{"id":"68cdfc7a-dae6-4ea3-91e5-04cfd6e4de2c","arxiv_id":"2504.14334","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A data-driven event-triggered controller for discrete-time nonlinear systems is designed by lifting them into a higher-dimensional linear space via Koopman operators, with stability conditions posed as LMIs.","lead":"This paper combines Koopman operator theory with event-triggered control to design stabilizing controllers for discrete-time nonlinear systems directly from data. Simulations show the method uses fewer communication updates than traditional periodic control, though the stability certificate is limited to an approximate lifted model.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The stability guarantee is proved only for an assumed-exact Koopman model; no closure-error bound transfers it to the original nonlinear plant, so the central claim is unsupported.","rationale":"The reader's weakest_assumption identifies the same load-bearing premise: the finite-dimensional Koopman model (12) is treated as an exact representation of the nonlinear plant even though the closure error is nonzero. My stress-test agrees with that assessment and sharpens it: the stability analysis is carried out entirely in the lifted data-driven loop (15), whose derivation via Lemma 1 uses the equality Z1 = [A B] [Z0; U0]. This equality holds for an exact linear representation, but not for an approximate dictionary. The residual R_k is absent from every Lyapunov inequality and from the S-procedure in Theorem 2, so no condition in the paper prevents R_k from violating the decay rate alpha. This is not merely a missing refinement; it is a gap between the object being analyzed and the object being controlled. Even if the LMI algebra in Theorem 1 were repaired, the transfer problem would remain, because the controller is applied to the original nonlinear plant while the certificate is computed for the approximate lifted model. The paper's own discussion of the closure problem in Section 4.2 and the qualitative Remark 1 confirm that the authors are aware of the issue but do not quantify it or include it in the guarantee. The trigger-law ambiguity in Section 2.2 and the footnote further weaken the link between theory and simulation, but it is secondary to the missing approximation-error bound. I therefore agree with the reader's REJECT verdict: the central claim of guaranteed exponential stability for the original nonlinear system is not supported by the presented analysis. A direct residual-based check on Example 2 would settle whether the concern manifests numerically in at least the paper's own demonstration.","tokens_in":28064,"tokens_out":9212,"duration_ms":87109,"concrete_test":"Re-run Example 2 with the published dictionary and data length, apply Algorithm 1 to obtain K and gamma, and simulate the original plant (24) under the event condition (11) with lifted error e_k = xi(x_{k_i}) - xi(x_k). Along the closed-loop trajectory, compute L and N from (14), set S = (Z0 G1)^{-1}, and evaluate both the residual R_k = xi(x_{k+1}) - (Z1 L xi(x_k) + Z1 N e_k) and the Lyapunov ratio V(z_{k+1})/V(z_k). If R_k is nonzero and any ratio exceeds alpha, or if the residual dominates the decay margin (1-alpha) z^T S z, then stability of the approximate lifted loop (15) does not transfer to the true plant, and the central claim is falsified for that example.","verdict_should_be":"REJECT","load_bearing_attack":"The central theorem pair is about the lifted data-driven loop (15), which is exact only if the dictionary is closed under the dynamics, i.e., if there exists (A,B) such that Z1 = A Z0 + B U0 on the observed data. For a generic dictionary this is false, as Example 2 itself acknowledges in Section 4.2 as the closure problem. The residual R_k = xi(x_{k+1}) - (Z1 L xi(x_k) + Z1 N e_k) is nonzero in that example, yet it does not appear in (15), (16), Theorem 1, or Theorem 2. The actual closed loop under the designed controller is z_{k+1} = Z1 L z_k + Z1 N e_k + R_k, and the Lyapunov inequality V(z_{k+1}) <= alpha V(z_k) can be destroyed by R_k unless a bound on the residual is included. Remark 1 only says a good dictionary 'could be of great benefit'; no formal approximation error bound is given anywhere. Thus exponential stability of (15) does not imply stability of the original nonlinear plant (1), so the stated guarantee 'ensuring exponential stability in Lyapunov sense' is not established. The paper's own footnote and the shift from (2)/(4) to (11) additionally leave it ambiguous whether the implemented control and triggering law are in x or in xi(x); if the implemented loop differs from the analyzed one, Theorems 1 and 2 do not even apply to the simulated system.