Pith. sign in

REVIEW 3 major objections 5 minor 50 references

The $\theta$-Symmetric SRG with Applications to Stability of Cactus Dynamic Networks

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The θ-symmetric scaled relative graph records signed phase, not just magnitude, and its submultiplicative and subadditive structure yields necessary and sufficient robust stability tests for cyclic and cactus feedback networks.

desk verdict A genuinely new SRG variant with a solid algebraic core, but Theorem 7's sufficiency proof is missing the key encirclement argument. read the letter →

arxiv 2608.12591 v1 pith:27E442IK submitted 2026-08-12 eess.SY cs.SYmath.OC

Xiaokan Yang , Wei Chen , Li Qiu This is my paper · ORCID
classification eess.SYcs.SYmath.OC MSC 93D0993D0593C8015A60
keywords θ-symmetricscaledrelativegraphphaseleadandlagNyquistplotcactusnetworkcyclicinterconnectionrobuststabilitysemidefiniteprogrammingreturndifferencematrix
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

The paper tries to establish that a phase-aware redefinition of the scaled relative graph (SRG)—the device that records how a matrix rotates and scales each vector direction—captures both gain and the sign of phase for multivariable systems, and that this one object cleanly certifies feedback stability of cyclic cascades and, recursively, of entire cactus networks. The authors first prove that the graph's phase aspect equals a scalar norm-minimization problem, so the phase interval is computable by semidefinite programming. They then prove submultiplicative and subadditive inequalities for the graph, which convert nonsingularity of product- and sum-type return difference matrices into the geometric condition that $-1$ avoid a certain region. For cyclic interconnections with symmetric gain-phase uncertainty they prove a necessary and sufficient robust stability condition, and they extend it to cactus networks by absorbing petal loops into their hubs layer by layer. A sympathetic reader would take this as a compositional MIMO generalization of the Nyquist plot: subsystem-level regions, not eigenloci, decide stability.

What carries the argument

The load-bearing object is the θ-symmetric scaled relative graph, $\mathrm{SRG}_\theta(C)=\{\frac{\|Cx\|}{\|x\|}e^{j(\theta\pm\angle_\theta(x,Cx))}:x\neq0\}$, with $\angle_\theta(x,y)=\arccos\frac{\operatorname{Re}\langle x,e^{-j\theta}y\rangle}{\|x\|\|y\|}$. The rotation $e^{-j\theta}$ before the inner product is what restores phase sign: a scalar matrix $zI$ now has $\mathrm{SRG}_{\angle z}(zI)=\{z\}$ instead of the conjugate pair $\{z,\bar z\}$. The argument then runs on three rails: Theorem 1 ties the phase interval to $\arcsin\min_{\gamma>0}\|\gamma e^{-j\theta}C-I\|$ (hence SDP-computable), Theorem 3 gives subadditivity and submultiplicativity under θ-chord and θ-arc hull over-approximations, and Proposition 5—the single-frequency interpolation of a matrix with prescribed SRG by a stable real-rational transfer function—converts the region-avoidance certificate into a necessary and sufficient robust stability test.

What would settle it

Check Proposition 5 numerically for the symmetric case: pick $\beta=30^\circ$, $\gamma=1$ and a random matrix $\Delta$ with $\mathrm{SRG}(\Delta)\subset R[\beta,\gamma]$, then attempt to construct a real stable rational $P$ with $P(j\omega_0)=\Delta$ as in the proof. If for some such $\Delta$ no real-rational stable $P$ exists, the necessity direction of Theorem 7 cannot be carried through and the iff claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 7: for the feedback interconnection of a cascade of stable systems $P_1,\dots,P_N$, with one nominal system $P_k$ and the rest uncertain in $P[\beta_i,\gamma_i]$ (sets whose 0-symmetric SRG lies in the gain-phase region $R[\beta_i,\gamma_i]$ at every frequency), the closed loop is robustly stable for all admissible plants if and only if there exists a frequency-dependent angle $\theta(\omega)$ such that $-1\notin \mathrm{SRG}_{\theta(\omega)}(P_k(j\omega))\prod_{i\in K'}R[\beta_i,\gamma_i]$ for all $\omega$. The 'only if' is the sharp part: it uses a constructive interpolation (Proposition 5) to turn a single-frequency violation into an actual destabilizing plant. Section VIII then lifts this cyclic test to cactus networks through a hierarchical 'absorb the leaves' procedure, replacing the global return-difference inverse by a sequence of local loop checks.

