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

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Pith's one-line read This paper develops a complete H2 theory for bilinear systems with quadratic outputs (BQO), giving a computable norm, output error bounds, and a projection algorithm whose converged reduced models provably satisfy first-order optimality con

desk verdict Solid H2 framework for BQO model reduction, but a wrong convergence condition in Theorem 3.1 undermines the existence assumptions; fixable and worth refereeing. read the letter →

arxiv 2607.19507 v1 pith:IRDXWTHW submitted 2026-07-21 math.NA cs.NAmath.OC

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.NAmath.OC MSC 93A1593B4093C1093C15
keywords H2modelorderreductionbilinearsystemsquadraticoutputsoptimalityconditionsPetrov–GalerkinprojectiongeneralizedLyapunovequationsVolterrakernels
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 develops a complete H2 theory for bilinear systems with quadratic outputs (BQO), a class where both the state dynamics and the measured output depend on quadratic state terms. The central claim is that every sufficiently stable BQO system has an H2 norm that can be computed from finite matrix equations rather than an infinite Volterra series — explicitly, the squared norm equals tr(CPC^T) plus a sum of trace terms involving the quadratic-output matrices, and also equals tr(B^T Q B), where P and Q solve generalized Lyapunov equations. On this foundation the authors build output-error bounds, first-order necessary optimality conditions for the best H2 reduced model, and an iterative two-sided projection algorithm (BQO-TSIA) that, on convergence, provably satisfies those conditions. If the theory holds, it unifies the separate H2 model-reduction frameworks for bilinear systems and for linear systems with quadratic outputs, and gives large-scale simulations a principled way to replace a BQO model by a much smaller one with error control.

What carries the argument

The load-bearing identity is the generalized Sylvester/Lyapunov trace duality: tr(E^T Y) = tr(F^T X) whenever X and Y solve the paired equations (3.10). It converts the infinite Volterra-kernel definition of the H2 inner product into finite trace formulas involving a cross-Gramian X and a dual variable Pi. The algorithm's engine is the two-sided projection update V = orth(X), W = orth(Pi), where X and Pi are the solutions of (5.6a)–(5.6b), so that the reduced matrices are built as (W^T V)^{-1} W^T A V, W^T B, C V, W^T N_k V, V^T M_j V.

What would settle it

Compute the H2 norm of a small BQO system both by numerical quadrature of the Volterra-kernel series and by the closed-form trace formula (4.3); a mismatch for any system satisfying the stated bounds would disprove Theorem 4.2. More sharply, run BQO-TSIA on a system satisfying the hypotheses and check whether the converged reduced matrices satisfy (5.2a)–(5.2e); a violation would disprove Theorem 5.3.

Watch

Extended reading notes

Core claim

The paper's central discovery is an explicit, computable formula for the H2 inner product between two BQO systems, and hence the H2 norm of one. The norm equals ||S||^2_{H2} = tr(CPC^T) + sum_{j=1}^p tr(PM_j P M_j) = tr(B^T Q B), where P is the reachability Gramian solving AP + PA^T + sum N_k P N_k^T + BB^T = 0 and Q solves the dual equation A^T Q + Q A + sum N_k^T Q N_k + sum M_j P M_j + C^T C = 0. This makes the norm finite and computable. From the same formulas the authors derive first-order necessary optimality conditions for an H2-optimal reduced BQO system, and show that any reduced system meeting them is equivalent to a Petrov–Galerkin projection using V = X P-hat^{-1} and W = Pi Psi^

Load-bearing premise

The whole theory rests on the assumption that the bilinear coupling matrices N (and the analogous reduced-system matrices) are small enough that the infinite series defining the Gramians and the H2 norm converge — explicitly ||N||, ||N^T|| < 2 alpha / beta^2 — otherwise the generalized Lyapunov equations have no unique solution and the H2 norm simply does not exist.