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a data-driven event-triggered control method for discrete-time nonlinear systems. It lifts the state via a dictionary of observable functions xi(x), assumes the lifted dynamics are approximately linear, z_{k+1} = A z_k + B u_k, and then designs a static feedback gain K and a relative triggering threshold gamma by solving LMIs derived from Lyapunov-based arguments. The main results are Theorem 1 (controller gain from data), Theorem 2 (threshold computation), and Algorithm 1, validated on three simulation examples including a nonlinear polynomial system. The paper claims exponential stability in the Lyapunov sense with reduced communication compared to time-triggered control.","tokens_in":28315,"tokens_out":9903,"duration_ms":80927,"significance":"If the stability guarantee could be transferred to the original nonlinear plant, the contribution would be a useful direct data-driven event-triggered design for nonlinear systems. The paper demonstrates the idea in simulations with up to 40% communication reduction and gives a concrete algorithm. The strengths are the problem formulation, the explicit LMI-based procedure, and the attempt to combine Koopman lifting with event-triggered data-driven control. However, as written, the analysis applies only to the lifted linear model and has several technical gaps, including an incorrect LMI in Theorem 1, a reversed S-procedure step in Theorem 2, and no approximation-error bound to relate the lifted model to the original plant. The paper does not provide code or machine-checked proofs, so the value depends on correcting these gaps.","major_comments":[{"comment":"The data-driven representation identifies z_{k+1} = Z1 L z_k + Z1 N e_k by using the equality [A B][Z0; U0] = Z1 in the proof. This equality holds only if the finite-dimensional Koopman model (12) is exact on the data. For a generic observable dictionary, the per-step residual R_k = xi(x_{k+1}) - (A xi(x_k) + B u_k) is nonzero, as Example 2's closure problem (Section 4.2) concedes. The actual closed loop under the designed controller is z_{k+1} = Z1 L z_k + Z1 N e_k + R_k, and no bound on R_k is given anywhere in Sections 3.1-3.4. Consequently, the Lyapunov inequality (16) and Theorems 1-2 do not imply exponential stability of the original nonlinear plant (1). The authors need to state an explicit bound on the lifting/closure error and incorporate it into the stability analysis, or explicitly restrict the paper's claims to the exact-lifting case.","section":"Section 3.2, Lemma 1 and Eq. (15)"},{"comment":"The LMI in Theorem 1 misses the decay rate alpha in the (1,1) block. With P = S^{-1} = Z0 G1, the Schur complement of G1^T Z1^T P^{-1} Z1 G1 - alpha P <= 0 is [[alpha P, G1^T Z1^T], [Z1 G1, P]] >= 0, not [[Z0 G1, G1^T Z1^T], [Z1 G1, Z0 G1]] >= 0. As written, (18) only enforces a neutral (alpha = 1) decrease condition, and the derivation's step 'pre- and post-multiply by S^{-1}' is consistent with the corrected LMI rather than with the displayed matrix. The statement that the resulting K guarantees decay rate alpha is therefore unsupported.","section":"Section 3.3, Theorem 1, Eq. (18)"},{"comment":"The S-procedure condition has the wrong sign. To deduce the Lyapunov decrease from the triggering condition v^T M v <= 0 with M = diag(-gamma^2 I, I), one requires an eta >= 0 such that Psi - eta M <= 0 (equivalently eta M - Psi >= 0). The proof writes eta M <= Psi, which is the reverse and cannot be used to conclude that (4) implies (16). The subsequent Schur-complement substitution then produces a different inequality, and it is unclear whether the LMI (21) is the correct condition. A corrected derivation is needed before the threshold gamma* can be trusted.","section":"Section 3.4, Theorem 2 and Eq. (21)"},{"comment":"The control and triggering laws are specified inconsistently across coordinate systems. Eq. (2) gives u_k = K x_{k_i}, while the footnote and the rest of Section 3 use u_k = K xi(x_{k_i}); Eq. (4) uses ||e_k|| <= gamma ||x_k|| in original coordinates, but Eq. (11) switches to ||e_xi_k|| <= gamma ||xi(x_k)||, and the closed-loop analysis (13) reuses e_k without redefining it in the lifted space. If the implemented controller uses x_k, then Theorems 1 and 2 do not apply to the simulated system; if it uses xi(x_k), then Eq. (2) and the plots in Section 4 need to be updated. The paper should consistently analyze the lifted loop or the original loop and clearly state which one is implemented.","section":"Sections 2.2 and 3.1"}],"minor_comments":[{"comment":"The notation U(a,b) is said to represent a normal distribution, but the text and examples use it as a uniform distribution; please correct the definition.","section":"Section 2.2"},{"comment":"The controller gain K is introduced as R^{m x n}, but after lifting it becomes m x p. Please adjust the dimension statements and ensure the error vector e_k is defined consistently in both original and lifted coordinates.","section":"Section 2.2 and Eq. (4)"},{"comment":"In the S-procedure matrix, the off-diagonal blocks are written as L^T Z1^T S Z1 L in both positions; the second should read L^T Z1^T S Z1 N (and its transpose). This is likely a typo, but it obscures