Load-bearing premise

The 'only if' direction of the main robust-stability theorem rests on the interpolation claim that every matrix whose SRG lies in the allowed symmetric gain-phase region at one frequency can be realized by a stable real-rational transfer matrix, a claim the paper proves only for symmetric phase bounds and explicitly leaves open for asymmetric θ-symmetric bounds.

Editorial extensions

If this is right

  • $-1$ avoidance conditions like (20) and Theorem 7 can be checked frequency by frequency from each subsystem's SRG region, so stability verification does not require computing or plotting MIMO eigenloci.
  • For SISO systems the frequency-wise canonical SRG coincides exactly with the Nyquist plot, so all the new tests specialize to the classical picture.
  • The θ-segmental phase is computed by a semidefinite program, giving a concrete computational route for the previously open problem of computing segmental phase.
  • Cactus network stability can be certified by a recursive leaf-absorption algorithm, so the cost and the insight scale with local loops rather than with the global return difference matrix.
  • In the symmetric uncertainty description the cyclic robust stability test is exact (if and only if), eliminating the conservatism of sufficient-only gain-phase separation methods.

Reading between the lines

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

  • Editorial inference: the same structure suggests a phase-domain analogue of structured singular value analysis: choosing the $\theta_i$ that maximize the distance from $-1$ to the product of SRG regions (Remark 5) would automate robustness certificates, much as $D$-scaling tightens $\mu$ bounds.
  • Editorial inference: if the open asymmetric real-coefficient interpolation problem (Remark 6) is solved, Theorem 7 should extend to uncertainties with different upper and lower phase bounds, which matters because physical plant uncertainty is rarely symmetric in phase.
  • Editorial inference: the leaf-absorption procedure applies to any network whose simple-loop intersection graph is a tree, so quantifying the approximation error for networks slightly denser than cactus is a natural next test.
  • Editorial inference: because the original SRG was defined for nonlinear operators, the θ-symmetric version may carry the same compositional stability checks to nonlinear multivariable systems; the paper leaves this, and controller design, to future work.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript introduces the θ-symmetric scaled relative graph (SRG), a phase-parameterized variant of the SRG designed to retain the sign of phase information for complex matrices and systems. It develops the basic properties of this object, including a spectrum-containment lemma and inverse/rotation rules, and gives an LMI/norm-minimization characterization of the associated θ-segmental phase. The paper then proves subadditive and submultiplicative properties for sums and products of matrix sets, derives nonsingularity criteria for sum-type and product-type return-difference matrices, and states a necessary and sufficient robust-stability condition for cyclic interconnections under mixed gain-phase uncertainty. The final section extends the framework to cactus networks, with a detailed sunflower-graph theorem and an algorithmic hierarchical procedure for general cactus graphs.

Significance. If the results hold, the paper provides a genuinely compositional graphical method for MIMO feedback stability that generalizes the classical Nyquist plot and handles mixed gain-phase uncertainty in a less conservative way than separate gain or phase conditions. The SDP characterization of segmental phase in Theorem 1 and the interpolation construction in Proposition 5 are nontrivial and likely to be of independent use. The submultiplicative/subadditive calculus of θ-symmetric SRGs is clean and well suited to network analysis. The paper does not provide code or machine-checked proofs, but the main derivations are self-contained except for the specific gaps identified below.

