{"id":"231c0feb-80e1-42c7-a6f0-c0245a38cb70","arxiv_id":"2511.03002","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"ROM-based MPC with a scalar error-bounding system from filtered peak-to-peak analysis guarantees robust constraint satisfaction for large linear systems with far less conservatism than prior error bounds.","lead":"This paper designs a predictive controller that uses a small reduced-order model of a large linear system, together with a scalar 'error-bounding' system that guarantees the true system stays near the model's prediction. The result is a controller that can enforce safety constraints for the full high-dimensional system while only computing with the small model.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Feasibility of filtered LMI (17) implicitly demands exact DC-gain matching, which the paper assumes but never states or verifies; Theorem 1's guarantee is conditional on this.","rationale":"The paper explicitly flags 'we suppose that Problem (17) admits a feasible solution' (Sec. V) and 'Normalization and frequency were chosen heuristically' (Sec. VIII). These are admitted gaps, not artifacts of my reading. My analysis shows the gap is load-bearing: without feasibility of (17), Lemma 1 is unavailable, so Theorem 1 has no basis. Feasibility is not automatic because the high-pass filter makes rψ vanish for constant r, and the LMI then forces the steady-state error ze to vanish, which is a nontrivial DC-gain matching condition on the ROM. The reader's weakest_assumption captures exactly this. No other step in the proof seems questionable: the comparison lemma for V(t)≤δ(t), the Schur complement for the output bound, and the disturbance handling with the direct feedthrough structure in (14)–(15) are all internally consistent. Therefore the appropriate posture is to keep the CONDITIONAL verdict; the paper should either prove (or give sufficient conditions for) feasibility of (17), or restrict the claims accordingly. The open-source code is useful but does not address the analytical gap.","tokens_in":12876,"tokens_out":14408,"duration_ms":133950,"concrete_test":"Using the released code (https://github.com/KohlerJohannes/ROM_MPC_ECC), replace the 8-mode ROM in Sec. VIII with a 7-mode ROM (or otherwise perturb the projection to introduce a small DC-gain mismatch), keeping the same first-order high-pass filter. Re-solve Problem (17). If it becomes infeasible, the concern is confirmed: feasibility silently enforces DC-gain matching. If it remains feasible, run the closed-loop MPC (24) and check whether δz(t) still bounds the true error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central guarantee (Theorem 1) depends on the scalar error-bounding system (21) being valid, which requires Proposition 3, which in turn requires feasibility of the filtered LMI (17). The paper only 'supposes' this (Sec. V, after Eq. (17)). This is not a harmless regularity condition. For the high-pass filter used in Section VIII, rψ(t)→0 for any constant input r(t). If (17) were feasible, the dissipation inequality λχVχ + V̇χ ≤ γχ‖rψ‖² (which follows from (17b) by congruence) would imply, at the steady state reached for a constant r, that Vχ→0 and hence, via (17c), that ze→0. Thus feasibility of (17) with a high-pass filter entails that the ROM's steady-state output error vanishes for all constant inputs — an exact DC-gain matching condition on the filtered channels. The paper neither states this condition nor provides a design rule to achieve it; the filter is chosen heuristically (Sec. VIII). The numerical example may satisfy the condition by construction, but the paper offers no evidence that the approach is generally applicable. The theorem is conditionally sound, but the headline conservatism improvement rests on this unverified feasibility/DC-matching requirement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a ROM-based MPC scheme for stable linear continuous-time systems with hard input/output constraints and bounded disturbances. The full-order system is decomposed into a Petrov–Galerkin ROM and an error dynamics driven by the ROM state and input. The authors derive a scalar error-bounding system whose constants are obtained offline from peak-to-peak LMIs, first with the raw excitation and then with a dynamically filtered excitation. This bound is embedded in a reduced-order optimal control problem with tightened constraints, and Theorem 1 states that any feasible input of this problem is feasible for the full-order robust optimal control problem with a guaranteed cost bound. The