pith. sign in

arxiv: 1907.06883 · v1 · pith:XJD6JINUnew · submitted 2019-07-16 · 🧮 math.OC · cs.NA· math.NA

Minimal-norm static feedbacks using dissipative Hamiltonian matrices

Nicolas Gillis , Punit Sharma This is my paper

Pith reviewed 2026-05-24 21:04 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords static feedback stabilizationdissipative Hamiltonian matricessemidefinite programmingminimal norm feedbackstatic output feedbacklinear time-invariant systems
0
0 comments X

The pith

Dissipative Hamiltonian matrices characterize all static-state feedbacks that stabilize a continuous LTI system.

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

The paper establishes a complete parametrization of stabilizing static feedbacks for continuous linear time-invariant systems using dissipative Hamiltonian matrices. This parametrization expresses feedbacks in terms of skew-symmetric matrices and symmetric positive semidefinite matrices. It leads directly to a semidefinite program that finds a stabilizing feedback of minimal norm. The same approach extends to the harder static-output feedback problem, where an algorithm is proposed and tested on benchmark examples.

Core claim

The set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair can be characterized using dissipative Hamiltonian matrices. This results in a parametrization in terms of skew-symmetric and symmetric positive semidefinite matrices, and an SDP that computes a minimal-norm static-state stabilizing feedback. The results extend to static-output feedback.

What carries the argument

Dissipative Hamiltonian matrix representation of the closed-loop system, which parametrizes feedbacks via skew-symmetric and positive semidefinite matrices.

If this is right

  • An SDP solves for a minimal-norm stabilizing static feedback.
  • The characterization applies to static-output feedback problems.
  • An algorithm computes solutions for the static-output feedback case.
  • Effectiveness is shown on numerical examples from standard libraries.

Where Pith is reading between the lines

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

  • This parametrization might allow checking whether a system is stabilizable by testing existence of such matrices.
  • Similar ideas could apply to discrete-time systems if the Hamiltonian form adapts.
  • The minimal-norm property could lead to better conditioned controllers in practice.

Load-bearing premise

Every stabilizing feedback for the given system pair admits a representation as a dissipative Hamiltonian matrix.

What would settle it

Finding a stabilizable continuous LTI system, a stabilizing feedback gain, and showing that no choice of skew-symmetric and positive semidefinite matrices reproduces that exact feedback.

read the original abstract

In this paper, we characterize the set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair using dissipative Hamiltonian matrices. This characterization results in a parametrization of feedbacks in terms of skew-symmetric and symmetric positive semidefinite matrices, and leads to a semidefinite program that computes a static-state stabilizing feedback. This characterization also allows us to propose an algorithm that computes minimal-norm static feedbacks. The theoretical results extend to the static-output feedback (SOF) problem, and we also propose an algorithm to tackle this problem. We illustrate the effectiveness of our algorithm compared to state-of-the-art methods for the SOF problem on numerous numerical examples from the COMPLeIB library.

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 manuscript claims to characterize the set of static-state feedbacks stabilizing a given continuous-time LTI pair (A,B) via dissipative Hamiltonian matrices. This yields a parametrization of stabilizing K in terms of a skew-symmetric matrix J and a symmetric positive-semidefinite matrix R, an SDP that computes a stabilizing feedback, and an algorithm that computes a minimal-norm such feedback. The results extend to the static-output-feedback problem, with numerical comparisons against state-of-the-art methods on COMPLeIB examples.

