{"id":"156bfc3f-b27a-43ad-acca-70f44ec24592","arxiv_id":"2507.03684","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Balanced truncation for bilinear systems with quadratic outputs is developed using new algebraic Gramians and generalized Lyapunov equations.","lead":"This paper develops Gramian-based model reduction for bilinear systems with quadratic outputs, defining several possible Gramians and testing them on circuit and heat examples. It gives engineers a balanced truncation route for large weakly nonlinear systems whose outputs measure variance or energy.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Numerical experiments do not report whether the Gramian-existence smallness bounds of Theorem 4.1 actually hold; if they fail, the full-Gramian BT comparisons are outside the proven regime.","rationale":"I read the paper in good faith. The core derivations—the Volterra-series Gramians, the recursion for QS_i, the block-row outer product identity, and the Lyapunov equations (3.6), (3.19), and (3.30)—are mathematically sound. I checked the step in the proof of Theorem 3.4 where [A B][A B]^T is expanded; the cross terms do not arise because the product of a block row with its transpose gives A A^T + B B^T exactly, so that potential concern is unfounded. The existence proofs are consistent with the stated smallness conditions, although these conditions are sufficient rather than necessary. The reader's CONDITIONAL verdict is appropriate. My most load-bearing concern is neither an internal inconsistency nor a mathematical error: it is that the numerical experiments are described as rescaled to satisfy Theorem 4.1, but the actual values of Γ_P, Γ_QS, Γ_QA, α, and β are never reported, so the reader cannot tell whether the demonstrated behavior occurs inside the regime where the Gramians are proven to exist. This is directly load-bearing because the central claim concerns balanced truncation using these particular Gramians. A secondary concern, also noted by the reader, is that no error bound or stability guarantee is provided for the reduced BQO model; the output tracking is supported only empirically. I therefore recommend keeping the CONDITIONAL verdict, with the added condition that the authors verify the smallness bounds for the reported experiments and state them explicitly in the paper.","tokens_in":27424,"tokens_out":29858,"duration_ms":298973,"concrete_test":"For each numerical example, estimate the stability constants α, β such that ||e^{At}|| ≤ β e^{-αt}, and compute Γ_P = ||N||^2, Γ_QS = max{Γ_P, ||N^T||^2, ||M||^2}, and Γ_QA = max{Γ_P, ||M||^2} for both γ values. Check whether all three are strictly less than 2α/β^2. Then rerun the RC example with γ chosen to push Γ_QS above the bound (e.g., the unscaled γ=1) and compare the full-Gramian BT output error; if the reduced output still tracks the full output, the smallness condition is not a genuine boundary of the framework, while if the Lyapunov solvers fail or the error grows, it is confirmed essential.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The existence and uniqueness results for the Gramians (Theorems 4.1 and 4.4) are conditional on the smallness bounds Γ_P = ||N||^2 < 2α/β^2, Γ_QS = max{Γ_P, ||N^T||^2, ||M||^2} < 2α/β^2, and Γ_QA = max{Γ_P, ||M||^2} < 2α/β^2. In Section 7, the authors introduce a scaling γ so that these conditions are 'easier to hold', but they never report the actual values of Γ_P, Γ_QS, Γ_QA, α, or β, nor do they verify the inequalities. The central claim that balanced truncation produces a reduced output tracking the full output rests on using the Volterra-series Gramians of Definitions 3.1, 3.3, and 3.7; if the bounds are violated, the algebraic solutions of the Lyapunov equations may still exist, but they are not proven to equal those Gramians, and the reduced model's output may not track. The comparisons in Figures 1–6 among QS, QA, and the truncated variants are only meaningful inside the proven regime. Without verification, the numerical support for the central claim is not anchored to the theory.