major comments (3)
  1. [Section VII-B, Theorem 7 (sufficiency)] The statement 'The sufficiency follows from Theorem 6' is not justified as written. Theorem 6 requires the ray avoidance condition [−∞,−1] ∩ ∏ SRG_{θ_i(ω)}(P_i(jω)) = ∅ for every ω, whereas Theorem 7 assumes only −1 ∉ SRG_{θ(ω)}(P_k(jω)) ∏ R[β_i, γ_i]. The gap is repairable: each R[β_i, γ_i] is star-shaped about 0 and contains 0; finite products of star-shaped sets are star-shaped; and multiplication of a star-shaped set by any fixed complex set preserves star-shapedness, so the set S(ω) = SRG_{θ(ω)}(P_k(jω)) ∏ R[β_i, γ_i] is star-shaped. If S(ω) met (−∞,−1], it would contain −1. The authors should add this argument explicitly, and should also state that the uncertain subsystems are assigned θ_i(ω) = 0 with over-approximations R[β_i, γ_i], before invoking Theorem 6.
  2. [Section VII-B, Theorem 7 (necessity) and Appendix I, proof of Theorem 5] The necessity direction invokes Theorem 5 but does not specify which variant of Theorem 5 is being used. The uncertainty set P[β, γ] in Theorem 7 is defined by the 0-symmetric SRG, i.e., by SRG_0(P(jω)) ⊂ R[β, γ], which corresponds to the M_c[−β, β, γ] variant of Theorem 5, not to the M[−β, β, γ] variant whose proof is supplied. The appendix only proves the M-case, saying that the M_c case follows similarly. Since the entire necessity argument depends on the M_c case, the paper should either provide the M_c proof or give a precise reduction to the M-case; as written, this is an omitted proof in a load-bearing position.
  3. [Section VIII-B, Algorithm 1] No correctness theorem is stated or proved for the general cactus-network algorithm. Theorem 8 covers sunflower graphs only; the text asserts that the recursive decomposition extends naturally and then presents Algorithm 1, but there is no proof that the local checks plus loop absorption certify stability of every cactus network, nor a statement of conservatism. Since the abstract and introduction claim a unified framework for general cactus networks, this is a load-bearing omission: either add a formal theorem with an induction proof over the loop tree, or restrict the claim to the sunflower case.
minor comments (5)
  1. [Section VIII-A, Theorem 8] The notation [1, e^{2jθ_i(ω)}] is used without definition; presumably it denotes the line segment between 1 and e^{2jθ_i(ω)}, but the text should define it and explain how it arises from SRG_{θ_i(ω)}(I + Σ_q L_q(jω)).
  2. [Example 7] The region Ω is described as {r e^{jθ} : |r| ≤ 0.6 or −15° ≤ θ ≤ 10°}; the use of 'or' and the relationship between r and θ are ambiguous, and the polar-coordinate specification should be rewritten for clarity.
  3. [Section VIII-A, Theorem 8 proof] In the equation for the equivalent central loop, the product over i ∈ V_1 is described as being taken 'in the order consistent with the connection along the central loop'; this ordering should be made explicit, since cyclic permutations of the factors and the accompanying phase parameters affect the formal statement.
  4. [Section VII-A, Theorem 6 proof] The proof states that condition (20) implies [−∞,−1] ∩ Λ(P_1(jω)⋯P_N(jω)) = ∅; this is correct because the product of the SRGs contains the spectrum of the product, but the step should be stated explicitly so that the role of spectrum containment is transparent.
  5. [Section VIII-B, Algorithm 1] The output line 'Stable if all verification steps pass' should name the exact inequalities verified at each layer (e.g., the analogues of (26) and (27)) and the admissible over-approximation choices, so that the algorithm is reproducible without reconstructing the derivation from Theorem 8.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the theta-symmetric SRG stability conditions are genuine graph-separation criteria, with only minor non-load-bearing self-citations and a non-circular proof gap.