Editorial extensions

If this is right

  • For any BQO system satisfying the stability conditions, the H2 norm and H2 distance become computable quantities — just solve two generalized Lyapunov equations — so error measurement no longer requires time-domain simulation of the full model.
  • The output bound of Theorem 4.7 gives a rigorous worst-case guarantee: if a reduced model is close to the full system in H2 norm, its output error is provably small for bounded L2 inputs, which licenses H2-optimal reduction as the right objective.
  • The optimality conditions (5.2) provide an explicit target: a locally H2-optimal reduced BQO system must satisfy the five matrix equations; any algorithm that reaches them is provably at a first-order necessary optimum.
  • The framework recovers the bilinear and LQO H2 theories as special cases (M_j = 0 or N_k = 0), so both research lines become instances of one BQO theory with a single algorithm.
  • Numerical tests on a Carleman-bilinearized RC circuit and a heat-transfer MIMO problem show the algorithm can out-perform balanced truncation in accuracy and speed, and the global-GMRES variant is the fastest of the four methods compared.

Reading between the lines

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

  • The structure of the optimality conditions (5.2) closely parallels the generalized tangential interpolation conditions of linear and bilinear H2 model reduction, so one may be able to prove a matching interpolation theorem for BQO systems without needing the fixed-point iteration — a direct, non-iterative construction of optimal reduced models.
  • The existence bound ||N|| < 2 alpha / beta^2 is the real bottleneck: the theory cannot certify systems with strong bilinear coupling, and the input-rescaling trick used in the experiments only shifts the effective coupling, so extending the norm definition (e.g., through finite-horizon or density-based arguments) would substantially broaden the class of tractable systems.
  • Since the final projection is determined by X and Pi alone, a per-iteration complexity analysis of BQO-TSIA would likely show a cost of O(n r^2 m) for solving the Sylvester equations via fixed-point iteration, plus the cost of the projection — the paper does not analyze this, but it suggests the method can scale to the large n typical of discretized PDEs.
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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

2 major / 3 minor

Summary. The paper develops an H2 framework for bilinear systems with quadratic outputs (BQO). It defines an H2 inner product and norm via Volterra kernels, derives explicit formulas in terms of solutions of generalized Lyapunov/Sylvester equations, proves output bounds in terms of the H2 norm, derives first-order necessary optimality conditions for H2-optimal model reduction, and proposes an iterative projection algorithm (BQO-TSIA) that satisfies these conditions upon convergence. The framework is intended to unify existing H2 theories for bilinear systems and for linear systems with quadratic outputs. Two numerical examples compare the proposed algorithm with balanced truncation.

Significance. If the existence hypotheses are corrected, this is a valuable contribution. The explicit H2 inner-product/norm formulas (Theorem 4.2), the output bounds (Theorem 4.7, Corollary 4.8), and the first-order optimality conditions (Theorem 5.1) extend and unify prior work on bilinear and LQO systems in a natural way. The trace identities and Lagrange-gradient computations are carefully performed and appear sound. The projection-based characterization (Theorem 5.2) and the BQO-TSIA algorithm are elegant. The paper also provides reproducible code on ZENODO, which is a strength. However, the central existence premise, stated in Theorem 3.1/Remark 3.2 and propagated through Theorem 3.4 and Theorem 4.2, is quantitatively incorrect as written; this currently undermines the H2 norm, the optimality conditions, and the algorithm's theoretical guarantees.

major comments (2)
  1. [Theorem 3.1 / Remark 3.2] The stated sufficiency condition Γ := max(||N||_2, ||N^T||_2) < 2α/β^2 does not ensure existence of a PSD reachability Gramian. For the scalar SISO BQO system A=-1, B=1, N=1.5, M=0 with α=β=1, the condition holds (1.5<2), yet Eq. (3.2a) has unique solution P=-4, and the Volterra series P=ΣP_i with P_i=(N^2/2)^{i-1}/2 diverges. The correct contraction condition for the fixed-point map is β^2||N||_2^2/(2α)<1, i.e. ||N||_2<√(2α)/β, not the linear condition. This also affects Corollary 3.3, Remark 3.2, and the looser bound in Remark 3.5. Please correct the hypothesis throughout; the scalar example shows that without this correction Theorem 4.2 would give a negative H2 norm via (4.3).
  2. [Theorem 3.4] The proof asserts that ||L^{-1}_{A,\hat A}Π_{N,\hat N}||<1 follows from the stated assumptions. This is not implied. Using the standard bound ||L^{-1}||≤β\hatβ/(α+\hatα) and ||Π||≤||N||_2||\hat N||_2, the hypotheses ||N||_2<2α/β^2 and ||\hat N||_2<2\hatα/\hatβ^2 only give an upper bound of order 4α\hatα/(α+\hatα), which is generally not less than 1. Thus the fixed-point contraction in the proof of Eq. (3.5) is not established, and uniqueness of X in (3.9a) — and hence Corollary 3.6 and Theorem 4.2 — is not guaranteed. A sufficient condition would be β\hatβ||N||_2||\hat N||_2/(α+\hatα)<1, which is satisfied, for example, if ||N||_2<√(2α)/β and ||\hat N||_2<√(2\hatα)/\hatβ.
minor comments (3)
  1. [Theorem 5.3 / Algorithm 5.1] The convergence of Algorithm 5.1 is assumed rather than proved. This is stated explicitly in Theorem 5.3, but the abstract and introduction describe the method as an 'algorithmic solution' without this caveat. Please make the conditional nature of the convergence claim more prominent.
  2. [Section 4.1] The notation for the multi-indexed input u^{(t_1,...,t_l)}_k(t) in the Volterra series expansion is dense and not fully defined. A short explanation or a reference for this notation would improve readability.
  3. [Section 6] The figures and tables are clear, but the text states that BQO_TSIA has a convergence issue for order 8 without further discussion. A brief comment on the cause, or on whether this is related to the existence conditions in Theorem 3.1, would be useful.

