pith. sign in

arxiv: 1907.01684 · v1 · pith:TJTWMVMDnew · submitted 2019-07-03 · 📡 eess.SY · cs.SY· eess.SP

Solvents based model reduction of linear systems

Karim Cherifi , Kamel Hariche This is my paper

Pith reviewed 2026-05-25 10:38 UTC · model grok-4.3

classification 📡 eess.SY cs.SYeess.SP
keywords model order reductionMIMO linear systemssolventsblock polesmatrix transfer functionMATLAB implementation
0
0 comments X

The pith

Two solvent-based methods reduce the order of MIMO linear systems given in matrix transfer function form.

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

The paper presents two approaches to approximate high-order MIMO linear systems with lower-order equivalents by removing solvents, also known as block poles, from the system matrix. These techniques apply directly when the system is expressed as a matrix transfer function, with one method removing solvents sequentially and the other removing several at once. A reader would care because the methods aim to simplify dynamical systems while keeping essential input-output behavior, and the authors supply MATLAB implementations to make the process systematic and repeatable.

Core claim

Model order reduction for MIMO linear systems proceeds by identifying and eliminating solvents from the matrix transfer function; the first procedure removes them individually while the second removes multiple solvents simultaneously, both yielding reduced-order models that preserve the original input-output characteristics, with the procedures coded in MATLAB for direct application.

What carries the argument

Solvents (block poles), which serve as matrix factors of the transfer function that can be selectively removed to lower system order.

If this is right

  • Systems in matrix transfer function form can be reduced by sequential solvent removal for fine control over the final order.
  • Multiple solvents can be eliminated together to achieve larger order reductions in a single step.
  • The MATLAB implementations turn the solvent removal process into an automated computational procedure.
  • The approach is positioned as particularly convenient when the model is already available in transfer-function matrix form rather than state-space form.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If a conversion step from state-space to transfer-function matrix is added, the methods could apply to a broader class of models.
  • The preservation of input-output maps suggests the reduced models remain usable for controller design or simulation where exact internal dynamics are secondary.
  • Direct numerical comparison of the resulting reduced models against established techniques such as balanced truncation on the same examples would quantify relative accuracy.

Load-bearing premise

The system is already written as a matrix transfer function and has well-defined solvents that can be removed without changing essential input-output behavior.

What would settle it

A concrete MIMO transfer function where removing one or more identified solvents produces a reduced model whose step response or frequency response deviates measurably from the original beyond what the method claims to preserve.

read the original abstract

Model order reduction is the approximation of dynamical systems into equivalent systems with smaller order. Model reduction has been studied extensively for different types of systems. In this paper, we present two methods for multi input multi output linear systems. These methods are based on solvents, also called block poles. These methods are particularly suitable if the given system is in matrix transfer function form. The first method eliminates solvents one by one whereas, the second method can eliminate multiple solvents at the same time. The two presented methods are implemented in MATLAB in order to provide a systematic method for the model order reduction of MIMO linear systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims to present two methods for model order reduction of MIMO linear systems based on solvents (block poles). The first eliminates solvents sequentially and the second eliminates multiple solvents simultaneously; both are implemented in MATLAB and are intended for systems supplied in matrix transfer function form.

Significance. If the procedures correctly factor and deflate the matrix polynomial while matching the original system on the retained poles, the work would supply a direct polynomial-matrix route to reduction that avoids state-space conversion, which is uncommon in the literature.

major comments (2)
  1. [Abstract] Abstract: the central claim of existence, utility, and MATLAB implementation of the two methods rests on assertion; the manuscript supplies no derivation steps, error analysis, stability proofs, or numerical examples to support that the reduced-order rational matrix preserves essential input-output behavior.
  2. The precondition that the given system already possesses well-defined solvents removable without destroying input-output behavior is stated but never tested or illustrated, leaving the practical scope of both algorithms unexamined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive comments. We address each major comment below and will revise the manuscript to incorporate additional supporting material where the current version is lacking.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of existence, utility, and MATLAB implementation of the two methods rests on assertion; the manuscript supplies no derivation steps, error analysis, stability proofs, or numerical examples to support that the reduced-order rational matrix preserves essential input-output behavior.

    Authors: The manuscript describes the two solvent-elimination procedures (sequential and simultaneous) for systems given in matrix transfer-function form and states that they are implemented in MATLAB. We agree that the current version does not include explicit derivation steps, error bounds, stability arguments, or numerical examples showing preservation of input-output behavior. In the revised manuscript we will add these elements, including at least one worked numerical example with error metrics. revision: yes

  2. Referee: [—] The precondition that the given system already possesses well-defined solvents removable without destroying input-output behavior is stated but never tested or illustrated, leaving the practical scope of both algorithms unexamined.