method is illustrated on a 100-dimensional mass–spring–damper system, reporting a reduction in conservatism of four orders of magnitude relative to existing ROM-based approaches.","tokens_in":13240,"tokens_out":10049,"duration_ms":105440,"significance":"If the result holds, the paper is a valuable contribution to ROM-based MPC: it provides pointwise, input-dependent prediction error bounds rather than uniform bounds, it replaces hand-tuned error-bounding constants with LMI-based optimization, and it demonstrates a filter extension that can make the bound vanish at steady state. The theoretical structure is clear and the proofs are largely standard. The paper also ships open-source code, which supports reproducibility. The main reservation is that the headline improvement relies on feasibility of the filtered LMI (17), and the paper does not analyze the structural condition that this feasibility implies for high-pass filters; as a result, the general applicability of the approach is not fully established.","major_comments":[{"comment":"Problem (17) is only 'supposed' to be feasible, but the paper does not analyze when this is true. For the high-pass filter used in Sec. VIII, rψ(t) tends to 0 for any constant input r. At the resulting steady state, the dissipation inequality from (17b) gives λχVχ(χ∞) ≤ 0, so Vχ(χ∞)=0; the output LMI (17c) then implies C_zχ∞=0, i.e., the steady-state error ze vanishes for all constant inputs. Feasibility of (17) with a high-pass filter therefore entails exact DC-gain matching between the ROM and the full-order model on the filtered channels. This condition is neither stated nor verified, and no design rule is given to enforce it; the filter is chosen heuristically in Sec. VIII. Since the Peak-filter result of Fig. 4 and Theorem 1 both depend on this LMI being feasible, the general applicability of the method is not established. Please state the feasibility condition explicitly, explain h","section":"Sec. V, Eq. (17); Lemma 1 and Theorem 1"},{"comment":"The dynamic filter Ψ is a key enabler of the reported improvement, but its selection is entirely heuristic: a first-order high-pass filter with nψ=nr+nu is chosen, and the normalization and frequency are said to be 'chosen heuristically'. The paper claims a systematic framework, yet gives no guidance on choosing Ψ or on how the result depends on that choice. A sensitivity study or an optimization-based filter selection would substantially strengthen the contribution. Without it, the four-orders-of-magnitude improvement cannot be separated from a favorable filter choice.","section":"Sec. V and Sec. VIII, filter design"},{"comment":"The printed condition (15) is unclear and, as written, only states that the first nw components of rψ equal w. The proof of Lemma 1, however, uses the stronger structural identity rψ(t)=¯rψ(t)+[I,0]^T w(t), which also requires that the remaining components of rψ be independent of w. The latter property is needed for the bound ∥rψ∥² ≤ ∥¯rψ∥²+¯w² used in (21c). Please rewrite (15) in clear component form and include the full structural argument in the proof of Lemma 1.","section":"Eq. (15) and Lemma 1"}],"minor_comments":[{"comment":"Typo: 'an (iii)' should be 'and (iii)'.","section":"Abstract"},{"comment":"The matrix expression in (15) is garbled in the typeset version; please fix the formatting and define all blocks explicitly.","section":"Sec. V, Eq. (15)"},{"comment":"Typo: 'polyotpic' should be 'polytopic'.","section":"Sec. VII"},{"comment":"The numerical example considers no disturbances (w=0). The theoretical guarantee covers disturbances, but this aspect is not demonstrated. A small disturbance case would strengthen the validation.","section":"Sec. VIII"},{"comment":"Minor grammar issue: 'the large difference is primarily due the choice' should be 'due to the choice'.","section":"Sec. VIII"},{"comment":"Typo: 'We except that' should be 'We expect that'.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The core derivation is sound and the paper is likely to be a useful contribution after revision. The central issue is the feasibility of LMI (17) and its hidden DC-gain-matching requirement for high-pass filters; this should be addressed explicitly. If the authors can provide a feasibility analysis or a design rule, I would be willing to accept a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid paper and a real step forward for ROM-based MPC. The filtered peak-to-peak error-bounding system is new relative to the scalar, hand-tuned bounds in [18] and the Lipschitz-based ones in [19], and the main theorem is conditionally sound. What you should check before building on it is the feasibility of LMI (17): with a high-pass filter it implicitly requires exact DC-gain matching, and the paper neither states nor verifies that.