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript treats bilinear systems with quadratic output, of the form \\dot x = Ax + \\sum N_k x u_k + Bu, y = Cx + [x^T M_j x]. It defines a reachability Gramian P and several observability Gramians QS, QP, QA, QM from Volterra and dual-system expansions, derives the generalized Lyapunov equations (3.6), (3.19), (3.30), (3.31), proves existence and uniqueness under explicit smallness conditions, shows QS = QP, introduces truncated Gramians that require only standard Lyapunov solves, and presents a balanced-truncation algorithm with two numerical examples. The stated goal is a structure-preserving reduced BQO model whose output tracks the full output.","tokens_in":27676,"tokens_out":24981,"duration_ms":273058,"significance":"The algebraic core is a useful extension of existing bilinear and LQO Gramian theories to MIMO BQO systems without the (n+n^2) lifting transformation. Strong points include explicit smallness hypotheses, a clean derivation of the Lyapunov equations from Volterra kernels, the identification QS = QP, and efficient truncated algorithms with an available implementation and reproducible code. No fitted constants enter the derivations; the user choices gamma and phi are declared as such. The numerical comparison of QS, QA, and QM is informative. The main deficiency is that the balanced-truncation step is not accompanied by any stability, error, or energy bound, and the experiments do not certify the hypotheses of Theorem 4.1, so the central reduction claim currently rests on heuristics and numerics alone.","major_comments":[{"comment":"The balanced truncation framework is presented without a stability or error theorem for the reduced BQO model (1.3). Algorithm 6.1 constructs \\hat A, \\hat B, \\hat C, \\hat N_k, \\hat M_j, but nowhere is it proved that \\hat A is stable, that the reduced system remains balanced in the same sense, or that \\|y - \\hat y\\| is controlled by discarded Hankel singular values. Theorem 6.1 characterizes only the kernels of P, QS, and QA; it does not imply a reduction error bound. Since the abstract and Section 8 claim that the reduced output approximates the full output, this is a load-bearing gap. Please add such a result, even under the smallness conditions of Theorem 4.1, or explicitly label the reduction step as heuristic and supported only by experiments.","section":"Section 6, Algorithm 6.1"},{"comment":"The numerical experiments never report the constants needed to check the hypotheses of Theorems 4.1 and 4.4. After the gamma-scaling in Section 7, the paper states that the existence conditions are 'easier to hold', but it does not give alpha, beta, Gamma_P, Gamma_QS, or Gamma_QA for either example. If any of these inequalities fail, the algebraic solutions of (3.6), (3.19), and (3.30) may still exist, but they are not proven to coincide with the Volterra-series Gramians of Definitions 3.1, 3.3, and 3.7; consequently, the reductions depicted in Figures 1-6 and Tables 1-2 may lie outside the proven regime. Please verify the inequalities and report the values, or state explicitly that the algebraic Gramians are used by definition in the numerical part.","section":"Section 7"},{"comment":"The truncated Gramians PT, QS_T, QP_T, and QA_T are proposed as cheaper surrogates, but the paper gives no perturbation bound such as \\|P - P_T\\| or \\|QS - QS_T\\| in terms of the discarded terms P_j and QS_j. The statements in Section 5 and the Conclusion that the truncated variants 'approximate the full ones effectively' are supported only by the two examples. A bound using the smallness constants of Theorem 4.1 would make the computational shortcut rigorous; otherwise this claim should be presented as empirical.","section":"Section 5"}],"minor_comments":[{"comment":"The symbol alpha is used both for the exponential decay rate in Theorem 4.1 and for the input-energy bound in the observability functional (6.2), which is confusing; please use two different symbols.","section":"Section 2 and Section 6"},{"comment":"In the proof of Theorem 3.4, the notation N^T_T (presumably the transpose of the block row N^T) is never defined; please introduce a symbol such as \\tilde N = [N_1; ...; N_m] once and use it consistently.","section":"Theorem 3.4 proof"},{"comment":"The text states that the discrete-time output satisfies Y_i \\approx y(t_i) \\in \\mathbb{R}^2 and Y \\in \\mathbb{R}^{2\\times N_t}, but the RC example is SISO; these should be \\mathbb{R} and \\mathbb{R}^{1\\times N_t}.","section":"Section 7.1"},{"comment":"The sentence 'the HSVs of the truncated variants match the HSVs of the methods using the truncated variants' appears to be a typo; presumably it should compare the truncated variants with the full-Gramian variants.","section":"Section 7.1"},{"comment":"The captions of Figures 3 and 6 do not identify which line style corresponds to which method; please add a legend or describe the line styles in the captions.","section":"Figures 3 and 6"},{"comment":"There is an unbalanced parenthesis in the title of Definition 3.7: '(Alternative) Observability Gramian QA)' should read 'Alternative Observability Gramian QA'.","section":"Definition 3.7"},{"comment":"Step 2 of Algorithm 5.2 says 'Solve ... for \\hat Q_S = Q_S^1 + Q_S^2'; this should be phrased as 'Solve for \\hat Q_S, and set \\hat Q_S = Q_S^1 + Q_S^2'.","section":"Algorithm 5.