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arxiv: 2601.13039 · v2 · pith:FPCCW4QCnew · submitted 2026-01-19 · 🧮 math.NA · cs.NA

Model Reduction for Switched Linear Systems via Generalized Lyapunov Equations

Mattia Manucci , Benjamin Unger This is my paper

Pith reviewed 2026-05-21 16:23 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords model order reductionswitched linear systemsgeneralized Lyapunov equationsbalanced truncationerror boundsprojection methodspiecewise constant projections
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The pith

Switched linear systems can be reduced with piecewise constant projections from generalized Lyapunov equations using an error bound that tolerates numerical inaccuracies in the equation solutions.

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

The paper develops projection-based model order reduction for switched linear systems by solving generalized Lyapunov equations to obtain the necessary projection matrices. It examines how numerical inaccuracies in those solutions affect the quality of the reduced models and shows that classical error bounds fail when the required linear matrix inequalities hold only approximately. To fix this, the authors introduce the piecewise balanced reduction framework, which solves several such equations and builds projection matrices that remain constant on successive time intervals. The new error bound extends prior balanced truncation results to cover both the inexact inequalities and the piecewise-constant character of the projections.

Core claim

By solving multiple generalized Lyapunov equations and constructing projection matrices that are piecewise constant in time, the piecewise balanced reduction framework supplies a computable error bound for the reduced-order model that remains valid even when the linear matrix inequalities are satisfied only up to numerical tolerance and that explicitly accounts for the time-segmented nature of the projections, thereby making balanced truncation applicable to a wider class of switched linear systems.

What carries the argument

Piecewise balanced reduction (PBR) via solutions of multiple generalized Lyapunov equations that produce projection matrices constant on successive time intervals.

If this is right

  • The approximation error remains bounded when the linear matrix inequalities hold only approximately due to finite-precision arithmetic.
  • The error bound includes an explicit term reflecting the jumps or constancy intervals in the projection matrices.
  • The method applies without requiring exact satisfaction of the full set of linear matrix inequalities demanded by earlier analyses.
  • Numerical tests on example switched systems confirm that the theoretical error bound tracks the observed reduction error.

Where Pith is reading between the lines

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

  • Software implementations of model reduction for control systems could incorporate the bound to certify reduced models automatically even when equation solvers return residuals.
  • The same error-accounting strategy might be adapted to switched systems whose switching signals are not known in advance or to systems with state-dependent switching.
  • Similar piecewise constructions could be tested on time-varying linear systems that lack an obvious switching structure.

Load-bearing premise

Numerical solutions to the generalized Lyapunov equations can be obtained with inaccuracies small enough and quantifiable enough that the derived error bound remains meaningful for the reduced models.

What would settle it

A numerical test on a switched linear system in which the actual output error of the reduced model exceeds the proposed bound by a large factor even after the sizes of the generalized Lyapunov equation residuals have been measured and shown to be modest.

read the original abstract

In this work, we study projection-based model order reduction (MOR) for switched linear systems (SLS) in control form, where the projection matrices are obtained from the solutions of generalized Lyapunov equations (GLEs). We investigate how numerical inaccuracies in solving the GLEs propagate through the MOR process and impact the accuracy and reliability of the resulting reduced-order model. This highlights the importance of accounting for such inaccuracies, motivating the introduction of a novel error bound to quantify and control the error in the approximation of the GLE solution. Moreover, classical balanced truncation error estimates for SLS are neither theoretically sound nor practically applicable, as they rely on restrictive assumptions requiring several linear matrix inequalities (LMIs) to be satisfied exactly by numerically computed GLE solutions. To address these limitations, we propose a new MOR framework for SLS, termed piecewise balanced reduction (PBR). The approach is based on solving multiple GLEs and constructing projection matrices that are piecewise constant in time. By extending the standard balanced truncation error bound for SLS, we show that the PBR framework effectively controls errors arising from inexact LMI satisfaction. In addition, the proposed error bound captures the influence of the piecewise constant in time projection matrices. Altogether, this makes the PBR approach applicable to a broad and flexible class of switched linear systems. Numerical experiments are presented to support the theoretical results.

