{"id":"a772b0b1-a8b5-433f-8d0b-ba2502879942","arxiv_id":"1908.01654","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The paper presents sufficient dissipativity-based conditions under which two-dimensional feedback systems remain L2 stable after sampling, quantization, and event-triggered network communication.","lead":"Two-dimensional systems, where variables depend on two coordinates such as space and time, are analyzed for stability when their inputs and outputs travel over a digital network. The paper gives mathematical conditions for sampling, quantization, and event-triggered transmission to preserve stability, illustrated on a heat exchanger model.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's sampled discrete model is not the exact sampled model of the continuous 2-D Roesser system, so the LMI-based dissipativity certificate may not apply to the actual sampled plant.","rationale":"The reader identified the bounded spatial domain as the weakest assumption, but there is a more fundamental correctness issue: the sampling theorem that feeds the example appears to be for a different model. Theorems 3 and 4 are conditional statements that may be correct; however, the paper's overall claim that sampling, quantization, and event-triggered communication preserve closed-loop L2 stability relies on Theorem 1 to connect the continuous 2-D plant to the discrete IF-OFP plant. Because the discrete recurrence (8) is demonstrably not the exact sample of the continuous Roesser model, the computed passivity levels in Section IV and the consequent quantized/event-triggered stability conclusion are not connected to the actual continuous plant. This does not invalidate the abstract IF-OFP framework, but it requires either replacing Theorem 1 with an exact discretization or explicitly bounding the mismatch between the model used and the true sampled system. The reader's CONDITIONAL verdict therefore remains appropriate, though the stated reason should shift from scope limitations to this internal correctness gap.","tokens_in":23471,"tokens_out":22020,"duration_ms":220885,"concrete_test":"Compute the exact solution of (2) for A11=A22=0, A12=A21=1, B=0, C=[1 0], with boundary data xh(0,z2)=e^{z2}, xv(z1,0)=e^{z1}; the exact grid values are xh(ih1,jh2)=xv(ih1,jh2)=e^{ih1+jh2}. Evaluate the recurrence in (8) from (0,0): it predicts xh(h1,0)=1+h1 and xv(0,h2)=1+h2, while exact values are e^{h1}, e^{h2}. If these differ, Theorem 1's discrete model is not the sampled continuous system, and the LMI certificate (7) does not apply to the true plant. A second check: re-run the Section IV example using an exact sampled-data model (or fine-grid numerical solution of the PDE) and compare the IF-OFP levels and closed-loop response.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step for the sampling part of the central claim is Theorem 1, whose proof asserts that the exact sampled dynamics of the continuous Roesser model (2) are given by the discrete recurrence with Ad, Bd in (8). The block-diagonal exponentials e^{A11 h1}, e^{A22 h2} with cross terms (e^{A11 h1}-I)A11^{-1}A12 and (e^{A22 h2}-I)A22^{-1}A21 treat the other state as constant during one step. This is not exact. Counterexample: take A11=A22=0, A12=A21=1, B=0, C=[1 0]. The exact solution from boundary xh(0,z2)=e^{z2}, xv(z1,0)=e^{z1} is xh=xv=e^{z1+z2}. At (0,0), the claimed recurrence gives xh(h1,0)=xh(0,0)+h1 xv(0,0)=1+h1, whereas the exact value is e^{h1}. These differ for h1>0. Hence the LMI (7) certifies QSR-dissipativity of a different discrete system, not of the sampled continuous plant. Since the example in Section IV computes the plant IF-OFP levels (rho_p=-1.317) from exactly this model, the subsequent application of Theorems 3 and 4 may not guarantee stability of the true sampled feedback loop. The abstract QSR/event-triggered framework may survive if IF-OFP levels are independently available, but the paper's claim of a verifiable sampled-system certificate is not established.