full rationale

The derivation chain is not circular. The uncertainty regions R[beta,gamma] are defined directly as subsets of the complex plane, and the matrix/system uncertainty sets M[alpha,beta,gamma] and P[beta,gamma] are then defined via SRG inclusions; the robust-stability conditions in Theorems 5 and 7 are stated in terms of the same R, but this is a standard graph-separation formulation analogous to a small-gain bound, not a restatement of the definitions. The frequency-dependent functions theta(omega) are free decision variables, not fitted parameters, and no data are fitted anywhere in the paper. Theorem 1's norm-minimization formula for the theta-segmental phase is proved from Proposition 1 rather than assumed, and the submultiplicative/subadditive properties in Theorem 3 are proved in the text. The only self-citations are to preliminary conference papers [34],[35] and to [38] for the elementary proofs of Lemma 2; these are not load-bearing for the central claims. The main caveat is a correctness gap, not circularity: Theorem 7's sufficiency is asserted to follow from Theorem 6, but Theorem 6 requires avoidance of the whole ray [-infinity,-1], while Theorem 7 only gives avoidance of -1; and Remark 6 explicitly leaves open the asymmetric interpolation needed for full necessity. These are proof-completeness concerns, not instances of a prediction being equivalent to its input by construction. Hence the score is 2 rather than 0 only because of the minor self-citations.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No new physical or ontological entities are introduced; the θ-symmetric SRG is a defined mathematical object and is accounted for through the free phase parameters and the stated axioms. The most fragile input is the interpolation premise (Proposition 5), recorded as an ad_hoc_to_paper axiom.

free parameters (1)
  • Phase-center functions θ_i(ω), ξ_k(ω), η_i(ω)
    The stability tests require existence of user-chosen phase functions; the examples hand-select these (e.g., ξ_k in Section VIII-A), but no data fitting or hidden constants are involved.
assumptions (6)
  • standard math Generalized Nyquist criterion with τ-homotopy applies to stable rational transfer matrices
    Used in Theorem 6 to reduce stability to nonsingularity of I+τP1...PN for τ∈[0,1].
  • standard math Schur complement and LMI equivalence
    Used throughout Proposition 1 and Theorem 1 to convert phase conditions into matrix inequalities.
  • standard math Standard rational matrix interpolation results, including small-gain interpolation
    Used in Proposition 5 to interpolate single-frequency matrix data by stable transfer matrices.
  • domain assumption Systems are LTI, stable, proper, and square, with uncertainty described by SRG inclusions
    The framework is confined to RH∞ systems and the robust results quantify only over uncertainty sets defined by SRG bounds.
  • domain assumption Cactus graph is strongly connected, each pair of simple loops shares at most one vertex, and exactly one edge per loop carries a negative sign
    Definition 1 and the sign convention in Section VIII; more general topologies such as overlapping loops are not addressed.
  • ad hoc to paper Single-frequency mixed gain and phase data can be interpolated by a stable real-rational transfer matrix
    Proposition 5 is proven only by reduction to small-gain interpolation; for asymmetric θ-symmetric bounds the paper admits in Remark 6 that this is nontrivial.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The $\theta$-Symmetric SRG with Applications to Stability of Cactus Dynamic Networks." pith.science (2026). https://pith.science/paper/27E442IK

@misc{pith2026260812591,
  author       = {Pith},
  title        = {Pith review of: The $\theta$-Symmetric SRG with Applications to Stability of Cactus Dynamic Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27E442IK}},
  note         = {Machine review of arXiv:2608.12591}
}
abstract