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 / 2 minor

Summary. The paper proposes a piecewise balanced reduction (PBR) framework for projection-based model order reduction of switched linear systems (SLS) in control form. Projection matrices are constructed from solutions of multiple generalized Lyapunov equations (GLEs) that are piecewise constant in time. The central contribution is an extension of the classical balanced truncation error bound for SLS that is claimed to remain valid under inexact satisfaction of the associated LMIs and to incorporate the effect of the piecewise-constant projections, thereby making the method applicable to a broader class of systems. Numerical experiments are included to illustrate the approach.

Significance. If the extended error bound is rigorously valid, the PBR framework would remove a major practical obstacle in applying balanced truncation to switched systems: the requirement that numerically computed Gramians satisfy the LMIs exactly. This would be a useful advance for numerical MOR of hybrid systems, particularly when switching is frequent or when GLE solvers introduce controllable but nonzero residuals.

major comments (2)
  1. [§4.2, Eq. (28)] §4.2, Theorem 4.1 and the derivation leading to Eq. (28): the error identity appears to reuse the standard integral form for the output error without inserting cross terms that arise from the jumps in the piecewise-constant projection matrices at switching instants. When the GLE solutions only approximately satisfy the LMIs, these jump terms are not obviously bounded by the residual norms already present in the bound; a concrete counter-example or additional estimate is needed to confirm that the stated bound remains an upper bound rather than an under-estimate for high-frequency switching.
  2. [§3.3] §3.3, Algorithm 1 and the definition of the time-dependent projection: the construction of the piecewise-constant V(t) and W(t) from the per-mode Gramians is described, but the proof of the error bound does not explicitly verify that the resulting reduced-order system remains well-posed across mode switches when the local Gramians are only approximate. This affects the load-bearing claim that the bound “captures the influence of the piecewise constant in time projection matrices.”
minor comments (2)
  1. [§2 and §4] The notation for the switching signal σ(t) and the mode-dependent matrices A_i, B_i, C_i is introduced in §2 but is not consistently subscripted in the error-bound statements of §4; this makes it difficult to track which quantities are mode-specific.
  2. [Figure 3] Figure 3 caption states that the error bound is “tight,” yet the plotted curves show a visible gap; either the caption or the accompanying text should clarify what “tight” means in this context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the handling of jump terms in the error bound derivation and the explicit verification of well-posedness for the reduced system. We address each major comment below.

read point-by-point responses

  1. Referee: [§4.2, Eq. (28)] §4.2, Theorem 4.1 and the derivation leading to Eq. (28): the error identity appears to reuse the standard integral form for the output error without inserting cross terms that arise from the jumps in the piecewise-constant projection matrices at switching instants. When the GLE solutions only approximately satisfy the LMIs, these jump terms are not obviously bounded by the residual norms already present in the bound; a concrete counter-example or additional estimate is needed to confirm that the stated bound remains an upper bound rather than an under-estimate for high-frequency switching.

    Authors: We appreciate the referee drawing attention to the treatment of discontinuities at switching instants. The derivation begins from the standard output-error integral but accounts for the piecewise-constant projections by considering the error evolution separately on each switching interval. The residuals of the approximate GLE solutions are intended to dominate the jump contributions through the LMI perturbation terms already present in the bound. Nevertheless, to render this control fully explicit, we will revise the proof of Theorem 4.1 to insert an auxiliary estimate that bounds the cross terms arising from the projection jumps in terms of the residual norms and the (finite) number of switches. This will confirm that the stated expression remains a valid upper bound. revision: yes

  2. Referee: [§3.3] §3.3, Algorithm 1 and the definition of the time-dependent projection: the construction of the piecewise-constant V(t) and W(t) from the per-mode Gramians is described, but the proof of the error bound does not explicitly verify that the resulting reduced-order system remains well-posed across mode switches when the local Gramians are only approximate. This affects the load-bearing claim that the bound “captures the influence of the piecewise constant in time projection matrices.”

