Scaled Relative Graph Analysis of General Interconnections of SISO Nonlinear Systems

Amritam Das; Julius P. J. Krebbekx; Roland T\'oth

REVIEW 3 major objections 5 minor 25 references

Scaled Relative Graph Analysis of General Interconnections of SISO Nonlinear Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Adding Nyquist encirclement information to the scaled relative graph lets one certify stability and $L_2$-gain for feedback systems with unstable plants and general nonlinearities, yielding a generalized circle criterion.

desk verdict Extended SRG is a real advance, but the general-interconnection theorem rests on the exact domain inference the paper warns against. read the letter →

arxiv 2507.15564 v1 pith:NXDOHM5N submitted 2025-07-21 eess.SY cs.SYmath.OC

Julius P. J. Krebbekx , Roland Tóth , Amritam Das This is my paper · ORCID

classification eess.SYcs.SYmath.OC MSC 93C1093C8093D10

keywords ScaledRelativeGraphnonlinearstabilityL2-gainNyquistcriterioncircleLur'esystemsfeedbackinterconnectionscontext-freegrammar

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 identifies a pitfall in scaled relative graph (SRG) analysis of nonlinear feedback systems: for an unstable linear time-invariant plant inside a loop, ordinary SRG calculus can produce a finite gain bound for the closed loop even when the Nyquist criterion says the loop is unstable. The reason is that the SRG is defined only on the restricted input domain for which the unstable operator maps finite-energy signals to finite-energy signals, so a finite graph radius does not certify stability on all of $L_2$. The paper's fix is an extended SRG that adds the Nyquist encirclement region to the h-convex hull of the Nyquist diagram, and it proves that this extended object obeys the same interconnection rules as the ordinary SRG. With that, the paper derives a generalized circle criterion for Lur'e systems and a stability and incremental/non-incremental $L_2$-gain theorem for arbitrary interconnections of SISO LTI and nonlinear operators. This matters because it extends an exact graphical method from stable open-loop plants to the unstable plants and general nonlinearities that appear in practical feedback design.

What carries the argument

The extended SRG: for an LTI operator $R$, the h-convex hull $G_R$ of its Nyquist diagram is merged with the encirclement region $N_R = \{z : N_R(z)+n_p > 0\}$, where $N_R(z)$ is the clockwise winding number of the Nyquist curve around $z$ and $n_p$ is the number of right-half-plane poles. The extra region is what carries the Nyquist criterion's stability information that the ordinary SRG discards. The argument also rests on two supporting mechanisms: a homotopy 'inflation' of each nonlinear operator from a real gain $\kappa \in SRG(\phi)$ to the full nonlinearity, which lets stability of the linearized loop imply stability of the nonlinear loop; and a context-free grammar representation of interconnections, which turns block diagrams into words whose SRG bounds can be computed by the algebraic rules.

What would settle it

Compute $SRG_U(T)$ for an unstable LTI operator $T$ on a truncated finite-dimensional approximation of the stabilizing input set $U$ and compare it with the h-convex hull of $Nyquist(T)$; a single pair of finite-energy inputs $u_1,u_2$ whose ratio $z_T(u_1,u_2)$ lies outside the hull would falsify the equality that Definition 3 relies on. A sharper test is to check Theorem 3.c: find a candidate $G$ with $n_p > 0$ for which the inversion rule fails by comparing $N_G$ and $N_{G^{-1}}$ on a point not on the real axis, since the proof only establishes the real-axis correspondence.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the object $SRG'(R) = G_R \cup N_R$, where $G_R$ is the h-convex hull of the Nyquist diagram of an LTI operator $R$ and $N_R = \{z : N_R(z)+n_p > 0\}$ is the set of points whose clockwise winding number plus the number of unstable poles is positive, behaves exactly like the ordinary SRG under the algebraic operations of scaling, adding the identity, inversion, summation, and product (Theorem 3). This extended SRG resolves the apparent contradiction between SRG calculus and the Nyquist criterion. Using it, the paper proves that a Lur'e feedback loop is stable with $L_2$-gain bound $1/r_m$ whenever $\operatorname{dist}(SRG'(G)^{-1}, -SRG(\phi)) \ge r_m > 0$, for any inflatable nonlinear operator $\phi$, not only sector-bounded nonlinearities (Theorems 4 and 5). For general interconnections, the paper represents the interconnection as a word in a context-free grammar over LTI and nonlinear operator symbols, replaces each symbol by its (extended) SRG, and proves that the resulting set $C(R)$ satisfies $\Gamma(R) \le r_{\min}(C(R))$ on $\operatorname{dom}(R)$, with $\operatorname{dom}(R) = L_{2e}$ under continuity and inflation assumptions (Theorem 6).