In this paper, we systematically study a variant of the scaled relative graph (SRG), referred to as the $\theta$-symmetric SRG, and apply it to the stability analysis of cactus networks. Compared with the previous SRG definition, the $\theta$-symmetric SRG enables the characterization of phase lead and lag behaviors, and serves as a more natural multivariable extension of the classical Nyquist plot. We first analyze the gain and phase aspects of $\theta$-symmetric SRG separately and build a connection between $\theta$-segmental phase and a norm minimization problem. This connection makes it possible to compute $\theta$-segmental phase via semidefinite programming. We further derive the submultiplicative and subadditive properties of $\theta$-symmetric SRG. These algebraic properties are crucial to determine the nonsingularity of product-type and sum-type return difference matrices, which topologically correspond to the cyclic and parallel-feedback extreme cases of cactus networks. By taking the cyclic interconnection as the fundamental starting point, we establish necessary and sufficient conditions for its robust stability. Integrating this with the parallel case, we synthesize a unified stability framework for general multi-loop cactus networks. The $\theta$-symmetric SRG framework is less conservative and provides a more intuitive geometric interpretation of system behaviors compared with some existing approaches. Several examples are included to demonstrate the effectiveness of the proposed methods.

Figures

Figures reproduced from arXiv: 2608.12591 by the authors.

Figure 1
Figure 1. Illustrations of (a) Chord(z), (b) Arc+(z) and (c) Arc−(z). Let Chord(z) denote the line segment connecting point z ∈ C and its complex conjugate z, as shown in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. An illustration of SRG(C) (gray region) and SRGθ(C) (green region) with θ = 50◦, where the red points are the eigenvalues of C. Example 2: Consider a matrix C =     1 + 2j 0 2 0 0 1 + j −1 1 −j 1 1.5 0 2 −j 1 0     . Let θ = 50◦ , then an illustration of SRG(C) and SRGθ(C) is shown in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. (a) An illustrations of W(T) (gray region), where the parabola is defined by a 2 = b cos2 δ, the blue line denotes the tangent line 2µa = cos δ(b + µ 2 ) and the red point marks the point of tangency. (b) An illustration of D[θ, µ] (gray region), where the red point represents the center of D[θ, µ]. Now we are ready to prove Theorem 1. Proof of Theorem 1: We first show (i). Since Γθ(C) < π 2 , it follows from Propos… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: (a) Parallel and (b) cascade interconnections under unit negative feedback. Based on the subadditive and submultiplicative properties of θ-symmetric SRG, we can derive graphical criteria for the nonsingularity of the sum-type and product-type return difference matrices…
Figure 5
Figure 5. Figure 5: Illustrations of (a) R[α, β, γ] (gray region), and (b) S[α, β, γ] (gray region), where θ = 1 2 (β + α) and the red point represents the complex scalar γejθ . Accordingly, graphically define the matrix set via its canon￾ical SRG as M[α, β, γ] = {C ∈ C n×n : SRGθ ⋆ (C) ⊂…
Figure 6
Figure 6. Figure 6: An illustration of the frequency-wise canonical SRG of G(s) in (18), where the red curves denote the eigenloci of G(s). For the SISO case, the frequency-wise canonical SRG degenerate precisely to the classical Nyquist plot. The proof follows directly from the definitio…
Figure 7
Figure 7. Figure 7: Feedback interconnection of a cascade of systems. By fixing θi(ω) = 0, Theorem 6 reduces to the following 0-symmetric SRG result. The proof is omitted for brevity. Corollary 5: The feedback interconnection of a cascade of systems P1, P2, . . . , PN ∈ RHm×m ∞ in [PITH_…
Figure 8
Figure 8. Figure 8: Without loss of generality, let loop 1 be the central loop and V1 denote its vertex set. Let Vhub ⊂ V1 be the set of hubs shared with petal loops. For each hub i ∈ Vhub, let Qi denote the index set of the petal loops containing vertex i [PITH_FULL_IMAGE:figures/full_f…
Figure 8
Figure 8. Figure 8: A sunflower graph with a central loop 1 → 2 → 3 → 1 and petal loops attached at hubs 2 and 3. With these over-approximations in place, we are now ready to characterize the individual subsystems in the sunflower graph. Definition 2: For given ξk(ω) ∈ R, k = 1, 2, . . . …
Figure 10
Figure 10. Figure 10: An illustration of the choices of ξk(ω), k = 2, 3, . . . , 7. When the gain and phase are considered independently, Theorem 8 reduces to the following decoupled stability criteria for sunflower graphs. Corollary 9: A sunflower graph of systems P1, P2, . . . , PN ∈ RHm…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 43 canonical work pages