Circularity Check

0 steps flagged · score 1.0 of 10

Core derivations are self-contained; only minor load-bearing reliance on the authors' prior [12] for the Gramian existence condition, which is not circular but is a stated assumption/premise.

full rationale

The paper's central project—defining an H2 inner product/norm for BQO systems, deriving output bounds, and obtaining first-order optimality conditions—is carried out within the manuscript itself. Definition 4.1 defines the H2 inner product via Volterra kernels; Theorem 4.2 computes it by substituting kernel definitions and applying Lemma 3.7 to the Sylvester equations, a direct computation rather than a renaming. The output bounds (Theorems 4.5–4.7, Corollary 4.8) are obtained from explicit integral estimates, not from assuming the conclusion. Theorem 5.1 derives the optimality conditions through a genuine constrained-optimization / Lagrange multiplier argument, and Theorem 5.3 verifies that Algorithm 5.1's convergence implies those conditions. The only prerequisites imported from elsewhere are the existence/uniqueness of Gramians and Sylvester solutions, taken from the authors' companion paper [12, Theorems 4.1/4.4] (Theorem 3.1 here). This is a hypothesis/infrastructure citation, not a result whose conclusion is the target result; the H2 theorems are not reduced to [12] by construction. There is a self-citation component (Gugercin co-authors [12] and [28], and [28] contributes the stopping criterion and initialization), but it is not load-bearing for the derivation of the optimality conditions. The reader's/critical worry that the stated sufficient condition in Theorem 3.1/3.4 (||N||_2 < 2α/β^2) may be quantitatively too weak is a correctness/soundness concern about a cited assumption, not evidence that the present derivations are circular: the H2 results are conditional on that existence premise, not obtained by assuming the H2 results themselves. Therefore no specific circular step can be exhibited, and the score remains low.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The core theory introduces no free parameters; the H2 norm and optimality conditions are determined by the system data. The listed free parameters are numerical-experiment heuristics. The axioms are the usual stability/smallness assumptions for bilinear Gramians and the conditional convergence assumption for the algorithm.

free parameters (3)
  • input scaling γ = 0.1
    Section 6: the input is rescaled u→u/γ with γ=0.1 to force the smallness conditions for Gramian existence; this changes the system being reduced and is chosen by hand.
  • initial reduced spectrum placement = log-spaced diagonal entries
    Section 6: initial  is diagonal with logarithmically spaced entries and B̂/Ĉ are initial columns/rows of the identity; this heuristic affects the local minimum reached.
  • stopping tolerance for τ^{(j)} = 1e-6
    Section 6: stopping criterion |τ^{(j+1)}−τ^{(j)}|/|τ^{(1)}| < 10^{-6}; a standard algorithmic tolerance.
assumptions (5)
  • domain assumption A and  are stable (eigenvalues in left half-plane)
    Throughout; ensures exponential decay of e^{At} and existence of Gramians.
  • domain assumption Smallness conditions ||N||₂, ||N^T||₂ < 2α/β² and reduced analogues hold
    Theorem 3.1, Remark 3.2; guarantees unique solutions to the generalized Lyapunov/Sylvester equations and convergence of the infinite series defining P, Q, X, Y and the H2 norm.
  • domain assumption The infinite series in Definition 4.1 converge
    Definition 4.1, Theorem 4.2; the H2 norm is defined only when the Volterra-kernel series converge, which the smallness conditions aim to ensure.
  • domain assumption P̂ and Ψ are invertible
    Theorem 5.2; needed to define V=X P̂^{-1} and W=Π Ψ^{-1} for the Petrov-Galerkin construction of the optimal reduced system.
  • ad hoc to paper Algorithm 5.1 converges
    Theorem 5.3 is stated as 'Assume Algorithm 5.1 converges'; no convergence proof is given, and Section 6.1 reports non-convergence for RC order 8.