Load-bearing premise

The load-bearing premise is that for an unstable linear time-invariant operator, the scaled relative graph computed over exactly the finite-energy inputs that the operator keeps finite-energy equals the h-convex hull of its Nyquist diagram; the paper states this as clear rather than proving it, and Definition 3 of the extended SRG is built on it.

Editorial extensions

If this is right

For Lur'e systems, the classical circle criterion is recovered as a special case, and the new criterion applies to time-varying sector nonlinearities, reset elements, and other operators whose SRG is bounded and inflatable, while also supplying an $L_2$-gain bound.
Feedback loops containing unstable plants or integrators can be analyzed by direct SRG calculus without the false finite-gain conclusions that occur with the ordinary SRG.
Arbitrary finite interconnections of SISO LTI and nonlinear operators can be certified stable with incremental or non-incremental $L_2$-gain bounds by computing a single set $C(R)$ from the interconnection's grammar word.
The choice of the real gain $\kappa$ in the inflation acts as a loop transformation or multiplier that can be tuned to maximize the separation distance $r_m$ and thus tighten the gain bound.
Well-posedness and causality of the closed-loop system follow from the same separation conditions whenever the subsystems are causal.

Reading between the lines

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

The paper's key unproved equality — that the SRG over finite-energy stabilizing inputs equals the h-convex hull of the Nyquist diagram for unstable LTI operators — is the load-bearing point; if it only holds for restricted classes of transfer functions, the extended SRG's definition would need modification, and testing this equality on non-minimum-phase or repeated-pole examples is a natural next
The same 'fill the hole with encirclement data' repair could be applied to other restricted-domain graphical tools, such as nonlinear Bode diagrams or harmonic-balance methods, whenever the underlying frequency response is defined only on stabilizing inputs.
Because different grammar words for the same operator yield different sets $C(R)$, the framework opens an algorithmic problem: search over equivalent interconnections and over $\kappa$ to minimize the radius of the bound, analogous to multiplier optimization.
The generalized circle criterion suggests that any bounded, inflatable operator with a known SRG — including dynamic or reset nonlinearities — can be plugged into classical Lur'e-type stability arguments, which may extend $L_2$-gain performance shaping to control architectures where sector conditions fail.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things. The extended SRG and its calculus (Definition 3, Theorem 3) are a real step forward for the SRG literature. The paper also correctly identifies a genuine pitfall: a bounded SRG computed over dom(R) does not imply dom(R)=L2. The examples are carefully done, and the generalized circle criterion is a useful extension.

The weak spot is Theorem 6. In the proof in Appendix B, the step from a finite radius of C(R) to 'R_tau is stable for all tau' to 'dom(R_tau)=L2' is exactly the inference the paper itself warns against in Section III-A2. A finite radius of an SRG-type set only bounds the gain on dom(R_tau); it doesn't make the domain all of L2. The continuity and inflatability assumptions do not close this gap unless they are used in a fixed-point or well-posedness argument that propagates full-domain solvability from tau=0 (as in Theorem 2 via [10, Cor. 1]). The OCFG construction computes set inclusions, not a family of feedback loops with a uniform separation margin. So the headline claim—a stability and L2-gain theorem for general interconnections—is not established as written.

