{"id":"e7ce26cd-6201-4a8d-b051-1a7dbe2cfbc2","arxiv_id":"2505.16939","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A delayed dynamic-feedback controller assigns imaginary transmission zeros and optimizes stability to suppress multiple vibration frequencies at a distant target.","lead":"This paper designs a controller with intentionally delayed feedback to cancel vibrations at several frequencies at once, using measurements away from the target mass. The method is demonstrated in simulation on a four-mass system, yielding a stable closed loop with suppressed oscillations at the target.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing step is the elimination in Eqs. (13)/(16); the paper assumes R and P are invertible without analyzing how often that fails, so the reduced optimization problem may not represent the original zero-location constraints.","rationale":"The reader's weakest assumption identified the invertibility of R(omega_k,K_L) and P(K_L) in the constraint elimination. My stress test converges to the same point but sharpens why it is load-bearing: this is not merely a numerical robustness issue; it determines whether the reduced problem (17) is equivalent to the constrained problem (11). Since the paper's claimed result is that the optimized controller simultaneously places four transmission zeros and stabilizes the closed loop, an elimination that changes the feasible set invalidates the verification of the central claim. The concern is a genuine gap, not an internally inconsistent argument: the reported simulation values may be correct, but they are not enough to establish the method because the relation between the reduced objective and the original constraints is not shown to hold at the optimum. The reader already judged the paper CONDITIONAL, and this concern supports that verdict: the paper should be accepted only if the authors provide controller parameters, a feasibility check against the original determinant constraints, and an analysis of the singular cases in the elimination. Therefore I recommend no change to the reader's verdict. No ad hominem is intended; the critique is about the mathematical reduction and the absence of evidence linking the computed solution to the stated zero-location constraints.","tokens_in":8390,"tokens_out":11365,"duration_ms":126594,"concrete_test":"Recompute the original determinant h_k(K) from Eq. (8) at the optimized controller K reported in Section 4.2 for each of the four frequencies omega_k, after the authors release the controller parameters or a script reproducing the optimization. If any |h_k| exceeds, say, 1e-6, the optimized K is infeasible for the original zero-location constraints, and the central claim fails. If all |h_k| are at machine precision, also check det R(omega_k,K_L) and det P(K_L); if either is near zero while h_k is still satisfied, the elimination is unreliable and should be re-derived without the invertibility assumption to determine whether singular solutions are lost.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends entirely on the reduction in Section 3.4: solving the unconstrained problem (17) over K_L must be equivalent to minimizing the spectral abscissa over all controllers satisfying the original zero-location constraints h_k(K)=0 in Eq. (8). That equivalence is established through two matrix inversions. First, Eq. (13) is obtained from Eq. (8) via the Weinstein-Aragozajn identity only when R(omega_k,K_L) in Eq. (14) is invertible; the text says 'assuming R is invertible' but provides no condition or safeguard for the singular case. Second, Eq. (16) requires the matrix P(K_L) to be invertible, with only the parenthetical 'provided the matrix P(K_L) is invertible'. The issue is substantive: R(omega_k,K_L) is the system matrix of the partially controlled plant at the target frequency, so det R=0 is a genuine possibility for some K_L; P(K_L) can become singular exactly when the 2m dependent gains cannot simultaneously enforce all prescribed zero pairs. At any singular point, Eqs. (13) and (16) no longer follow from the original determinant condition, so the reduced feasible set is an open subset of the original algebraic variety h_k(K)=0. A 'successful' optimization could therefore terminate at a parameter set whose original determinants h_k(K) are not zero, meaning the controller does not actually place the four transmission zeros and the reported alpha<0 and silenced oscillation do not support the central claim. Because no controller parameters, code, or post-hoc feasibility check are provided, the reader cannot tell whether the reported solution satisfies the original constraints. This is the softest load-bearing point in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a