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Pith. "Pith review of Bilinear Systems with Quadratic Outputs: $\mathcal{H}_2$ Analysis, Optimality Conditions for Model Reduction, and Algorithmic Solutions." pith.science (2026). https://pith.science/paper/IRDXWTHW

@misc{pith2026260719507,
  author       = {Pith},
  title        = {Pith review of: Bilinear Systems with Quadratic Outputs: $\mathcalH_2$ Analysis, Optimality Conditions for Model Reduction, and Algorithmic Solutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRDXWTHW}},
  note         = {Machine review of arXiv:2607.19507}
}
abstract

Bilinear systems with quadratic outputs (BQO) have recently emerged as an important system class, arising naturally in applications where both the dynamics and the quantities of interest depend nonlinearly on the state. Despite the growing interest in this class of systems, a systematic $\mathcal{H}_2$ framework for BQO systems has been lacking. In this paper, we develop such a framework by establishing an $\mathcal{H}_2$ inner product and norm for BQO systems, deriving output bounds in terms of the $\mathcal{H}_2$ norm, and obtaining first-order optimality conditions for $\mathcal{H}_2$ optimal model reduction. Building on these theoretical foundations, we propose an algorithm that computes a reduced BQO system satisfying these optimality conditions, and thus generalizing existing $\mathcal{H}_2$ optimal methods for bilinear and linear quadratic output systems. The effectiveness of the proposed framework is demonstrated on two numerical examples.

Figures

Figures reproduced from arXiv: 2607.19507 by the authors.

Figure 6
Figure 6. for ascending reduced orders. We observe that the balancing algorithms [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 6.1
Figure 6.1. for ascending reduced orders. We observe that the balancing algorithms BT BQO(P, Q) and BT BQO(PT , QT ) show identical results and provide a better output error for smaller orders from 2 to 3. But for higher orders, the H2 inspired methods produce a slightly smaller output error. Comparing these H2 inspired methods, we see small differences but the same tendency which can be explained by converging to only local mi… view at source ↗
Figure 6.2
Figure 6.2. Heat equation example, Left plot: Relative [PITH_FULL_IMAGE:figures/full_fig_p022_6_2.png] view at source ↗

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Works this paper leans on

78 extracted references · 30 canonical work pages

  1. [10]

    Balanced truncation for model order reduction of linear dynamical systems with quadratic outputs , author=. SIAM J. Sci. Comput. , volume=. 2019 , publisher=

    2019

  2. [1]

    Antoulas, A. C. and Beattie, C. A. and Güğercin, S. , title =. 2020 , doi =

    2020

  3. [2]

    Optimal. Int. J. Numer. Anal. Model. , author =. 2011 , pages =

    2011

  4. [3]

    Global. Appl. Numer. Math. , author =. 1999 , keywords =. doi:10.1016/S0168-9274(98)00094-4 , number =

    work page doi:10.1016/s0168-9274(98)00094-4 1999

  5. [4]

    Multipoint. SIAM J. Matrix Anal. Appl. , author =. 2015 , keywords =. doi:10.1137/130947830 , number =

    work page doi:10.1137/130947830 2015

  6. [5]

    Willems, J. L. , title =

  7. [6]

    Extending balanced truncation for model reduction of bilinear systems with quadratic outputs , author =

  8. [7]

    Bilinear Quadratic Output Systems and Balanced Truncation

    Faßbender, Heike and Gugercin, Serkan and Peters, Till , month = jul, year =. Bilinear. doi:10.48550/arXiv.2507.03684 , publisher =

    work page Pith review arXiv doi:10.48550/arxiv.2507.03684

Show all 78 references
  1. [8]

