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Bilinear Quadratic Output Systems and Balanced Truncation

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Pith's one-line read Bilinear systems with quadratic outputs can be reduced by balanced truncation using newly defined algebraic Gramians and generalized Lyapunov equations.

desk verdict 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. read the letter →

arxiv 2507.03684 v1 pith:R2EXQWQE submitted 2025-07-04 math.NA cs.NA

Heike Fa{ss}bender (1) , Serkan Gugercin (2) , Till Peters (1) ((1) Institute for Numerical Analysis , TU Braunschweig , (2) Department of Mathematics , Division of Computational Modeling , Data Analytics , Academy of Data Science
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Virginia Tech)
This is my paper · ORCID
classification math.NAcs.NA MSC 93A1593B0593B0793C1093C15
keywords modelorderreductionbalancedtruncationbilinearsystemsquadraticoutputGramiansgeneralizedLyapunovequationsVolterraseriesMIMO
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

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.

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 (3)
  1. [Section 6, Algorithm 6.1] 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.
  2. [Section 7] 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.
  3. [Section 5] 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.
minor comments (7)
  1. [Section 2 and Section 6] 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.
  2. [Theorem 3.4 proof] 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.
  3. [Section 7.1] 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}.
  4. [Section 7.1] 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.
  5. [Figures 3 and 6] 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.
  6. [Definition 3.7] There is an unbalanced parenthesis in the title of Definition 3.7: '(Alternative) Observability Gramian QA)' should read 'Alternative Observability Gramian QA'.
  7. [Algorithm 5.2] 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'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Gramians are defined from Volterra/dual expansions and their Lyapunov equations are derived by direct manipulation, with no fitted parameter relabeled as a prediction and no load-bearing self-citation.

full rationale

The paper derives the reachability Gramian P from the Volterra series solution of the BQO state equation (Definition 3.1 and Theorem 3.2), and the observability Gramians QS, QP, QM, QA from explicit dual-system Volterra constructions (Definitions 3.3, 3.5, 3.7, Theorem 3.11). The Lyapunov equations (3.6), (3.19), (3.30), and (3.31) are obtained by substituting these definitions into the integrals and using the recurrence relations for the Volterra kernels; they are not imposed as the target. The fact that QS is built using reachability kernels Pi is not circular: the dual system in (3.14) genuinely couples z to the primal trajectory x, so the observability mapping necessarily contains Pi terms, and the resulting P-dependent Lyapunov equation is a computed consequence rather than an assumption. The equality QS = QP is not assumed; it follows from Corollary 4.5 because both satisfy the same generalized Lyapunov equation whose unique symmetric positive semidefinite solution is established in Theorem 4.4 using external uniqueness theorems (Theorem 4.3). The existence conditions in Theorem 4.1 are proven by explicit norm bounds, not by citing the conclusion. The scaling parameter gamma and the mixed parameter phi in Section 7 are user-chosen and are reported as such; they are not fitted to the output data and then relabeled as predictions. The numerical experiments compare reduced outputs to full outputs, which is a standard empirical validation of the balancing construction, not a circular use of the Gramians. The only caveat worth noting is that the numerical examples do not explicitly verify that the Theorem 4.1 smallness bounds (Gamma_P, Gamma_QS, Gamma_QA < 2 alpha / beta^2) hold for the reported systems; this is a validation-gap or correctness-risk issue, not a circularity. There is no self-citation chain carrying a load-bearing argument: the cited works [10], [28], [8], [7], and [16] are independent prior results applied as standard tools. Overall, the derivation is self-contained in the sense that the claimed Lyapunov equations and existence results are derived from the definitions rather than being equivalent to the inputs by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central framework is derived rather than fitted: no unknown constants are tuned to match numerical outputs. The only user-chosen quantities are the standard input scaling gamma and the optional mixing coefficients phi in the demonstration. The main structural assumption is the smallness condition guaranteeing Gramian existence, plus the standard dual-system definition of observability Gramians.