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of math.NA, and the algebraic Gramian derivations appear sound. The main concern is that the central reduction claim needs either a stability/error theorem or a clearly delimited empirical statement, and the numerics must certify the stated smallness hypotheses. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this is a credit-worthy MIMO generalization of balanced truncation for bilinear systems with quadratic outputs, with clean derivations and reproducible numerics. Second, it does not prove any error or stability bound for the reduced model, and the numerical section never checks that the smallness conditions of Theorem 4.1 actually hold.\n\nWhat's genuinely new: the paper extends Padhi's SISO thesis to the MIMO case, introduces two additional observability Gramians (QA and QM) alongside the standard QS, proves their existence and uniqueness under explicit spectral-norm conditions, and shows QS = QP under uniqueness. The Lyapunov equations are derived from Volterra expansions rather than imposed, so there's no circularity. The truncated Gramians are a practical contribution: they reduce the cost to a few standard Lyapunov solves. The numerical examples include a large-scale (n = 40200) RC circuit and a MIMO heat equation, with code on Zenodo. That counts as real evidence.\n\nSoft spots. The main limitation is the absence of stability and error bounds for the reduced BQO model. The paper is honest about this and frames the numerics as demonstrations, but it means the practical claim rests on heuristics. The stress-test note is correct: Section 7 never reports ΓP, ΓQS, ΓQA, α, or β, so the reader can't verify that the Gramians used in the experiments are the ones the existence theorems cover. This is a reporting gap rather than a fatal flaw, since the scaling parameter γ is standard and the theory explicitly conditions on the bounds. A referee should ask the authors to include those checks.\n\nThe finding that QA underperforms QS is an honest, useful negative result. Comparisons among the proposed variants are fair, though a baseline from another reduction method would strengthen the story.\n\nWho this is for: researchers working on model reduction for bilinear or quadratically output systems. It is a solid incremental contribution, not a paradigm shift. It deserves a serious referee: the math is internally consistent, the literature is well covered, and the code makes the numerics reproducible.\n\nRecommendation: send it to peer review with a request to verify the smallness conditions numerically and to temper the reliability claims accordingly.","headline":"Solid MIMO extension of balanced truncation for bilinear quadratic output systems, with clean Gramian theory and reproducible numerics, but no error bounds and unverified smallness conditions.","tokens_in":28232,"tokens_out":3207,"would_cite":true,"duration_ms":33529,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93A15","93B05","93B07","93C10","93C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Bilinear systems with quadratic outputs can be reduced by balanced truncation using newly defined algebraic Gramians and generalized Lyapunov equations.","keywords":["model order reduction","balanced truncation","bilinear systems","quadratic output systems","Gramians","generalized Lyapunov equations","Volterra series","MIMO systems"],"falsifier":"Take a stable $A$ with known decay constants $\\alpha$ and $\\beta$ and choose $N_1$ and $M_1$ with norms satisfying $\\Gamma \\ge 2\\alpha/\\beta^2$; if the fixed-point iteration for $P$ or $Q_S$ still converges to a symmetric positive semidefinite solution, or if balanced truncation still matches the full output, then the stated threshold is not the true boundary of the theory.","tokens_in":27263,"feed_emoji":"📉","tokens_out":5332,"duration_ms":58898,"temperature":0.7,"pith_summary":"This paper extends balanced