    Authors: The paper states the precondition that the original system must admit removable solvents without loss of essential dynamics, but provides no numerical test or illustration of this condition. We accept that this leaves the practical scope unclear. The revision will include concrete examples that verify the precondition holds and demonstrate the resulting reduced-order models. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper describes two explicit algorithmic procedures (one-by-one and simultaneous solvent elimination) for reducing the order of a MIMO system already supplied in matrix transfer function form. No equations, parameters, or performance metrics are fitted to data; the central claim is that the procedures correctly factor and deflate the matrix polynomial while preserving retained poles. No self-citations, ansatzes, or uniqueness theorems are invoked in the provided text, and the suitability condition is stated explicitly as a precondition rather than derived. The derivation chain is therefore self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or additional axioms are stated beyond the domain assumption that the system is supplied in transfer-function form.

axioms (1)
  • domain assumption The system is linear, MIMO, and supplied in matrix transfer-function form with well-defined solvents.
    Explicitly stated as the setting for which the methods are suitable.

pith-pipeline@v0.9.0 · 5621 in / 1145 out tokens · 33102 ms · 2026-05-25T10:38:13.842430+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    A Survey of Model Reduction by Balanced Truncation and Some New Results,

    S. Gugercin and A. C. Antoulas, "A Survey of Model Reduction by Balanced Truncation and Some New Results," International Journal of Control, vol. 77, no. 8, pp. 748-766, 2004

    work page 2004

  2. [2]

    A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems,

    P. Benner, S. Gugercin, and K. Willcox, "A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems," SIAM Review, vol. 57, no. 4, pp. 483-531, 2015

    work page 2015

  3. [3]

    An overview of approximation methods for large-scale dynamical systems,

    A. C. Antoulas, "An overview of approximation methods for large-scale dynamical systems," Annual Reviews in Control, vol. 29, no. 2, pp. 181- 190, 2005

    work page 2005

  4. [4]

    Model reduction by phase matching,

    M. Green and B. D. O. Anderson, "Model reduction by phase matching," Mathematics of Control, Signals and Systems, vol. 2, no. 3, pp. 221-263, 1989

    work page 1989

  5. [5]

    System reduction via truncated Hankel matrices,

    C. K. Chui, X. Li, and J. D. Ward, "System reduction via truncated Hankel matrices," Mathematics of Control, Signals and Systems, vol. 4, no. 2, pp. 161-175, 1991

    work page 1991

  6. [6]

    On the embedding of state space 14 realizations,

    Y. Genin and A. Vandendorpe, "On the embedding of state space 14 realizations," Mathematics of Control, Signals, and Systems, vol. 19, no. 2, pp. 123-149, 2007

    work page 2007

  7. [7]

    On The Order Reduction of MIMO Large Scale Systems Using Block-Roots of Matrix Polynomials,

    B. Bekhiti, A. Dahimene, B. Nail, and K. Hariche, "On The Order Reduction of MIMO Large Scale Systems Using Block-Roots of Matrix Polynomials," ed

    work page

  8. [8]

    On solvents-based model reduction of MIMO systems,

    H. Zabot and K. Hariche, "On solvents-based model reduction of MIMO systems," International Journal of Systems Science, vol. 28, no. 5, pp. 499-505, 1997

    work page 1997

  9. [9]

    Gohberg, P

    I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials (Classics in Applied Mathematics). Society for Industrial and Applied Mathematics, p. 423, 2009

    work page 2009

  10. [10]

    Interpolation theory and λ-matrices,

    K. Hariche and E. D. Denman, "Interpolation theory and λ-matrices," Journal of Mathematical Analysis and Applications, vol. 143, no. 2, pp. 530-547, 1989

    work page 1989

  11. [11]

    On solvents and Lagrange interpolating λ- matrices,

    K. Hariche and E. D. Denman, "On solvents and Lagrange interpolating λ- matrices," Applied Mathematics and Computation, vol. 25, no. 4, pp. 321- 332, 1988

    work page 1988

  12. [12]

    On solvents of matrix polynomials,

    M. Yaici and K. Hariche, "On solvents of matrix polynomials," international journal of modeling and optimization, vol. 4, no. 4, pp. 273- 277, 2014

    work page 2014

  13. [13]

    W. H. Schilders, H. A. Van Der Vorst, and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications (The European Consortium for Mathematics in Industry). Springer-Verlag Berlin Heidelberg, 2008, pp. XI, 471

    work page 2008

  14. [14]