    Mohler, R.R. , year=

  2. [9]

    afer-Bung, C and Th\

    Hartmann, C. and Sch\"afer-Bung, C and Th\"ons-Zueva, A. , title =. SIAM J. Control Optim. , volume =

  3. [11]

    Energy-based model order reduction for linear stochastic

    Pulch, Roland , journal=. Energy-based model order reduction for linear stochastic. 2023 , publisher=

    2023

  4. [12]

    Accelerating optimization of parametric linear systems by model order reduction , author=. SIAM J. Optim. , volume=. 2013 , publisher=

    2013

  5. [13]

    IFAC-PapersOnLine , volume=

    Interpolatory model reduction of dynamical systems with root mean squared error , author=. IFAC-PapersOnLine , volume=. 2025 , publisher=

    2025

  6. [14]

    Internat

    Model reduction for dynamical systems with quadratic output , author=. Internat. J. Numer. Methods Engrg. , volume=. 2012 , publisher=

    2012

  7. [15]

    Control of port-

    Mehrmann, Volker and Unger, Benjamin , journal=. Control of port-. 2023 , publisher=

    2023

  8. [16]

    van der Schaft, Arjan , booktitle=. Port-. 2006 , organization=

    2006

  9. [17]

    Structure-

    Mlinaric, Petar , year =. Structure-

  10. [18]

    IEEE Trans

    An. IEEE Trans. Automat. Control , author =

  11. [20]

    , journal=

    Moore, B. , journal=. Principal component analysis in linear systems: Controllability, observability, and model reduction , year=

  12. [21]

    and Roberts, R

    Mullis, C. and Roberts, R. , journal=. Synthesis of minimum roundoff noise fixed point digital filters , year=

  13. [22]

    , year =

    Rugh, Wilson J. , year =. Nonlinear

  14. [23]

    IFAC Proceedings Volumes , author =

    Energy. IFAC Proceedings Volumes , author =. 1998 , pages =. doi:10.1016/S1474-6670(17)40318-1 , language =

    1998 doi

  15. [24]

    Systems & Control Letters , author =

    Balancing for nonlinear systems , volume =. Systems & Control Letters , author =. 1993 , keywords =. doi:10.1016/0167-6911(93)90117-O , number =

    1993 doi

  16. [25]

    Linear Algebra Appl

    A projection method for model reduction of bilinear dynamical systems , volume =. Linear Algebra Appl. , author =. 2006 , keywords =. doi:10.1016/j.laa.2005.04.032 , number =

    2006 doi

  17. [26]

    Direct methods and. Numer. Linear Algebra Appl. , author =. 2008 , keywords =. doi:10.1002/nla.603 , language =

    2008 doi

  18. [27]

    2024 , eprint=

    Interpolatory model order reduction of large-scale dynamical systems with root mean squared error measures , author=. 2024 , eprint=

    2024

  19. [28]

    Saak, J. and K\". doi:10.5281/zenodo.7701424 , key =

    doi

  20. [29]

    Adrianova, L. Ya. , Title =. 1995 , Publisher =

    1995

  21. [30]

    COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , author =

    Nonlinear systems – algebraic gramians and model reduction , volume =. COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , author =. 2005 , pages =. doi:10.1108/03321640510571147 , language =

    2005 doi

  22. [31]

    , title =

    Antoulas, Athanasios C. , title =. 2005 , doi =

    2005

  23. [32]

    Modellreduktion:

    Benner, Peter and Faßbender, Heike , year =. Modellreduktion:. doi:10.1007/978-3-662-67493-2 , note =

    doi

  24. [33]

    and Johnson, Charles R

    Horn, Roger A. and Johnson, Charles R. , Title =. 1994 , Publisher =

    1994

  25. [34]

    and Johnson, Charles R

    Horn, Roger A. and Johnson, Charles R. , year =. Matrix analysis , isbn =

  26. [35]

    Sorensen, D. C. and Antoulas, A. C. , month = aug, year =. The. doi:10.1016/S0024-3795(02)00283-5 , journal =

    doi

  27. [36]

    Padhi, Reetish , month = may, year =. Model

  28. [37]

    and Van Loan, Charles F

    Golub, Gene H. and Van Loan, Charles F. , title =. 2013 , doi =

    2013

  29. [38]

