Dissipativity Analysis of Nonlinear Systems: A Linear--Radial Kernel-based Approach

TL;DR

This paper introduces a kernel-based method using Koopman operators and RKHS to estimate dissipativity in nonlinear systems from data, generalizing quadratic forms and enabling convex optimization.

Contribution

It proposes a novel approach combining Koopman operator theory and RKHS with linear-radial kernels for data-driven dissipativity analysis of nonlinear systems.

Findings

Kernel quadratic forms can express dissipativity inequalities as linear operator inequalities.

The method reduces dissipativity estimation to a convex optimization problem.

A statistical learning bound guarantees probabilistic correctness of the estimates.

Abstract

Estimating the dissipativity of nonlinear systems from empirical data is useful for the analysis and control of nonlinear systems, especially when an accurate model is unavailable. Based on a Koopman operator model of the nonlinear system on a reproducing kernel Hilbert space (RKHS), the storage function and supply rate functions are expressed as kernel quadratic forms, through which the dissipative inequality is expressed as a linear operator inequality. The RKHS is specified by a linear--radial kernel, which inherently encode the information of equilibrium point, thus ensuring that all functions in the RKHS are locally at least linear around the origin and that kernel quadratic forms are locally at least quadratic, which expressively generalize conventional quadratic forms including sum-of-squares polynomials. Based on the kernel matrices of the sampled data, the dissipativity estimation can be posed as a finite-dimensional convex optimization problem, and a statistical learning bound can be derived on the kernel quadratic form for the probabilistic approximate correctness of dissipativity estimation.