methodology for designing a delayed dynamic-feedback controller for multi-frequency non-collocated vibration suppression. The authors remodel the system as a delay-differential algebraic equation (DDAE) and formulate the design as a constrained optimization problem that minimizes the spectral abscissa subject to zero-location constraints, i.e., placing transmission zeros of the disturbance-to-target transfer function at prescribed imaginary frequencies. Using the Weinstein-Aragozajn identity, the constraints are eliminated by expressing 2m dependent controller gains as a function of the remaining independent parameters, yielding an unconstrained nonsmooth nonconvex optimization problem that is solved with the HANSO solver. A case study on a four-mass system with four harmonic frequencies (4, 8, 12, 16 Hz) reports negative spectral abscissa values for controller orders 0 through 3, with slightly improved values at higher orders, and a time-domain simulation shows silencing of the target-mass oscillation after the controller is activated.","tokens_in":8658,"tokens_out":10168,"duration_ms":81033,"significance":"The problem addressed is relevant, and the DDAE-based reformulation combined with constraint elimination is a clever extension of the authors' earlier work on delay-based controllers for single-frequency and static-feedback cases. If the zero-location constraints are indeed satisfied at the optimized solutions, the method would provide a systematic single-input design procedure for multi-frequency non-collocated vibration suppression with dynamic controllers, which is currently lacking in the literature. The paper gives a plausible derivation and reports a sensible qualitative trend of improving spectral abscissa with controller order. However, the central claim rests on a reduction step whose validity requires invertibility of the matrices R(omega_k,K_L) and P(K_L), and the paper neither analyzes when these conditions hold nor verifies numerically that the final solution satisfies the original zero-location constraints. The omission of controller parameters and the lack of a pole-zero plot also prevent independent verification of the results. These gaps are load-bearing for the stated claims, so the paper requires revision before it can be accepted.","major_comments":[{"comment":"The equivalence between the original zero-location constraints h_k(K)=0 in Eq. (8) and the reduced equations (15)-(16) that lead to the unconstrained problem (17) depends on the invertibility of R(omega_k,K_L) and P(K_L). The text only states 'assuming R is invertible' and 'provided the matrix P(K_L) is invertible' without any analysis of when these conditions hold or any safeguard for singular cases. If R(omega_k,K_L) is singular for some K_L, the determinant h_k(K) can vanish even when the reduced scalar equation (13) is not satisfied, so the reduced feasible set is not equivalent to the original constraint set. Similarly, if P(K_L) becomes singular, the dependent gains g cannot be computed, and the unconstrained problem (17) is not defined. Since the central claim that the method assigns transmission zeros at the prescribed frequencies rests on this reduction, the paper must provide sufficient conditions for invertibility, or include a numerical mechanism to detect and avoid singular R or P during the optimization, or verify at the computed solution that the original determinants h_k(K) are zero to numerical precision and that R and P are well conditioned. Without such analysis or verification, the reported alpha values and the simulation do not establish that the designed controller actually places the transmission zeros.","section":"3.4, Eqs. (13)-(16)"},{"comment":"The case study does not provide the controller parameters, which are stated as omitted due to limited space, nor a pole-zero plot verifying that the closed-loop transmission zeros are located at the four prescribed frequencies. The caption of Fig. 3 claims 'the desired zeros are assigned correctly', but the figure itself is not included in the manuscript, so the reader cannot confirm that claim. Table 2 reports only the spectral abscissa values alpha, which are necessary but not sufficient: a controller that fails to place the zeros could still yield a negative alpha in the time-domain simulation because the closed-loop system might be stable even if the prescribed zeros are not exactly assigned. To support the central claim, the paper should include the explicit controller gains (at least the dependent gains g and the independent parameters K_L) or provide a supplementary file, together with a numerical check of |h_k(K)| at the optimized parameter vector, and ideally a plot of the zero