\n\nThe core idea is clean. They write the full-order error dynamics as a linear system driven by the ROM state, input, and disturbance, then use a peak-to-peak LMI to get a scalar bounding ODE. The generalization in Sec. V replaces the raw input r by a filtered signal r_psi, so the bound can be small when the excitation is smooth—exactly where ROMs are usually accurate. I checked the Schur complements in Props. 2 and 3 and the proof of Theorem 1; they go through. The qualitative comparison to IQC-based methods in Appendix A is honest: their filtered norm is not a special case of IQC filtering, which is worth saying.\n\nThe soft spots are real but addressable. Problem (17) is 'supposed' feasible; feasibility is not automatic. For a high-pass filter, r_psi tends to zero at constant inputs, so the dissipation inequality forces the steady-state error to zero. That means the ROM must exactly match the full-order DC gain on the filtered channels. The example apparently satisfies this, but the paper gives no design rule and doesn't even state the condition. Its absence matters because the headline four-orders-of-magnitude result rests on this filter working. Relatedly, the filter itself (frequency, normalization) is chosen heuristically, and the conservatism comparison is done on a single 100-dimensional example. For a methods paper that's acceptable, but both points should be made explicit and at least one more example would help.\n\nI'd send this to peer review. The conditional guarantee is not a fatal flaw, but the authors need to characterize feasibility of (17) or give a verifiable procedure for choosing Psi, and state the DC-gain condition. If that gets fixed, the contribution is solid and useful.","headline":"A genuine improvement in ROM-based MPC error bounds, but the filtered-peak LMI smuggles in a DC-gain matching condition that needs to be stated.","tokens_in":13676,"tokens_out":3722,"would_cite":true,"duration_ms":37394,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C05","93D09","93B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a reduced-order model, paired with a scalar filter-driven error bound, can guarantee constraint satisfaction in model predictive control.","keywords":["model predictive control","reduced-order models","error-bounding system","peak-to-peak gain","filtered signals","robust constraint satisfaction","linear matrix inequalities","large-scale systems"],"falsifier":"Solve the LMI (17) for a chosen filter and ROM on a system where the ROM's DC gain differs from the full-order system's DC gain on the filtered channels. If the LMI is feasible yet a worst-case simulation violates ∥zr(t) − z(t)∥ ≤ δz(t), then the peak-to-peak bound (22) fails; if the LMI is infeasible, the proposed scheme cannot be applied at all.","tokens_in":12787,"feed_emoji":"🎛️","tokens_out":3514,"duration_ms":37381,"temperature":0.7,"pith_summary":"Model-predictive control of large linear systems normally requires solving a high-dimensional optimal control problem online. The paper shows that a small reduced-order model can run the MPC safely if it is accompanied by a scalar 'error-bounding system' that, driven by a filtered version of the predicted trajectory, provides a pointwise upper bound on the difference between the reduced-order output and the full-order output. This bound is computed offline via a peak-to-peak gain LMI, and online it is used to tighten constraints and to bound the cost. On a 100-dimensional mass-spring-damper example, the method reduces conservatism by over four orders of magnitude compared with existing ROM-based MPC bounds, and it enables operation close to the constraint boundary.","feed_headline":"Tiny error bound makes reduced-order MPC safe","feed_subtitle":"A scalar filter-driven bound shrinks prediction error by four orders of magnitude, enabling safe control of a 100-dimensional system.","key_machinery":"Peak-to-peak analysis of filtered signals: a stable linear dynamic filter Ψ (state ψ, output rψ) transforms the lumped error input r into a filtered signal; the augmented error-filter dynamics (16) are analyzed through the LMI (17), which minimizes the output-to-output gain γχ. The associated Lyapunov function Vχ and decay rate λχ yield the scalar error-bounding system (18), whose trajectory δχ(t) gives the pointwise bound (19) and hence the