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Detecting Destabilizing Nonlinearities in Absolute Stability Analysis of Discrete-Time Feedback Systems

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Pith's one-line read A rank-one solution of the dual LMI furnishes a constructive proof that the system is not absolutely stable.

desk verdict Solid conditional result: dual-LMI rank-one plus a sign condition constructively certifies instability in discrete-time, but the unquantified reach of that condition limits the practical claim. read the letter →

arxiv 2503.05875 v1 pith:CUGANZLB submitted 2025-02-02 math.OC

Hibiki Gyotoku , Tsuyoshi Yuno , Yoshio Ebihara , Dimitri Peaucelle , Sophie Tarbouriech , Victor Magron This is my paper · ORCID
classification math.OC MSC 93C5593D0593D0990C22
keywords absolutestabilitydiscrete-timefeedbacksystemsslope-restrictednonlinearitiesintegralquadraticconstraintsO'Shea-Zames-Falbmultipliersduallinearmatrixinequalitiesdestabilizingnonlinearitydetectionrank-onesolution
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

This paper turns a previously inconclusive outcome in absolute stability analysis into a constructive proof of instability. The standard IQC-based test for a discrete-time feedback system with slope-restricted nonlinearities is a linear matrix inequality (LMI); if the LMI is feasible, stability follows, but if it is infeasible, nothing could be concluded. The authors analyze the dual LMI, which is feasible exactly when the primal LMI is infeasible, and show that a rank-one dual solution, together with a sign condition, lets them build a specific slope-restricted nonlinearity that makes the closed-loop system have a nonzero equilibrium. That equilibrium proves the system is not absolutely stable over the assumed nonlinearity class.

What carries the argument

The engine is the dual of the IQC-stability LMI, obtained via Lagrange duality. On that dual feasible set, a rank-one factor $H = [h_1;\,h_2][h_1;\,h_2]^T$ supplies a candidate equilibrium $h_1$ and a candidate input-output pair $(z^*, w^*) = (Ch_1+Dh_2,\, h_2)$. The constraint $(Ah_1+Bh_2)(Ah_1+Bh_2)^T = h_1 h_1^T$ forces $Ah_1+Bh_2 = \pm h_1$, and the sign condition $P_d((Ah_1+Bh_2)h_1^T) \ge 0$ selects the plus branch so that $h_1$ is a fixed point. The second dual equality collapses to $w^*(z^*-w^*)^T = 1_m f^T + g 1_m^T + X$ with $f,g \ge 0$ and $X$ a Z-matrix with zero diagonal; that identity is exactly what guarantees the interpolation points $(z^*_i, w^*_i)$ have slopes between 0 and 1 (and, in the odd case, the symmetry), so an admissible destabilizing nonlinearity can be pieced together.

What would settle it

Take any system $(A,B,C,D)$ with $A$ Schur stable and $\|D\|<1$ where the dual LMI (6) is feasible with a rank-one $H$ satisfying the sign condition, and verify by independent means (e.g., a full-block IQC test or exhaustive simulation) whether the closed-loop system with every slope[0,1] repeated nonlinearity is globally asymptotically stable; if such a system is found to be absolutely stable, Theorem 6's construction of a destabilizing nonlinearity would be contradicted.

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Extended reading notes

Core claim

The paper's central claim is Theorem 6 (and its odd-symmetry counterpart, Theorem 8). Suppose the dual LMI (6) (or (13)) has a feasible solution H of rank one, written as $H = [h_1;\,h_2][h_1;\,h_2]^T$, and suppose the entrywise condition $P_d((Ah_1+Bh_2)h_1^T) \ge 0$ holds. Then $h_1$ is nonzero, and there is a nonlinearity $\varphi_{wc} \in \mathrm{slope}[0,1]$ (odd in the second case) that interpolates the data $\varphi_{wc}(Ch_1+Dh_2) = h_2$. The feedback system with this nonlinearity has $h_1$ as a nonzero equilibrium point, so it is not globally asymptotically stable; because $\varphi_{wc}$ lies in the assumed class of slope-restricted repeated nonlinearities, the original system is never absolutely stable over that class. The dual LMI's feasibility therefore upgrades LMI infeasibility from 'inconclusive' to 'definitively not absolutely stable' whenever a rank-one solution with the sign condition exists.