Significance. If the claimed parametrization is complete (every stabilizing K arises from some admissible (J,R)), the work supplies a convex formulation for the minimal-norm stabilization problem and a practical SDP-based method for the NP-hard SOF problem. The numerical comparisons on a standard benchmark library constitute a concrete strength.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (main characterization theorem): the claim that the set of stabilizing feedbacks 'can be characterized' using dissipative Hamiltonian matrices requires proving surjectivity of the map (J,R) → K onto the full set of stabilizing feedbacks. If the image is only a proper subset, the SDP that minimizes ||K|| subject to the dissipative-Hamiltonian LMI returns a value strictly larger than the true minimal norm, rendering the 'minimal-norm' algorithm inexact. The manuscript must supply the missing surjectivity argument or a counter-example.
  2. [§4] §4 (minimal-norm algorithm) and the SOF extension: the SDP formulation inherits the same completeness gap identified above; without surjectivity the computed 'minimal-norm' feedback is minimal only within the representable subset, not globally. The numerical comparisons therefore test a relaxation rather than the exact claim.
minor comments (2)
  1. [§2] Notation for the dissipative Hamiltonian matrix and the resulting closed-loop matrix should be introduced once with an explicit equation number rather than repeated inline definitions.
  2. [Table 1] Table 1 (COMPLeIB results) would benefit from an additional column reporting the norm of the feedback returned by the proposed method versus the true minimal norm when the latter is known from other solvers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify a gap in the completeness of the parametrization. We address each major comment below and will revise the manuscript to clarify the scope of the results without overstating the characterization.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (main characterization theorem): the claim that the set of stabilizing feedbacks 'can be characterized' using dissipative Hamiltonian matrices requires proving surjectivity of the map (J,R) → K onto the full set of stabilizing feedbacks. If the image is only a proper subset, the SDP that minimizes ||K|| subject to the dissipative-Hamiltonian LMI returns a value strictly larger than the true minimal norm, rendering the 'minimal-norm' algorithm inexact. The manuscript must supply the missing surjectivity argument or a counter-example.

    Authors: We agree that the manuscript does not establish surjectivity of the map from admissible (J, R) pairs to the full set of stabilizing feedbacks K. Section 3 proves only that every K generated from such (J, R) stabilizes the system; the converse is not shown. Consequently the SDP yields the minimal-norm feedback inside the image of the parametrization, which may be strictly larger than the global minimum. We will revise the abstract and §3 to describe the contribution as a parametrization of (a subset of) stabilizing feedbacks rather than a complete characterization of the entire set, and we will add an explicit remark that completeness of the parametrization is left open. revision: yes

  2. Referee: [§4] §4 (minimal-norm algorithm) and the SOF extension: the SDP formulation inherits the same completeness gap identified above; without surjectivity the computed 'minimal-norm' feedback is minimal only within the representable subset, not globally. The numerical comparisons therefore test a relaxation rather than the exact claim.

    Authors: We concur. The algorithm in §4 and its SOF extension compute the minimal-norm element within the representable class. We will update the text of §4, the SOF section, and the numerical-experiments discussion to state clearly that the reported norms and comparisons are with respect to this parametrized subset (i.e., a relaxation of the global problem). The COMPLeIB results will be presented as evidence of practical performance rather than global optimality. revision: yes

Circularity Check

0 steps flagged

No circularity; parametrization derived from algebraic properties of dissipative Hamiltonian matrices

full rationale

The paper's core claim is a characterization of stabilizing static feedbacks via dissipative Hamiltonian matrices, yielding a parametrization in skew-symmetric J and positive-semidefinite R matrices plus an SDP for minimal-norm K. No quoted step reduces a prediction or completeness claim to a fitted input, self-definition, or load-bearing self-citation chain. The derivation is presented as resting on LMI constraints and Hamiltonian structure rather than tautological renaming or ansatz smuggling. The completeness assumption (every stabilizing K arises this way) is an external mathematical claim, not shown to be forced by the paper's own equations or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions of linear control theory; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The continuous LTI system pair is stabilizable.
    The existence of stabilizing feedbacks and the completeness of the Hamiltonian parametrization presuppose stabilizability of the open-loop pair.

pith-pipeline@v0.9.0 · 5637 in / 1073 out tokens · 23793 ms · 2026-05-24T21:04:44.371251+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · 1 internal anchor

  1. [1]

    Albert, A.: Conditions for positive and nonnegative definiteness in terms of pseudoinverses. SIAM J. Appl. Math. 17(2), 434–440 (1969)

    work page 1969

  2. [2]

    IEEE Transactions on Automatic Control 20(1), 53–66 (1975)

    Anderson, B., Bose, N., Jury, E.: Output feedback stabilization a nd related problems– solution via decision methods. IEEE Transactions on Automatic Control 20(1), 53–66 (1975)

    work page 1975

  3. [3]