  1. [1]

    Qiu and K

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

    work page 2009

  2. [2]

    K. Zhou, J. C. Doyle, and K. Glover,Robust and Optimal Control. New Jersey: Prentice Hall, 1996

    work page 1996

  3. [3]

    The generalized Nyquist stability criterion and multivariable root loci,

    A. G. J. MacFarlane and I. Postlethwaite, “The generalized Nyquist stability criterion and multivariable root loci,”Int. J. Control, vol. 25, no. 1, pp. 81–127, 1977

    work page 1977

  4. [4]

    On the generalized Nyquist stability criterion,

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

    work page 1980

  5. [5]

    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 J. Control Optim., vol. 62, no. 2, pp. 1235–1260, 2024

    work page 2024

  6. [6]

    Robustness under bounded uncertainty with phase information,

    A. L. Tits, V . Balakrishnan, and L. Lee, “Robustness under bounded uncertainty with phase information,”IEEE Trans. Autom. Control, vol. 44, no. 1, pp. 50–65, 1999

    work page 1999

  7. [7]

    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, 2025

    2025

  8. [8]

    Wielandt,Topics in the Analytic Theory Matrices

    H. Wielandt,Topics in the Analytic Theory Matrices. Lecture Notes, Dept. of Mathematics, Univ. of Wisconsin, Madison, 1967

    work page 1967

Show all 50 references
  1. [9]

    The angle of an operator and positive operator products,

    K. Gustafson, “The angle of an operator and positive operator products,” Bull. Amer. Math. Soc., vol. 74, pp. 488–492, 1968

    1968

  2. [10]

    Angular localization of the spectrum of a multiplicative integral in a Hilbert space,

    M. G. Krein, “Angular localization of the spectrum of a multiplicative integral in a Hilbert space,”Funct. Anal. Appl., vol. 3, no. 1, pp. 73–74, 1969

    1969

  3. [11]

    Antieigenvalues,

    K. Gustafson, “Antieigenvalues,”Linear Algebra Appl., vol. 208, pp. 437–454, 1994

    1994

  4. [12]

    A cyclic small phase theorem,

    C. Chen, W. Chen, D. Zhao, J. Chen, and L. Qiu, “A cyclic small phase theorem,”IEEE Trans. Autom. Control, vol. 71, no. 3, pp. 1676–1691, 2026

    2026

  5. [13]

    B. D. O. Anderson and S. V ongpanitlerd,Network Analysis and Syn- thesis: A Modern Systems Theory Approach. Upper Saddle River, NJ: Prentice Hall, 1973

    1973

  6. [14]

    Feedback control of negative-imaginary systems,

    I. R. Petersen and A. Lanzon, “Feedback control of negative-imaginary systems,”IEEE Control Syst. Mag., vol. 30, no. 5, pp. 54–72, 2010

    2010

  7. [15]

    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 J. Matrix Anal. Appl., vol. 30, no. 2, pp. 560–581, 2008

    2008

  8. [16]

    Feedback stability under mixed gain and phase uncertainty,

    J. Liang, D. Zhao, and L. Qiu, “Feedback stability under mixed gain and phase uncertainty,”IEEE Trans. Autom. Control, vol. 70, no. 2, pp. 1008–1023, 2024

    2024

  9. [17]

    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,”arXiv preprint arXiv:2507.19918, 2025. 16 IEEE TRANSACTIONS AND JOURNALS TEMPLATE

    2025

  10. [18]

    The dynamics of feedback control circuits in biochemical pathways,

    J. J. Tyson and H. G. Othmer, “The dynamics of feedback control circuits in biochemical pathways,”Prog. Theor. Biol., vol. 5, pp. 1–62, 1978

    1978

  11. [19]