    Interpolatory methods for model reduction of large-scale dynamical systems , doi =

    Breiten, Tobias , year =. Interpolatory methods for model reduction of large-scale dynamical systems , doi =

  30. [39]

    and Bettayeb, M

    Al-Baiyat, S.A. and Bettayeb, M. , month = dec, year =. A new model reduction scheme for k-power bilinear systems , doi =. Proceedings of 32nd

  31. [40]

    \. SIAM J. Matrix Anal. Appl. , author =. 2018 , pages =. doi:10.1137/16M1098280 , language =

    2018 doi

  32. [41]

    Interpolation-. SIAM J. Matrix Anal. Appl. , author =. 2012 , keywords =. doi:10.1137/110836742 , number =

    2012 doi

  33. [42]

    Matrix Anal

    SIAM J. Matrix Anal. Appl. , author =. 2008 , pages =. doi:10.1137/060666123 , number =

    2008 doi

  34. [43]

    Bilinear. SIAM J. Control Optim. , author =. 2021 , keywords =. doi:10.1137/19M1304106 , language =

    2021 doi

  35. [44]

    Advances in Computational Mathematics , author =

    Structure-preserving interpolation of bilinear control systems , volume =. Advances in Computational Mathematics , author =. 2021 , keywords =. doi:10.1007/s10444-021-09863-w , language =

    2021 doi

  36. [45]

    Error bounds for port-

    Breiten, Tobias and Morandin, Riccardo and Schulze, Philipp , month = jul, year =. Error bounds for port-

  37. [46]

    Lyapunov. SIAM J. Control Optim. , author =. 2011 , keywords =. doi:10.1137/09075041X , number =

    2011 doi

  38. [47]

    International Journal of Systems Science , author =

    Convergence of. International Journal of Systems Science , author =. 1991 , pages =. doi:10.1080/00207729108910824 , number =

    1991 doi

  39. [48]

    Automatica , author =

    On. Automatica , author =. 2002 , keywords =. doi:10.1016/S0005-1098(01)00204-7 , number =

    2002 doi

  40. [49]

    Systems & Control Letters , author =

    Effort- and flow-constraint reduction methods for structure preserving model reduction of port-. Systems & Control Letters , author =. 2012 , keywords =. doi:10.1016/j.sysconle.2011.12.008 , number =

    2012 doi

  41. [50]

    2002 , issn =

    Hamiltonian realizations of nonlinear adjoint operators , journal =. 2002 , issn =. doi:https://doi.org/10.1016/S0005-1098(02)00079-1 , author =

    2002 doi

  42. [51]

    IEEE Trans

    Gramians,. IEEE Trans. Automat. Control , author =. 2022 , keywords =. doi:10.1109/TAC.2021.3086319 , number =

    2022

  43. [52]

    Balanced truncation for quadratic-bilinear control systems , volume =. Adv. Comput. Math. , author =. 2024 , keywords =. doi:10.1007/s10444-024-10186-9 , language =

    2024 doi

  44. [53]

    Model order reduction of linear and nonlinear systems in the

    Gosea, Ion Victor , month = feb, year =. Model order reduction of linear and nonlinear systems in the

  45. [54]

    and Powell, J

    Franklin, Gene F. and Powell, J. David and Workman, Michael , month = jan, year =. Digital control of dynamic systems , isbn =

  46. [55]

    IMA Journal of Mathematical Control and Information , author =

    On the. IMA Journal of Mathematical Control and Information , author =. 1986 , pages =. doi:10.1093/imamci/3.2-3.61 , language =

    1986 doi

  47. [56]

    Applied Mathematics Letters , author =

    The. Applied Mathematics Letters , author =. 2016 , pages =. doi:10.1016/j.aml.2015.10.011 , language =

    2016 doi

  48. [57]

    Introduction to

    Starr, Gregory , year =. Introduction to

  49. [58]

    SIAM Review , author =

    Barycentric. SIAM Review , author =. 2004 , pages =. doi:10.1137/S0036144502417715 , language =

    2004 doi

  50. [59]

    Systems & Control Letters , author =

    Identification of port-. Systems & Control Letters , author =. 2020 , pages =. doi:10.1016/j.sysconle.2020.104741 , language =

    2020

  51. [60]