the derivation.","section":"Section 3.4, proof of Theorem 2"},{"comment":"The dictionary listing includes identical entries 'x3 1 x3 2 x3 1 x3 2', which seems to be a typo for distinct monomials; please clarify.","section":"Section 4.2 and Fig. 7"},{"comment":"In Eq. (25) the third control term is written as K3 x2 2; based on the observable x_1^2 this should be K3 x_1^2. The same typo appears in Section 5.2.","section":"Section 4.1 and Eq. (25)"},{"comment":"There are multiple typos, including 'Lyaponuv' instead of 'Lyapunov' and inconsistent capitalization of 'Koopman' in places. A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper has potential, but the theoretical core is not yet sound. The missing closure-error bound is the most serious issue: without it, the theorems do not establish stability of the original nonlinear system. The LMI errors and the reversed S-procedure step are concrete and fixable, but correcting them may require a substantial revision of Section 3. The simulations are encouraging but are not a substitute for a valid proof. I would not reject outright, because the framework is plausible and the contribution, if properly repaired, could be of interest to the event-triggered control community. However, the authors should be strongly encouraged to either add a rigorous residual analysis or clearly downgrade the claimed guarantees."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The combination is new — Koopman lifting with direct data-driven event-triggered control — and the paper is honestly written. The simulations are useful, especially Example 2, which admits that no exact Koopman linearization exists and shows how steady-state error varies with the polynomial degree. The authors also cite the relevant prior work (Digge and Pasumarthy, De Persis and Tesi) and openly call the observable-selection question open. That is real value.\n\nThe central claim, however, is not established. The stability theorems are about the data-driven lifted loop (15), which is exact only if the dictionary is closed under the dynamics. The paper itself acknowledges the closure problem in Section 4.2, but the residual R_k simply never appears in the analysis. Without a bound on that residual, exponential stability of (15) does not imply stability of the original nonlinear plant (1). This is not a minor omission; it is the load-bearing gap in the guarantee.\n\nThere is also a concrete algebraic error in Theorem 1. The LMI as printed has Z0G1 in the (1,1) block, but the derivation needs α Z0G1 there, and the Schur complement step replaces (Z0G1)^{-1} with Z0G1. This is fixable, but as written the proof is wrong. Additionally, the footnote about uk = Kξ(x_ki) versus uk = Kx_ki leaves it ambiguous which loop is analyzed and which is simulated; if they differ, the theorems do not even apply to the simulations.\n\nProportionately: the paper deserves credit for the empirical study and the honest discussion of limitations. The simulations do show plausible behavior with reduced communication. But until the LMI is corrected and an approximation-error bound is supplied, the advertised Lyapunov stability guarantee is unsupported.\n\nI would not cite this as a rigorous result in the next year, though I might cite it as a useful empirical example. It is not a reject-without-review: the topic is timely and the framework is worth referee time. Send it to peer review with instructions that the referee focus on the LMI correction and, more importantly, on any closed-loop error analysis connecting the lifted design to the true plant. A serious editor could ask for major revision rather than desk rejection.","headline":"A legitimate empirical combination with a load-bearing theoretical gap: the Lyapunov guarantee is proved only for an assumed-exact lifted model, and the printed LMI has a concrete fixable error.","tokens_in":28917,"tokens_out":2675,"would_cite":false,"duration_ms":25652,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C65","93C55","93B30","37N35","93C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Using one dataset, two LMIs, and a Koopman lift, the paper designs a state-feedback gain and an event-triggering threshold that make the lifted closed loop of a discrete-time nonlinear system exponentially stable in the Lyapunov sense.","keywords":["event-triggered control","data-driven control","Koopman operator","discrete-time nonlinear systems","Lyapunov stability","linear matrix inequalities","zero-order hold"],"falsifier":"Take the polynomial plant of Example 2 with a degree-6 monomial dictionary (or any dictionary with visible closure error), learn $K$ and $\\gamma$ by the algorithm, simulate the original nonlinear plant with the event-triggered zero-order hold, and record $V(k+1)/V(k)$ along the trajectory; a single time instant at which this ratio exceeds $\\alpha$ while $\\|e_k\\|\\le\\gamma\\|z_k\\|$ would falsify the transfer of stability from the lifted model to the