pith. machine review for the scientific record. sign in

arxiv: 2604.12960 · v1 · submitted 2026-04-14 · 📡 eess.SY · cs.SY· math.OC· math.RA

Recognition: unknown

Symmetry Is Almost All You Need: Robust Stability with Uncertainty Induced by Symmetric SRG Regions

Ding Zhang , Di Zhao , Philipp Braun , Jianqi Chen
Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:30 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OCmath.RA
keywords robust stabilityscaled relative graphmirror symmetrymatrix robust nonsingularityMIMO LTI systemsDavis-Wielandt shelluncertainty regionsangle-gain profile
0
0 comments X

The pith

Mirror symmetry of uncertainty regions makes scaled relative graph separation necessary and sufficient for matrix robust nonsingularity.

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

The paper seeks graphical conditions that guarantee a feedback system stays stable when all possible uncertainties lie inside a given region of the complex plane. It establishes that mirror symmetry of the region about the theta axis converts the separation between a variant of the system's scaled relative graph and that region into a necessary and sufficient test for matrix robust nonsingularity. This test then yields sufficient conditions for robust stability of MIMO linear systems and produces state-space characterizations plus a visual profile for systems bounded in both gain and angle. The approach recovers classical small-gain and small-angle results as special cases when the regions take familiar shapes. A reader would care because the symmetry assumption turns an otherwise conservative or intractable check into an exact geometric criterion.

Core claim

Whenever the uncertainty-inducing region is mirror symmetric about the theta-axis, the separation between a specific variant of the SRG and the region provides a necessary and sufficient condition for MRN. When the region is asymmetric, necessity generally fails. An additional theta-circular connectivity property is required to obtain necessary and sufficient conditions when no prior information on theta is available. These MRN results then supply sufficient conditions for robust stability of MIMO LTI systems under frequencywise symmetric uncertainties, together with state-space characterizations for angle-bounded and mixed gain-angle-bounded systems and a theta-angle-gain profile that shows

What carries the argument

The Davis-Wielandt shell together with its connection to variants of the scaled relative graph, which converts mirror symmetry of the uncertainty region into a separation condition for matrix robust nonsingularity.

If this is right

  • The small-gain condition, small-angle conditions, and sectored-disc conditions become necessary at the matrix level when symmetry holds.
  • Sufficient conditions for robust stability of MIMO LTI systems follow directly from the matrix robust nonsingularity results.
  • State-space characterizations become available for angle-bounded and mixed gain-angle-bounded systems.
  • A theta-angle-gain profile visualizes the system's feedback robustness against conic and sectorial uncertainties.

Where Pith is reading between the lines

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

  • The symmetry requirement suggests a practical design step of shaping uncertainty descriptions to restore mirror symmetry when possible.
  • The extra connectivity property for the no-prior-theta case points to a testable check on whether the uncertainty set remains path-connected after rotation.
  • The graphical profile may allow direct comparison of competing controllers by overlaying their angle-gain curves on the same plot.
  • The method could be extended to discrete-time or sampled-data systems by adapting the frequencywise symmetry assumption to the unit circle.

Load-bearing premise

The uncertainty-inducing region must be mirror symmetric about the theta-axis.

What would settle it

A concrete matrix example in which the uncertainty region is mirror symmetric about the theta-axis, the scaled relative graph variant stays separated from the region, yet the closed-loop matrix becomes singular for some point inside the region.

Figures

Figures reproduced from arXiv: 2604.12960 by Ding Zhang, Di Zhao, Jianqi Chen, Philipp Braun.

