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

TL;DR

This paper introduces a unifying way to characterize the frequency response of nonlinear systems using the invariance principle, enabling graphical Bode-like diagrams for nonlinear performance analysis.

Contribution

It extends the concept of frequency response to nonlinear systems, including gain, phase, and distortion, with a graphical representation for analysis and loop-shaping.

Findings

Defined a complex-valued frequency response function for nonlinear systems.

Introduced a nonlinear Bode diagram for gain, phase, and distortion.

Enabled nonlinear performance analysis and loop-shaping techniques.

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.