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arxiv: 2604.13842 · v1 · submitted 2026-04-15 · 📡 eess.SY · cs.SY

Frequency Response of Nonlinear Systems: Notions, Analysis, and Graphical Representation

Alessio Moreschini , Matteo Scandella This is my paper

Pith reviewed 2026-05-10 12:45 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords nonlinear systemsfrequency responsephasor formdistortion functionBode diagramssteady-state analysiscontrol design
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The pith

Nonlinear systems under periodic excitations have a frequency response defined as a phasor with gain, phase and distortion functions.

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

The paper shows that frequency response remains definable for nonlinear systems driven by nonlinear periodic inputs through a complex phasor representation of the steady-state output. Alongside gain and phase, a distortion function is introduced to quantify how the output deviates from a pure sinusoid. This matters because it creates a complete picture of nonlinear effects in the frequency domain. If the claim holds, engineers gain graphical tools to plot these quantities against both frequency and amplitude, supporting analysis and design of systems that must handle nonlinear behavior.

Core claim

For systems under nonlinear periodic excitations, the frequency response is defined as a complex-valued function in phasor form. This is completed by gain and phase functions plus a distortion function that quantifies the alteration introduced by the system in the steady-state output, allowing the full characterization to be shown in diagrams over input frequency and amplitude.

What carries the argument

the phasor representation of the steady-state output completed by a distortion function

If this is right

  • Gain, phase, and distortion can be plotted as functions of both input frequency and amplitude.
  • The resulting diagrams support performance analysis of nonlinear systems.
  • The loop-shaping design problem can be stated directly for nonlinear systems using the new representation.

Where Pith is reading between the lines

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

  • The distortion function may serve as a practical metric to decide when a nonlinear system is close enough to linear for standard design tools to apply.
  • This approach could extend frequency-domain methods to systems with harmonic-rich inputs without needing separate time-domain analysis.
  • Numerical computation of the three functions for specific nonlinear models would allow direct comparison of different system designs in the same format.

Load-bearing premise

That the steady-state output of a nonlinear system under periodic inputs settles into a pattern fully captured by one complex phasor plus a separate distortion term.

What would settle it

A concrete nonlinear system driven by a periodic input whose steady-state output waveform cannot be summarized by a single phasor value and one distortion metric would disprove the proposed characterization.

Figures

Figures reproduced from arXiv: 2604.13842 by Alessio Moreschini, Matteo Scandella.

Figure 5
Figure 5. Figure 5: Specifically, Figure 5a shows that the ω-gain is consistent with that of the linearized system until the effect of saturation becomes visible and dominates the linear effect. This is consistent with the shape of the ω-radius in Figure 5c, which proves that, at high amplitudes, the system bends the shape of the input, achieving high nonlinearity at high frequencies. Finally, Figure 5b confirms that the phas… view at source ↗
read the original abstract

The invariance principle, through which the steady-state behavior of nonlinear systems was introduced by Isidori and Byrnes, is leveraged in this article to bring forth a unifying characterization of the frequency response of nonlinear systems. We show that, for systems under nonlinear periodic excitations, the frequency response can still be defined as a complex-valued function in a phasor form. However, together with suitable notions of gain and phase functions, we show the existence of another function that completes the frequency response and allows quantifying the distortion introduced by the system in the steady-state output. This nonlinear characterization enabled the representation over input frequency and amplitude of the gain, phase, and distortion produced by the system, via a nonlinear enhancement of the Bode diagrams. This graphical representation of the frequency response is well-suited to performance analysis of a nonlinear system and, furthermore, allows for the formulation of the loop-shaping problem for nonlinear systems.

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 leverages the Isidori-Byrnes invariance principle to characterize the frequency response of nonlinear systems under nonlinear periodic excitations. It defines this response as a complex-valued phasor function, accompanied by gain and phase functions, and introduces a distortion function to quantify nonlinearity-induced effects in the steady-state output. This framework enables nonlinear Bode diagrams that represent gain, phase, and distortion over ranges of input frequency and amplitude, supporting performance analysis and the formulation of loop-shaping problems for nonlinear systems.