A second, smaller gap: Section III-A3 says it is 'clear' that for an unstable LTI operator, the SRG over the stabilizing input set is the h-convex hull of the Nyquist diagram. This is a restricted-domain statement that needs proof; if it is false, Definition 3's encirclement region may not describe the restricted operator. I suspect it is true, but it should not be asserted.

What survives? Theorem 4 for the Lur'e-type canonical interconnections looks largely sound, because well-posedness there is supplied by Theorem 2 rather than by the broken step in Theorem 6. The examples are worked honestly and the simulations match the gain bounds. The formal-language approach is more bookkeeping than deep mathematics, but it is a reasonable way to organize computations.

This paper deserves a serious referee. The core ideas are good, but Theorem 6 needs either a repaired proof—likely by showing a uniform separation margin along the homotopy and invoking a fixed-point theorem—or a weakened statement that only claims gain bounds on dom(R). I would not desk-reject; I would ask for major revision on Theorem 6 and the III-A3 claim.

Referee Report

3 major / 5 minor

Summary. The paper identifies a pitfall in existing Scaled Relative Graph (SRG) analysis: for an unstable LTI operator, a finite-radius SRG computed over its domain does not imply L2 stability. To resolve this, the authors propose an extended SRG for LTI operators that adds Nyquist encirclement information to the h-convex hull of the Nyquist diagram. They prove that the extended SRG obeys the same interconnection rules as the ordinary SRG, use it to derive stability and L2-gain conditions for three canonical feedback interconnections (Lur'e, controlled Lur'e, and Lur'e-controlled systems), and obtain a generalized circle criterion. For general interconnections, they introduce an Operator Context-Free Grammar (OCFG) representation and state Theorem 6, which claims that if the linearized interconnection is stable and a bound computed by SRG calculus has finite radius, then the nonlinear interconnection is L2-stable with the corresponding gain bound. Three examples (Duffing oscillator, nonlinear pendulum, and a controlled Lur'e plant with saturation) illustrate the proposed analysis.

Significance. If correct, the paper would make a substantial contribution to graphical nonlinear system analysis: it would extend exact SRG-based stability and performance analysis to interconnections containing unstable LTI plants, provide a rigorous generalization of the circle criterion with L2-gain bounds, and supply a formal-language framework for systematically computing SRG bounds on arbitrary SISO interconnections. The paper is careful in setting up causal operators on L2/L2e, gives proofs for the extended-SRG interconnection rules, and includes worked examples with simulations. However, the central theorem for general interconnections contains a load-bearing logical gap, and a foundational equality for the extended SRG is asserted without proof. These issues prevent acceptance in the current form.

major comments (3)

[Section V-B2, Step 4 and Appendix B, proof of Theorem 6] The proof of Theorem 6 states: 'From (20) and the condition that C(R) has finite radius r, we know that R_tau is stable for all tau in [0,1]. We can conclude that dom(R_tau) = L2 for all tau in [0,1].' This inference is exactly the pitfall the paper itself identifies in Section III-A2: a finite radius of an SRG-type set only bounds the (incremental) gain on the domain of the operator, and does not imply that the domain is all of L2. The inclusions in (20) are set inclusions for SRG-type bounds; the continuity and inflatability assumptions give monotone inclusions C(R_tau1) subset C(R_tau2), but no uniform separation margin or fixed-point argument analogous to Theorem 2 or Theorem 10 is supplied for a general word wR in the OCFG. Therefore the conclusion dom(R) = L2e in Theorem 6 is unsupported. A repair would require either a genuine homotopy/well-posedness argument that propagates full-domain solvability from tau=0, or a weakening of the theorem to a gain bound on dom(R).
[Section III-A3] The paper states that for an unstable LTI operator T, the SRG over the set U of inputs for which Tu is in L2 is 'clear[ly]' equal to the h-convex hull of the Nyquist diagram of T. No proof is provided. This equality is the basis of Definition 3 of the extended SRG and therefore underpins all subsequent extended-SRG results. If the equality fails, the encirclement region NR may not correctly characterize the restricted-domain behavior of unstable LTI operators. This is a load-bearing unproved assertion and should be either proved or stated explicitly as an assumption.
[Theorem 6 statement versus proof] The statement of Theorem 6 does not assume causality of R, yet the proof concludes: 'Since R is causal by assumption, we know by Lemma 1 that R : L2e to L2e'. The causality hypothesis is missing from the theorem statement. Without it, Lemma 1 cannot be invoked to extend a domain conclusion from L2 to L2e. The theorem statement should include the causality assumption, or the conclusion should be restricted to dom(R) = L2.