Figure 1
Figure 1. Figure 1: System diagram for the robust stability analysis. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The θ-SRG can be obtained as a two-step projection of the DW shell. Note that the DW shell always resides in epi P1, which we refer to as the DW space. Each point in the DW space is mapped by Πθ to a θ-conjugate pair in the SRG plane, and ΠθDW(M) = SRGθ(M). Readers may refer to [24], [36], [39], [40] for an expository study on the DW shells and their connection to, e.g., quadratic constraints, SRGs, and nu… view at source ↗
Figure 3
Figure 3. Figure 3: Precise characterisations of the DW shell unions of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The θ-symmetric part and cover of a region X (red). cover of X (as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: The θ-circular hull (orange) of a broken heart R (red). and Tab. III] together with the closedness of the disc, conic, and sectorial regions. Next, we move on to the case of general uncertainties as defined in Eq. (10). Before showing the main results, we introduce the notion of θ-circular connectedness and the associated circular hull operation that will streamline our statements. Definition 4 (θ-circular… view at source ↗
Figure 7
Figure 7. Figure 7: Using gain-, angle-, mixed-type halfspaces (discs) [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Gain-angle robustness profile of a transfer matrix. [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Dissect the θ-SRG region with concentric discs centered at the origin, and then project back to the DW space. A. Proof of Proposition 2 We first show the formula for the θ-uncertainty set. Let σ(∆) and σ(∆) denote the smallest and largest singular values of ∆, respectively. For an arbitrary ∆ ∈ Xθ, by definition SRGθ(∆) ⊆ Xinf θ . Then it follows that the height [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

This paper investigates the robust stability problem of a feedback system in the presence of uncertainties induced by graphical regions in the plane where the scaled relative graphs (SRGs) reside. Our main results are developed using a novel and intuitive concept, the Davis-Wielandt shell, together with its connection to SRGs and related variants. We first study a matrix robust nonsingularity (MRN) problem for two types of graphically induced uncertainty sets: one with prior information on $\theta$ and one without. In the former case, we show that, whenever the uncertainty-inducing region is mirror symmetric about the $\theta$-axis, the separation between a specific variant of the SRG and the region provides a necessary and sufficient condition for MRN. When the region is asymmetric, the necessity generally fails. This recovers the necessity of the small gain condition, and reveals the necessity of small angle conditions and sectored-disc conditions at the matrix level. In the latter case, we show that an additional $\theta$-circular connectivity property is required to obtain necessary and sufficient conditions. Building on these MRN results, we then derive sufficient conditions for robust stability of multi-input multi-output (MIMO) linear time-invariant (LTI) systems under frequencywise symmetric uncertainties. In addition, connections with existing system characteristics such as disc-boundedness are discussed and exploited to obtain state-space characterisations for angle-bounded and mixed gain-angle-bounded systems. Based on these results, we construct a $\theta$-angle-gain profile of a system that provides an intuitive visualisation of its feedback robustness against conic and sectorial uncertainties.

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 claims that mirror symmetry of the uncertainty-inducing region about the θ-axis makes separation between a specific variant of the scaled relative graph (SRG) and the region a necessary and sufficient condition for matrix robust nonsingularity (MRN). It notes that necessity fails for asymmetric regions and requires an additional θ-circular connectivity property when no prior information on θ is available. These MRN results are lifted to sufficient conditions for robust stability of MIMO LTI systems under frequencywise symmetric uncertainties. The work introduces the Davis-Wielandt shell to connect SRGs and variants, provides state-space characterizations for angle-bounded and mixed gain-angle-bounded systems, and constructs a θ-angle-gain profile for visualizing feedback robustness against conic and sectorial uncertainties.

Significance. If the necessity and sufficiency claims hold, the paper offers a meaningful unification of graphical robust stability methods by showing how symmetry enables tight necessary-and-sufficient conditions at the matrix level, recovering the small-gain condition and small-angle conditions. The Davis-Wielandt shell and θ-angle-gain profile provide new intuitive visualization tools, while the state-space characterizations for angle-bounded systems add practical value for analysis and design in control theory.

