{"id":"d5989732-c436-45fb-91ea-b34093510e41","arxiv_id":"1908.07803","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that heterogeneous nonlinear multi-agent systems on strongly connected directed graphs can achieve asymptotic output synchronization with event-triggered, Zeno-free distributed controllers under a two-step design.","lead":"This paper designs event-triggered controllers that let a group of different nonlinear agents synchronize their outputs while agents exchange messages only at event times. It splits the task into a consensus step for linear reference models and an output regulation step for each nonlinear agent, and claims both Zeno-free behavior and no continuous communication.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem IV.1's Zeno-free claim is not established for the perturbed case: because µ jumps at consensus events, q(t) is discontinuous, so the equality-based rule (50) can be violated without a prior equality; the transfer from [32] is not automatic.","rationale":"The paper's two-step plan is coherent, and Theorem III.1 appears internally consistent: the consensus loop has an explicit positive lower bound b and a self-contained Lyapunov analysis. The fragile point is the regulation step. Theorem IV.1 is the only place where the nonlinear agents are connected to the reference models, and its proof leans on an unpublished companion manuscript [32] for the existence of κ, the ISS/IOS gains, and, explicitly, for the Zeno-free lower bound. Moreover, the claim that the Zeno proof in [32] transfers unchanged to the perturbed system is not merely a missing citation: the perturbation enters q(t) through µ(t), which is discontinuous at consensus events, so the equality-based triggering rule (50) is not well matched to the actual signal regularity. This can break the standing inequality (55) at jump instants. A reviewer cannot certify the strongest claim without either the missing derivation or a modified rule that triggers on violations and a Zeno proof that handles jumps. The numerical example is suggestive but is not a proof of (53). I therefore agree with the reader's CONDITIONAL verdict; my stress-test makes the underlying concern sharper by pointing at a specific mechanism, so I mark agreement as partial.","tokens_in":17848,"tokens_out":22588,"duration_ms":236430,"concrete_test":"Re-derive the regulation inter-event lower bound for (47) with \\bar{µ}(t) piecewise constant and jumps at consensus times (which are separated by at least b from Theorem III.1), using an event rule that triggers at the first violation ||ϖ(t)|| > σ(||q(t)||). Show that the lower bound is uniform in k. If the bound requires continuity of q or depends on the consensus jump schedule, then the omitted step in Theorem IV.1 is essential and the proof must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem IV.1 the regulation event rule is (50): t_{k+1} = inf{t > t_k | ||ϖ(t)|| = σ(||q(t)||)}. The proof then uses (55), ||ϖ(t)|| ≤ σ(||q(t)||), as if it always holds between events. In the coupled closed loop, q(t) = dκ(\\bar{x}(t))/dt depends on \\dot{\\bar{x}}, and the latter contains \\bar{µ}(t), which is a linear function of µ(t). The signal µ(t) is piecewise constant and jumps at the consensus event times of Theorem III.1, so q(t) is discontinuous at those instants. In contrast, ϖ(t), being the integral of -q, is continuous. Immediately after a downward jump of ||q(t)||, one can have ||ϖ(t)|| > σ(||q(t)||) without the equality in (50) ever being attained, so (55) can fail and the next event is not triggered. The paper says the remaining Zeno proof is referred to in [32] and that the additional \\bar{µ} is irrelevant; this is precisely the assertion that needs proof, because the Zeno lower-bound argument must handle discontinuous q and an event rule that should trigger on violations, not only on equality. The same companion paper [32] is also the source for the smooth κ, the ISS/IOS inequalities (49), and the local Lipschitz property of γ; none of these are derived for the perturbed case in the present preprint. Therefore the central claim of asymptotic synchronization with Zeno-free behavior is not fully supported within the preprint.