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a dissipativity-based framework for analyzing L2 stability of two-dimensional (2-D) feedback systems interconnected over a digital network. It introduces QSR-dissipativity definitions for continuous and discrete 2-D Roesser systems, then treats three network-induced effects sequentially: sampling with zero-order hold, logarithmic quantization, and event-triggered transmission. For linear sampled 2-D systems it provides an LMI condition (Theorem 1); for nonlinear systems it gives a sampling-error bound and a dissipativity degradation condition (Lemma 3 and Theorem 2). It then analyzes how logarithmic quantization degrades IF-OFP passivity levels (Theorem 3) and proposes an event-triggering rule that preserves a finite L2-gain on bounded spatial domains (Theorem 4). A hyperbolic PDE example is used to illustrate the conditions and the closed-loop behavior.","tokens_in":23794,"tokens_out":13932,"duration_ms":135854,"significance":"The topic is timely: networked control of 2-D systems is relatively unexplored, and the modular decomposition into sampling, quantization, and event-triggering is a sensible program. The quantization analysis in Theorem 3 and the event-triggering scheme, once its typographical inconsistencies are fixed, provide explicit and verifiable sufficient conditions, and the running PDE example is instructive. I credit the authors for stating the bounded-domain assumption explicitly in Assumption 2 and for making the triggering rule implementable as an online check of condition (22). However, the two sampling results at the foundation of the paper are not established as stated: Theorem 1's discrete model is an approximation rather than the exact sampled model, and the proof of Theorem 2 uses an inequality that fails for negative-definite Qp_hat. Because the example's passivity levels are computed from Theorem 1, the numerical validation does not currently back the advertised guarantees. The contribution can be salvaged, but the manuscript needs substantive correction.","major_comments":[{"comment":"The discrete model in (8) is not the exact sampled model of the continuous Roesser system (2). In the singular case covered by Remark 1, take A11=A22=0, A12=A21=1, B=0. With boundary data xh(0,z2)=e^{z2} and xv(z1,0)=e^{z1}, the exact solution is xh(z1,z2)=xv(z1,z2)=e^{z1+z2}, so xh(h1,0)=e^{h1}; the recurrence (8) gives xh(h1,0)=1+h1. These differ for h1>0. Therefore the LMI (7) certifies QSR-dissipativity of a different, approximate discrete system rather than of the actual sampled plant, and the IF-OFP levels computed in Section IV from this model do not by themselves certify stability of the true sampled feedback loop. This point needs to be fixed by deriving an exact sampled model, or by explicitly treating the discretization as an approximation with a separate error analysis.","section":"Section III-A, Theorem 1; Appendix D, Eq. (8)"},{"comment":"The proof's inequality Δy_p^T Qp_hat Δy_p ≥ |λ_min(Qp_hat)| |Δy_p|^2 is valid only when Qp_hat is positive semidefinite. For example, if Qp_hat = -I, the left side is -|Δy_p|^2, which is not bounded below by |λ_min(Qp_hat)| |Δy_p|^2 = |Δy_p|^2. Since Theorem 2 does not require Qp_hat ⪰ 0 and condition (13) uses the absolute value, the theorem as stated is not proved. In addition, the second inequality in (13) writes 2(α1 h1 + α2 h2), while the proof in Appendix C uses Lemma 3's 2(α1^2 h1^2 + α2^2 h2^2); the statement and proof need to be reconciled, and the correct form should follow from Lemma 3.","section":"Section III-A, Theorem 2 and Appendix C"},{"comment":"The displayed definition of q2 in Theorem 4 is not the same as the q2 used in the proof and in Eq. (32): the theorem statement has |ν_c| and δ_p^2 where the proof has |ν_p| and δ_c^2. This is more than a typo because the threshold in (22) depends on q1 and q2 through ϵ^2; a reader implementing the printed condition obtains a different event-triggering rule. The theorem's conclusion should also be stated as finite L2-gain on a bounded spatial domain (Definition 4), which is weaker than the global L2 stability mentioned in the abstract.","section":"Section III-C, Theorem 4 statement and proof"}],"minor_comments":[{"comment":"The abstract and introduction state global L2 stability, but Theorem 4 only gives finite L2-gain on a bounded spatial domain (Definition 4); please add an explicit scope statement where the main results are summarized.","section":"Abstract and Section I"},{"comment":"The example writes D=∅; it should be D=0 or the corresponding zero matrix of appropriate dimensions, since the output equation y = C x + D u is otherwise underspecified.","section":"Section IV, Eq. (26)"},{"comment":"Theorem 1 assumes A11 and A22 are nonsingular 'for simplicity', but Remark 1 already extends the formulas to the singular case via series; please state in the theorem whether the nonsingularity assumption is part of the hypothesis or whether Remark 1 formally extends the statement.","section":"Section III-A, Theorem 1"}],"recommendation":"major_revision","confidential_remarks":"The sampling results are central and currently incorrect as stated; however, the quantization and event-triggering framework has independent value and could be publishable after a substantive revision that corrects Theorem 1 and Theorem 2. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper extends the well-worn 1-D passivity/dissipativity toolkit to 2-D Roesser systems with sampling, quantization, and event-triggered communication. That is genuinely new; I don't know of prior dissipativity-based network analysis for 2-D. Lemma 3's sampling-error bound, the quantizer degradation conditions, and the event-trigger rule in Theorem 4 are real contributions. The writing is clear and the proofs are mostly detailed. If the results are correct, designers get verifiable stability certificates for 2-D plants over digital links.\n\nBut there is a load-bearing problem with Theorem 1. The claimed exact sampled model is not exact. The proof uses block-diagonal exponentials e^{A11 h1}, e^{A22 h2} and cross terms that treat the orthogonal state as constant during each step. The exact solution of the continuous Roesser model does not reduce to that recurrence. A simple counterexample: A11=A22=0, A12=A21=1, boundary conditions e^{z2}, e^{z1}; the exact state at (h1,0) is e^{h1}, but the recurrence gives 1+h1. So the LMI in (7) certifies dissipativity of an approximate discrete model, not of the actual sampled plant. The example's rho_p=-1.317 is computed from that model, so the subsequent closed-loop claims for the true sampled system are not established. The abstract IF-OFP/quantization/event-trigger framework may still hold if accurate passivity levels are available independently, but the paper's central 'sampled certificate' story doesn't.\n\nTwo smaller issues. In Theorem 2's proof, the lower bound on Delta y^T Qp_hat Delta y uses |lambda_min(Qp_hat)|, which is only valid if Qp_hat is positive semidefinite; the theorem doesn't state that. And Theorem 4's stated q2 formula is inconsistent with the appendix: it has |nu_c| and delta_p^2 where it should have |nu_p| and delta_c^2. The second is a typo, but in a theorem statement it matters.\n\nSo: the conceptual framework is sound and the flaws are fixable, but Theorem 1 needs honest rework—either replace exact-sampling with an approximation with quantified error, or recompute the example using a genuinely exact or conservative sampled model. This deserves serious peer review, but I'd ask for major revision before publication. I wouldn't cite it in its current form.\n\nBest.","headline":"A new 2-D dissipativity extension with real ideas, but Theorem 1's 'exact' sampling is not exact, and the example leans on it.","tokens_in":24388,"tokens_out":5318,"would_cite":false,"duration_ms":45559,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C35","93D25","93C57"],"pacs":[],"model":"deepseek-v4-flash","headline":"2-D feedback systems can stay L2-stable over digital networks","keywords":["two-dimensional systems","dissipativity","passivity","networked control systems","event-triggered control","logarithmic quantization","L2 stability","Roesser model"],"falsifier":"A single numerical counterexample—a plant-controller pair satisfying the passivity inequalities of Theorem 3 and the conditions of Theorem 4 whose sampled, quantized, event-triggered closed-loop output diverges—would refute the central claim; the Section IV heat-exchanger example with $h_1=h_2=0.1$, $\\delta_p=\\delta_c=0.04$, $K=3$, and $N_1=40$ is the concrete configuration to test.","tokens_in":23223,"feed_emoji":"⚙️","tokens_out":5442,"duration_ms":54714,"temperature":0.7,"pith_summary":"This paper extends dissipativity and passivity tools from one-dimensional control to two-dimensional systems, whose dynamics depend on two independent coordinates such as space and time, and shows how to keep such systems stable when the feedback loop is closed over a digital network. It provides checkable conditions, mostly in the form of scalar inequalities and matrix inequalities, under which three network effects—sampling, logarithmic quantization, and event-triggered transmission—do not destroy the closed-loop $\\mathcal{L}_2$ stability. If the conditions hold, a designer can guarantee stability using only passivity levels rather than a full plant model, which matters for applications such as thermal processes, image processing, and repetitive