major comments (2)
  1. [MRN results section] The central MRN theorem (with prior θ information): the necessity direction for mirror-symmetric regions is load-bearing for the main claim, yet the abstract provides no derivation sketch or explicit counter-example showing an asymmetric region where SRG-region separation holds but MRN fails. Including such a concrete counter-example would confirm that symmetry is essential rather than an artifact of the chosen SRG variant.
  2. [MRN without prior θ information] The θ-circular connectivity property (no prior θ case): this property is introduced to restore necessity and sufficiency, but its definition appears closely tailored to the SRG separation condition. A concrete test or independent example where the property holds naturally (outside the paper's constructions) would reduce the risk that it is circular with the desired result.
minor comments (2)
  1. [Abstract] The abstract refers to 'a specific variant of the SRG' without naming or defining it; early clarification of which variant (e.g., via a short table or diagram) would improve readability.
  2. [Figures and visualization section] Figure captions and the θ-angle-gain profile visualization would benefit from explicit labeling of the mirror-symmetry axis and the separation distance to make the geometric arguments easier to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive suggestions. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [MRN results section] The central MRN theorem (with prior θ information): the necessity direction for mirror-symmetric regions is load-bearing for the main claim, yet the abstract provides no derivation sketch or explicit counter-example showing an asymmetric region where SRG-region separation holds but MRN fails. Including such a concrete counter-example would confirm that symmetry is essential rather than an artifact of the chosen SRG variant.

    Authors: We agree that an explicit counter-example would strengthen the exposition of why symmetry is necessary. Although the manuscript explains that necessity fails in general for asymmetric regions and recovers the small-gain theorem as a special case, we will add a simple concrete counter-example (e.g., a 2-by-2 matrix with an asymmetric uncertainty region where the SRG does not intersect the region but the matrix can be singular) to the MRN results section in the revised manuscript. revision: yes

  2. Referee: [MRN without prior θ information] The θ-circular connectivity property (no prior θ case): this property is introduced to restore necessity and sufficiency, but its definition appears closely tailored to the SRG separation condition. A concrete test or independent example where the property holds naturally (outside the paper's constructions) would reduce the risk that it is circular with the desired result.

    Authors: The θ-circular connectivity property has an independent interpretation as ensuring that the uncertainty region permits continuous variation in the angle θ without disconnected components that would allow bypassing the separation condition. It is not defined in a circular manner but derived from the topological requirements for necessity. To address the concern, we will include an independent natural example, such as a standard sector uncertainty set that satisfies the property, in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained from matrix/graph definitions

full rationale

The paper's core claims rest on introducing the Davis-Wielandt shell and its links to SRG variants, then proving that mirror symmetry of the uncertainty region about the θ-axis yields necessary and sufficient separation conditions for MRN. These steps are presented as direct consequences of the symmetry assumption and standard matrix nonsingularity arguments, recovering known small-gain and small-angle results without redefining inputs in terms of outputs. The additional θ-circular connectivity property is explicitly stated as an extra requirement when prior θ information is absent, rather than being fitted or smuggled to force necessity. Lifting to MIMO robust stability and state-space characterizations follows from the MRN results via frequencywise application, with no load-bearing self-citations or ansatz redefinitions visible in the abstract or described chain. The derivation chain remains independent of the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on the novel Davis-Wielandt shell concept, standard properties of complex-plane regions, and the assumption that symmetry about the θ-axis is sufficient to restore necessity.

axioms (2)
  • domain assumption The Davis-Wielandt shell connects SRGs to matrix robust nonsingularity
    Invoked to obtain the separation condition for MRN.
  • domain assumption Mirror symmetry of the uncertainty region about the θ-axis
    Required for necessity and sufficiency of the separation test.
invented entities (2)
  • Davis-Wielandt shell no independent evidence
    purpose: To link scaled relative graphs to robust nonsingularity conditions
    Described as a novel concept developed for this work.
  • θ-circular connectivity property no independent evidence
    purpose: To obtain necessary and sufficient conditions when no prior θ information is given
    Introduced as an additional requirement for the no-prior-information case.

pith-pipeline@v0.9.0 · 5607 in / 1551 out tokens · 37875 ms · 2026-05-10T14:30:41.546345+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

46 extracted references · 10 canonical work pages · 1 internal anchor

  1. [1]

    Vinnicombe,Uncertainty and Feedback:H ∞ Loop-Shaping and theν-Gap Metric

    G. Vinnicombe,Uncertainty and Feedback:H ∞ Loop-Shaping and theν-Gap Metric. London: Imperial College Press, 2001

    2001

  2. [2]

