{"id":"69f76590-a7e4-4dd6-bc86-0f1f3488c671","arxiv_id":"2608.10246","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A convex co-design framework jointly synthesizes the controller gain and event-triggering matrices for multivariable extremum seeking, guaranteeing exponential convergence and Zeno-free operation with full-matrix gains outperforming diagonal ones.","lead":"An event-triggered extremum seeking controller is co-designed with its triggering rule through a convex optimization problem, allowing full-matrix gains that exploit Hessian coupling. A generalist might read this because it shows how prior curvature information can reduce communication in model-free real-time optimization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's averaging bridge is unproven: state-dependent event times do not fit the cited differential-inclusion averaging theorem, and the claim that the impulse average vanishes is not justified.","rationale":"The paper's strongest claim is the exponential convergence bound (42) for the actual event-triggered ES system. The LMI construction in Lemma 1 is internally consistent, but it certifies only the average event-triggered system (30)-(31) with (34). The step from average to actual is made in Theorem 1's proof by invoking the averaging theorem for differential inclusions [16] and by asserting that the impulse average vanishes. This is the softest point because the system (26)-(27) has resets at state-dependent event times, the average system is itself a hybrid system, and the impulse contribution is nonzero with I_k=[0,0,-e^⊤]. The reader's weakest_assumption identifies exactly this gap, and my reading agrees. The concern is not that the co-design idea is wrong; the numerical example supports its qualitative advantage. Rather, the proof as written does not establish that an LMI solution guarantees the bound (42) for the implemented sampled-data controller. This is an internal rigor gap, not a disagreement with consensus. Because the gap is addressable by a careful hybrid averaging theorem with verified hypotheses, or by deriving the averaging error bound directly, the existing CONDITIONAL verdict is appropriate; my read does not change it.","tokens_in":10792,"tokens_out":15776,"duration_ms":159669,"concrete_test":"For the scalar case n=1 with H*=1, K=-1, deduce the event-time map t_k(G,e) defined by (23)-(24) and check whether it meets the hypotheses of the averaging theorem in [16] for the augmented inclusion (26)-(27). In particular, verify (i) the time-to-trigger function is continuous in the state, and (ii) the number of events per excitation period is O(1/ω) uniformly in a neighborhood of the origin. If either condition fails, or if the impulse-average term in (29) turns out not to be O(1/ω) uniformly, then the bound (46) in Theorem 1 is not justified and the averaging transfer to (42) must be replaced by a rigorous hybrid averaging argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main result, the exponential bound (42) for the implemented event-triggered system, depends on transferring stability from the average system (30)-(31) to the actual hybrid system (26)-(27). That transfer is made by citing Plotnikov's averaging theorem for differential inclusions [16] and by the assertion after (29) that the impulse average 'does not contribute' because the impulse map resets the error to zero. Both steps are unsupported. The system (26)-(27) is not an ordinary differential inclusion: the reset times t_k(y) are state-dependent, defined implicitly by the trigger condition in (23)-(24). The cited theorem, as used, concerns inclusions of the form ẋ ∈ F(t,x) with a prescribed time average; it is not shown to cover resets at state-dependent event times. Moreover, the average system (30)-(31)+(34) is itself an event-triggered hybrid system, not an autonomous ODE, so comparing it to the actual system requires a hybrid averaging theorem with verified dwell-time and regularity conditions, which is absent. The impulse average is not zero: the reset jump is -e(t_k^-), since I_k=[0,0,-e^⊤] in (27), and e_av in (32) is not small. The sentence 'over a period, the average error tends to zero' would require a uniform bound on the measure or frequency of event times; no such bound is established in the proof. Without the averaging bridge, Lemma 1 only certifies the average event-triggered system, not the sampled-data controller. Thus the central claim (42) is, as written, unproven.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11110,"tokens_out":10108,"duration_ms":99724,"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":[{"comment":"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.","section":"Section II.D, Eqs. (26)-(29) and Theorem 1"},{"comment":"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).","section":"Section II.D, Eq. (29) and the paragraph after it"},{"comment":"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.","section":"Section II.D, Eq. (27)"},{"comment":"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.","section":"Theorem 1 proof and Proposition 1"}],"minor_comments":[{"comment":"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.","section":"Section II.D, Eq. (29)"},{"comment":"There is a small typographical error near the end of the proof: 'with where kappa = ...' should read 'where kappa = ...'.","section":"Lemma 1 proof"},{"comment":"In (54) the expression '|y(t) - Q*| = <=' contains a typo; it should be '|y(t) - Q*| <='.","section":"Theorem 1, Eq. (54)"},{"comment":"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.","section":"Theorem 1 statement"},{"comment":"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.","section":"Section IV, simulation results"}],"recommendation":"major_revision","confidential_remarks":"The main concern is whether the authors can replace the unsupported averaging bridge with a rigorous hybrid averaging theorem or an alternative direct proof. If they cannot, the central claim (42) is not established. The LMI co-design result itself appears sound, and the numerical study is informative, so the paper could become acceptable after a substantial revision that supplies the missing analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this is the first joint synthesis of the controller gain and trigger matrices for event-triggered extremum seeking under polytopic Hessian uncertainty, including full non-diagonal gains, and the LMI in Lemma 1 is correctly derived. The trace-minimization surrogate for reducing communication is a nice practical touch. If the average-system result is all you need, the paper is in good shape.