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses output synchronization of heterogeneous nonlinear multi-agent systems under directed communication graphs using a two-step event-triggered design. In Problem 1, linear reference models are driven to consensus by an event-triggered controller with an explicit minimum inter-event interval; in Problem 2, each nonlinear agent solves an event-triggered perturbed output regulation problem using an ISS/IOS construction, and the two problems are combined in Theorem II.1 to yield asymptotic output synchronization with Zeno behavior excluded. The main technical contents are Theorem III.1 for consensus of the reference models and Theorem IV.1 for perturbed output regulation, the latter relying substantially on the authors' companion manuscript [32].","tokens_in":18185,"tokens_out":9650,"duration_ms":92509,"significance":"The two-step decomposition in Theorem II.1 is clean, and Problem 1 is a meaningful contribution: it gives an explicit event-triggering rule for general linear reference models on a directed graph with a positive lower bound on inter-event times, extending earlier undirected-graph results. The numerical example demonstrates the design. If the gaps identified below are repaired, the paper would provide a useful framework for event-triggered output synchronization of heterogeneous nonlinear multi-agents without continuous neighbor monitoring and with a milder actuator-disturbance ISS condition. As it stands, however, the central nonlinear claim is not self-contained and the Zeno-free proof for the perturbed case has a concrete technical gap.","major_comments":[{"comment":"The triggering interval τ_ik in (20) is defined through the future signal w_i(τ) = ‖BB^T P ∑_{j∈N_i} a_ij g_j p_j^c(τ)‖ over the future interval τ ∈ [t_ik^c, t_ik^c + t]. Since p_j^c(τ) is the piecewise-constant broadcast value of neighbor j and can change at neighbor triggering instants inside that interval, agent i cannot compute τ_ik at time t_ik^c from information available to it. Thus the rule (24) is not a causal event-triggering law, and the claim in Problem 1 that continuous monitoring is avoided is not supported. The authors need to replace τ_ik by a computable self-triggered bound using only currently available broadcast data, or add and prove an assumption on future neighbor behavior.","section":"Lemma III.2 / Eq. (20)"},{"comment":"The proof of Lemma III.2 contains a circular step. From the differential inequality d/dt‖ϵ_i(t)‖ ≤ ‖A‖‖ϵ_i(t)‖ + w_ik + w_i(t), the paper concludes ‖ϵ_i(t)‖ ≤ s_ik by comparing with the integral in (20). However, (20) uses the constant s_ik inside the integral, whereas the differential inequality contains ‖ϵ_i(τ)‖. The step therefore presupposes the bound ‖ϵ_i(τ)‖ ≤ s_ik on the whole interval, which is exactly what is being proved. A bootstrap or comparison argument is needed to establish the bound; without it, the key inequality (21) and the subsequent lower bound (35) are not rigorously justified.","section":"Lemma III.2"},{"comment":"The core of Problem 2 is delegated to the authors' unpublished submitted manuscript [32]. The existence of the smooth function κ, the ISS/IOS estimates (49), the local Lipschitz property of γ, and the Zeno-exclusion proof are all asserted by reference to [32], with only a brief statement that κ can be constructed 'following the recursive technique given in [1]'. Since Theorem IV.1 is the load-bearing component of the output-synchronization claim, the paper is not self-contained. The authors should provide complete proofs of (49), of the local Lipschitz property, and of the Zeno-free property for the perturbed system, or replace [32] by a published reference whose stated results can be verified.","section":"Theorem IV.1"},{"comment":"The claim that the event rule (50) guarantees ‖ϖ(t)‖ ≤ σ(‖q(t)‖) for all t between events is not valid in the present closed loop. In (47), q(t) = dκ(\\bar{x}(t))/dt inherits the argument \\bar{µ}, which contains the consensus input µ(t) from Problem 1; µ is piecewise constant and jumps at the consensus event times of Theorem III.1, so q(t) can jump downward at those instants. The signal ϖ(t), being the integral of −q, is continuous. Immediately after such a downward jump of σ(‖q(t)‖), it is possible that ‖ϖ(t)‖ > σ(‖q(t)‖) without the equality in (50) ever being attained, so (55) can fail and the next regulation event need not occur. The statement in the proof that the Zeno analysis is 'irrelevant to the additional \\bar{µ}' is therefore incorrect. The event rule and the Zeno/IOS analysis must be modified to handle discontinuities in q, for example by triggering on ‖ϖ(t)‖ ≥ σ(‖q(t)‖) and proving that this still excludes Zeno.","section":"Eq. (50) / Eq. (55) / Zeno proof"}],"minor_comments":[{"comment":"There are duplicated words in Section V: 'the the event-triggered perturbed output regulation problem' and 'the the IOS gain function' should be corrected.","section":"Section V"},{"comment":"Reference [32] is listed only as 'Submitted, 2018'. If it remains essential to the proofs, a preprint number or a published version should be provided; otherwise the manuscript should be made self-contained.","section":"Reference [32]"},{"comment":"The lower bound b in (25) depends on b1 and b2, which involve the largest eigenvalue λ_LG of LG and the norm ‖BB^T P‖; the statement in Remark II.1 that the two designs are 'completely distributed' would benefit from a clarification of which quantities are global and how they are obtained in a distributed implementation.","section":"Theorem III.1"},{"comment":"In Problem 2, the notation ‖e_i[t,∞)‖ and ‖µ_i[t,∞)‖ is used before being defined; a brief definition of the restricted sup-norm would improve readability.","section":"Eq. (11)"}],"recommendation":"major_revision","confidential_remarks":"The paper relies very heavily on the authors' own unpublished submission [32] for the nonlinear ISS/IOS construction and Zeno-free proof, and the perturbed version has an additional discontinuity issue in q(t) that the current text does not address. I would recommend asking the authors to make the proofs self-contained or to supply the published version of [32], and to explicitly handle the combined event-triggered dynamics before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Khan-Chen-Yan preprint on event-triggered output synchronization. The headline: the paper has a real extension—fixed-timer event-triggered consensus for general linear reference models under strongly connected directed graphs—and a plausible two-step design for heterogeneous nonlinear agents. The consensus theorem is mostly self-contained and the numerical example is consistent. But the output-regulation half leans on an unpublished companion paper [32], and the transfer to the perturbed networked case has a discontinuity gap that is not cosmetic. I would not take the Zeno-free claim as established from this preprint.\n\nWhat is new: extending the undirected fixed-timer consensus machinery to directed topology with an explicit triggering formula is genuinely useful for the event-triggered MAS subfield. The Lyapunov argument for Problem 1 is mostly coherent, and the lower bound b in (25) does give a positive inter-event interval if the triggering definition is implementable. The paper is also honest about what [32] supplies.\n\nWhere it wobbles. First, τ_ik in Lemma III.2 is defined through an integral of w_i(τ), which depends on neighbors' future event times. At time t_ik, agent i does not know when neighbor j will trigger; so the mechanism (24) is not causal as written. This is a fixable but nontrivial omission: either a true self-triggering computation using only current broadcast values, or a re-evaluation protocol, needs to be stated before the consensus claim is fully supported. The proof of Lemma III.2 also skips a short bootstrap to justify replacing ||ε(τ)|| by s_ik inside the integral; that one is minor.\n\nSecond, and more serious: Theorem IV.1 imports the ISS/IOS construction and the Zeno exclusion from [32], and says the additional perturbation µ is irrelevant. The stress-test note lands here. The event rule (50) is equality-based, and in the coupled closed loop q(t) contains dκ(\\bar{x})/dt, which inherits jumps from µ(t) at consensus events. q can drop discontinuously, so ||ω(t)|| can exceed σ(||q(t)||) without the equality being attained. Then (55) can fail and the small-gain argument in the proof collapses. For the theorem as stated, this needs a proof or a changed rule (e.g., trigger on the inequality). The authors do not provide either for the perturbed case.\n\nThe citation pattern is fair; self-citing [32] is not itself a problem, but here it is load-bearing. If [32] is solid, the consensus-plus-regulation idea might be salvaged with a modified event rule. As a preprint, the central claim is not fully supported.\n\nMy take: this is worth a serious referee, because the problems are addressable and the directed-topology consensus step is a real contribution. But I would not cite the result as it stands, and I would tell the authors to fix the causal definition of the consensus trigger and prove the event rule for discontinuous q. A referee's report should be conditional: major revision, then re-review.","headline":"Real extension and plausible two-step design, but the Zeno-free claim for the perturbed regulation half rests on an unproven transfer from an unpublished companion paper.","tokens_in":18704,"tokens_out":4798,"would_cite":false,"duration_ms":97173,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Under Assumptions 1–6, the paper's two-step design achieves asymptotic output synchronization with Zeno-free event triggering and no continuous monitoring.","keywords":["event-triggered control","multi-agent systems","output synchronization","output regulation","nonlinear systems","input-to-state stability","Zeno behavior","directed graphs"],"falsifier":"Construct or simulate a single nonlinear agent satisfying Assumptions 3–6 whose event rule (50) is driven by a bounded but persistent $\\mu$ (for example, a small sinusoid), and measure the inter-event intervals. If the intervals have no positive lower bound, or if the output error fails to converge to zero with $\\mu$ present, the central claim fails; the numerical example in Section V is the natural place to run this test.","tokens_in":17646,"feed_emoji":"🔄","tokens_out":7260,"duration_ms":256571,"temperature":0.7,"pith_summary":"This paper claims that a group of heterogeneous nonlinear