    ETNA - Electronic Transactions on Numerical Analysis , author =

    A non-intrusive method to inferring linear port-. ETNA - Electronic Transactions on Numerical Analysis , author =. 2022 , pages =. doi:10.1553/etna_vol56s102 , language =

    2022 doi

  52. [61]

    SIAM Journal on Scientific Computing , author =

    Data-. SIAM Journal on Scientific Computing , author =. 2017 , pages =. doi:10.1137/16M1094750 , language =

    2017 doi

  53. [62]

    Linear Algebra and its Applications , author =

    A framework for the solution of the generalized realization problem , volume =. Linear Algebra and its Applications , author =. 2007 , pages =. doi:10.1016/j.laa.2007.03.008 , language =

    2007 doi

  54. [63]

    , year =

    Larimore, W.E. , year =. Canonical variate analysis in identification, filtering, and adaptive control , doi =. 29th

  55. [64]

    Automatica , author =

    Identification of the deterministic part of. Automatica , author =. 1994 , keywords =. doi:10.1016/0005-1098(94)90229-1 , number =

    1994 doi

  56. [65]

    2017 , keywords =

    Model reduction and approximation: theory and algorithms , isbn =. 2017 , keywords =

    2017

  57. [66]

    Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems , language =

    Ionita, Antonio , month = nov, year =. Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems , language =

  58. [67]

    Acta Numerica , author =

    Control of port-. Acta Numerica , author =. 2023 , pages =. doi:10.1017/S0962492922000083 , language =

    2023 doi

  59. [68]

    Automatica , author =

    Structure-preserving tangential interpolation for model reduction of port-. Automatica , author =. 2012 , keywords =. doi:10.1016/j.automatica.2012.05.052 , number =

    2012 doi

  60. [69]

    Reiter, Sean and Duff, Igor Pontes and Gosea, Ion Victor and Gugercin, Serkan , month = may, year =

  61. [70]

    Advances in Computational Mathematics , author =

    Interpolatory model reduction of quadratic-bilinear dynamical systems with quadratic-bilinear outputs , volume =. Advances in Computational Mathematics , author =. 2023 , keywords =. doi:10.1007/s10444-023-10096-2 , language =

    2023 doi

  62. [71]

    Volume 1 System- and Data-Driven Methods and Algorithms , editor =

    Tobias Breiten and Tatjana Stykel , title =. Volume 1 System- and Data-Driven Methods and Algorithms , editor =. 2021 , lastchecked =. doi:doi:10.1515/9783110498967-002 , isbn =

    2021 doi

  63. [72]

    , Title =

    Sontag, Eduardo D. , Title =. 1998 , Publisher =

    1998

  64. [73]

    and Glover, Keith , Title =

    Zhou, Kemin and Doyle, John C. and Glover, Keith , Title =. 1996 , Publisher =

    1996

  65. [74]

    Truncated Gramians for Bilinear Systems and Their Advantages in Model Order Reduction

    Benner, Peter and Goyal, Pawan and Redmann, Martin. Truncated Gramians for Bilinear Systems and Their Advantages in Model Order Reduction. Model Reduction of Parametrized Systems. 2017. doi:10.1007/978-3-319-58786-8_18

    2017 doi

  66. [75]

    and Simoncini, Valeria and Szyld, Daniel B

    Shank, Stephen D. and Simoncini, Valeria and Szyld, Daniel B. , Title =. Numer. Math. , ISSN =. 2016 , Language =. doi:10.1007/s00211-015-0777-7 , Keywords =

    2016 doi

  67. [76]

    Benner, Peter and Breiten, Tobias , Title =. Numer. Math. , ISSN =. 2013 , Language =. doi:10.1007/s00211-013-0521-0 , Keywords =

    2013 doi

  68. [77]

    Model Reduction and Approximation: Theory and Algorithms , chapter =

    Peter Benner and Tobias Breiten , title =. Model Reduction and Approximation: Theory and Algorithms , chapter =. 2017 , publisher =

    2017

  69. [78]

    International Journal of Control , volume=

    A survey of model reduction by balanced truncation and some new results , author=. International Journal of Control , volume=. 2004 , doi =

    2004

  70. [79]

    Proceedings of the Institution of Electrical Engineers , volume=

    Optimum solution of model-reduction problem , author=. Proceedings of the Institution of Electrical Engineers , volume=. 1970 , organization=

    1970

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