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REVIEW 4 major objections 5 minor 23 references

Towards Co-Designed Event-Triggered Extremum Seeking

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

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Pith's one-line read A single LMI can co-design the controller gain and the event-triggering rule of multivariable extremum seeking, and full-matrix gains beat diagonal gains on communication.

desk verdict A genuinely useful co-design LMI for event-triggered extremum seeking, with a clean average-system proof, but the bridge to the implemented hybrid system is a sketch, not a proof. read the letter →

arxiv 2608.10246 v1 pith:5J7WNLTI submitted 2026-08-10 math.OC cs.SYeess.SY

Roberto Luo , Pedro Henrique Silva Coutinho , Victor Hugo Pereira Rodrigues , Tiago Roux Oliveira , Miroslav Krstic This is my paper · ORCID
classification math.OCcs.SYeess.SY MSC 93C6593B5293D3090C2293C40
keywords extremumseekingevent-triggeredcontrolco-designlinearmatrixinequalitypolytopicuncertaintyaveragingtheoryZeno-freenessmultivariableoptimization
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

The paper claims that the feedback gain and the event-triggering rule of a multivariable extremum-seeking controller can be designed in one convex step, instead of fixing the gain first and tuning the trigger afterward. The design is encoded in a linear matrix inequality whose feasibility, given only a polytope of possible Hessians, certifies exponential convergence of the averaged closed loop at a prescribed rate while enlarging the admissible transmission error. A sympathetic reader would care because the co-design exposes extra degrees of freedom: a full-matrix gain can exploit Hessian cross-coupling, and the numerical results show that no diagonal gain reaches the same triggering threshold and decay rate. The paper also proves that the actual sampled-data system converges to the optimizer up to an $O(a+1/\omega)$ residual and that executions are Zeno-free.

What carries the argument

The load-bearing object is the co-design LMI (37) over decision variables $W, Z, \tilde{Q}_G, \tilde{Q}_e$; the controller gain is recovered as $K=ZW^{-1}$, the trigger matrices as $Q_G=\tilde{Q}_G^{-1}$ and $Q_e=W^{-1}\tilde{Q}_e W^{-1}$. Feasibility at every vertex $H_i$ of the Hessian polytope makes one Lyapunov function $V=\hat{G}_{av}^\top P\hat{G}_{av}$, with $P=W^{-1}$, work for every $H^*$ in the polytope. The trigger condition $\hat{G}^\top Q_G \hat{G} < e^\top Q_e e$ cancels the cross term in the Lyapunov derivative, so feasibility becomes the differential inequality $\dot{V} \le -2\eta V$. Averaging for differential inclusions carries the discontinuous sampled-data system to the smooth average system, and a comparison bound on $\phi=\beta\|e\|/\|\hat{G}\|$ yields the uniform positive lower bound on inter-event times.

What would settle it

Simulate the actual event-triggered loop (21)-(22) with the co-designed gain and trigger for several initial conditions, and compare the measured decay rate and minimum inter-event time with the predictions of Theorem 1 and Proposition 1: any run whose decay rate falls clearly below $\eta$ for large $\omega$, or whose inter-event time drops below $\tau^*$, would falsify the claimed transfer from the average system to the sampled-data system.

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Extended reading notes

Core claim

The central claim is Theorem 1 together with Lemma 1: if the LMI (37) is feasible with a prescribed decay rate $\eta$ for every vertex of the Hessian polytope, then the average event-triggered system is exponentially stable at the optimizer, and for sufficiently large probing frequency $\omega$ and small initial conditions the implemented sampled-data system satisfies $\|\theta(t)-\theta^*\| \le \kappa_\theta e^{-\eta t}\|\theta(0)-\theta^*\| + O(a+1/\omega)$. Thus convergence to the optimum is exponential down to a practical residual set by the probe amplitude and frequency. A further structural claim is supported by the simulations: within the same co-design procedure, a full gain matrix achieves triggering thresholds and decay rates that diagonal gains cannot match, and more Hessian uncertainty monotonically raises the communication cost.

Load-bearing premise

The result depends on the assumption that replacing the fast-oscillating event-triggered system by its smoothed average is legitimate even though the event times themselves depend on the state; if that assumption fails, the proof guarantees the smoothed model, not necessarily the implemented system.

Editorial extensions

If this is right

  • A full-matrix gain solves the co-design optimization with a lower objective than a diagonal gain for the same decay rate, so the joint design can permit larger transmission errors and fewer updates while keeping the guaranteed convergence.
  • Larger Hessian uncertainty polytopes produce higher co-design cost, so more accurate prior curvature information translates directly into fewer control transmissions.
  • The implemented event-triggered controller converges to the optimizer with the prescribed decay rate up to an $O(a+1/\omega)$ residual, so reducing probe amplitude and raising probe frequency shrinks the final error.
  • Inter-event times are uniformly bounded below by the positive bound in (56), so the zero-order-hold implementation never exhibits Zeno behavior.
  • In the reported two-vertex example, the upper-bound Hessian vertex requires roughly 104 updates versus 38 for the lower-bound vertex, yet convergence is guaranteed for any Hessian in the polytope.

