{"id":"fb2930de-eaf8-4d8b-9070-aef6881e181d","arxiv_id":"2502.00930","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Event-triggered Newton-based extremum seeking makes the convergence rate user-assignable and independent of the unknown Hessian for scalar static maps.","lead":"This paper combines two existing control ideas: Newton-based extremum seeking, which estimates the unknown curvature of a function, and event-triggered control, which updates the input only when a condition fails. The result is a controller whose convergence speed can be chosen by the user and that sends fewer, smaller control updates for scalar static maps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Averaging step is not justified: after the claimed time-scaling \\bar t = \\omega t, the vector field still contains O(omega) terms, so Plotnikov's theorem does not apply and the O(1/omega) closeness estimates underlying Theorem 1 are unsupported.","rationale":"Good-faith reading: the paper proposes a plausible and interesting combination of Riccati-filter-based Newton extremum seeking with static event triggering, and the average dynamics (42)-(44) are reasonable. The claimed Hessian-independent rate follows from the average Lyapunov analysis. The reader's conditional verdict is appropriate. My stress-test focused on the bridge between the discontinuous non-autonomous system (31) and the average system (32)-(33). This bridge is the most load-bearing step: without it, none of the exponential bounds (50)-(52) is established. The time-scaling in Section 4 is not merely a typo: if \\bar t = omega t, then cos(2omega t) must become cos(2\\bar t), not cos(2omega \\bar t). The paper's version makes the 'fast time' \\bar t itself depend on omega, making T=2pi/omega the averaging period in that already-fast time, which is inconsistent. More importantly, the right-hand side after the 1/omega scaling is not uniformly O(1) in omega; terms like -3a^2 omega H*/8 cos(3omega \\bar t) in (27) contribute to (1/omega)F an O(1) forcing that does not vanish as omega tends to infinity. A standard averaging theorem for \\dot x = epsilon f(t,x,epsilon) requires f to be bounded in epsilon; hence Plotnikov (1980) cannot be invoked as written. The O(1/omega) closeness of \\hat G(t) to \\hat G_av(t) claimed in (A.16) is especially suspect because \\hat G(t) = a sin(omega t) y(t) itself contains O(a) sinusoidal terms. Even if the final theta-bound (50) might survive with a different proof, the current proof does not establish it. I therefore see no reason to change the reader's CONDITIONAL verdict, but the requested revision should specifically require a correct averaging derivation or a direct perturbation argument that respects the O(omega) coefficients and the state-dependent event times.","tokens_in":13987,"tokens_out":13833,"duration_ms":136280,"concrete_test":"Take the original equations (18)-(19), substitute \\bar t = omega t correctly so that cos(2omega t) becomes cos(2\\bar t), cos(3omega t) becomes cos(3\\bar t), and sin(omega t) becomes sin(\\bar t), and rewrite the result as dX/d\\bar t = epsilon F(tau,X,epsilon) with epsilon = 1/omega. Compute sup_{tau in [0,2pi]} |F(tau,X,epsilon)| for representative states (e.g., the steady-state values used in Section 6) and for omega = 10, 10^2, 10^4. If this sup norm grows like omega rather than remaining O(1), the boundedness hypothesis of Plotnikov's theorem fails, so the averaging step and the O(1/omega) bounds in (A.16)-(A.18) are invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem rests on applying Plotnikov's averaging theorem to (31), written as dX/d\\bar t = (1/omega)F(\\bar t,X,1/omega) after the transformation \\bar t = omega t. The transformation is applied inconsistently: terms that are cos(2omega t), cos(3omega t), sin(omega t) in (18)-(19) become cos(2omega \\bar t), cos(3omega \\bar t), sin(omega \\bar t) in (27)-(28), whereas they should become cos(2\\bar t), cos(3\\bar t), sin(\\bar t). This alone breaks the claimed T=2pi/omega periodicity in \\bar t. More substantively, even if the arguments were corrected, the vector field F in (27)-(29) contains terms proportional to omega, e.g., -3a^2 omega H*/8 cos(3omega \\bar t) and a^2 omega H* sin(2omega \\bar t) \\tilde theta. Hence (1/omega)F has O(1) components, not O(1/omega). Plotnikov's theorem for discontinuous averaging requires the right-hand side to be of the form epsilon f(tau,x,epsilon) with f uniformly bounded in epsilon, where epsilon = 1/omega. Because f inherits O(1/epsilon) coefficients, the hypotheses are not verified. Consequently, the O(1/omega) approximation in (A.16)-(A.18), which converts the average-system exponential decay into the real-system bound (50), is unsupported. A related symptom is that the actual \\hat G(t) in (8) contains a term a Q* sin(omega t) of order O(a), so the event-trigger condition (20) sees O(a) oscillations rather than the averaged signal (47); the claimed Zeno-free inter-event times are computed from the average trigger and do not control the actual trigger when \\hat G crosses zero every half-period.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14336,"tokens_out":18746,"duration_ms":182551,"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":[{"comment":"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.","section":"Section 4, Eqs. (24)-(31) and (27)-(28)"},{"comment":"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.","section":"Appendix B, Eqs. (A.26)-(A.38)"},{"comment":"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.","section":"Appendix A, Eq. (A.2)"}],"minor_comments":[{"comment":"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.","section":"Section 6"},{"comment":"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.","section":"Equation (29)"},{"comment":"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.","section":"Appendix B, Eqs. (A.27)-(A.30)"},{"comment":"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.","section":"Appendix B, Eq. (A.38)"},{"comment":"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.