    Existence criteria of periodic oscil- lations in cyclic gene regulatory networks,

    Y . Hori, T.-H. Kim, and S. Hara, “Existence criteria of periodic oscil- lations in cyclic gene regulatory networks,”Automatica, vol. 47, no. 6, pp. 1203–1209, 2011

    2011

  12. [20]

    Diagonal stability of a class of cyclic systems and its connection with the secant criterion,

    M. Arcak and E. D. Sontag, “Diagonal stability of a class of cyclic systems and its connection with the secant criterion,”Automatica, vol. 42, no. 9, pp. 1531–1537, 2006

    2006

  13. [21]

    Passivity gains and the “secant condition

    E. D. Sontag, “Passivity gains and the “secant condition” of stability,” Syst. Control Lett., vol. 55, no. 3, pp. 177–183, 2006

    2006

  14. [22]

    A generalisation of the secant criterion,

    R. Pates, “A generalisation of the secant criterion,”IFAC-PapersOnLine, vol. 56, no. 2, pp. 9141–9146, 2023

    2023

  15. [23]

    Lyapunov formulation of ISS cyclic- small-gain in continuous-time dynamical networks,

    T. Liu, D. J. Hill, and Z.-P. Jiang, “Lyapunov formulation of ISS cyclic- small-gain in continuous-time dynamical networks,”Automatica, vol. 47, no. 9, pp. 2088–2093, 2011

    2011

  16. [24]

    Secant and Popov-like conditions in power network stability,

    N. Monshizadeh and I. Lestas, “Secant and Popov-like conditions in power network stability,”Automatica, vol. 101, pp. 258–268, 2019

    2019

  17. [25]

    Scaled relative graph,

    R. Hannah, E. K. . Ryu, and W. Yin, “Scaled relative graph,”UCLA CAM Report, 2016

    2016

  18. [26]

    Scaled relative graphs: nonexpan- sive operators via 2D Euclidean geometry,

    E. K. Ryu, R. Hannah, and W. Yin, “Scaled relative graphs: nonexpan- sive operators via 2D Euclidean geometry,”Math. Program., vol. 194, no. 1, pp. 569–619, 2022

    2022

  19. [27]

    The scaled relative graph of a linear operator,

    R. Pates, “The scaled relative graph of a linear operator,”arXiv preprint arXiv:2106.05650, 2021

    2021 arXiv

  20. [28]

    Graphical nonlinear system analysis,

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

    2023

  21. [29]

    On phase in scaled graphs,

    S. van den Eijnden, C. Chen, K. Scheres, T. Chaffey, and A. Lanzon, “On phase in scaled graphs,” inProc. 64th IEEE Conf. Decis. Control, 2025, pp. 3595–3600

    2025

  22. [30]

    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,” inProc. 64th IEEE Conf. Decis. Control, 2025, pp. 2720–2725

    2025

  23. [31]

    Graphical analysis of nonlinear mul- tivariable feedback systems,

    J. Krebbekx, R. T ´oth, and A. Das, “Graphical analysis of nonlinear mul- tivariable feedback systems,”arXiv preprint arXiv:2507.16513, 2025

    2025 arXiv

  24. [32]

    A dissipativity frame- work for constructing scaled graphs,

    T. de Groot, W. Heemels, and S. van den Eijnden, “A dissipativity frame- work for constructing scaled graphs,”arXiv preprint arXiv:2507.08411, 2025

    2025 arXiv

  25. [33]

    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,” inProc. Eur. Control Conf., 2025, pp. 114–119

    2025

  26. [34]

    A cascade of systems and the product of theirθ-symmetric scaled relative graphs,

    X. Yang, D. Zhang, W. Chen, and L. Qiu, “A cascade of systems and the product of theirθ-symmetric scaled relative graphs,” inIFAC- PapersOnLine, 2026, to appear

    2026

  27. [35]

    Further study on robust stability of cyclic systems viaθ-symmetric scaled relative graphs,