plant.","tokens_in":27820,"feed_emoji":"📡","tokens_out":9553,"duration_ms":79069,"temperature":0.7,"pith_summary":"This paper is trying to establish a direct data-driven route to event-triggered control for discrete-time nonlinear systems whose equations are unknown. Instead of identifying the plant, the method lifts measured states through a chosen dictionary of observables, $\\Xi(x)$, so that the dynamics look linear in a higher-dimensional Koopman space, and then designs both the feedback gain and the triggering threshold from a single dataset by solving LMIs. Theorems 1 and 2 state that the resulting controller exponentially stabilizes the lifted closed loop in the Lyapunov sense with a prescribed rate $\\alpha$, while the event-triggering policy reduces communication instances relative to time-triggered control. If true, this would turn event-triggered control for a class of nonlinear plants into a convex, data-only design problem, which is why the paper matters.","feed_headline":"One dataset sets both gain and trigger for nonlinear control","feed_subtitle":"Koopman lifting makes the plant look linear; two LMIs then fix the feedback law and the trigger rule that saves transmissions.","key_machinery":"The load-bearing construction is the Koopman-lifted, data-driven closed-loop representation of Lemma 1. With the Koopman operator understood as the linear evolution $K_t\\xi=\\xi\\circ f$ on observable functions, the paper chooses a finite dictionary $\\Xi(x)$, so the controlled plant is approximated by $z_{k+1}=Az_k+Bu_k$; then, under the full-row-rank assumption, the identities $\\begin{bmatrix} I \\\\ K \\end{bmatrix}=\\begin{bmatrix} Z_0 \\\\ U_0 \\end{bmatrix}L$ and $\\begin{bmatrix} 0 \\\\ K \\end{bmatrix}=\\begin{bmatrix} Z_0 \\\\ U_0 \\end{bmatrix}N$ let the event-triggered loop be written as $z_{k+1}=Z_1Lz_k+Z_1Ne_k$, with no $\\{A,B\\}$ appearing. This representation carries the argument because it converts both tasks into LMIs: Theorem 1 finds $G_1$ through a Schur-complement condition to obtain $K$, and Theorem 2 uses the S-procedure to maximize $\\gamma$ subject to the Lyapunov condition, with everything expressed directly in the collected data matrices.","core_discovery":"The central claim is that the pair of design objects can be computed straight from data, with no explicit identification of $A$ and $B$: after collecting $U_0$, $Z_0=\\Xi(X_0)$, and $Z_1=\\Xi(X_1)$, one solves the LMI in Theorem 1 for $G_1$ to get $K=U_0G_1(Z_0G_1)^{-1}$, and then solves the LMI in Theorem 2 for the largest threshold $\\gamma$ that keeps the Lyapunov inequality satisfied whenever $\\|e_k\\|\\le\\gamma\\|z_k\\|$. The closed loop is written in the data-driven form $z_{k+1}=Z_1Lz_k+Z_1Ne_k$, and Theorem 2 guarantees exponential stability with rate $\\alpha$ along event-triggered trajectories. The paper presents three simulations: a slow-manifold system that admits an exact Koopman linearization, a polynomial system that exhibits the closure problem with finite dictionaries, and a linear system that recovers the existing data-driven event-triggered result as a special case.","pith_inferences":["Going beyond the paper: the proven exponential stability is for the lifted model; turning it into a guarantee for the original plant would require an explicit bound on the Koopman closure residual $z_{k+1}-(Az_k+Bu_k)$, and such a bound could turn the method into a robust data-driven event-triggered control design.","A natural testable extension is to compute a guaranteed positive minimum inter-event time from the data matrices and enforce it, since the paper does not analyze dwell time between transmissions.","Because the trigger is evaluated in lifted coordinates, $\\|e^\\xi_k\\|\\le\\gamma\\|\\xi(x_k)\\|$, a designer choosing this scheme implicitly weights observable components rather than original states; an alternative threshold on $\\|x_k-x_{k_i}\\|$ would be a different policy and could be compared experimentally on the same data."],"forward_implications":["If the lifted model is faithful, no identification step is needed: one dataset fixes both the control law and the scheduling rule.","The triggering rule preserves the exponential decay rate $\\alpha$ of the lifted closed loop while updating $u_k$ only when the relative lifted-state error crosses $\\gamma$.","In the exactly linearizable test case the method matches known Koopman-control behavior and cuts communication events by roughly 40 percent in simulation.","In the approximate (closure-error) case, the steady-state error depends systematically on the dictionary degree, decreasing from degree 1 through 5 and rising again at degree 6, so dictionary size alone does not guarantee better performance.","With identity observables the algorithm reduces to the existing data-driven event-triggered design for linear systems, giving a consistency check."],"supporting_citations":[{"why":"Supplies the data-driven event-triggered closed-loop representation that Lemma 1 adapts to the lifted system, and