    Qiu and K

    L. Qiu and K. Zhou,Introduction to Feedback Control, 1st ed. Upper Saddle River, N.J: Prentice Hall, Mar. 2009

    2009

  3. [3]

    Dissipative dynamical systems part I: General theory,

    J. C. Willems, “Dissipative dynamical systems part I: General theory,” Archive for Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321– 351, Jan. 1972

    1972

  4. [4]

    System analysis via integral quadratic constraints,

    A. Megretski and A. Rantzer, “System analysis via integral quadratic constraints,”IEEE Transactions on Automatic Control, vol. 42, no. 6, pp. 819–830, Jun. 1997

    1997

  5. [5]

    K. Zhou, J. C. Doyle, and K. Glover,Robust and Optimal Control. Prentice Hall, 1995

    1995

  6. [6]

    A system theory criterion for positive real matrices,

    B. D. O. Anderson, “A system theory criterion for positive real matrices,”SIAM Journal on Control, vol. 5, p. 12, 1967

    1967

  7. [7]

    Stability robustness of a feedback inter- connection of systems with negative imaginary frequency response,

    A. Lanzon and I. R. Petersen, “Stability robustness of a feedback inter- connection of systems with negative imaginary frequency response,” IEEE Transactions on Automatic Control, vol. 53, no. 4, pp. 1042– 1046, May 2008

    2008

  8. [8]

    Phase IQC for the hierarchical performance analysis of uncertain large scale systems,

    K. Laib, A. Korniienko, G. Scorletti, and F. Morel, “Phase IQC for the hierarchical performance analysis of uncertain large scale systems,” in 2015 54th IEEE Conference on Decision and Control (CDC), Dec. 2015, pp. 5953–5958

    2015

  9. [9]

    A phase theory of multi-input multi-output linear time-invariant systems,

    W. Chen, D. Wang, S. Z. Khong, and L. Qiu, “A phase theory of multi-input multi-output linear time-invariant systems,”SIAM Journal on Control and Optimization, vol. 62, no. 2, pp. 1235–1260, Apr. 2024

    2024

  10. [10]

    The singular angle of nonlinear systems,

    C. Chen, D. Zhao, and S. Z. Khong, “The singular angle of nonlinear systems,”Automatica, vol. 181, p. 112515, Nov. 2025

    2025

  11. [11]

    A cyclic small phase theorem,

    C. Chen, W. Chen, D. Zhao, J. Chen, and L. Qiu, “A cyclic small phase theorem,”IEEE Transactions on Automatic Control, pp. 1–16, 2025

    2025

  12. [12]

    Robust stability analysis using LMIs: Beyond small gain and passivity,

    S. Gupta, “Robust stability analysis using LMIs: Beyond small gain and passivity,”International Journal of Robust and Nonlinear Control, vol. 6, no. 9-10, pp. 953–968, 1996

    1996

  13. [13]

    Conic-sector-based control to circumvent passivity violations,

    L. J. Bridgeman and J. R. Forbes, “Conic-sector-based control to circumvent passivity violations,”International Journal of Control, vol. 87, no. 8, pp. 1467–1477, Aug. 2014

    2014

  14. [14]

    A “mixed

    W. M. Griggs, B. D. O. Anderson, and A. Lanzon, “A “mixed” small gain and passivity theorem in the frequency domain,”Systems & Control Letters, vol. 56, no. 9-10, pp. 596–602, Sep. 2007

    2007

  15. [15]

    The scaled relative graph of a linear operator,

    R. Pates, “The scaled relative graph of a linear operator,” no. arXiv:2106.05650, Jun. 2021

    work page arXiv 2021

  16. [16]

    Graphical nonlinear system analysis,

    T. Chaffey, F. Forni, and R. Sepulchre, “Graphical nonlinear system analysis,”IEEE Transactions on Automatic Control, vol. 68, no. 10, pp. 6067–6081, Oct. 2023

    2023

  17. [17]

    Soft and Hard Scaled Relative Graphs for Nonlinear Feedback Stability

    C. Chen, S. Z. Khong, and R. Sepulchre, “Soft and hard scaled relative graphs for nonlinear feedback stability,” no. arXiv:2504.14407, Apr. 2025

    work page internal anchor Pith review arXiv 2025

  18. [18]