Significance. If the central derivations hold with the stated conditions, the work offers a principled extension of linear frequency-domain tools to nonlinear systems. The distortion function and resulting graphical representations provide concrete tools for analyzing steady-state behavior without full time-domain simulation, and the loop-shaping formulation could influence nonlinear controller design. The grounding in the established invariance principle is a clear strength, as is the explicit construction of amplitude- and frequency-dependent plots.

major comments (2)
  1. [Section 4] The main theorem (Section 4): the existence of the phasor representation and distortion function is asserted to follow directly from the invariance principle applied to the steady-state output, but the manuscript must state the precise hypotheses on the system class (e.g., relative degree, stability of the zero dynamics, and the form of the periodic excitation) that guarantee a unique attractive periodic response; without these, the claim that the phasor and distortion are well-defined for general nonlinear periodic inputs is not yet load-bearing.
  2. [Section 4] Definition of the distortion function (Eq. (12) or equivalent): the paper should demonstrate that this function is independent of the choice of coordinates on the invariant manifold and reduces to zero for linear systems; the current sketch leaves open whether the distortion term is uniquely determined or could be absorbed into a redefinition of the phasor.
minor comments (2)
  1. [Section 3] Notation: the symbols for the gain, phase, and distortion functions should be introduced with a single consistent table or list early in the paper to avoid later confusion when they appear in the nonlinear Bode plots.
  2. [Abstract] The abstract states the existence result but does not mention the required system hypotheses; adding one sentence on the class of systems would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment, the recommendation for minor revision, and the constructive comments on Section 4. We address each major comment below and will incorporate the requested clarifications and explicit statements into the revised manuscript.

read point-by-point responses

  1. Referee: [Section 4] The main theorem (Section 4): the existence of the phasor representation and distortion function is asserted to follow directly from the invariance principle applied to the steady-state output, but the manuscript must state the precise hypotheses on the system class (e.g., relative degree, stability of the zero dynamics, and the form of the periodic excitation) that guarantee a unique attractive periodic response; without these, the claim that the phasor and distortion are well-defined for general nonlinear periodic inputs is not yet load-bearing.

    Authors: We appreciate the referee highlighting the need for explicit hypotheses. The derivations in Section 4 rely on the standard conditions from the Isidori-Byrnes invariance principle under which a unique attractive periodic response exists. In the revised manuscript we will state these hypotheses explicitly at the beginning of Section 4: the plant has well-defined relative degree, the zero dynamics are asymptotically stable, and the periodic excitation is generated by a neutrally stable exosystem. These conditions guarantee the existence of a unique invariant manifold and therefore well-defined phasor and distortion functions. We will also add a direct reference to the relevant theorem in Isidori and Byrnes. revision: yes

  2. Referee: [Section 4] Definition of the distortion function (Eq. (12) or equivalent): the paper should demonstrate that this function is independent of the choice of coordinates on the invariant manifold and reduces to zero for linear systems; the current sketch leaves open whether the distortion term is uniquely determined or could be absorbed into a redefinition of the phasor.

    Authors: We agree that the distortion function requires an explicit demonstration of coordinate independence and uniqueness. The distortion is defined intrinsically as the L2-norm of the residual between the steady-state output waveform and its fundamental harmonic component (extracted via the phasor). Because this definition operates directly on the scalar output signal, it is independent of any particular coordinate chart on the invariant manifold. For linear systems the steady-state output is purely sinusoidal at the driving frequency, so the residual vanishes and the distortion is identically zero. In the revision we will add a short remark (or appendix paragraph) showing that the phasor is the unique Fourier coefficient of the fundamental frequency on the manifold; the distortion is then the orthogonal complement in L2 and cannot be absorbed into a redefinition of the phasor. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation begins from the external Isidori-Byrnes invariance principle (cited as prior literature, not self-authored) to establish existence of a unique attractive steady-state periodic output under nonlinear periodic forcing. From this foundation the authors introduce phasor-form frequency response together with explicit gain, phase, and distortion functions, then construct the associated nonlinear Bode representation. No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the central steps remain independent once the cited invariance result is granted. The construction is therefore self-contained against an external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The characterization depends on the invariance principle as background and on the assumption that periodic excitations admit a well-defined steady-state phasor representation; no free parameters or new invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The invariance principle of Isidori and Byrnes applies to the steady-state behavior of the nonlinear system under periodic excitation.
    Invoked in the abstract as the foundation for defining the frequency response.

pith-pipeline@v0.9.0 · 5453 in / 1332 out tokens · 19094 ms · 2026-05-10T12:45:35.460338+00:00 · methodology

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