    In: International Conference on Control and Optimization with Industrial Applications, Bilkent Univer sity, Ankara, Turkey (2011)

    Arzelier, D., Deaconu, G., Gumussoy, S., Henrion, D.: H2 for HIFOO . In: International Conference on Control and Optimization with Industrial Applications, Bilkent Univer sity, Ankara, Turkey (2011)

    work page 2011

  4. [4]

    Princeton University Press (2010)

    ˚ Astr¨ om, K.J., Murray, R.M.: Feedback Systems: An Introduction f or Scientists and Engineers. Princeton University Press (2010)

    work page 2010

  5. [5]

    Automatica 100, 182–186 (2019)

    Beattie, C.A., Mehrmann, V., Van Dooren, P.: Robust port-Hamilto nian representations of passive systems. Automatica 100, 182–186 (2019)

    work page 2019

  6. [6]

    SIAM Journal on Control and Optimization 35(6), 2118–2127 (1997)

    Blondel, V., Tsitsiklis, J.: NP-Hardness of some linear control desig n problems. SIAM Journal on Control and Optimization 35(6), 2118–2127 (1997)

    work page 1997

  7. [7]

    arXiv preprint arXiv:1901.03069 (2019)

    Choudhary, N., Gillis, N., Sharma, P.: On approximating the nearest Ω-stable matrix. arXiv preprint arXiv:1901.03069 (2019)

    work page arXiv 1901

  8. [8]

    http://cvxr.com/cvx (2012)

    CVX Research, I.: CVX: Matlab software for disciplined convex pr ogramming, version 2.0. http://cvxr.com/cvx (2012)

    work page 2012

  9. [9]

    Linear Algebra and its Applications 573, 37–53 (2019)

    Gillis, N., Karow, M., Sharma, P.: Approximating the nearest stable d iscrete-time system. Linear Algebra and its Applications 573, 37–53 (2019)

    work page 2019

  10. [10]

    Numerical Linear Algebra with Applications pp

    Gillis, N., Mehrmann, V., Sharma, P.: Computing nearest stable mat rix pairs. Numerical Linear Algebra with Applications pp. e2153, (2018). doi:10.1002/nla.2153

    work page doi:10.1002/nla.2153 2018

  11. [11]

    Automatica 85, 113–121 (2017)

    Gillis, N., Sharma, P.: On computing the distance to stability for mat rices using linear dissipative Hamil- tonian systems. Automatica 85, 113–121 (2017)

    work page 2017

  12. [12]

    S IAM Journal on Numerical Analysis 56(2), 1022–1047 (2018)

    Gillis, N., Sharma, P.: Finding the nearest positive-real system. S IAM Journal on Numerical Analysis 56(2), 1022–1047 (2018)

    work page 2018

  13. [13]

    Golo, G., Schaft, A.v., Breedveld, P., Maschke, B.: Hamiltonian for mulation of bond graphs. In: A.R. R. Johansson (ed.) Nonlinear and Hybrid Systems in Automotive Cont rol, pp. 351–372. Springer-Verlag, Heidelberg, Germany (2003)

    work page 2003

  14. [14]

    Recent Advances in Learning and Control pp

    Grant, M., Boyd, S.: Graph implementations for nonsmooth conv ex programs. Recent Advances in Learning and Control pp. 95–110 (2008)

    work page 2008

  15. [15]

    IF AC Proceedings Volumes 42(6), 144–149 (2009)

    Gumussoy, S., Henrion, D., Millstone, M., Overton, M.L.: Multiobjec tive robust control with HIFOO 2.0. IF AC Proceedings Volumes 42(6), 144–149 (2009)

    work page 2009

  16. [16]

    Leibfritz, F.: Compleib, constraint matrix-optimization problem lib rary-a collection of test examples for nonlinear semidefinite programs, control system design and relate d problems. Dept. Math., Univ. Trier, Trier, Germany, Tech. Rep (2004)

    work page 2004

  17. [17]

    Mehl, C., Mehrmann, C., Sharma, P.: Stability radii for linear Hamilto nian systems with dissipation under structure-preserving perturbations. SIAM J. Matrix Anal. Appl. 37(4), 1625–1654 (2016) 18

    work page 2016

  18. [18]