processes.","feed_headline":"2-D feedback loops stay stable across digital networks","feed_subtitle":"New dissipativity conditions cover sampling, quantization, and event-triggered transmission in one framework.","key_machinery":"The argument is carried by a 2-D version of QSR-dissipativity with a separable storage function $V(x_h,x_v)=V_h(x_h)+V_v(x_v)$, and by the reduced passivity indices called IF-OFP levels $(\\rho,\\nu)$. Sampling is handled through a bound on the sampling error $\\Delta y_p$ in terms of the sampling periods $h_1,h_2$ and gain constants $\\alpha_1,\\alpha_2$; logarithmic quantization is handled through the sector bound $|\\Delta v|\\le \\delta|v|$; and the event-triggering scheme compares the accumulated output error against a threshold derived from the same passivity levels, so that the combined supply rate remains negative definite.","core_discovery":"The paper's central claim is that the $\\mathcal{L}_2$ stability of a 2-D feedback interconnection is preserved under the main effects of a digital link—sampling, logarithmic quantization, and event-triggered transmission—provided explicit dissipativity inequalities hold. For quantization, the closed-loop map is $\\mathcal{L}_2$-stable whenever the passivity surplus of each subsystem dominates the degradation introduced by the quantizer, as expressed by the two inequalities $\\rho_p + \\nu_c > (\\delta_p^2+2\\delta_p)|\\nu_c| + (1+\\beta_2^2)\\delta_p^2 + \\frac{1}{2\\beta_1}$ and its mirror image. For event-triggered communication, the paper shows that triggering at times selected by condition (22) yields finite $\\mathcal{L}_2$-gain on a bounded spatial domain, so transmissions can be reduced without destroying stability.","pith_inferences":["The paper's Remark 4 suggests the bounded-spatial-domain assumption can be relaxed by gridding the domain and applying the same trigger rule on each interval; if that extension holds, the method would apply to long spatial extents such as pipelines or convection processes, but the extension is not proved as a theorem.","The separable storage function $V_h+V_v$ is intentionally conservative; using less conservative 2-D Lyapunov functions could sharpen the passivity degradation bounds and allow less restrictive triggering thresholds.","The explicit dependence of the stability conditions on $\\delta_p,\\delta_c$ suggests a co-design problem: communication rate (quantizer density) and event-trigger thresholds can be traded directly against controller passivity levels.","The framework treats the plant and controller asymmetrically only through their passivity levels and quantizer densities, so the same conditions could be adapted to other 2-D architectures such as iterative learning control or distributed spatial systems."],"forward_implications":["If Theorem 3 is right, a designer can guarantee $\\mathcal{L}_2$ stability by tuning the quantizer densities and the auxiliary constants $\\beta_1,\\beta_2$, using only the plant and controller IF-OFP levels.","If Theorem 4 is right, event-triggered 2-D control can reduce transmissions along the vertical coordinate while preserving a finite $\\mathcal{L}_2$-gain on bounded spatial domains.","If Theorems 1 and 2 are right, sampling periods $h_1,h_2$ can be chosen from the LMI or from the inequalities to keep the sampled system dissipative.","Lemma 2 extends the classical small-gain style passivity theorem to 2-D IF-OFP systems, giving a simple sufficient condition $\\nu_2+\\rho_1>0$ and $\\nu_1+\\rho_2>0$ for feedback stability."],"supporting_citations":[{"why":"Willems' dissipative systems theory supplies the QSR-dissipativity notion that Definition 1 generalizes to two dimensions.","marker":"[13]"},{"why":"Khalil's nonlinear systems text supplies the standard L2 stability and passivity relations used in Lemmas 1 and 2.","marker":"[14]"},{"why":"Ahn, Shi, and Basin's 2-D dissipative control and filtering work provides the 2-D QSR-dissipativity framework that Definition 1 builds on with non-zero boundary conditions.","marker":"[15]"},{"why":"Garcia and Antsaklis' model-based event-triggered control with quantization supplies the logarithmic quantizer and event-triggered networked-control setup.","marker":"[19]"},{"why":"Oishi's passivity degradation under discretization motivates Assumption 1, the bound on how fast the 2-D output can change along each coordinate.","marker":"[26]"},{"why":"Xia, Antsaklis, Gupta, and Zhu's passivity/dissipativity analysis of a system and its approximation is the basis for