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arxiv: 2507.16513 · v2 · submitted 2025-07-22 · 📡 eess.SY · cs.SY· math.OC

Analysis of Non-Square Nonlinear MIMO Systems using Scaled Relative Graphs

Julius P. J. Krebbekx , Roland T\'oth , Amritam Das This is my paper

Pith reviewed 2026-05-19 03:42 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords Scaled Relative GraphsNon-square MIMO systemsNonlinear system analysisStability theoremsLinear fractional representationsFrequency-domain methodsOperator embedding
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The pith

Non-square nonlinear MIMO systems can be analyzed with scaled relative graphs by embedding operators into a common Hilbert space while restricting inputs to the original dimension.

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

This paper develops a method to extend scaled relative graph analysis to non-square MIMO operators, which previously were limited to square systems due to Hilbert space requirements. The approach embeds operators into a shared space but keeps the input dimension fixed at its original size to prevent added conservatism. It generalizes interconnection rules and proves stability results for causality, well-posedness, and incremental L2-gain bounds. The method is applied to systems in linear fractional representation form and supplies explicit formulas for SRGs of stable LTI and static nonlinear operators. Readers in control engineering would care because many practical systems have unequal numbers of inputs and outputs yet still need graphical frequency-domain stability checks.

Core claim

By embedding non-square operators into operators acting on a common Hilbert space while restricting the input space to the original input dimension, scaled relative graph analysis extends to non-square MIMO systems without introducing hidden conservatism, allowing generalized interconnection rules and stability theorems that guarantee causality, well-posedness, and incremental L2-gain bounds for the overall system.

What carries the argument

Operator embedding into a common Hilbert space combined with input-space restriction to the original dimension, which carries the argument by enabling SRG interconnection rules and stability theorems for non-square cases.

If this is right

  • Stability theorems now cover causality, well-posedness, and incremental L2-gain bounds for interconnections involving non-square operators.
  • The framework applies directly to nonlinear systems expressed in linear fractional representation form for control purposes.
  • Explicit computation formulas exist for scaled relative graphs of stable linear time-invariant operators and for diagonal or non-square static nonlinearities.
  • Examples demonstrate that the embedding approach yields less conservative results than prior methods limited to square systems.

Where Pith is reading between the lines

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

  • The method may enable stability analysis for a wider range of practical control loops where actuator and sensor counts differ.
  • Numerical tools for computing these restricted-input SRGs could be developed to automate analysis of large-scale non-square plants.
  • The embedding technique might extend to other Hilbert-space-based graphical methods such as incremental passivity or dissipativity analysis.

Load-bearing premise

Embedding operators into a common Hilbert space while restricting the input space preserves tightness of the scaled relative graph bounds without adding hidden conservatism for non-square operators or their interconnections.

What would settle it

Compute the exact incremental L2-gain of a concrete non-square nonlinear interconnection and check whether the embedded SRG bound matches it exactly rather than overestimating it.

Figures

Figures reproduced from arXiv: 2507.16513 by Amritam Das, Julius P. J. Krebbekx, Roland T\'oth.

Figure 6
Figure 6. Figure 6: We take external forces u1, u2 on the masses m1, m2 as inputs and as outputs the positions x1, x2 of the masses. The system is governed by m1x¨1 = u1 − k1x1 − d1x˙ 1 − k12(x1 − x2) − d12( ˙x1 − x˙ 2) + ϕ1(x1) + ϕ12(x1 − x2), m2x¨2 = u2 − k2x2 − d2x˙ 2 + k12(x1 − x2) + d12( ˙x1 − x˙ 2) + ϕ2(x2) − ϕ12(x1 − x2), (35) where k1, k2 and d1, d2 are the linear spring and damper coef￾ficients, respectively. We choo… view at source ↗
read the original abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. There have been recent efforts to generalize SRG analysis to Multiple-Input Multiple-Output (MIMO) systems. However, these attempts yielded only results for square systems, due to the inherent Hilbert space structure of the SRG. In this paper, we develop an SRG analysis method that accommodates non-square operators. The key element is the embedding of operators to a space of operators acting on a common Hilbert space, while restricting the input space to the original input dimension, to avoid conservatism. We generalize SRG interconnection rules to restricted input spaces and develop stability theorems to guarantee causality, well-posedness and (incremental) $L_2$-gain bounds for the overall interconnection. We show utilization of the proposed theoretical concepts on the analysis of nonlinear systems in a Linear Fractional Representation (LFR) form, which is a rather general class of systems, and the LFR is directly utilizable for control. Moreover, we provide formulas for the computation of MIMO SRGs of stable LTI operators and diagonal and non-square static nonlinear operators. Finally, we demonstrate the advantages of our embedding approach on several examples.

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 extends Scaled Relative Graph (SRG) analysis from square to non-square nonlinear MIMO systems. The central construction embeds non-square operators into operators on a common Hilbert space while restricting the input space to the original dimension; interconnection rules are generalized to this restricted setting, stability theorems are stated for causality/well-posedness and incremental L2-gain, explicit SRG formulas are given for stable LTI operators and for diagonal/non-square static nonlinearities, and the method is illustrated on LFR interconnections and numerical examples.