\n\nThe soft spots are where the average system meets the actual event-triggered system. The averaging step in Theorem 1 invokes Plotnikov's theorem for differential inclusions, but the closed loop is a hybrid system with state-dependent event times and resets. The theorem is not shown to cover that case. The sentence that the impulse average does not contribute is not justified: the reset jumps are of order 1/omega, but the number of jumps per period is not uniformly bounded, and the statement that the average error tends to zero needs a measure bound that is not there. So the exponential bound (42) for the actual implemented system is unproven as written. The Zeno-freeness Proposition 1 also has an algebraic mismatch: solving the differential inequality (59) gives an inter-event time in scaled time of order 1, which means order 1/omega in original time, not the O(1) bound in (56). This is fixable, but it should be corrected. Also, the claim that diagonal gains cannot achieve the same thresholds is based on one numerical example, fine as an illustration but too strong as stated.\n\nI would not desk-reject this. The core co-design idea is sound and the LMI is usable as is for the average system. A serious referee could require a proper hybrid averaging argument (or a re-stated theorem that only claims the average system) and a corrected Zeno bound. That is a substantial revision but not a change of direction.","headline":"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.","tokens_in":11665,"tokens_out":5352,"would_cite":true,"duration_ms":47127,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C65","93B52","93D30","90C22","93C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["extremum seeking","event-triggered control","co-design","linear matrix inequality","polytopic uncertainty","averaging theory","Zeno-freeness","multivariable optimization"],"falsifier":"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.","tokens_in":10577,"feed_emoji":"📉","tokens_out":11338,"duration_ms":99946,"temperature":0.7,"pith_summary":"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.","feed_headline":"Co-design of trigger and gain cuts communication in extremum seeking","feed_subtitle":"A full-matrix gain plus a jointly tuned trigger achieves the same decay rate with fewer updates than diagonal designs.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"provides the local quadratic-map approximation in which the quadratic gradient term is neglected for the local analysis.","marker":"[2]"},{"why":"supplies the multivariable probing and demodulation signal structure and the nominal Hessian used in the simulation.","marker":"[6]"},{"why":"gives the comparison-function argument that yields the positive minimum inter-event time bound in Proposition 1.","marker":"[7]"},{"why":"supplies the comparison lemma and Rayleigh-Ritz inequalities used to turn the Lyapunov inequality into exponential decay.","marker":"[9]"},{"why":"provides the averaging theorem invoked to transfer stability from the average system to the original system.","marker":"[10]"},{"why":"provides the averaging method for differential inclusions that connects the discontinuous event-triggered system to the average system.","marker":"[16]"},{"why":"supplies the event-triggered extremum-seeking baseline that the co-design is contrasted with.","marker":"[17]"},{"why":"supplies the event-triggered Newton extremum-seeking design and the parameter choices used for comparison and simulation.","marker":"[18]"},{"why":"inspires the trace-minimization form of the co-design optimization that maximizes inter-event time.","marker":"[4]"}],"fun_headline_variants":["Co-design cuts communication in extremum seeking","Full-matrix gains enable leaner event-triggered extremum seeking","Co-designed gain and trigger boost communication efficiency","Joint gain-trigger design trims communication in extremum seeking"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Co-design cuts communication in extremum seeking","Full-matrix gains enable leaner event-triggered extremum seeking","Co-designed gain and trigger boost communication efficiency","Joint gain-trigger design trims communication in extremum seeking"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001185,"raw_usage":{"total_tokens":4858,"prompt_tokens":873,"completion_tokens":3985,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":3918}},"tokens_in":489,"tokens_out":3985,"duration_ms":26453,"temperature":1.0,"reasoning_tokens":3918,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:12:25.514282+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the local quadratic-map approximation in which the quadratic gradient term is neglected for the local analysis."},{"cited_title":"Multivariable Newton- based extremum seeking,","cited_arxiv_id":null,"evidence_quote":"supplies the multivariable probing and demodulation signal structure and the nominal Hessian used in the simulation."},{"cited_title":"Dynamic triggering mechanism for event-triggered control,","cited_arxiv_id":null,"evidence_quote":"gives the comparison-function argument that yields the positive minimum inter-event time bound in Proposition 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the comparison lemma and Rayleigh-Ritz inequalities used to turn the Lyapunov inequality into exponential decay."},{"cited_title":"Overview of V.A. Plotnikov’s research on averaging of differential inclu- sions,","cited_arxiv_id":null,"evidence_quote":"provides the averaging theorem invoked to transfer stability from the average system to the original system."},{"cited_title":"A veraging of differential inclusions,","cited_arxiv_id":null,"evidence_quote":"provides the averaging method for differential inclusions that connects the discontinuous event-triggered system to the average system."},{"cited_title":"Event-triggered and periodic event-triggered extremum seeking control,","cited_arxiv_id":null,"evidence_quote":"supplies the event-triggered extremum-seeking baseline that the co-design is contrasted with."},{"cited_title":"Event-triggered Newton extremum seeking for multivariable optimization,","cited_arxiv_id":null,"evidence_quote":"supplies the event-triggered Newton extremum-seeking design and the parameter choices used for comparison and simulation."},{"cited_title":"Codesign of dynamic event-triggered gain-scheduling control for a class of nonlinear systems,","cited_arxiv_id":null,"evidence_quote":"inspires the trace-minimization form of the co-design optimization that maximizes inter-event time."}],"review_version":1}