Technical Report for Dissipativity Learning in Reproducing Kernel Hilbert Space

TL;DR

This paper introduces a nonparametric RKHS-based framework for learning dissipativity properties of unknown nonlinear systems directly from data, enabling stability and performance certification without explicit models.

Contribution

It proposes a novel operator-based approach representing dissipativity as Hilbert Schmidt operators, formulated as a convex kernel-based optimization problem with theoretical generalization guarantees.

Findings

Successfully identifies nonlinear dissipative behavior from data

Provides confidence bounds on dissipation rate and L2 gain

Demonstrates effectiveness in model-free control analysis

Abstract

This work presents a nonparametric framework for dissipativity learning in reproducing kernel Hilbert spaces, which enables data-driven certification of stability and performance properties for unknown nonlinear systems without requiring an explicit dynamic model. Dissipativity is a fundamental system property that generalizes Lyapunov stability, passivity, and finite L2 gain conditions through an energy balance inequality between a storage function and a supply rate. Unlike prior parametric formulations that approximate these functions using quadratic forms with fixed matrices, the proposed method represents them as Hilbert Schmidt operators acting on canonical kernel features, thereby capturing nonlinearities implicitly while preserving convexity and analytic tractability. The resulting operator optimization problem is formulated in the form of a one-class support vector machine and reduced, via the representer theorem, to a finite dimensional convex program expressed through kernel Gram matrices. Furthermore, statistical learning theory is applied to establish generalization guarantees, including confidence bounds on the dissipation rate and the L2 gain. Numerical results demonstrate that the proposed RKHS based dissipativity learning method effectively identifies nonlinear dissipative behavior directly from input output data, providing a powerful and interpretable framework for model free control analysis and synthesis.