locations. This would also allow the reader to assess whether the small improvements in alpha with increasing controller order (from -0.5218 to -0.5347) are meaningful and not merely numerical noise.","section":"4.2, Table 2, Fig. 3"}],"minor_comments":[{"comment":"Please provide a reference for the Weinstein-Aragozajn identity, as it may not be familiar to all readers of the journal.","section":"3.4, Eq. (13)"},{"comment":"The construction of the matrix P(K_L) and the vector Q in Eq. (16) is not explained in detail; please clarify the dimensions and how P is assembled from the vector z(omega_k,K_L) defined just below Eq. (15).","section":"3.4, Eqs. (15)-(16)"},{"comment":"The sentence 'the resulting optimum solution achieved with the higher-order controller can be no lower than the optimum computed for the previous lower-order controller' is logically reversed; since alpha is minimized, a higher-order controller should achieve an optimum that is no larger (i.e., no worse) than the lower-order optimum.","section":"4.2, after Table 2"},{"comment":"Equation (7) contains a typographical error: the determinant expression is not properly typeset, with '−~B1K~C1 − ~B' followed by '~C 0' in a way that obscures the Rosenbrock matrix structure; please rewrite the matrix explicitly with brackets and define all symbols.","section":"3.2, Eq. (7)"},{"comment":"The statement 'we would require at least eight free controller parameters, one for each zero' is imprecise: each complex conjugate zero pair requires two real equations, so eight free parameters for four pairs is the correct count, but the phrasing should be clarified.","section":"4.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is written in the style of a short IFAC conference paper, and the length constraints may explain the omission of controller parameters and detailed verification. However, the technical gap on invertibility of R and P is substantive and should be addressed. The paper also relies heavily on the authors' own prior publications (Saldanha et al. 2022, 2023, 2024) for the elimination procedure; while self-citation is not a problem, the current paper should summarize the key steps sufficiently for the method to be used without consulting those references. The case study is limited to a single simulation and would benefit from a frequency-domain or persistence analysis to confirm that all four target frequencies are suppressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Adrian and coauthors have extended their earlier delayed static-feedback framework to dynamic feedback controllers, using a DDAE remodeling to keep the optimization close to their previous work. The new element is real: with dynamic feedback, the zero-location constraints become polynomial rather than affine, and the paper shows how to eliminate the dependent parameters using the Weinstein–Aragozajn identity, then minimize the spectral abscissa with a nonsmooth solver. The case study, with four target frequencies, a single actuator, and only non-collocated sensors, is a good stress test, and the reported monotonic improvement in spectral abscissa as controller order increases is internally consistent.\n\nWhere the paper is soft is exactly the elimination step. Equations (13) and (16) require R(ω_k, K_L) and P(K_L) to be invertible, and the paper just states “assuming R is invertible” without analyzing when that fails. If either matrix becomes singular, the reduced constraints no longer represent the original determinant conditions h_k(K)=0. Since the controller gains are not reported, the reader cannot check whether the claimed solution actually places the four zero pairs. This is a reproducibility gap and a load-bearing one. The paper should at least add a post-hoc verification of the original constraints, and ideally discuss conditions under which the elimination is valid. The omission of controller parameters and code makes this hard to evaluate.\n\nA second, smaller point: the benefit of dynamic feedback in the example is modest. The static controller with four delays already gives α = −0.5218, and the order-3 controller only improves this to −0.5347. That does not invalidate the method, but it means the added complexity buys little in this particular case, and the paper would be stronger with a baseline comparison or a problem where the dynamic part is essential.