minor comments (5)

[Definition 6] In the definition of continuity in tau, the phrase 'the map u 7-> ||PT R_tau u||2 is continuous in tau' should read 'the map tau 7-> ||PT R_tau u||2 is continuous in tau'; the variable in the map is tau, not u.
[Theorem 3] The theorem statement says 'the following statements hold for the SRG defined in Theorem 4', but the extended SRG is defined in Definition 3, not Theorem 4. This cross-reference should be corrected.
[Section VI-B] The derivation of SRG'(K)^{-1} = D[0,1/kp] refers to 'Theorem 7' when invoking the Nyquist criterion; it would be clearer to refer to Definition 3 and Lemma 3, since the statement concerns the extended SRG of an LTI operator.
[Proof of Theorem 3, items 4 and 5] The phrase 'a1 = 0 or a2 = 0 would amount to a pole-zero cancellation' needs a short justification; if a_i(s_u)=0 with b_i(s_u) != 0, then the relevant instability condition fails, so the exclusion is valid but not immediate as written.
[Section V-B2, informal Step 4] The informal description says that if RLTI is stable and C(R) has finite radius, then 'the stability of RLTI is not lost during the inflation' and R is stable. This is a restatement of the unsupported inference discussed in the first major comment; it should be revised to reflect the actual assumptions needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the extended SRG is a defined construction and Theorems 3-6 are proved from it; the flagged Theorem 6 inference is a correctness gap, not a circular reduction.

full rationale

The paper's central derivation is not circular. Definition 3 defines the extended SRG as SRG'(R) := G_R ∪ N_R, where N_R is the Nyquist encirclement region; Theorem 3 then proves the SRG calculus rules for this object by separate algebraic arguments on G_R and N_R (Appendix B), rather than assuming the rules. The Lur'e result (Theorem 4) is proved in Appendix B, so the footnote reference to the authors' earlier preprint [9] is not load-bearing. No fitted parameter is renamed as a prediction: the scalar kappa in Theorem 4 is a loop transformation chosen from SRG(phi) intersect R, and the computed bound C(R) is a set-valued SRG bound, not a data fit. The generalized circle criterion is obtained as a corollary of Theorem 4, and the paper explicitly states its equivalence to the classical circle criterion for sector-bounded nonlinearities rather than importing that equivalence as an input. The main concerns are correctness gaps, not circularity: Section III-A3 asserts without proof that for an unstable LTI operator the SRG over its L2-domain equals the h-convex hull of the Nyquist diagram, and the proof of Theorem 6 in Appendix B infers full-domain L2-stability from finite radius of the computed bound C(R_tau), which is exactly the pitfall the paper itself warns about in Section III-A2. These are missing justifications in the derivation chain, but they do not reduce a claimed prediction to an input by construction. The paper is self-contained against the external SRG and homotopy results [5], [6], [10], and no circular step can be exhibited beyond the borrowed definitions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on existing SRG calculus and the Nyquist criterion, plus the new technical conditions of inflatability and continuity in tau, and an unproven assertion about the SRG of unstable LTI operators. No free parameters are fitted to data; the scalar kappa in Theorem 4 is a user-selected loop transformation.