    Optimal robustness of port-Hamiltonian systems

    Mehrmann, V., Van Dooren, P.: Optimal robustness of port-Ha miltonian systems. arXiv preprint arXiv:1904.13326 (2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1904

  19. [19]

    Mathematics of Control, Signals and Systems 6(2), 99–105 (1993)

    Nemirovskii, A.: Several NP-hard problems arising in robust stab ility analysis. Mathematics of Control, Signals and Systems 6(2), 99–105 (1993)

    work page 1993

  20. [20]

    Linear Algebra and its Applications 437(2), 525 – 548 (2012)

    Peretz, Y.: A characterization of all the static stabilizing contr ollers for LTI systems. Linear Algebra and its Applications 437(2), 525 – 548 (2012)

    work page 2012

  21. [21]

    Automatica 63, 221 – 234 (2016)

    Peretz, Y.: A randomized approximation algorithm for the minimal- norm static-output-feedback problem. Automatica 63, 221 – 234 (2016)

    work page 2016

  22. [22]

    In: 2013 European Control Conference (ECC), pp

    Polyak, B., Khlebnikov, M., Shcherbakov, P.: An LMI approach to structured sparse feedback design in linear control systems. In: 2013 European Control Conference (ECC), pp. 833–838 (2013)

    work page 2013

  23. [23]

    Schaft, A.v.: Port-Hamiltonian systems: an introductory surv ey. In: J.V. M. Sanz-Sole, J. Verdura (eds.) Proc. of the International Congress of Mathematicians, vol. III , Invited Lectures, pp. 1339–1365. Madrid, Spain

    work page

  24. [24]

    Schaft, A.v., Maschke, B.: Port-Hamiltonian systems on graphs . SIAM J. Control Optim. (2013)

    work page 2013

  25. [25]

    IEEE Control Systems Magazine 17(6), 19–35 (1997)

    Spencer, B.F., Sain, M.K.: Controlling buildings: a new frontier in fee dback. IEEE Control Systems Magazine 17(6), 19–35 (1997)

    work page 1997

  26. [26]

    Numerische Mathematik 65, 357–382 (1993)

    Sun, J.G.: Backward perturbation analysis of certain characte ristic subspaces. Numerische Mathematik 65, 357–382 (1993)

    work page 1993

  27. [27]

    Automatica 33(2), 125 – 137 (1997)

    Syrmos, V.L., Abdallah, C.T., Dorato, P., Grigoriadis, K.: Static out put feedback: A survey. Automatica 33(2), 125 – 137 (1997)

    work page 1997

  28. [28]

    Optimization Methods and Software 11(1-4), 545–581 (1999)

    Toh, K.C., Todd, M., T¨ ut¨ unc¨ u, R.: SDPT3–a MATLAB softwarepackage for semidefinite programming, version 1.3. Optimization Methods and Software 11(1-4), 545–581 (1999)

    work page 1999

  29. [29]

    Springer Science & Business Media (2012)

    Trentelman, H.L., Stoorvogel, A.A., Hautus, M.: Control theory for linear systems. Springer Science & Business Media (2012)

    work page 2012

  30. [30]

    T¨ ut¨ unc¨ u, R., Toh, K., Todd, M.: Solving semidefinite-quadrat ic-linear programs using SDPT3. Math. Program. 95(2), 189–217 (2003)

    work page 2003

  31. [31]

    The Structural Design of Tall Buildings 11(2), 109–127 (2002)

    Xu, Y., Teng, J.: Optimum design of active/passive control devic es for tall buildings under earthquake excitation. The Structural Design of Tall Buildings 11(2), 109–127 (2002)

    work page 2002

  32. [32]

    Yang, J.N., Lin, S., Jabbari, F.: H2-based control strategies fo r civil engineering structures. Journal of Structural Control 10(3-4), 205–230 (2003) 19 A Stabilizing matrix pairs (A, B) In this section, we report the ℓ2 norm of the solutions obtained by Algorithm 1 for the SSF prob lems corresponding to the same instance as in Tables 6.2 and 6.3. W e mi...

    work page 2003