Matteo Scandella | Searcharxiv

arXiv2026

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

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.

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arXiv2024

Kernel-Based Learning of Stable Nonlinear Systems

Learning models of dynamical systems characterized by specific stability properties is of crucial importance in applications. Existing results mainly focus on linear systems or some limited classes of nonlinear systems and stability notions, and the general problem is still open. This article proposes a kernel-based nonlinear identification procedure to directly and systematically learn stable nonlinear discrete-time systems. In particular, the proposed method can be used to enforce, on the learned model, bounded-input-bounded-state stability, asymptotic gain, and input-to-state stability properties, as well as their incremental counterparts. To this aim, we build on the reproducing kernel theory and the Representer Theorem, which are suitably enhanced to handle stability constraints in the kernel properties and in the hyperparameters' selection algorithm. Once the methodology is detailed, and sufficient conditions for stability are singled out, the article reviews some widely used kernels and their applicability within the proposed framework. Finally, numerical results validate the theoretical findings showing, in particular, that stability may have a beneficial impact in long-term simulation with minimal impact on prediction.

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