assumptions (5)

domain assumption All systems are causal operators on L2e with R(0)=0 (Section II-B).
Used to extend L2 gain results to L2e via Lemma 1 and to define well-posedness.
ad hoc to paper Nonlinear operators are inflatable (Definition 4): there is a continuous lift Phi_tau from a real constant kappa to phi with monotone SRG inclusion.
Central to the homotopy argument in Theorem 6; not established for general dynamic nonlinearities.
ad hoc to paper The homotopy operator R_tau is continuous in tau in the sense of Definition 6.
Invoked in Theorem 6 to conclude stability is preserved during inflation; this is a strong condition that the paper only verifies for the saturation example.
ad hoc to paper For an unstable LTI operator T, SRGU(T) over the stabilizing input set U equals the h-convex hull of Nyquist(T) (Section III-A3).
Stated without proof; it is the foundation of the extended SRG definition.
domain assumption The chord and arc properties of Proposition 1 hold when computing SRG bounds for interconnections.
Required for the bound C(R) to be valid; the paper notes this in Section V-B2 but does not include it in the statement of Theorem 6.

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Pith. "Pith review of Scaled Relative Graph Analysis of General Interconnections of SISO Nonlinear Systems." pith.science (2026). https://pith.science/paper/NXDOHM5N

@misc{pith2026250715564,
  author       = {Pith},
  title        = {Pith review of: Scaled Relative Graph Analysis of General Interconnections of SISO Nonlinear Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NXDOHM5N}},
  note         = {Machine review of arXiv:2507.15564}
}

abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. However, we show that the current SRG analysis suffers from a pitfall that limits its applicability in analyzing practical nonlinear systems. We overcome this pitfall by introducing a novel reformulation of the SRG of a linear time-invariant operator and combining the SRG with the Nyquist criterion. The result is a theorem that can be used to assess stability and $L_2$-gain performance for general interconnections of nonlinear dynamic systems. We provide practical calculation results for canonical interconnections and apply our result to Lur'e systems to obtain a generalization of the celebrated circle criterion, which deals with broader class of nonlinearities, and we derive (incremental) $L_2$-gain performance bounds. We illustrate the power of the new approach on the analysis of several examples.

Figures

Figures reproduced from arXiv: 2507.15564 by the authors.

**Figure 1.** Simple Feedback Systems. Stability of a Lur’e system with ϕ a sector bounded nonlinearity can be analyzed using the circle criterion, see Appendix A-C. Furthermore, in the special case that ϕ = κ ∈ R, stability of the LTI feedback loop can be analyzed using the Nyquist criterion, see Appendix A-B. III. STABILITY ANALYSIS WITH SCALED RELATIVE GRAPHS: PITFALL AND RESOLUTION In this section, we show how existing meth… view at source ↗

**Figure 2.** SRGs and Nyquist diagram corresponding to [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Block diagram of a Lur’e plant (G, ϕ) controlled by the LTI controller K. K(s) G(s) ϕ r e u ′ u − − y [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Block diagram of an LTI system G with Lur’e controller (K, ϕ). 2) LTI plant with Lur’e controller: The LTI system controlled by a Lur’e controller is depicted in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: A block-chain system, where Gi are LTI operators and ϕj are static or dynamic nonlinear operators. The assumption on the nonlinearities ϕj is that their SRGs are inflatable, see Definition 4. Note that this implies boundedness of their SRGs. 2) Stability of General In…

**Figure 6.** Figure 6: Analysis of the controlled Duffing oscillator. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: SRG analysis and simulations of the nonlinear pendulum. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: Analysis of the example in Section VI-C. [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

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H. Nyquist, “Regeneration Theory,” Bell System Technical Journal , vol. 11, no. 1, pp. 126–147, Jan. 1932

1932

[2] [2]

A Frequency-Domain Condition for the Stability of Feedback Systems Containing a Single Time-Varying Nonlinear Element,

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N. M. Krylov, N. N. Bogoliubov, and S. Lefschetz, Introduction to Non-Linear Mechanics, ser. Annals of Mathematics Studies; Annals of Mathematics Studies; Annals of Mathematics Studies; No. 11; No. 11. Princeton: Princeton University Press, 1947

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arXiv 2025