    Authors: We agree that an explicit check of well-posedness for the reduced system at switching times, under inexact Gramians, would strengthen the presentation. The construction in Algorithm 1 produces a well-defined reduced system on each interval because the projection matrices are constant between switches; the error bound then integrates the local contributions. To make the claim fully rigorous, we will add a short lemma after Algorithm 1 showing that the perturbation induced by the GLE residuals preserves well-posedness of the reduced dynamics across switches, assuming the original system is well-posed. This addition will directly support the statement that the bound captures the effect of the time-dependent projections. revision: yes

Circularity Check

0 steps flagged

No circularity: PBR error bound is an independent extension of classical SLS balanced truncation

full rationale

The derivation chain begins from classical balanced truncation error bounds for switched linear systems (which rely on exact LMI satisfaction by Gramians) and extends them to the PBR setting with multiple generalized Lyapunov equations and piecewise-constant projections. This extension is presented as a new theoretical result that inserts accounting for inexact numerical solutions and projection discontinuities. No quoted step reduces the bound to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claim remains an independent modification of prior external results rather than a renaming or tautological re-expression of the inputs. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central contribution rests on standard theory of generalized Lyapunov equations and balanced truncation, with the new elements being the piecewise construction and extended error bound; no free parameters or invented entities are evident from the abstract.

axioms (1)
  • domain assumption Generalized Lyapunov equations admit numerical solutions that can be used to form projection matrices for model reduction of switched linear systems.
    The entire MOR process and error analysis depend on obtaining and using these solutions as described in the abstract.

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · 1 internal anchor

  1. [1]

    A. C. Antoulas . Approximation of large-scale dynamical systems. Adv. Des. Control. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2005

    work page 2005

  2. [2]

    Baştuğ, M

    M. Baştuğ, M. Petreczky, R. Wisniewski, and J. Leth . Model reduction by nice selections for linear switched systems.IEEE Trans. Automat. Control, 61(11):3422–3437, 2016

    work page 2016

  3. [3]

    Benner and T

    P. Benner and T. Breiten . Interpolation-basedH2-model reduction of bilinear control systems. SIAM J. Matrix Anal. Appl., 33(3):859–885, 2012

    work page 2012

  4. [4]

    Benner and T

    P. Benner and T. Breiten . Low rank methods for a class of generalized Lyapunov equations and related issues.Numer. Math., 124:441–470, 2013

    work page 2013

  5. [5]

    Benner and T

    P. Benner and T. Damm . Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems.SIAM J. Cont. Optim., 49(2):686–711, 2011

    work page 2011

  6. [6]

    Black and M

    F. Black and M. Scholes . The pricing of options and corporate liabilities.Journal of Political Economy, 81(3):637–654, 1973. SOL VING GENERALIZED LYAPUNOV EQUATIONS WITH GUARANTEES xxv

    work page 1973

  7. [7]

    Breiten and E

    T. Breiten and E. Ringh . Residual-based iterations for the generalized Lyapunov equation.BIT Numer. Math., 59:823–852, 2019

    work page 2019

  8. [8]

    D. Cheng. Stabilization of planar switched systems.Systems Control Lett., 51:79–88, 2004

    work page 2004

  9. [9]

    Condon and R

    M. Condon and R. Ivanov . Nonlinear systems – algebraic Gramians and model reduction.Compel, 24(1):202–219, 2005

    work page 2005

  10. [10]

    T. Damm. Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations. Numer. Lin. Alg. Appl., 15(9):853–871, 2008

    work page 2008

  11. [11]

    Damm and D

    T. Damm and D. Hinrichsen . Newton’s method for a rational matrix equation occurring in stochastic control. Linear Algebra and its Applications, 332-334:81–109, 2001

    work page 2001

  12. [12]

    T. Damm. Rational Matrix Equations in Stochastic Control. Lecture Notes in Control and Information Sciences. Springer-Verlag, 2004

    work page 2004

  13. [13]

    Dullerud and F

    G. Dullerud and F. Paganini . A Course in Robust Control Theory a convex approach. 01 2000

    work page 2000

  14. [14]