    Graphical dominance analysis for linear systems: A frequency-domain approach,

    C. Chen, T. Chaffey, and R. Sepulchre, “Graphical dominance analysis for linear systems: A frequency-domain approach,” no. arXiv:2504.14394, Apr. 2025

    work page arXiv 2025

  19. [19]

    Mixed small gain and phase theorem: A new view using scale relative graphs,

    E. Baron-Prada, A. Anta, A. Padoan, and F. D ¨orfler, “Mixed small gain and phase theorem: A new view using scale relative graphs,” no. arXiv:2503.13367, Mar. 2025

    work page arXiv 2025

  20. [20]

    Large scale heterogeneous networks, the Davis-Wielandt shell, and graph separation,

    I. Lestas, “Large scale heterogeneous networks, the Davis-Wielandt shell, and graph separation,”SIAM Journal on Control and Optimiza- tion, vol. 50, no. 4, pp. 1753–1774, Jan. 2012

    2012

  21. [21]

    When small gain meets small phase,

    D. Zhao, W. Chen, and L. Qiu, “When small gain meets small phase,” arXiv:2201.06041 [eess.SY], Jan. 2022

    work page arXiv 2022

  22. [22]

    Local stability of congestion control protocols: A MIMO gain and phase perspective,

    D. Zhang, I. Lestas, and L. Qiu, “Local stability of congestion control protocols: A MIMO gain and phase perspective,”Automatica, vol. 179, p. 112435, Sep. 2025

    2025

  23. [23]

    Feedback stability under mixed gain and phase uncertainty,

    J. Liang, D. Zhao, and L. Qiu, “Feedback stability under mixed gain and phase uncertainty,”IEEE Transactions on Automatic Control, vol. 70, no. 2, pp. 1008–1023, Feb. 2025

    2025

  24. [24]

    The phantom of Davis- Wielandt shell: A unified framework for graphical stability analysis of MIMO LTI systems,

    D. Zhang, X. Yang, A. Ringh, and L. Qiu, “The phantom of Davis- Wielandt shell: A unified framework for graphical stability analysis of MIMO LTI systems,” no. arXiv:2507.19918, Jul. 2025

    work page arXiv 2025

  25. [25]

    Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,

    E. K. Ryu, R. Hannah, and W. Yin, “Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,”Mathematical Pro- gramming, vol. 194, no. 1, pp. 569–619, Jul. 2022

    2022

  26. [26]

    On phase in scaled graphs,

    S. van den Eijnden, C. Chen, K. Scheres, T. Chaffey, and A. Lanzon, “On phase in scaled graphs,” no. arXiv:2504.21448, May 2025

    work page arXiv 2025

  27. [27]

    The small phase condition is necessary for symmetric systems,

    X. Yang, W. Chen, and L. Qiu, “The small phase condition is necessary for symmetric systems,” no. arXiv:2507.06617, Jul. 2025

    work page arXiv 2025

  28. [28]

    Gain and phase type multipliers for feedback robustness,

    A. Ringh, X. Mao, W. Chen, L. Qiu, and S. Z. Khong, “Gain and phase type multipliers for feedback robustness,”IEEE Transactions on Automatic Control, pp. 1–16, 2025

    2025

  29. [29]

    Scaled relative graph,

    R. Hannah, E. K. Ryu, and W. Yin, “Scaled relative graph,” University of California, Los Angeles, Los Angeles, CA, USA, UCLA CAM Report, 2016

    2016

  30. [30]

    Exploiting structure in MIMO scaled graph analysis,

    T. de Groot, T. Oomen, and S. van den Eijnden, “Exploiting structure in MIMO scaled graph analysis,” no. arXiv:2504.10135, Apr. 2025

    work page arXiv 2025

  31. [31]

    Stability results for MIMO LTI systems via scaled relative graphs,

    E. Baron-Prada, A. Anta, A. Padoan, and F. D ¨orfler, “Stability results for MIMO LTI systems via scaled relative graphs,” no. arXiv:2503.13583, Mar. 2025

    work page arXiv 2025

  32. [32]

