arXiv subjects

Natsuki Katayama

Publications and source records attributed to Natsuki Katayama.

4 recordsLinked to original sources

arXiv2026

Global Linearization of Parameterized Nonlinear Systems with Stable Equilibrium Point Using the Koopman Operator

The Koopman operator framework enables global analysis of nonlinear systems through its inherent linearity. This study aims to clarify spectral properties of the Koopman operators for nonlinear systems with control inputs. To this end, we treat the inputs as parameters throughout this paper. We then introduce the Koopman operator for a parameterized dynamical system with a globally exponentially stable equilibrium point and analyze how eigenfunctions of the operator depend on the parameter. As a main result, we obtain a global linearization, which enables one to transform the nonlinear system into a finite-dimensional linear system, and we show that it depends continuously on the parameter. Subsequently, for a control-affine system, we investigate a condition under which the transformation providing a global bilinearization does not depend on the parameter. This provides the condition under which the global bilinearization for the control-affine system is independent of the parameter.

math.DS↗

arXiv2026

On Koopman Resolvents and Frequency Response of Nonlinear Systems

This paper proposes a novel formulation of frequency response for nonlinear systems in the Koopman operator framework. This framework is a promising direction for the analysis and synthesis of systems with nonlinear dynamics based on (linear) Koopman operators. We show that the frequency response of a nonlinear plant is derived through the Laplace transform of the output of the plant, which is a generalization of the classical approach to LTI plants and is guided by the resolvent theory of Koopman operators. The response is a complex-valued function of the driving angular frequency, allowing one to draw the so-called Bode plots, which display the gain and phase characteristics. Sufficient conditions for the existence of the frequency response are presented for three classes of dynamics.

eess.SY↗

arXiv2024

Koopman Operators for Global Analysis of Hybrid Limit-Cycling Systems: Construction and Spectral Properties

This paper reports a theory of Koopman operators for a class of hybrid dynamical systems with globally asymptotically stable periodic orbits, called hybrid limit-cycling systems. We leverage smooth structures intrinsic to the hybrid dynamical systems, thereby extending the existing theory of Koopman operators for smooth dynamical systems. Rigorous construction of an observable space is carried out to preserve the inherited smooth structures of the hybrid dynamical systems. Complete spectral characterization of the Koopman operators acting on the constructed space is then derived where the existence and uniqueness of their eigenfunctions are ensured. Our results facilitate global analysis of hybrid dynamical systems using the Koopman operator.

math.DS↗

arXiv2024

Koopman Analysis of the Singularly-Perturbed van der Pol Oscillator

The Koopman operator framework holds promise for spectral analysis of nonlinear dynamical systems based on linear operators. Eigenvalues and eigenfunctions of the Koopman operator, so-called Koopman eigenvalues and Koopman eigenfunctions, respectively, mirror global properties of the system's flow. In this paper we perform the Koopman analysis of the singularly-perturbed van der Pol system. First, we show the spectral signature depending on singular perturbation: how two Koopman {principal} eigenvalues are ordered and what distinct shapes emerge in their associated Koopman eigenfunctions. Second, we discuss the singular limit of the Koopman operator, which is derived through the concatenation of Koopman operators for the fast and slow subsystems. From the spectral properties of the Koopman operator for the {singularly}-perturbed system and the singular limit, we suggest that the Koopman eigenfunctions inherit geometric properties of the singularly-perturbed system. These results are applicable to general planar singularly-perturbed systems with stable limit cycles.

math.DS↗