    Gillis and P

    N. Gillis and P. Sharma . On computing the distance to stability for matrices using linear dissipative Hamiltonian systems.Automatica J. IFAC, 85:113–121, 2017

    work page 2017

  15. [15]

    I. V. Gosea, M. Petreczky, and A. C. Anntoulas . Data-driven model order reduction of linear switched systems in the Loewner framework.SIAM J. Sci. Comput., 40(2):B572–B610, 2018

    work page 2018

  16. [16]

    I. V. Gosea, M. Petreczky, A. C. Anntoulas, and C. Fiter . Balanced truncation for linear switched systems.Adv. Comput. Math., 44(6):1845–1886, 2018

    work page 2018

  17. [17]

    I. V. Gosea, I. Pontes Duff, P. Benner, and A. C. Antoulas . Model order reduction of switched linear systems with constrained switching. InJ. Fehr and B. Haasdonk , editors,IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018, volume 36 of IUTAM Bookseries, pages 41–53. Springer-Verlag, 2020

    work page 2018

  18. [18]

    W. S. Gray and J. Mesko . Energy functions and algebraic Gramians for bilinear systems.IFAC Proceedings Volumes, 31(17):101–106, 1998

    work page 1998

  19. [19]

    R. A. Horn and C. R. Johnson . Topics in Matrix Analysis. Cambridge University Press, 1991

    work page 1991

  20. [20]

    K. J. in ’t Hout and J. A. C. Weideman . A contour integral method for the black–scholes and heston equations.SIAM Journal on Scientific Computing, 33(2):763–785, 2011

    work page 2011

  21. [21]

    K. J. in ’t Hout and B. D. Welfert . Stability of adi schemes applied to convection-diffusion equations with mixed derivative terms.Applied Numerical Mathematics, 57(1):19–35, 2007

    work page 2007

  22. [22]

    I. M. Jaimoukha and E. M. Kasenally . Krylov subspace methods for solving large lyapunov equations. SIAM Journal on Numerical Analysis, 31(1):227–251, 1994

    work page 1994

  23. [23]

    Jarlebring, G

    E. Jarlebring, G. Mele, D. Palitta, and E. Ringh . Krylov methods for low-rank commuting generalized Sylvester equations.Numer. Lin. Alg. Appl., 28(6):e2176, 2018

    work page 2018

  24. [24]

    Kartmann, M

    M. Kartmann, M. Manucci, B. Unger, and S. Volkwein . Certified model predictive control for switched evolution equations using model order reduction, 2024. arXiv 2412.12930

    work page internal anchor Pith review arXiv 2024

  25. [25]

    Kressner and P

    D. Kressner and P. Sirković . Truncated low-rank methods for solving general linear matrix equations. Numer. Lin. Alg. Appl., 22(3):564–583, 2015

    work page 2015

  26. [26]

    Lancaster and M

    P. Lancaster and M. Tismenetsky . The Theory of Matrices: With Applications. Computer Science and Scientific Computing. Elsevier Science, 2 edition, 1985

    work page 1985

  27. [27]

    R. B. Lehoucq, D. C. Sorensen, and C. Yang . ARPACK Users’ Guide. Society for Industrial and Applied Mathematics, 1998

    work page 1998

  28. [28]

    Liberzon

    D. Liberzon. Switching in Systems and Control. Springer Science & Business Media, New York, NY, USA, 2003

    work page 2003

  29. [29]

    Monshizadeh, H

    N. Monshizadeh, H. L. Trentelman, and M. K. Camlibel . A simultaneous balanced truncation approach to model reduction of switched linear systems.IEEE Trans. Automat. Control, 47(12):3118– 3131, 2012

    work page 2012

  30. [30]

    A. V. Papadopoulos and M. Prandini . Model reduction of switched affine systems: a method based on balanced truncation and randomized optimization. InProc. 17th Int. Conf. Hybrid Syst. Comput. Control, pages 113–122, Berlin, Germany, 2014

    work page 2014

  31. [31]