free parameters (2)
  • Input scaling factor gamma = 0.1 or 0.5 in the numerical examples
    Chosen by hand to rescale B and N_k so the sufficient existence conditions in Theorem 4.1 hold. It is a standard scaling trick in bilinear model reduction, not fitted to output data.
  • Mixed Gramian weights phi_k = 0, 0.1, 0.5, 1 considered in experiments
    Interpolate between the observability Gramians QS and QA in the primal-dual formulation (3.13). They are not fitted to the output; the paper leaves their optimization to future research.
assumptions (4)
  • standard math Stability of A with decay rate constants beta and alpha such that ||e^{At}|| <= beta e^{-alpha t} for t >= 0.
    Used throughout Theorem 4.1 to bound the Volterra kernels and in Theorem 4.4 for Lyapunov solution bounds. This is a standard property of stable matrices.
  • standard math Theorem 4.3 from references [7,16], characterizing existence and uniqueness for generalized Lyapunov equations via the spectrum of L_A + Pi and the spectral radius of L_A^{-1} Pi.
    Invoked in Theorem 4.4 to prove that the generalized Lyapunov equations have unique positive semidefinite solutions under the stated conditions.
  • domain assumption The Volterra series expansion (3.2) of the bilinear state equation converges when the smallness conditions hold.
    This expansion defines the reachability Gramian P and is the basis for term-wise derivation of the Lyapunov equations. Convergence is asserted under Gamma < 2 alpha / beta^2.
  • domain assumption The observability Gramian is defined as the reachability Gramian of the dual system, following [10,17].
    This is the central modeling choice that produces QS, QA, and QM. Its validity is only partially justified through the energy-functionals in Theorem 6.1 under extra hypotheses such as P > 0 or zero input.

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Pith. "Pith review of Bilinear Quadratic Output Systems and Balanced Truncation." pith.science (2026). https://pith.science/paper/R2EXQWQE

@misc{pith2026250703684,
  author       = {Pith},
  title        = {Pith review of: Bilinear Quadratic Output Systems and Balanced Truncation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R2EXQWQE}},
  note         = {Machine review of arXiv:2507.03684}
}
read the original abstract

Dynamical systems with quadratic outputs have recently attracted significant attention. In this paper, we consider bilinear dynamical systems, a special class of weakly nonlinear systems, with a quadratic output. We develop various primal-dual formulations for these systems and define the corresponding system Gramians. Conditions for the existence and uniqueness of these Gramians are established, and the generalized Lyapunov equations they satisfy are derived. Using these Gramians and their truncated versions, which are computationally more efficient, we construct a balanced truncation framework for bilinear systems with quadratic outputs. The proposed approach is demonstrated through two numerical examples.

Figures

Figures reproduced from arXiv: 2507.03684 by the authors.

Figure 1
Figure 1. RC example, full Gramians using γ = 0.1 (first row) and γ = 0.5 (second row) Next we present the results using the truncated Gramians, with γ = 0.1 and omit￾ting the QM-Gramians. As shown in [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. RC example, full vs truncated Gramians 5 10 15 20 25 30 35 40 10−4 10−3 10−2 Order r ∥Y − Y (r)∥F ∥Y ∥F Output Frobenius norm error, n = 40200, γ = 0.1 BT BQO(P, QS) BT BQO(P, QA) BT BQO(PT , QP T ) BT BQO(PT , QS T ) BT BQO(PT , QA T ) [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. RC example, full vs truncated Gramians [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Heat equation example, full vs truncated Gramians, [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: Heat equation example, full vs truncated Gramians, [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]
Figure 6
Figure 6. Figure 6: Heat equation example, full vs truncated Gramians, [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Bilinear Systems with Quadratic Outputs: $\mathcal{H}_2$ Analysis, Optimality Conditions for Model Reduction, and Algorithmic Solutions

    math.NA 2026-07 conditional novelty 6.0 of 10

    An H2 inner product, norm, output bounds, and first-order optimality conditions are derived for bilinear systems with quadratic outputs, and an iteration (BQO-TSIA) is shown to meet the conditions upon convergence.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.