Load-bearing premise

The load-bearing premise is that the dual LMI's feasible set actually contains a rank-one solution $H = [h_1;\,h_2][h_1;\,h_2]^T$ that also satisfies the sign condition $P_d((Ah_1+Bh_2)h_1^T) \ge 0$; the theorems say nothing when no such solution exists, and the paper offers no algorithm or guarantee that one will be present.

Editorial extensions

If this is right

  • Whenever the dual LMI (6) is feasible with a rank-one $H$ satisfying the sign condition, the system is provably not absolutely stable over slope[0,1] repeated nonlinearities, not merely possibly unstable.
  • The same rank-one certificate for the odd-symmetry dual LMI (13) proves the system is not absolutely stable even over the smaller class of odd slope-restricted nonlinearities.
  • The constructed $\varphi_{wc}$ is explicit piecewise-linear and can be extracted from the dual solution, so the certificate is constructive: it yields both a witness nonlinearity and a witness nonzero equilibrium.
  • Because the dual LMI is feasible if and only if the primal LMI is infeasible, the theorem provides a rigorous way to interpret primal infeasibility in terms of actual instability, closing the gap left by the sufficiency-only primal condition.
  • The restriction to slope[0,1] and $\|D\|<1$ is used to keep the nonlinearities nonexpansive and the feedback well-posed; the authors state that extending to general slope[$\mu$,$\nu$] remains open.

Reading between the lines

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

  • If a rank-one dual solution can be found systematically by the solver (for instance by adding a rank penalty), the result becomes an automated instability test that runs whenever the primal LMI fails, turning every LMI infeasibility into either a constructive instability certificate or a reason to refine the multiplier class.
  • The same Lagrange-duality technique could be applied to other IQC-based sufficient conditions (e.g., with dynamic multipliers or sector-bounded nonlinearities), where a rank-one dual solution might similarly expose a destabilizing element in the assumed class.
  • The sign condition $P_d((Ah_1+Bh_2)h_1^T) \ge 0$ is the discrete-time analogue of a condition that was automatically satisfied in continuous time; understanding when it holds may reveal which systems are prone to having hidden equilibria at exactly the boundary of the multiplier class.
  • A testable extension is to check numerically whether, for random stable (A,B,C,D), the set of rank-one dual solutions with the sign condition is nonempty whenever the primal LMI is infeasible; if gaps exist, the detection test is conservative.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies absolute stability of discrete-time LTI feedback systems with repeated slope-restricted nonlinearities, using static O'Shea-Zames-Falb multipliers in an IQC/LMI framework. It derives primal LMIs for absolute stability together with their duals, and the main results (Theorems 6 and 8) state that if the dual LMI admits a rank-one solution H = [h1; h2][h1; h2]^T satisfying the sign condition Pd((Ah1+Bh2)h1^T) >= 0, then there exists a scalar nonlinearity phi_wc in slope[0,1] (odd in Theorem 8) interpolating the data phi_wc(z*_i) = w*_i, and the feedback system with Phi_wc = diag_m(phi_wc) has the nonzero equilibrium h1. The proofs use algebraic identities from the dual equality constraints and an interpolation argument. Numerical examples for both the non-odd and odd cases illustrate the construction.

Significance. If the main result holds, it offers a constructive way to convert infeasibility of a sufficient LMI certificate into a falsifying nonlinearity and an explicit nonzero equilibrium, which is valuable for absolute stability analysis of neural-network and other nonlinear feedback loops. The proof of the conditional theorem is sound, self-contained, and the interpolation argument is elegant. The main caveat is that the result requires a rank-one dual solution with an additional sign condition; the paper gives no guarantee or algorithm for finding such a solution, and Remark 9 explicitly states that the sign condition is under investigation. This limits the practical reach of the detection method but does not invalidate the conditional theorem.