implementable bound (22). The ROM is a standard Petrov-Galerkin projection with residual operator I − V W^T, and the filter is a design choice that can encode frequency-domain knowledge about where the ROM is accurate.","core_discovery":"The central claim (Theorem 1) is that any feasible input of the reduced-order optimal control problem (24) is feasible for the original full-order constrained optimal control problem (4), and its cost is no larger than the reduced-order cost. This is carried by Lemma 1: the scalar system (21) predicts δz(t) with ∥zr(t) − z(t)∥ ≤ δz(t) for all admissible disturbances, provided the filter-based peak-to-peak analysis in (17) is feasible. The novel step is the use of a dynamic filter Ψ on the lumped input so that the error bound reflects the frequency content of the trajectory. A high-pass filter makes the bound vanish at steady state, which is why the numerical example reaches zero conservatism","pith_inferences":["The filter is chosen heuristically; a joint offline optimization of the ROM and the filter could further shrink the error bound and is not explored in the paper.","The feasibility of the offline LMI (17) is assumed, but for high-pass filters it implicitly requires the ROM's steady-state gain to match the full-order system on the filtered channels; this condition is not verified and should be checked before deployment.","The IQC reformulation in Appendix A suggests a unified framework where filters describe both model error and peak-to-peak weighting; unifying these two filter roles could lower the LMI dimension and scale to even larger systems.","The method is presented for linear continuous-time systems, but the same filtered-peak-to-peak machinery could likely be adapted to discrete-time or mildly nonlinear ROMs, though the paper does not claim this."],"forward_implications":["The online MPC complexity depends on the ROM dimension nr plus the filter dimension nψ plus one scalar, not on the full order nf, enabling real-time control of large-scale systems.","Every feasible solution of the reduced-order problem (24) satisfies the full-order constraints for all admissible disturbances and yields a cost no larger than the reduced-order cost (Theorem 1).","Because the error bound depends on the optimized trajectory, the optimizer inherently favors inputs that keep the prediction error small, allowing tight operation near constraints.","The high-pass filter choice makes the error bound decay to zero at steady state, eliminating the conservative offset that uniform bounds retain even at equilibrium.","A receding-horizon implementation can be obtained by adapting standard terminal conditions from the ROM-based MPC literature, as noted in Remark 2."],"fun_headline_variants":["Filtered signals slash ROM-MPC conservatism by four orders","Peak-to-peak analysis of filtered signals: safe ROM-MPC","Scalar error system guarantees ROM-MPC constraint satisfaction","Reduced-order MPC with dynamic filter: zero steady-state conservatism","ROM-MPC via peak-to-peak filtered bound: 100-dim safe"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire result hinges on the offline LMI (17) being feasible for the chosen filter, which the paper simply supposes; with a high-pass filter this requires the reduced-order model's steady-state gain to match the full-order system on the filtered channels, a condition that is neither stated nor verified.","fun_headline_variants_meta":{"raw":{"variants":["Filtered signals slash ROM-MPC conservatism by four orders","Peak-to-peak analysis of filtered signals: safe ROM-MPC","Scalar error system guarantees ROM-MPC constraint satisfaction","Reduced-order MPC with dynamic filter: zero steady-state conservatism","ROM-MPC via peak-to-peak filtered bound: 100-dim safe"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000364,"raw_usage":{"total_tokens":1772,"prompt_tokens":694,"completion_tokens":1078,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":988}},"tokens_in":438,"tokens_out":1078,"duration_ms":11799,"temperature":1.0,"reasoning_tokens":988,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T00:02:19.839692+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the LMI (17) for a chosen filter and ROM on a system where the ROM's DC gain differs from the full-order system's DC gain on the filtered channels. If the LMI is feasible yet a worst-case simulation violates ∥zr(t) − z(t)∥ ≤ δz(t), then the peak-to-peak bound (22) fails; if the LMI is infeasible, the proposed scheme cannot be applied at all.","supporting_citations":[],"review_version":1}