    X. Yang, W. Chen, and L. Qiu, “Further study on robust stability of cyclic systems viaθ-symmetric scaled relative graphs,” inProc. 65th IEEE Conf. Decis. Control, 2026, to appear

    2026

  28. [36]

    Scaled relative graph of normal matrices,

    X. Huang, E. K. Ryu, and W. Yin, “Scaled relative graph of normal matrices,”arXiv preprint arXiv:2001.02061, 2019

    2001 arXiv

  29. [37]

    K. E. Gustafson and D. K. Rao,Numerical Range: The Field of Values of Linear Operators and Matrices. Springer, 1997

    1997

  30. [38]

    A cascade of systems and the product of theirθ-symmetric scaled relative graphs,

    X. Yang, D. Zhang, W. Chen, and L. Qiu, “A cascade of systems and the product of theirθ-symmetric scaled relative graphs,”arXiv preprint arXiv:2510.06583, 2025

    2025

  31. [39]

    Sectorial operators and normalized numerical range,

    W. Auzinger, “Sectorial operators and normalized numerical range,” Appl. Numer. Math., vol. 45, no. 4, pp. 367–388, 2003

    2003

  32. [40]

    The normalized numerical range and the Davis-Wielandt shell,

    B. Lins, I. M. Spitkovsky, and S. Zhong, “The normalized numerical range and the Davis-Wielandt shell,”Linear Algebra Appl., vol. 546, pp. 187–209, 2018

    2018

  33. [41]

    Extending the Kantorovi ˇc inequality to normal matrices,

    C. Davis, “Extending the Kantorovi ˇc inequality to normal matrices,” Linear Algebra Appl., vol. 31, pp. 173–177, 1980

    1980

  34. [42]

    Antieigenvalues: method of estimation and calculation,

    B. Mirman, “Antieigenvalues: method of estimation and calculation,” Linear Algebra Appl., vol. 49, pp. 247–255, 1983

    1983

  35. [43]

    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,”Int. J. Appl. Comput. Math., vol. 1, no. 1, pp. 111–119, 2015

    2015

  36. [44]

    On interconnections of “mixed

    W. M. Griggs, S. S. K. Sajja, B. D. O. Anderson, and R. N. Shorten, “On interconnections of “mixed” systems using classical stability theory,” Syst. Control Lett., vol. 61, no. 5, pp. 676–682, 2012

    2012

  37. [45]

    J. A. Ball, I. Gohberg, and L. Rodman,Interpolation of Rational Matrix Functions. Basel, Germany: Birkh ¨auser, 1990

    1990

  38. [46]

    Diagonal stability on cactus graphs and application to network stability analysis,

    M. Arcak, “Diagonal stability on cactus graphs and application to network stability analysis,”IEEE Trans. Autom. Control, vol. 56, no. 12, pp. 2766–2777, 2011

    2011

  39. [47]

    Cactus-expandable graphs,

    S. Azuma, T. Kure, T. Yoshida, and T. Sugie, “Cactus-expandable graphs,”IEEE Trans. Control Netw. Syst., vol. 6, no. 2, pp. 775–788, 2019

    2019

  40. [48]

    Graph-theoretic stability conditions for Metzler matrices and monotone systems,

    X. Duan, S. Jafarpour, and F. Bullo, “Graph-theoretic stability conditions for Metzler matrices and monotone systems,”SIAM J. Control Optim., vol. 59, no. 5, pp. 3447–3471, 2021

    2021

  41. [49]

    Scalable decentralized robust stability certificates for networks of interconnected heterogeneous dynamical systems,

    I. Lestas and G. Vinnicombe, “Scalable decentralized robust stability certificates for networks of interconnected heterogeneous dynamical systems,”IEEE Trans. Autom. Control, vol. 51, no. 10, pp. 1613–1625, 2006

    2006

  42. [50]

    The quadratic eigenvalue problem,

    F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Review, vol. 43, no. 2, pp. 235–286, 2001

    2001

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.