major comments (2)
  1. [§3.2, Theorem 6 and §4.2, Theorem 8] The dual LMI is feasible whenever the primal LMI is infeasible, but the theorems require a rank-one feasible matrix H satisfying the additional sign condition Pd((Ah1+Bh2)h1^T) >= 0. The rank-one constraint is nonconvex and cannot be enforced by a standard SDP solver, and the paper provides no theoretical condition or algorithm guaranteeing that such a solution exists whenever the primal LMI is infeasible. Since the stated purpose is to draw a definite conclusion from infeasibility, the practical scope of the detection method is unquantified. The numerical examples only report that the solver returned a solution 'numerically verified' to be rank one, without solver details or tolerances. Please either provide a guarantee or a systematic search procedure for rank-one solutions, or substantially soften the claims about detecting destabilizing nonlinearities from infeasibility and clearly state the conditional nature in the abstract and introduction.
  2. [§3.4 and §4.4] The numerical demonstrations are load-bearing for the claim that the technical conditions are usable, but they omit the verification data needed to reproduce them. For each example, please report the SDP solver and settings, the numerical values of the second singular value of H (or an equivalent rank-one verification), and the computed margin of the sign condition Pd((Ah1+Bh2)h1^T) >= 0. Without these, the reader cannot distinguish a genuine rank-one solution from a numerically near-rank-one one, nor check that the sign condition actually holds.
minor comments (5)
  1. [§4.2, proof of Theorem 8(ii)] In the final sentence of the proof, the condition on phi_wc is written as phi_wc(-z*_i) = w*_i, but it should be phi_wc(-z*_i) = -w*_i to match the statement of the theorem and the preceding argument.
  2. [§3.2, proof of Theorem 6(i)] The phrase 'To prove that rank(H) = 1 can happen' is misleading: the paragraph actually proves H is nonzero and h1 is nonzero under the rank-one hypothesis. Please rephrase to describe what is being shown.
  3. [§4.2, Theorem 8(iii)] The proof of assertion (iii) is omitted with 'Omitted since the proof is exactly the same as the proof of (iii) in Theorem 6.' Since the paper aims to be self-contained, it would be better to include the short argument or at least spell out the identical steps.
  4. [Figures 2-5] The figure captions are minimal. Please state explicitly which curve is phi_wc and which is the identity line, and distinguish the trajectories from x(0) = h1 and from the other initial condition.
  5. [§1 and §4.3] The reference to Gyotoku et al. (2025) is an arXiv preprint; if a published version exists, please update the citation, and if not, consider noting that the continuous-time counterpart is a preprint.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorems are conditional constructions from dual-LMI certificates, and the only self-referential elements are non-load-bearing.

full rationale

The derivation chain in Theorems 6 and 8 is self-contained. The dual LMI (6)/(13) is the formal Lagrange dual of the sufficient stability LMI (4)/(11), and feasibility of the dual is equivalent to infeasibility of the primal by standard convex duality (Scherer 2006). The theorems add two explicit hypotheses: rank(H)=1 and the sign condition P_d((Ah1+Bh2)h1^T) >= 0. From the first equality in the dual, rank-one H and h1 != 0 give Ah1+Bh2 = ±h1 (an elementary fact cited to Ebihara 2012); the sign condition selects +h1. The dual equality (7) then shows the data (z*_i, w*_i) are interpolable by a slope-[0,1] function, and setting Phi_wc = diag(phi_wc) makes x=h1 an equilibrium by construction. This is a certificate-to-witness construction, not an assumption of the conclusion. The paper does not claim an algorithm or existence guarantee for rank-one sign-condition solutions; Remark 9 explicitly says the sign condition's interpretation and active use are under investigation, and the numerical examples only report that the returned H was numerically rank-one. That is a limitation on scope, not circularity. The citations to the authors' continuous-time predecessor (Gyotoku et al. 2025) and to Ebihara 2012 are not load-bearing: the discrete-time proof is carried out in the paper, and the Ebihara fact is elementary and independent of the paper's claims.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard background results (IQC stability, static OZF multipliers, SDP duality) and on explicit modeling restrictions (Schur A, ||D||<1, slope[0,1]). There are no fitted constants or invented entities; the dual solution is computed, not fit, and the constructed nonlinearity is a certificate, not a free parameter.