agents, subject to parameter uncertainty and communicating over a directed graph, can achieve asymptotic output synchronization using controllers that act only at discrete event times. The proof divides the task into two concurrent subproblems: event-triggered consensus of linear reference models, and event-triggered perturbed output regulation for each nonlinear agent. For the consensus step, the authors give an explicit controller and triggering rule with a positive lower bound on inter-event intervals, so Zeno behavior is impossible. For the regulation step, they use an input-to-state stability property from actuator disturbance to state to absorb the network influence as a perturbation, and a triggering rule based on the rate of the control signal that drives the output error to zero asymptotically. This matters because earlier event-triggered nonlinear regulation designs could only guarantee bounded steady-state tracking error, not exact asymptotic regulation.","feed_headline":"Nonlinear agents synchronize with event triggers, no Zeno behavior","feed_subtitle":"Two-step design gives exact output agreement on directed networks while agents communicate only at event times.","key_machinery":"The carrying object is the two-step composition of an event-triggered consensus law for the reference models and an event-triggered perturbed output regulation law for each agent. In the consensus step, the central identity is the transformed error $p=-(L\\otimes I_q)v$ with Lyapunov function $V(p)=\\tfrac12 p^T(GR\\otimes P)p$; Lemma III.1 supplies a weighted Laplacian estimate that handles the asymmetry of a directed graph through the left eigenvector $r$, and the controller gains come from the Riccati equation (22). The event rule (24) compares each agent's hold-error $\\epsilon_i$ with the local consensus error $p_i$, and the fixed timer $b$ in (25) supplies the minimum inter-event interval. In the regulation step, the central object is the perturbed closed loop $\\dot\\xi=f_c(\\xi,\\nu)+E(w)\\varpi+\\bar\\mu$ with dynamic actuator and sensor compensators, and the triggering rule (50) compares the holding error $\\varpi$ with the rate signal $q=d\\kappa(\\bar x)/dt$; the IOS gain $\\gamma$ is used to choose $\\sigma$ so that $\\gamma(\\sigma(s))=cs$ with $c<1$, which is the contraction that turns a practical bound into an asymptotic one. Together these two mechanisms produce the synchronization result with no continuous monitoring.","core_discovery":"The central claim is that output synchronization in the sense of $\\lim_{t\\to\\infty}\\|y_i(t)-y_\\infty(t)\\|=0$ holds for all agents when each agent runs two independent event-triggered mechanisms. In the first step, each agent maintains a linear reference model $\\dot v_i=Av_i+B\\mu_i$; the sampled controller $\\mu_i(t)=g_iKp_i(t^c_{ik})$ with $K=B^TP$ drives the reference models to consensus, and the triggering instants satisfy $t^c_{i,k+1}\\ge t^c_{ik}+b$ for an explicit positive constant $b$, under a strongly connected directed graph and a Riccati equation $PA+A^TP-\\lambda PBB^TP+\\beta I=0$. In the second step, each agent uses an event-triggered output regulation controller $\\bar u_i(t)=\\kappa_i(\\bar x_i(t^r_{ik}))$ built from dynamic actuator and sensor compensators; the closed loop is input-to-state stable from the holding error and from the reference-model input $\\mu_i$, and the triggering rule (50) uses the IOS gain to enforce $\\gamma(\\sigma(s))=cs$ with $c<1$, which makes the asymptotic error satisfy $\\lim_{t\\to\\infty}\\|e_i[t,\\infty)\\|\\le\\hat\\gamma_i(\\lim_{t\\to\\infty}\\|\\mu_i[t,\\infty)\\|)$. Since the first step drives $\\mu_i$ to zero, the second step drives $e_i=y_i-c(v_i)$ to zero, and the composition yields output synchronization. The proof also provides the explicit event-time lower bound (35) for the consensus step and refers to the companion paper for the Zeno-free proof of the regulation step.","pith_inferences":["The paper leaves open whether the two event clocks can be implemented on separate processors with different triggering laws; because each agent's design is decoupled, a natural test is to run the consensus step with a fixed timer and the regulation step with the event rule (50), or vice versa.","The lower bound $b$ depends on global quantities such as the largest eigenvalue of $LG$ and the second-smallest eigenvalue of $\\hat L$; a distributed extension would estimate these quantities locally, and the proof suggests the bound would degrade gracefully but this is not shown.","If the companion paper's Zeno-exclusion proof does not survive the extra perturbation $\\bar\\mu$, the asymptotic claim would collapse to practical synchronization with a residual error; injecting a small persistent sinusoidal $\\mu$ into the Section V example would test exactly that."],"forward_implications":["Each agent's two triggering sequences are independent, so the consensus events