truncation, a standard model-order-reduction tool for linear systems, to bilinear quadratic output (BQO) systems. The central project is to define algebraic reachability and observability Gramians for these systems and to show that, whenever they exist, they solve generalized Lyapunov equations. The authors prove that existence is guaranteed by a smallness condition on the bilinear and quadratic coupling matrices, and they construct three observability Gramians from different primal-dual formulations, showing that two of them coincide. They then build a balanced-truncation algorithm and demonstrate, on a large-scale nonlinear RC circuit and a MIMO heat-transfer model, that reduced models track the quadratic output while truncated Gramians cut computation cost substantially. A sympathetic reader would care because this gives a structure-preserving reduction path for a class of systems that currently require impractical lifting to much larger dimensions.","feed_headline":"Balanced truncation extended to bilinear systems with quadratic outputs","feed_subtitle":"New algebraic Gramians let reduced models track quadratic outputs, with existence guaranteed under a small-coupling condition.","key_machinery":"The central object is the Volterra-series expansion of the state $x(t)=\\sum_i x_i(t)$ and of the dual state, whose kernel matrices $\\bar{P}_i$ and $\\bar{Q}_i$ define the Gramians as infinite sums of integrals. The carrying identity is the generalized Lyapunov equation: $P$ satisfies $AP + PA^T + \\sum_{k=1}^m N_k P N_k^T + BB^T = 0$, $Q_S$ satisfies $A^T Q_S + Q_S A + \\sum_{k=1}^m N_k^T Q_S N_k + \\sum_{j=1}^p M_j P M_j + C^T C = 0$, and $Q_A$ satisfies $A^T Q_A + Q_A A + \\sum_{j=1}^p M_j P M_j + C^T C = 0$. Existence and uniqueness hinge on the operator-theoretic characterization in Theorem 4.3, applied to the linear operator $L_A(X)=AX+XA^T$ with a nonnegative perturbation $\\Pi$, and on the smallness condition $\\Gamma < 2\\alpha/\\beta^2$. Truncated Gramians replace the generalized equations by two or three standard Lyapunov equations.","core_discovery":"The paper's central claim is that a bilinear quadratic output system of the form (1.1) carries well-defined algebraic reachability and observability Gramians that are computable as limits of Volterra-series kernels and satisfy generalized Lyapunov equations. Specifically, the reachability Gramian $P$ solves $AP + PA^T + \\sum_{k=1}^m N_k P N_k^T + BB^T = 0$; the standard observability Gramian $Q_S$ solves $A^T Q_S + Q_S A + \\sum_{k=1}^m N_k^T Q_S N_k + \\sum_{j=1}^p M_j P M_j + C^T C = 0$; and the alternative Gramian $Q_A$ solves $A^T Q_A + Q_A A + \\sum_{j=1}^p M_j P M_j + C^T C = 0$. The authors prove that under the smallness condition of Theorem 4.1 these equations have unique symmetric positive semidefinite solutions, that $Q_S$ and the independently constructed $Q_P$ coincide, and that balanced truncation built from the square-root factors of $P$ and $Q$ produces a reduced BQO system whose output tracks the full output. The same Gramians recover bilinear and linear-quadratic-output systems as special cases.","pith_inferences":["If the smallness condition is nearly tight, rescaling inputs by $\\gamma<1$ (as done in the experiments) buys existence of the Gramians but changes the input-output map; one could quantify the trade-off between the chosen $\\gamma$ and the resulting reduction error.","The $Q_A$ Gramian's failure to capture the bilinear dynamics under nonzero input suggests that $Q_A$ may still be useful for systems with weak bilinear coupling, or as a cheap initial guess for iterative solves of the $Q_S$ equation.","The equivalence $Q_S=Q_P$ under uniqueness indicates that the two derivations are not competing, and future work could identify when the cheaper truncation $Q_T^P$ is safe, since the examples show it loses slightly at higher reduced orders.","The same dual-system construction could be tested directly on bilinear port-Hamiltonian systems, which are a special case of (1.1), to see whether the reduced model preserves passivity or port-Hamiltonian structure."],"forward_implications":["Reduced BQO models returned by Algorithm 6.1 keep the bilinear state equation and the quadratic output structure, so downstream tasks such as simulation, control, and optimization can treat the reduced model as a BQO system.","Truncated Gramians $P_T$, $Q_T^S$, and $Q_T^P$ require only two or three standard Lyapunov solves, giving substantial speedups