Reading between the lines

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

  • Beyond the paper, the same LMI template could be applied to non-quadratic maps through their local quadratic approximations; the residual bound would then need an extra term accounting for higher-order curvature, which the current analysis does not track.
  • A natural extension the paper does not explore is constraining $Z$ to be sparse or block-structured; re-solving (64) under such constraints would map the trade-off between communication savings and implementation simplicity.
  • The trigger geometry could be optimized against concrete network models, such as available bandwidth or packet rates, instead of the trace objective (64); this would turn the co-design into an engineering resource-allocation tool rather than a purely stability-oriented one.
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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

4 major / 5 minor

Summary. The paper proposes a co-design framework for event-triggered gradient-based extremum seeking on multivariable quadratic maps whose Hessian is known only through a polytopic uncertainty set. The controller gain and the event-triggering matrices are jointly synthesized by a convex LMI optimization that enforces a prescribed exponential decay rate for an averaged event-triggered system. The authors prove exponential stability of this average system in Lemma 1, claim in Theorem 1 that the original sampled-data event-triggered system inherits local exponential convergence up to O(a + 1/omega) via averaging, and give a Zeno-freeness result in Proposition 1. Numerical simulations compare full-matrix gains with diagonal gains, showing that full gains achieve smaller triggering costs.

Significance. If the main theorem were rigorously established, the paper would make a useful contribution: the co-design LMI is convex, it handles polytopic Hessian uncertainty, it covers both diagonal and full gain structures, and the numerical comparison illustrates a concrete benefit of full-matrix gains. The proof of Lemma 1 itself appears sound, and the optimization formulation is a clean design result. However, the transfer from the average event-triggered system to the actual sampled-data hybrid system is the load-bearing step, and that transfer is not justified as written. The significance of the paper is therefore conditional on closing the averaging bridge with a correct and verified hybrid averaging argument.

major comments (4)
  1. [Section II.D, Eqs. (26)-(29) and Theorem 1] The averaging theorem for differential inclusions from [16] is invoked to pass from the hybrid system (26)-(27) to the average system (30)-(31), but the hypotheses of that theorem are not verified. The system (26)-(27) is not an ordinary differential inclusion: the reset times t_k(y) are state-dependent, being defined implicitly by the triggering condition (23)-(24), and the reset map (27) is nonzero. The cited Plotnikov averaging theorem concerns inclusions of the form x_dot in F(t,x) with a prescribed time average; it does not cover resets at state-dependent event times. Consequently, the closeness estimates (46) and (48) in the proof of Theorem 1 are not established, and the central bound (42) does not follow from the given arguments.
  2. [Section II.D, Eq. (29) and the paragraph after it] The assertion that the impulse average does not contribute to the average dynamics is unsupported and, as stated, appears incorrect. The reset map in (27) is I_k = [0, 0, -e^top]^top, so the average contribution of the impulses over a period is proportional to the sum of -e(t_k^-) over events in that period. This is not zero merely because the reset sends the instantaneous error to zero; the pre-jump values e(t_k^-) are generically nonzero. The sentence 'over a period, the average error tends to zero' would require a uniform bound on the number or magnitude of events per period, and no such bound is proved. Without a rigorous estimate of the impulse contribution, the average system (30)-(31) cannot be accepted as the correct averaged model of (26)-(27).
  3. [Section II.D, Eq. (27)] The scaling of the reset map in (27) is dimensionally inconsistent. For the transmission error e(t) = G_hat(t_k) - G_hat(t), the reset at t = t_k sends e to 0, so the jump is -e(t_k^-), which is independent of omega. Writing the jump as (1/omega) I_k(y) artificially shrinks the jump amplitude to O(1/omega) and makes it vanish in the averaging limit. Since the actual jump in the implemented event-triggered system is not small in omega, the averaged model derived from (27) may omit a non-negligible effect of the resets.
  4. [Theorem 1 proof and Proposition 1] Even if the average system (30)-(31) were correctly derived, comparing it with the actual system requires controlling the mismatch between the actual event times t_k and the average event times ar t_k. The proof of Theorem 1 invokes the averaging theorem only for the differential equation (22) and does not account for the fact that the hold error e(t) and the event times are state-dependent and differ between the two systems. The same gap appears in Proposition 1, where the bound (62) on |phi(t) - phi_av(t)| is asserted by the averaging theorem without verifying that it applies to the event-triggered hybrid system. Additionally, the 'average system' used in Lemma 1 is itself a hybrid event-triggered system, so its exponential stability should be proved with a rigorous treatment of jumps and Zeno-freeness, rather than by treating it as an ordinary differential equation.
minor comments (5)
  1. [Section II.D, Eq. (29)] The notation in (29) is confusing: T is already defined as the common period in (25), while the limit is written as T -> infinity. The average should be taken over the common period in the scaled time variable, so the limit notation should be adjusted accordingly.
  2. [Lemma 1 proof] There is a small typographical error near the end of the proof: 'with where kappa = ...' should read 'where kappa = ...'.
  3. [Theorem 1, Eq. (54)] In (54) the expression '|y(t) - Q*| = <=' contains a typo; it should be '|y(t) - Q*| <='.
  4. [Theorem 1 statement] The theorem states that the equilibrium is 'locally exponentially stable,' but the bounds (42) and (43) contain a practical residual O(a + 1/omega). This is better described as practical exponential stability, or the statement should explicitly define the practical residual.
  5. [Section IV, simulation results] The statement that 'no diagonal gain can simultaneously satisfy the decay rate and the same triggering threshold achieved by the corresponding full-matrix gain' is presented as a general conclusion, but it is only demonstrated for the specific numerical instance with ar sigma = 0.6 and threshold bound ar J = 10. It should be phrased as a numerical observation for the tested cases.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the LMI co-design is derived from the averaged dynamics and standard ETM inequalities, not fitted to simulation outcomes.