\n\nOn balance, this is a solid incremental contribution to the time-delay vibration-control literature. The connection to the authors’ prior work is clear and honest, and the DDAE formulation is a sensible way to handle the added complexity. It deserves a serious referee, but the revision should address the invertibility assumptions and provide enough data to reproduce the result.","headline":"A sound but incremental extension of the authors' delayed-feedback framework to dynamic controllers, with the main soft spot being the unexamined invertibility assumptions in the constraint elimination.","tokens_in":9253,"tokens_out":4010,"would_cite":false,"duration_ms":30160,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C23","93B55"],"pacs":[],"model":"deepseek-v4-flash","headline":"By adding fixed output delays and a dynamic feedback block, a single actuator can place transmission zeros at four prescribed frequencies while simultaneously minimizing the closed-loop spectral abscissa.","keywords":["vibration suppression","time-delay systems","delay-differential algebraic equations","transmission-zero placement","spectral abscissa optimization","non-collocated control","dynamic output feedback","multi-frequency harmonic disturbance"],"falsifier":"Evaluate the full zero-location determinant h_k(K) at the optimizer returned for the four-mass system; if any of the four complex constraints is nonzero beyond numerical tolerance, the elimination did not preserve the original problem. The paper reports the tuned zeros are assigned correctly in the pole-zero plot, so a direct check on the unreduced determinant is the decisive test.","tokens_in":8167,"feed_emoji":"🎛️","tokens_out":5890,"duration_ms":45749,"temperature":0.7,"pith_summary":"The paper presents a controller-design method for canceling harmonic vibrations at a target point when the actuator cannot be placed there. In this non-collocated setting, each targeted frequency requires two free controller parameters, and a static feedback loop rarely has enough. The authors show that adding several fixed output delays and a low-order dynamic controller creates enough parameters, and that the resulting zero-placement constraints, although polynomial rather than affine, can be handled by rewriting the plant as a delay-differential algebraic equation and eliminating the dependent gains. The payoff is a single-input feedback law that assigns transmission zeros at four prescribed frequencies and at the same time minimizes the spectral abscissa, so the closed loop is stable with a damping margin. A four-mass benchmark simulation shows the target mass's oscillations vanish after a short transient.","feed_headline":"Delayed feedback silences four vibration frequencies at once","feed_subtitle":"One feedback loop places zeros at 4, 8, 12 and 16 Hz and keeps the closed loop stable.","key_machinery":"The central object is the delay-differential algebraic equation (DDAE) formed by introducing slack variables for the delayed control input and delayed measured outputs, and the reinterpretation of the dynamic feedback controller as a static output feedback gain K in that augmented model. This makes the zero-location equations polynomial in K and lets the paper recycle static output feedback design machinery. The elimination step is carried by the Weinstein–Aragozajn identity: for each targeted frequency ω_k, the constraint is reduced to 1 − g^T [C_g 0]R(ω_k,K_L)^{−1}[B_11;0] = 0, assuming R is invertible; stacking the m real and imaginary parts gives the linear system P(K_L)g = Q. The dependent parameters g are then solved for as g(K_L)=P(K_L)^{−1}Q, turning the constrained problem into the unconstrained spectral-abscissa minimization, solved with a quasi-Newton BFGS method with weak Wolfe line search because the objective is nonsmooth and nonconvex.","core_discovery":"The central claim is that multi-frequency non-collocated vibration suppression can be reduced to a constrained optimization problem with polynomial zero-location constraints and solved with static-output-feedback design tools, provided the feedback is enlarged with multiple fixed delays and a dynamic controller. The authors remodel the plant together with the delayed feedback and controller as a DDAE, which lets the delayed dynamic controller be reinterpreted as a static output feedback gain K. The zero-location constraints h_k(K)=0, k=1,...,m, force transmission zeros of the disturbance-to-target transfer function to lie at ±jω_k. Because these constraints are affine in a carefully chosen subset of entries of K, they can be eliminated: the dependent gains are expressed as g(K_L)=P(K_L)^{-1}Q, leaving an unconstrained nonsmooth, nonconvex optimization in the independent gains K_L, whose objective is the spectral abscissa α(K_L). In the case study, with four delays and controller orders zero through three, the closed-loop spectral abscissa reaches −0.5218 to −0.5347, and the target displacement settles to zero under a four-harmonic disturbance.","pith_inferences":["A natural extension the paper leaves implicit is