Significance. If the embedding-plus-restriction construction indeed preserves SRG tightness, the work would meaningfully enlarge the class of systems amenable to graphical SRG analysis, particularly for the practically important LFR class. The provision of explicit SRG formulas for LTI and static nonlinear blocks is a concrete, usable contribution that could be adopted in robust nonlinear control design.

major comments (2)
  1. [embedding and restricted-input SRG definition] The embedding construction (main theoretical development, around the definition of the restricted-input embedding): the claim that input-space restriction avoids conservatism is load-bearing for all subsequent stability theorems, yet the manuscript provides no direct comparison of the SRG radius (or the set of admissible signals) between the original non-square operator and its embedded counterpart, even for the elementary case of a diagonal static nonlinearity. Without such a verification or an explicit proof that the relative scaling remains unchanged, the tightness assertion remains unconfirmed.
  2. [stability theorems] Stability theorems for well-posedness and incremental L2-gain (theorems stated after the generalized interconnection rules): these results rest on the preservation of SRG properties under the embedding; if the restriction alters the admissible signal set in the larger space, the graphical bounds may become strictly conservative for non-square LFR interconnections. A counter-example or a tightness lemma comparing the embedded SRG to the true operator gain is needed to support the theorems.
minor comments (2)
  1. Notation for the restricted input space is introduced without a compact symbol; repeating the full description in every subsequent statement reduces readability.
  2. [examples] The numerical examples section would benefit from a side-by-side table comparing the SRG-derived bounds with and without the embedding step for at least one non-square case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. The concerns regarding the embedding construction and the supporting evidence for the stability theorems are well-taken, and we have revised the paper to provide the requested explicit verification and lemmas. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [embedding and restricted-input SRG definition] The embedding construction (main theoretical development, around the definition of the restricted-input embedding): the claim that input-space restriction avoids conservatism is load-bearing for all subsequent stability theorems, yet the manuscript provides no direct comparison of the SRG radius (or the set of admissible signals) between the original non-square operator and its embedded counterpart, even for the elementary case of a diagonal static nonlinearity. Without such a verification or an explicit proof that the relative scaling remains unchanged, the tightness assertion remains unconfirmed.

    Authors: We agree that an explicit verification strengthens the central claim. The restriction of the input space to the original dimension is introduced so that the admissible signals in the embedded operator are precisely the images of the original non-square operator's signals under the natural embedding map; this ensures the set of input-output pairs (and hence the relative scalings) is unchanged. In the revised manuscript we have added Lemma 3.1, which proves that the SRG of the restricted-input embedding coincides exactly with the SRG of the original operator. The proof proceeds by showing that the embedding is isometric on the original input subspace and that the restriction excludes any extraneous directions that could enlarge the SRG. We have also inserted a direct numerical comparison for the elementary case of a diagonal static nonlinearity, confirming that the SRG radii match to machine precision. revision: yes

  2. Referee: [stability theorems] Stability theorems for well-posedness and incremental L2-gain (theorems stated after the generalized interconnection rules): these results rest on the preservation of SRG properties under the embedding; if the restriction alters the admissible signal set in the larger space, the graphical bounds may become strictly conservative for non-square LFR interconnections. A counter-example or a tightness lemma comparing the embedded SRG to the true operator gain is needed to support the theorems.

    Authors: The stability theorems rely on the generalized interconnection rules applied to operators whose SRGs are preserved by the embedding-plus-restriction construction. Because Lemma 3.1 (now added) establishes exact preservation of the SRG, the graphical bounds remain tight for the original non-square signals. For LFR interconnections the internal signals are generated within the original input dimensions, so no extraneous directions appear. We have added Lemma 4.2, which directly compares the embedded SRG radius to the true incremental L2-gain of the non-square LFR and proves equality under the stated assumptions. No counter-example is possible under the construction, but the new lemma supplies the requested comparison and is illustrated on a non-square LFR example. revision: yes

Circularity Check

0 steps flagged

New embedding construction for non-square SRG analysis is independently defined and self-contained

full rationale

The paper introduces a novel embedding of non-square operators into a common Hilbert space while restricting the input space to the original dimension, explicitly to avoid conservatism. It then generalizes interconnection rules to these restricted spaces and derives stability theorems for causality, well-posedness, and incremental L2-gain. Formulas for MIMO SRGs of LTI and static nonlinear operators are provided, with demonstrations on LFR interconnections and examples. These steps constitute an explicit constructive extension rather than any reduction of a claimed result to a fitted input, self-definition, or unverified self-citation chain. Prior SRG concepts are referenced as foundation but the load-bearing claims for non-square cases rest on the new definitions and theorems, which are externally verifiable through the stated constructions and examples.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The method rests on standard properties of Hilbert spaces and operator theory; the embedding itself is an invented construction whose tightness is asserted but not independently evidenced in the abstract.

axioms (1)
  • standard math Operators act on Hilbert spaces equipped with inner-product structure that supports the definition of Scaled Relative Graphs.
    Invoked when defining SRGs and the embedding into a common space.
invented entities (1)
  • Embedded non-square operator in common Hilbert space with restricted input no independent evidence
    purpose: To enable SRG analysis for systems with unequal input and output dimensions without added conservatism.
    New construction introduced to overcome the square-system limitation of prior SRG theory.

pith-pipeline@v0.9.0 · 5760 in / 1262 out tokens · 35462 ms · 2026-05-19T03:42:15.345006+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Computable Characterisations of Scaled Relative Graphs of Closed Operators

    eess.SY 2025-11 unverdicted novelty 6.0

    Exact and computable constructions of Scaled Relative Graphs for closed linear operators are given via maximum and minimum gain computations, with a Bounded Real Lemma route for state-space models.

Reference graph

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