and regulation events can be scheduled separately without a shared clock.","The minimum inter-event interval in the consensus step is at least the explicit constant $b$ in (25), so infinite triggering in finite time cannot occur.","The output error converges to zero asymptotically, not just to a prescribed ball, even though the agents are nonlinear and uncertain and the network is directed.","The design weakens the usual ISS requirement from sensor-to-state to actuator-to-state, which is easier to satisfy for nonlinear systems.","The method extends the fixed-timer event-triggered/sampled-data consensus approach from undirected to directed strongly connected networks."],"supporting_citations":[{"why":"Supplies the event-triggered stabilization and output regulation technique, including the ISS/IOS construction and the Zeno-exclusion proof that Theorem IV.1 extends to the perturbed networked case.","marker":"[32]"},{"why":"Introduces the robust perturbed output regulation framework that the paper's two-step architecture follows.","marker":"[1]"},{"why":"Provides the general framework for robust output synchronization of heterogeneous nonlinear networked systems that motivates splitting the problem into reference consensus and output regulation.","marker":"[2]"},{"why":"Gives Lemma III.1, the weighted Laplacian inequality used to handle the asymmetry of the directed graph in the consensus step.","marker":"[34]"},{"why":"Guarantees existence of the Riccati equation solution $P$ under Assumption 1.","marker":"[35]"},{"why":"Provides the standard assumptions and internal-model structure for nonlinear output regulation used in Problem 2.","marker":"[36]"}],"fun_headline_variants":["Event-triggered control syncs nonlinear agents, no Zeno","Two-step event-triggered design yields output sync, avoids Zeno","Nonlinear multi-agents sync outputs with event triggers","Heterogeneous agents output-sync via event-triggered control","Event-triggered output sync: nonlinear agents, no Zeno"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire regulation step assumes that the companion paper [32] has already produced a smooth event-triggered controller making the unperturbed closed loop ISS/IOS with locally Lipschitz gain, and that its Zeno-exclusion proof remains valid when the perturbation $\\bar\\mu$ is present; the present paper defers that proof and says the details are ignored.","fun_headline_variants_meta":{"raw":{"variants":["Event-triggered control syncs nonlinear agents, no Zeno","Two-step event-triggered design yields output sync, avoids Zeno","Nonlinear multi-agents sync outputs with event triggers","Heterogeneous agents output-sync via event-triggered control","Event-triggered output sync: nonlinear agents, no Zeno"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000267,"raw_usage":{"total_tokens":1662,"prompt_tokens":1041,"completion_tokens":621,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":657,"completion_tokens_details":{"reasoning_tokens":528}},"tokens_in":657,"tokens_out":621,"duration_ms":6817,"temperature":1.0,"reasoning_tokens":528,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:57:26.173867+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct or simulate a single nonlinear agent satisfying Assumptions 3–6 whose event rule (50) is driven by a bounded but persistent $\\mu$ (for example, a small sinusoid), and measure the inter-event intervals. If the intervals have no positive lower bound, or if the output error fails to converge to zero with $\\mu$ present, the central claim fails; the numerical example in Section V is the natural place to run this test.","supporting_citations":[{"cited_title":"A new approach for event- triggered stabilization and output regulation of nonlinear systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the event-triggered stabilization and output regulation technique, including the ISS/IOS construction and the Zeno-exclusion proof that Theorem IV.1 extends to the perturbed networked case."},{"cited_title":"Robust perturbed output regulation and synchro- nization of nonlinear heterogeneous multiagents,","cited_arxiv_id":null,"evidence_quote":"Introduces the robust perturbed output regulation framework that the paper's two-step architecture follows."},{"cited_title":"Consensus of second-order heterogeneous multi-agent systems under a directed graph,","cited_arxiv_id":null,"evidence_quote":"Gives Lemma III.1, the weighted Laplacian inequality used to handle the asymmetry of the directed graph in the consensus step."},{"cited_title":"A contribution to matrix quadratic equations,","cited_arxiv_id":null,"evidence_quote":"Guarantees existence of the Riccati equation solution $P$ under Assumption 1."},{"cited_title":"Chen and J","cited_arxiv_id":null,"evidence_quote":"Provides the standard assumptions and internal-model structure for nonlinear output regulation used in Problem 2."}],"review_version":1}