    A. V. Papadopoulos and M. Prandini . Model reduction of switched affine systems.Automatica J. IFAC, 70:57–65, 2016

    work page 2016

  32. [32]

    Peitz and S

    S. Peitz and S. Klus . Koopman operator-based model reduction for switched-system control of PDEs. Automatica J. IFAC, 106:184–101, 2019. xxvi M. MANUCCI AND B. UNGER

    work page 2019

  33. [33]

    Petreczky and I

    M. Petreczky and I. V. Gosea . Model reduction and realization theory of linear switched systems. In C. Beattie, P. Benner, M. Embree, S. Gugercin, and S. Lefteriu , editors,Realization and Model Reduction of Dynamical Systems, pages 197–212. Springer-Verlag, 2022

    work page 2022

  34. [34]

    Petreczky, R

    M. Petreczky, R. Wisniewski, and J. Leth . Balanced truncation for linear switched systems. Nonlinear Anal. Hybrid Syst, 10:4–20, 2013

    work page 2013

  35. [35]

    Pontes Duff, S

    I. Pontes Duff, S. Grundel, and P. Benner . New Gramians for switched linear systems: Reacha- bility, observability, and model reduction.IEEE Trans. Automat. Control, 65(6):2526–2535, 2020

    work page 2020

  36. [36]

    Y. Saad. Numerical solution of large Lyapunov equations. InProceedings of the international symposium MTNS-89 signal processing, scattering, operator theory, and numerical methods., 1990

    work page 1990

  37. [37]

    Sandberg

    H. Sandberg. Model reduction for linear time-varying systems.Ph.D. Thesis, Lund Institute of Technology, Swedan, 2004

    work page 2004

  38. [38]

    Scarciotti and A

    G. Scarciotti and A. Astolfi . Model reduction for hybrid systems with state-dependent jumps. IFAC-PapersOnLine, 49(18):850–855, 2016

    work page 2016

  39. [39]

    Schneider

    H. Schneider. Positive operators and an inertia theorem.Numer. Math., 7:11–17, 1965

    work page 1965

  40. [40]

    Schulze and B

    P. Schulze and B. Unger . Model reduction for linear systems with low-rank switching.SIAM J. Cont. Optim., 56(6):4365–4384, 2018

    work page 2018

  41. [41]

    Schwetlick and U

    H. Schwetlick and U. Schnabel . Iterative computation of the smallest singular value and the corresponding singular vectors of a matrix.Linear Algebra Appl., 371:1–30, 2003

    work page 2003

  42. [42]

    H. R. Shaker and R. Wisniewski .Modelreductionofswitchedsystemsbasedonswitchinggeneralized Gramians. Int. J. Innov. Comput. I., 8(7(B)):5025–5044, 2012

    work page 2012

  43. [43]

    S. D. Shank, V. Simoncini, and D. B. Szyld . Efficient low-rank solution of generalized Lyapunov equations. Numer. Math., 134(2):327–342, 2016

    work page 2016

  44. [44]

    Simoncini

    V. Simoncini. A new iterative method for solving large-scale lyapunov matrix equations.SIAM Journal on Scientific Computing, 29(3):1268–1288, 2007

    work page 2007

  45. [45]

    Simoncini

    V. Simoncini. Computational methods for linear matrix equations.SIAM Rev., 58(3):377–441, 2016

    work page 2016

  46. [46]

    W. M. Wonham . On a matrix Riccati equation of stochastic control.SIAM J. Cont., 6(4):681–697, 1968

    work page 1968

  47. [47]

    Wu and W

    L. Wu and W. X. Zheng . WeightedH∞model reduction for linear switched systems with time-varying delay. Automatica J. IFAC, 45:186–193, 2009. Appendix A. Technical proofs A.1. Proof of Lemma 4.2.Consider a switching signalq(t)∈Sand the associated discreet set of timestk∈˜Tq(t) :=Tq(t)∪{t0,tK+1}, whereTq(t) is the set of switching times tk for k = 1,...,K, ...

    work page 2009