    Robust model order reduction of an electrical machine at startup through reduction error estimation,

    L. Montier, T. Henneron, S. Clénet, and B. Goursaud, "Robust model order reduction of an electrical machine at startup through reduction error estimation," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, vol. 31, no. 2, 2018, Art. no. e2277

    work page 2018

  15. [15]

    New strategies in model order reduction of trajectory piecewise-linear models,

    S. S. Mohseni, M. J. Yazdanpanah, and A. Ranjbar N, "New strategies in model order reduction of trajectory piecewise-linear models," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, vol. 29, no. 4, pp. 707-725, 2016

    work page 2016

  16. [16]

    Krylov model order reduction of finite element approximations of electromagnetic devices with frequency- dependent material properties,

    H. Wu and A. C. Cangellaris, "Krylov model order reduction of finite element approximations of electromagnetic devices with frequency- dependent material properties," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, vol. 20, no. 5, pp. 217-235, 2007

    work page 2007

  17. [17]

    Combining Krylov subspace methods and identification-based methods for model order reduction,

    P. J. Heres, D. Deschrijver, W. H. A. Schilders, and T. Dhaene, "Combining Krylov subspace methods and identification-based methods for model order reduction," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, vol. 20, no. 6, pp. 271-282, 2007

    work page 2007

  18. [18]

    Error bounds for reduction of multi-port resistor networks,

    M. V. Ugryumova, J. Rommes, and W. H. A. Schilders, "Error bounds for reduction of multi-port resistor networks," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, vol. 26, no. 5, pp. 464-477, 2013

    work page 2013

  19. [19]

    A computer-aided method for solvents and spectral factors of matrix polynomials,

    J. S. H. Tsai, C. M. Chen, and L. S. Shieh, "A computer-aided method for solvents and spectral factors of matrix polynomials," Applied Mathematics 15 and Computation, vol. 47, no. 2, pp. 211-235, 1992

    work page 1992

  20. [20]

    Matrix Equations and Model Reduction,

    P. Benner, T. Breiten, and L. Feng, "Matrix Equations and Model Reduction," in Matrix Functions and Matrix Equations, vol. Vol 19(Series in Contemporary Applied Mathematics, no. Vol 19): Co-published with HEP, pp. 50-75, 2015

    work page 2015

  21. [21]

    Block power method for computing solvents and spectral factors of matrix polynomials,

    J. S. H. Tsai, L. S. Shieh, and T. T. C. Shen, "Block power method for computing solvents and spectral factors of matrix polynomials," Computers & Mathematics with Applications, vol. 16, no. 9, pp. 683-699, 1988

    work page 1988

  22. [22]

    Algorithms for solvents and spectral factors of matrix polynomials,

    L. S. Shieh, Y. T. Tsay, and N. P. Coleman, "Algorithms for solvents and spectral factors of matrix polynomials," International Journal of Systems Science, vol. 12, no. 11, pp. 1303-1316, 1981

    work page 1981

  23. [23]

    Efficient Computation of Multivariable Transfer Function Dominant Poles Using Subspace Acceleration,

    J. Rommes and N. Martins, "Efficient Computation of Multivariable Transfer Function Dominant Poles Using Subspace Acceleration," IEEE Transactions on Power Systems, vol. 21, no. 4, pp. 1471-1483, 2006

    work page 2006

  24. [24]

    Fortuna, G

    L. Fortuna, G. Nunnari, and A. Gallo, Model Order Reduction Techniques with Applications in Electrical Engineering. Springer-Verlag London, 1992

    work page 1992

  25. [25]

    Helicopter flight control compensator design,

    M. Yaici, K. Hariche, and T. Clarke, "Helicopter flight control compensator design," AIP Conference Proceedings, vol. 1798, no. 1, p. 020174, 2017

    work page 2017

  26. [26]

    On eigenstructure assignment using block poles placement,

    M. Yaici and K. Hariche, "On eigenstructure assignment using block poles placement," European Journal of Control, vol. 20, no. 5, pp. 217-226, 2014

    work page 2014

  27. [27]

    On Block roots of matrix polynomials based MIMO control system design,

    B. Bekhiti, A. Dahimene, B. Nail, K. Hariche, and A. Hamadouche, "On Block roots of matrix polynomials based MIMO control system design," in 2015 4th International Conference on Electrical Engineering (ICEE), 2015. 16 Table 1. The complete set of solvents and their corresponding eigenvalues Name Solvent Eigenvalues of the solvent R1 0 0.8000 2.6000 1.6000...

    work page 2015