while matching full-Gramian accuracy in both numerical examples.","The MIMO treatment covers systems with multiple inputs and outputs, generalizing the previous SISO analysis, with bilinear and linear-quadratic-output systems appearing as special cases.","Because $Q_S$ satisfies a generalized Lyapunov equation while $Q_A$ solves a standard one, $Q_A$ is cheaper but provably yields smaller Hankel singular values and, in the examples, worse output fidelity.","The mixed parameter $\\phi$ in $Q_M$ interpolates between $Q_S$ and $Q_A$, and with $\\phi=\\gamma$ it can be interpreted as an input-scaling choice for the observability equation."],"supporting_citations":[{"why":"Supplies the Volterra-series derivation of the reachability Gramian and the dual-system template used to define observability Gramians.","marker":"[10]"},{"why":"Introduces the alternative $Q_P$ observability Gramian in the SISO case and the same $P$-dependent Lyapunov equation that the paper extends to MIMO.","marker":"[28]"},{"why":"Gives the bilinear-system Gramian and Lyapunov framework, including the existence proof for the reachability Gramian.","marker":"[42]"},{"why":"Provides the energy-functionals and reachability/observability interpretation used in Theorem 6.1, along with the input-scaling trick.","marker":"[8]"},{"why":"Motivates truncated Gramians computed from standard Lyapunov equations, which the paper adapts to $Q_T^S$ and $Q_T^P$.","marker":"[12]"},{"why":"Supplies Theorem 4.3, the operator-spectrum characterization used to prove existence and uniqueness of the Gramian solutions.","marker":"[7]"},{"why":"Provides the same uniqueness theorem for generalized Lyapunov equations cited as Theorem 4.3.","marker":"[16]"},{"why":"Gives the linear-quadratic-output special case whose Gramians and equations are recovered when all $N_k=0$.","marker":"[11]"}],"fun_headline_variants":["Balanced truncation is extended to bilinear systems with quadratic outputs","New algebraic Gramians enable balanced truncation for bilinear quadratic outputs","Small-coupling condition guarantees Gramians for bilinear quadratic outputs","Reduced bilinear quadratic output models track full outputs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes the bilinear and quadratic coupling matrices are small enough that the Volterra series for the state converges, quantified as $\\Gamma < 2\\alpha/\\beta^2$; if that condition fails, the Gramians, the Lyapunov equations, and the balancing argument are undefined and no alternative is offered.","fun_headline_variants_meta":{"raw":{"variants":["Balanced truncation is extended to bilinear systems with quadratic outputs","New algebraic Gramians enable balanced truncation for bilinear quadratic outputs","Small-coupling condition guarantees Gramians for bilinear quadratic outputs","Reduced bilinear quadratic output models track full outputs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000903,"raw_usage":{"total_tokens":3871,"prompt_tokens":917,"completion_tokens":2954,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":2883}},"tokens_in":533,"tokens_out":2954,"duration_ms":23730,"temperature":1.0,"reasoning_tokens":2883,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:06:06.646575+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a stable $A$ with known decay constants $\\alpha$ and $\\beta$ and choose $N_1$ and $M_1$ with norms satisfying $\\Gamma \\ge 2\\alpha/\\beta^2$; if the fixed-point iteration for $P$ or $Q_S$ still converges to a symmetric positive semidefinite solution, or if balanced truncation still matches the full output, then the stated threshold is not the true boundary of the theory.","supporting_citations":[{"cited_title":"Benner and P","cited_arxiv_id":null,"evidence_quote":"Supplies the Volterra-series derivation of the reachability Gramian and the dual-system template used to define observability Gramians."},{"cited_title":"Padhi, Model Order Reduction of Nonlinear Dynamical Systems , master’s thesis, Indian Institute of Science Education and Research Pune, Pashan, Pune India, May 2024","cited_arxiv_id":null,"evidence_quote":"Introduces the alternative $Q_P$ observability Gramian in the SISO case and the same $P$-dependent Lyapunov equation that the paper extends to MIMO."},{"cited_title":"Benner, P","cited_arxiv_id":null,"evidence_quote":"Gives the linear-quadratic-output special case whose Gramians and equations are recovered when all $N_k=0$."}],"review_version":1}