curve marks the desired optimum \theta^*=7. This inconsistency should be corrected.
  2. [Equation (29)] Equation (29) writes F_3 with t instead of \bar t in the arguments \Gamma(t) and \hat H(t); after the time-scaling, all occurrences should be expressed in the scaled variable \bar t before averaging.
  3. [Appendix B, Eqs. (A.27)-(A.30)] The derivative of \phi_av in (A.27) is computed without absolute-value signs and is only valid while e_av and \hat G_av have constant signs; the subsequent upper bound (A.30) should be derived using the subdifferential of the absolute value function or an alternative argument that does not assume sign invariance.
  4. [Appendix B, Eq. (A.38)] Solving (A.36) for the time at which \hat\phi reaches 1 gives a factor \beta/\sigma, not \beta^2/\sigma^2; the displayed lower bound in (A.38) contains this algebraic error and should be corrected.
  5. [Section 6] The simulation does not specify the trigger gain \beta, even though \beta appears in the trigger condition (20) and in the stability condition \beta>|H^*| used in the proof; this makes the numerical example incomplete as a verification of the theoretical assumptions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Newton-based convergence rate is obtained by direct Lyapunov algebra and standard averaging, not by fitting or by self-citation.

full rationale

The paper's central claim, inequality (50), follows from a Lyapunov analysis of the averaged system (42)-(44): V_av = G_av^2, the event-trigger bound |e_av| <= sigma/beta |G_av|, and the algebraic relation (45)-(46) that converts G_av-decay into theta_av-decay. This is a direct derivation, not a fit: no parameter is tuned to make the stated rate appear, and the Hessian-independence arises from cancellation of H* when (42)-(43) and (45) are combined. The Riccati inverse-Hessian estimator is taken from the independent prior work of Ghaffari et al., and the self-citations to Rodrigues et al. supply the baseline gradient event-triggered scheme, not the Newton result. No prediction is renamed, and no load-bearing uniqueness theorem is imported from the authors' own work. The main technical weakness is the application of Plotnikov's averaging theorem to (31): the time-scaling in Section 4 leaves O(omega) terms in F, so the hypotheses of the discontinuous averaging theorem may not be satisfied. That is a correctness/verification gap, not a circular reduction, because the theorem itself is an external mathematical result and the conclusion is not assumed as an input. Overall, no circular step is present.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The only ingredients are controller gains and the Riccati filter, which is a standard estimation device from prior work.

free parameters (1)
  • beta (trigger gain)
    The proof requires beta>|H*| in Appendix A, but H* is declared unknown in Assumption (A2) and no bound on |H*| is listed among the assumptions. beta is a design parameter, not fitted to data, but its selection needs information the paper does not state.
assumptions (4)
  • domain assumption The static map Q can be approximated locally by the quadratic Q(theta)=Q*+H*/2(theta-theta*)^2, with the Hessian H* constant.
    Equation (1); justified by Taylor expansion in Section 2. This is standard for local extremum seeking but restricts the result to a local neighborhood.
  • domain assumption The averaging theorem of Plotnikov (1980) applies to the discontinuous event-triggered closed loop and the approximation error is O(1/omega).
    Invoked in Section 4 and Appendix A around equations (A.16)-(A.18); the paper does not verify the hypotheses of the theorem for the sampled-data system with event times depending on fast signals.
  • ad hoc to paper The trigger gain beta can be chosen larger than |H*| even though H* is unknown.
    Proof after (A.2) needs beta>|H*| to turn the cross term into a useful bound; this is not stated in Assumptions (A1)-(A5) or in Theorem 1.
  • domain assumption The Riccati filter (13) is well-posed and converges to the equilibrium 1/H* for the allowed initial conditions.
    Section 2.2 and equation (36); for H*<0, Gamma(0) must have the same sign as 1/H* to avoid divergence, and the theorem only assumes Gamma_tilde(0) small. This local condition is standard but unstated.

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Cite this review

Pith. "Pith review of Event-Triggered Newton-Based Extremum Seeking Control." pith.science (2026). https://pith.science/paper/N5RBRITZ

@misc{pith2026250200930,
  author       = {Pith},
  title        = {Pith review of: Event-Triggered Newton-Based Extremum Seeking Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N5RBRITZ}},
  note         = {Machine review of arXiv:2502.00930}
}
read the original abstract

This paper proposes the incorporation of static event-triggered control in the actuation path of Newton-based extremum seeking and its comparison with the earlier gradient version. As in the continuous methods, the convergence rate of the gradient approach depends on the unknown Hessian of the nonlinear map to be optimized, whereas the proposed event-triggered Newton-based extremum seeking eliminates this dependence, becoming user-assignable. This is achieved by means of a dynamic estimator for the Hessian's inverse, implemented as a Riccati equation filter. Lyapunov stability and averaging theory for discontinuous systems are applied to analyze the closed-loop system. Local exponential practical stability is guaranteed to a small neighborhood of the extremum point of scalar and static maps. Numerical simulations illustrate the advantages of the proposed approach over the previous gradient method, including improved convergence speed, followed by a reduction in the amplitude and updating frequency of the control signals.

Figures

Figures reproduced from arXiv: 2502.00930 by the authors.

Figure 1
Figure 1. Static Event-Triggered Gradient-based Extremum [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Static Event-Triggered Newton-based Extremum [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

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Reference graph

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