Kernel Methods for the Approximation of Some Key Quantities of Nonlinear Systems
This work addresses challenges in analyzing nonlinear systems for control theory and dynamical systems researchers, offering a data-driven approach that extends linear theory, but it appears incremental as it builds on existing kernel method frameworks.
The paper tackles the problem of estimating key quantities like controllability and observability energy functions for nonlinear control and dynamical systems by mapping them into high-dimensional feature spaces using kernel methods, and demonstrates that the controllability estimator can approximate the invariant measure of stochastically forced ergodic systems.
We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of success - once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems, and study the ellipsoids they induce. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.