    On eigenvalues of sums of normal matrices,

    H. Wielandt, “On eigenvalues of sums of normal matrices,”Pacific Journal of Mathematics, vol. 5, no. 4, pp. 633–638, 1955

    1955

  33. [33]

    The shell of a Hilbert-space operator,

    C. Davis, “The shell of a Hilbert-space operator,”Acta Scientiarum Mathematicarum, vol. 29, no. 1-2, pp. 69–86, 1968

    1968

  34. [34]

    The shell of a Hilbert-space operator : II

    ——, “The shell of a Hilbert-space operator : II.”Acta Scientiarum Mathematicarum, vol. 31, no. 3-4, pp. 301–318, 1970

    1970

  35. [35]

    K. E. Gustafson and D. K. M. Rao,Numerical Range: The Field of Values of Linear Operators and Matrices. Springer Science & Business Media, Dec. 2012

    2012

  36. [36]

    Davis-Wielandt shells of operators,

    C.-K. Li, Y .-T. Poon, and N.-S. Sze, “Davis-Wielandt shells of operators,”Operators and Matrices, no. 3, pp. 341–355, 2008

    2008

  37. [37]

    A scalable robust stability criterion for systems with heterogeneous LTI components,

    U. J ¨onsson and C.-Y . Kao, “A scalable robust stability criterion for systems with heterogeneous LTI components,”IEEE Transactions on Automatic Control, vol. 55, no. 10, pp. 2219–2234, Oct. 2010

    2010

  38. [38]

    Characterization of robust stability of a class of interconnected systems,

    C.-Y . Kao, U. J ¨onsson, and H. Fujioka, “Characterization of robust stability of a class of interconnected systems,”Automatica, vol. 45, no. 1, pp. 217–224, Jan. 2009

    2009

  39. [39]

    On network stability, graph separation, interconnection structure and convex shells,

    I. Lestas, “On network stability, graph separation, interconnection structure and convex shells,” in2011 50th IEEE Conference on Decision and Control and European Control Conference, Dec. 2011, pp. 4257–4263

    2011

  40. [40]

    Eigenvalues of the sum of matrices from unitary similarity orbits,

    C.-K. Li, Y .-T. Poon, and N.-S. Sze, “Eigenvalues of the sum of matrices from unitary similarity orbits,”SIAM Journal on Matrix Analysis and Applications, vol. 30, no. 2, pp. 560–581, Jan. 2008

    2008

  41. [41]

    Antieigenvalues,

    K. Gustafson, “Antieigenvalues,”Linear Algebra and its Applications, vol. 208–209, pp. 437–454, Sep. 1994

    1994

  42. [42]

    Computation of antieigenvalues of bounded linear operators via centre of mass,

    K. Paul, G. Das, and L. Debnath, “Computation of antieigenvalues of bounded linear operators via centre of mass,”International Journal of Applied and Computational Mathematics, vol. 1, no. 1, pp. 111–119, Mar. 2015

    2015

  43. [43]

    On the generalized Nyquist stability criterion,

    C. A. Desoer and Y . T. Wang, “On the generalized Nyquist stability criterion,”IEEE Transactions on Automatic Control, vol. 25, no. 2, pp. 187–196, Apr. 1980

    1980

  44. [44]

    A comparative study of input– output stability results,

    L. J. Bridgeman and J. R. Forbes, “A comparative study of input– output stability results,”IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 463–476, Feb. 2018

    2018

  45. [45]

    Sectored real lemma and its integration with bounded real lemma,

    X. Yang, D. Zhang, W. Chen, S. Hara, and L. Qiu, “Sectored real lemma and its integration with bounded real lemma,” in2025 IEEE 64th Conference on Decision and Control (CDC), Dec. 2025, pp. 3678–3683

    2025

  46. [46]

    Generalized KYP lemma: Unified frequency domain inequalities with design applications,

    T. Iwasaki and S. Hara, “Generalized KYP lemma: Unified frequency domain inequalities with design applications,”IEEE Transactions on Automatic Control, vol. 50, no. 1, pp. 41–59, Jan. 2005

    2005