replacing the fixed delay set by tunable delays: treating τ_i as optimization variables could shrink the controller order needed or improve the margin, but it would break the affine structure the elimination relies on.","The robustness questions flagged in the discussion could be probed by computing the sensitivity of the assigned zeros to small parameter variations; if the zeros drift rapidly with mass or stiffness changes, the practical suppression bandwidth will be narrow even though the nominal design is exact.","The method's logic is not limited to mechanical vibration; the same zero-placement-plus-spectral-abscissa formulation applies to any system whose disturbance-to-output transfer function can be computed, such as acoustic ducts or flexible structures, as long as the delay structure is admissible."],"forward_implications":["A single voice-coil actuator can suppress more independent harmonics than it has measured outputs, because each added delay multiplies the number of free controller parameters.","The same DDAE reformulation extends the method to multi-input settings and to additional control objectives like tracking, since the elimination leaves surplus parameters when the controller order or delay count is high.","The zero-order controller already gives α=−0.5218, and increasing controller order improves the stability margin (α=−0.5347 at order three), so dynamic feedback buys damping without needing extra delays.","Because the constraint elimination is algebraic, the design runs in existing time-delay system software and inherits its numerical treatment of infinitely many characteristic roots."],"supporting_citations":[{"why":"Supplies the DDAE remodeling technique for moving input-output delays to state delays and for modeling dynamic feedback in the augmented system.","marker":"Gumussoy and Michiels (2011)"},{"why":"Provides the dependent/independent parameter splitting and elimination procedure for static output feedback with zero-location constraints that this work adapts.","marker":"Saldanha et al. (2022)"},{"why":"Prior single-frequency design using multiple fixed delays that this paper extends to the multi-frequency case with dynamic feedback.","marker":"Saldanha et al. (2023)"},{"why":"Earlier non-collocated controller design methodology whose constraint form and solution strategy the paper generalizes.","marker":"Saldanha et al. (2024)"},{"why":"Provides the nonsmooth quasi-Newton optimization method used to solve the resulting nonconvex unconstrained problem.","marker":"Lewis and Overton (2013)"},{"why":"Supplies the time-delay system analysis and controller-design software used to compute spectral abscissa and run the optimization.","marker":"Appeltans and Michiels (2023)"},{"why":"Establishes the spectral abscissa framework and the nonsmooth, nonconvex nature of stability optimization for time-delay systems.","marker":"Michiels and Niculescu (2007)"}],"fun_headline_variants":["Silence 4,8,12,16 Hz with one delayed feedback loop","One delayed loop suppresses four vibration frequencies","One delayed loop, four silenced frequencies","Optimized delay controller suppresses four vibration modes","Multi-frequency vibration control via delayed feedback"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The design assumes the matrix R(ω_k,K_L) and the elimination matrix P(K_L) remain invertible throughout the optimization; when either becomes singular, the reduced constraint set no longer represents the original zero-placement condition, so the returned controller may not put the zeros where promised.","fun_headline_variants_meta":{"raw":{"variants":["Silence 4,8,12,16 Hz with one delayed feedback loop","One delayed loop suppresses four vibration frequencies","One delayed loop, four silenced frequencies","Optimized delay controller suppresses four vibration modes","Multi-frequency vibration control via delayed feedback"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000983,"raw_usage":{"total_tokens":4148,"prompt_tokens":896,"completion_tokens":3252,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":3181}},"tokens_in":512,"tokens_out":3252,"duration_ms":20398,"temperature":1.0,"reasoning_tokens":3181,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:51:52.630645+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the full zero-location determinant h_k(K) at the optimizer returned for the four-mass system; if any of the four complex constraints is nonzero beyond numerical tolerance, the elimination did not preserve the original problem. The paper reports the tuned zeros are assigned correctly in the pole-zero plot, so a direct check on the unreduced determinant is the decisive test.","supporting_citations":[],"review_version":1}