the sampling degradation analysis and passivation methods referenced in Remark 3.","marker":"[27]"},{"why":"Zhu et al.'s passivity of switched systems under quantization justifies applying the scalar logarithmic quantizer sector bound component-wise to vector signals.","marker":"[31]"},{"why":"Matiakis, Hirche, and Buss provide the domain of passivity indices used in Lemma 4 to constrain the allowed $\\rho,\\nu$ pairs for IF-OFP systems.","marker":"[40]"}],"fun_headline_variants":["2-D control stable across networks via dissipativity","Sampled, quantized, event-triggered: 2-D still stable","Dissipativity conditions for 2-D networked feedback","L2 stability preserved in 2-D loops with digital links","Networked 2-D systems: stability under quantization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 2, which restricts the event-triggered result to 2-D systems with one coordinate on a bounded spatial domain: Theorem 4 proves finite $\\mathcal{L}_2$-gain only for that case, and the extension to infinite spatial domains is a sketched remark rather than a theorem.","fun_headline_variants_meta":{"raw":{"variants":["2-D control stable across networks via dissipativity","Sampled, quantized, event-triggered: 2-D still stable","Dissipativity conditions for 2-D networked feedback","L2 stability preserved in 2-D loops with digital links","Networked 2-D systems: stability under quantization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000289,"raw_usage":{"total_tokens":1659,"prompt_tokens":878,"completion_tokens":781,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":697}},"tokens_in":494,"tokens_out":781,"duration_ms":7712,"temperature":1.0,"reasoning_tokens":697,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:07:13.292299+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A single numerical counterexample—a plant-controller pair satisfying the passivity inequalities of Theorem 3 and the conditions of Theorem 4 whose sampled, quantized, event-triggered closed-loop output diverges—would refute the central claim; the Section IV heat-exchanger example with $h_1=h_2=0.1$, $\\delta_p=\\delta_c=0.04$, $K=3$, and $N_1=40$ is the concrete configuration to test.","supporting_citations":[{"cited_title":"Dissipative dynamical systems part ii: Linear systems with quadratic supply rates,","cited_arxiv_id":null,"evidence_quote":"Willems' dissipative systems theory supplies the QSR-dissipativity notion that Definition 1 generalizes to two dimensions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Khalil's nonlinear systems text supplies the standard L2 stability and passivity relations used in Lemmas 1 and 2."},{"cited_title":"Two-dimensional dissipative control and filtering for roesser model,","cited_arxiv_id":null,"evidence_quote":"Ahn, Shi, and Basin's 2-D dissipative control and filtering work provides the 2-D QSR-dissipativity framework that Definition 1 builds on with non-zero boundary conditions."},{"cited_title":"Model-based event-triggered control for systems with quantization and time-varying network delays,","cited_arxiv_id":null,"evidence_quote":"Garcia and Antsaklis' model-based event-triggered control with quantization supplies the logarithmic quantizer and event-triggered networked-control setup."},{"cited_title":"Passivity degradation under the discretization with the zero- order hold and the ideal sampler,","cited_arxiv_id":null,"evidence_quote":"Oishi's passivity degradation under discretization motivates Assumption 1, the bound on how fast the 2-D output can change along each coordinate."},{"cited_title":"Passivity and dissipativity analysis of a system and its approximation,","cited_arxiv_id":null,"evidence_quote":"Xia, Antsaklis, Gupta, and Zhu's passivity/dissipativity analysis of a system and its approximation is the basis for the sampling degradation analysis and passivation methods referenced in Remark 3."},{"cited_title":"Passivity and stability of switched systems under quantization,","cited_arxiv_id":null,"evidence_quote":"Zhu et al.'s passivity of switched systems under quantization justifies applying the scalar logarithmic quantizer sector bound component-wise to vector signals."},{"cited_title":"A novel input-output transfor- mation method to stabilize networked control systems independent of delay,","cited_arxiv_id":null,"evidence_quote":"Matiakis, Hirche, and Buss provide the domain of passivity indices used in Lemma 4 to constrain the allowed $\\rho,\\nu$ pairs for IF-OFP systems."}],"review_version":1}