provides the linear example used for comparison.","marker":"[20]"},{"why":"Provides the direct data-driven stabilization construction (gain from data via $K=U_0G_1(Z_0G_1)^{-1}$) on which Theorem 1 rests.","marker":"[6]"},{"why":"Gives the extended-state Koopman treatment of controlled systems that justifies approximating (1) as $z_{k+1}=Az_k+Bu_k$.","marker":"[40]"},{"why":"Underpins Assumption 1 by guaranteeing that a persistently exciting input makes the data matrix full row rank.","marker":"[35]"},{"why":"The S-procedure and Schur complements used in Theorem 2 to encode the trigger condition inside the Lyapunov inequality.","marker":"[50]"},{"why":"Source of the slow-manifold plant in Example 1, whose exact Koopman linearization the method uses as a proof-of-concept.","marker":"[51]"},{"why":"Motivates the closed-loop data-collection procedure used in Example 2 for systems identified under a stabilizing controller.","marker":"[54]"},{"why":"Supports the paper's discussion that dictionary choice controls the approximation quality of the lifted model, i.e., the closure problem.","marker":"[55]"}],"fun_headline_variants":["One dataset sets controller gain and event trigger via Koopman","Data-driven event-triggered control: Koopman linearization, no ID","From data to exponential stability: Koopman ETC with two LMIs","Koopman-based ETC: jointly design feedback and trigger from data","Skip system ID: Koopman lifting tunes gain and event condition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The guarantee transfers from the lifted linear model to the true nonlinear plant only if the chosen observables represent the plant well enough to make the data-computed Lyapunov certificate valid for the actual system, and the paper provides no quantitative error bound for that transfer.","fun_headline_variants_meta":{"raw":{"variants":["One dataset sets controller gain and event trigger via Koopman","Data-driven event-triggered control: Koopman linearization, no ID","From data to exponential stability: Koopman ETC with two LMIs","Koopman-based ETC: jointly design feedback and trigger from data","Skip system ID: Koopman lifting tunes gain and event condition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000279,"raw_usage":{"total_tokens":1625,"prompt_tokens":878,"completion_tokens":747,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":650}},"tokens_in":494,"tokens_out":747,"duration_ms":6343,"temperature":1.0,"reasoning_tokens":650,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:51:36.855858+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the polynomial plant of Example 2 with a degree-6 monomial dictionary (or any dictionary with visible closure error), learn $K$ and $\\gamma$ by the algorithm, simulate the original nonlinear plant with the event-triggered zero-order hold, and record $V(k+1)/V(k)$ along the trajectory; a single time instant at which this ratio exceeds $\\alpha$ while $\\|e_k\\|\\le\\gamma\\|z_k\\|$ would falsify the transfer of stability from the lifted model to the plant.","supporting_citations":[{"cited_title":"Data- driven event-triggered control for discrete-time LTI sys- tems","cited_arxiv_id":null,"evidence_quote":"Supplies the data-driven event-triggered closed-loop representation that Lemma 1 adapts to the lifted system, and provides the linear example used for comparison."},{"cited_title":"Formulas for data- driven control: Stabilization, optimality, and robustness","cited_arxiv_id":null,"evidence_quote":"Provides the direct data-driven stabilization construction (gain from data via $K=U_0G_1(Z_0G_1)^{-1}$) on which Theorem 1 rests."},{"cited_title":"Linear predictors for nonlin- ear dynamical systems: Koopman operator meets model predictive control","cited_arxiv_id":null,"evidence_quote":"Gives the extended-state Koopman treatment of controlled systems that justifies approximating (1) as $z_{k+1}=Az_k+Bu_k$."},{"cited_title":"Convex opti- mization","cited_arxiv_id":null,"evidence_quote":"The S-procedure and Schur complements used in Theorem 2 to encode the trigger condition inside the Lyapunov inequality."},{"cited_title":"Koopman invariant subspaces and fi- nite linear representations of nonlinear dynamical systems for control","cited_arxiv_id":null,"evidence_quote":"Source of the slow-manifold plant in Example 1, whose exact Koopman linearization the method uses as a proof-of-concept."},{"cited_title":"Practical Guidelines for Data-driven Identification of Lifted Linear Predictors for Control","cited_arxiv_id":"2408.01116","evidence_quote":"Motivates the closed-loop data-collection procedure used in Example 2 for systems identified under a stabilizing controller."},{"cited_title":"A data–driven approximation of the koopman operator: Extending dynamic mode decom- position","cited_arxiv_id":null,"evidence_quote":"Supports the paper's discussion that dictionary choice controls the approximation quality of the lifted model, i.e., the closure problem."}],"review_version":1}