full rationale

The derivation chain is self-contained. Lemma 1 constructs an LMI (37) whose feasibility implies a Lyapunov inequality for the average event-triggered system; the proof algebraically reduces (37) to (39)-(41) and uses the trigger condition (35) to obtain \dot V ≤ -2\eta V. This is a design certificate, not a prediction from fitted data. Theorem 1 then transfers the average stability to the actual sampled-data system via an averaging argument citing [10], [16]; whether that transfer is fully justified is a correctness/hypotheses gap rather than a circular step, because the cited theorem is external and the paper does not define its target quantities in terms of the conclusion. Self-citations [17], [18] are used only for context and for simulation parameter selection, and these parameters are not used as inputs to the LMI optimization in a way that would force the reported thresholds. The triggering matrices QG and Qe are decision variables synthesized from the LMI, and the performance comparison between diagonal and full gains follows from feasibility of the optimization, not from fitting. No step reduces by construction to its own inputs; hence no circularity is found.

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

The central result rests on standard ES local-analysis assumptions (quadratic map, frequency separation, known Hessian polytope) and on an unverified applicability of averaging for the event-triggered hybrid closed loop. No parameters are fitted to data; eta and sigma_bar are design choices in the numerical study.

assumptions (5)
  • domain assumption The unknown Hessian lies in a known polytope co{H1,...,HN} with known vertices (Assumption 2).
    Priors on curvature are the input to the robust LMI; without known vertices the co-design cannot be run.
  • domain assumption Perturbation frequencies satisfy the non-resonance conditions of Assumption 1.
    Needed for the demodulation and averaging step that extracts the Hessian-gradient term and cancels sinusoidal terms.
  • domain assumption The static map is exactly quadratic in theta locally (Eq. (1)).
    The analysis is local around theta*; higher-order terms are neglected via standard ES local arguments.
  • ad hoc to paper The Plotnikov averaging theorem for differential inclusions applies to the state-dependent event-triggered hybrid system (26)-(27).
    Invoked in Theorem 1 to pass from average to actual system without verifying its hypotheses for state-dependent event times and impulses.
  • ad hoc to paper The impulse average in (29) does not contribute to the average dynamics.
    States that over a period, the average error tends to zero, which is asserted rather than proved.

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Pith. "Pith review of Towards Co-Designed Event-Triggered Extremum Seeking." pith.science (2026). https://pith.science/paper/5J7WNLTI

@misc{pith2026260810246,
  author       = {Pith},
  title        = {Pith review of: Towards Co-Designed Event-Triggered Extremum Seeking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5J7WNLTI}},
  note         = {Machine review of arXiv:2608.10246}
}
read the original abstract

This paper studies event-triggered gradient-based multivariable extremum seeking for nonlinear maps with polytopic Hessian uncertainty. Unlike existing event-triggered extremum-seeking methods, which first fix the controller (typically diagonal) and then design the triggering mechanism, the proposed approach jointly synthesizes the controller and the triggering mechanism through a co-design framework that admits both diagonal and full controller gain matrices. The co-design problem is formulated as a convex optimization problem with linear matrix inequality constraints. Its solution guarantees exponential convergence of the average closed-loop system with a prescribed decay rate while maximizing the admissible triggering threshold to reduce communication. Lyapunov and averaging analyses establish exponential stability of the event-triggered system, and Zeno-freeness is proved to guarantee implementability. Numerical results illustrate that diagonal gains cannot achieve the same triggering thresholds and decay rates as the full controller gain matrices, highlighting the benefits of exploiting Hessian coupling information.

Figures

Figures reproduced from arXiv: 2608.10246 by the authors.

Figure 1
Figure 1. Block diagram of the event-triggered gradient-based [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Cost function of the optimization problem (64) solved for fff f [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Simulation results of the event-triggered ES using a full () [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Reference graph

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