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The paper is a combination of two established components, and the central claim is plausible. The main obstacles are technical: the time-scaling/averaging step and the Zeno analysis for the actual event-triggered system need substantial rework before the theorem can be trusted. I do not recommend rejection because the scheme may be salvageable, for example by introducing a filtered gradient estimate or by analyzing the actual inter-event times directly, but the current proof does not support the stated result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read Rodrigues et al. The core idea is sound and worth pursuing, but the proof as written has a load-bearing inconsistency in the averaging step. I'd send it to review but with a request for major revision.\n\nWhat's actually new: combining Ghaffari's Riccati-based Newton ES with the authors' earlier static event-triggered gradient scheme. The claimed benefit—convergence rate independent of the unknown Hessian, made user-assignable via K, a, sigma—is real if the stability proof goes through. The average system (42)-(44) and the Lyapunov argument in outline are plausible. The paper is honest about the scalar static-map local setting and doesn't oversell the simulations. The Zeno analysis, despite an algebra slip, has the right structure.\n\nThe soft spots, in order of severity. First, the time scaling in Section 4 is wrong. They define tau_bar = omega*t, then write F with cos(2*omega*tau_bar), sin(omega*tau_bar), cos(3*omega*tau_bar), and even O(omega) multiplicative terms. With that substitution, the right-hand side is not a T-periodic function in tau_bar with T=2pi/omega, and (1/omega)F contains O(1) components, so Plotnikov's averaging theorem does not apply. This isn't a cosmetic typo; the O(1/omega) closeness estimates in Appendix A rest on it. If the arguments were corrected to cos(2*tau_bar) etc., the O(omega) terms still need to be re-derived—I suspect they come from differentiating the demodulated gradient, and the averaging would need those terms to vanish or be treatable. As written, Theorem 1's proof doesn't go through.\n\nSecond, the trigger parameter beta is required in Appendix A to satisfy beta > |H*|, but H* is unknown and this condition never appears in the theorem statement or assumptions. That's a hidden verifiability problem. Third, the Zeno dwell-time calculation has an algebra error (the expression for tau* doesn't follow cleanly from the preceding bound), and the bound itself is derived from the averaged trigger, not the actual trigger, which sees O(a) oscillations that can fire events early. The simulation is one run; 43 vs 47 updates is not strong evidence, though clearly illustrative only.\n\nSo: the central argument is plausible but not yet rigorous. The flaws are correctable, but they need real work, not copyediting. I'd give this to a serious referee and ask for a rewritten proof of the averaging step, a fix of the beta condition, and a cleaner Zeno argument. The paper deserves that engagement.","headline":"Interesting combination of Newton ES and event-triggered control, but the averaging proof has a time-scaling error that undermines Theorem 1 as written.","tokens_in":14958,"tokens_out":2344,"would_cite":false,"duration_ms":22653,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C57","93D05","34C29","93C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["extremum seeking control","event-triggered control","Newton-based optimization","Riccati equation filter","averaging theory","discontinuous systems","scalar static maps","exponential stability"],"falsifier":"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.","tokens_in":13729,"feed_emoji":"🎯","tokens_out":11788,"duration_ms":105490,"temperature":0.7,"pith_summary":"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.","feed_headline":"Event-triggered Newton search converges at a user-set rate","feed_subtitle":"A Riccati inverse-Hessian filter makes the optimizer's speed independent of unknown map curvature.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduced the Riccati differential equation estimator for the inverse Hessian that makes continuous Newton-based extremum seeking converge at a rate independent of the unknown Hessian.","marker":"Ghaffari et al., 2012"},{"why":"Proposed the static event-triggered gradient-based extremum seeking scheme that serves as the baseline; its Hessian-dependent convergence rate motivates the Newton version.","marker":"Rodrigues et al., 2022"},{"why":"Provided the averaging theorem for discontinuous systems used to justify passing from the averaged system to the original event-triggered dynamics.","marker":"Plotnikov, 1980"},{"why":"Supplied the averaging and singular perturbation framework that legitimizes analyzing extremum seeking through its average system.","marker":"Krstić and Wang, 2000"},{"why":"Used for the comparison lemma, ISS arguments, and Lyapunov stability tools in the proof of Theorem 1.","marker":"Khalil, 2002"},{"why":"Justifies the local quadratic approximation of the unknown static map and the neglect of higher-order terms in the gradient estimate.","marker":"Ariyur and Krstić, 2003"},{"why":"Original event-triggered scheduling result that motivates updating control tasks only when a state-dependent condition fails.","marker":"Tabuada, 2007"}],"fun_headline_variants":["Event-triggered Newton: pick your own convergence speed","Newton ES drops Hessian dependence for user-set rate","Event-triggered Newton: speed that you choose, not curvature","User-assignable rate from event-triggered Newton ES","Event-triggered Newton: no Zeno, just your speed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Event-triggered Newton: pick your own convergence speed","Newton ES drops Hessian dependence for user-set rate","Event-triggered Newton: speed that you choose, not curvature","User-assignable rate from event-triggered Newton ES","Event-triggered Newton: no Zeno, just your speed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000246,"raw_usage":{"total_tokens":1544,"prompt_tokens":957,"completion_tokens":587,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":498}},"tokens_in":573,"tokens_out":587,"duration_ms":5743,"temperature":1.0,"reasoning_tokens":498,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T17:13:37.400909+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}