assumptions (7)
  • standard math IQC-based stability criterion (Proposition 1) from Megretski and Rantzer (1997).
    Used as the background certificate in Lemma 5 and Lemma 7 to connect multiplier existence to absolute stability.
  • standard math Static O'Shea-Zames-Falb multiplier positivity for slope-restricted repeated nonlinearities (Lemma 4, Fetzer and Scherer 2017).
    Supplies the DHD and DD multiplier sets that define the primal LMIs.
  • standard math SDP duality or Farkas-type lemma for LMIs, cited to Scherer (2006).
    Justifies the claim that dual LMI feasibility is equivalent to primal LMI infeasibility, used to motivate the dual formulations.
  • standard math If vectors u and v satisfy uu^T = vv^T, then u = v or u = -v.
    Invoked in the proofs of Theorems 6(iii) and 8(iii), cited to Ebihara (2012).
  • domain assumption A is Schur stable and ||D|| < 1.
    Explicitly assumed in Sections 3.2 and 4.2; Schur stability lets P > 0 be dropped from the reduced primal, and ||D|| < 1 is used to prove h1 != 0.
  • domain assumption Main theorems are restricted to slope-restricted nonlinearities with slope[0,1] (mu=0, nu=1).
    Stated before Theorems 6 and 8; Section 5 identifies extension to general slope[mu,nu] as future work.
  • domain assumption The nonlinearity is static, repeated, and of the form Phi = diag_m(phi).
    This is the class of nonlinearities studied throughout; the constructed Phi_wc is built within this class.

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Pith. "Pith review of Detecting Destabilizing Nonlinearities in Absolute Stability Analysis of Discrete-Time Feedback Systems." pith.science (2026). https://pith.science/paper/CUGANZLB

@misc{pith2026250305875,
  author       = {Pith},
  title        = {Pith review of: Detecting Destabilizing Nonlinearities in Absolute Stability Analysis of Discrete-Time Feedback Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUGANZLB}},
  note         = {Machine review of arXiv:2503.05875}
}
read the original abstract

This paper is concerned with the absolute stability analysis of discrete-time feedback systems with slope-restricted nonlinearities. By employing static O'Shea-Zames-Falb multipliers in the framework of integral quadratic constraints, we can obtain a certificate for the absolute stability in the form of a linear matrix inequality (LMI). However, since this LMI certificate is only a sufficient condition, we cannot draw any definite conclusion if the LMI turns out to be infeasible. To address this issue, we focus on the dual LMI that is feasible if and only if the original (primal) LMI is infeasible. As the main result, if the dual solution satisfies a certain rank condition, we prove that we can detect a destabilizing nonlinearity within the assumed class of slope-restricted nonlinearities as well as a non-zero equilibrium point of the resulting feedback system, thereby we can conclude that the system of interest is never absolutely stable. The effectiveness of the technical results is demonstrated through numerical examples.

Figures

Figures reproduced from arXiv: 2503.05875 by the authors.

Figure 1
Figure 1. Discrete-Time Nonlinear Feedback System Σ. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The input-output map of the detected φwc [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. State trajectories with x(0) = h1 and x(0) = [−0.5 − 0.5]T For this system, the dual LMI (6) turns out to be feasible, and the resulting dual solution H is numerically verified to be rank(H) = 1. The full-rank factorization of H as well as z ∗ , w∗ ∈ R 4 in Theorem 6 are obtained as h1 =  1.3898 1.0337 , h2 =    −0.0513 −0.0582 −0.0513 0.0151   , z∗ =    −0.9277 −1.2417 −0.4440 0.9210   , w∗ =    −0… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The input-output map of the detected φwc [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: State trajectories with x(0) = h1 and x(0) = [−0.5 − 0.5]T 4.4 Numerical Examples In this section, we demonstrate the effectiveness of the result in Theorem 8 by a numerical example. Let us consider the case where the coefficient matrices in (1) are given by A =  0.95…

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