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

Researchers Xiuzhen Ye and Wentao Tang have introduced a novel kernel-based approach for analyzing the dissipativity of nonlinear systems directly from empirical data, bypassing the need for accurate mathematical models. Their method, detailed in a paper submitted to the 65th IEEE Conference on Decision and Control, leverages Koopman operator theory within reproducing kernel Hilbert spaces (RKHS) specified by linear-radial kernels. These kernels inherently encode equilibrium point information, ensuring all functions in the RKHS are at least linear around the origin and that kernel quadratic forms generalize conventional quadratic forms like sum-of-squares polynomials.

By expressing storage and supply rate functions as kernel quadratic forms, the researchers transform the dissipative inequality into a linear operator inequality. This formulation enables dissipativity estimation to be posed as a finite-dimensional convex optimization problem using kernel matrices derived from sampled data. The approach includes derived statistical learning bounds that provide probabilistic guarantees for the approximate correctness of dissipativity estimates, offering a mathematically rigorous framework for stability analysis when system models are incomplete or unavailable.

The technique represents a significant advancement in data-driven control theory, particularly for complex systems where traditional modeling approaches fall short. By combining kernel methods with convex optimization, the researchers have created a practical tool for engineers working with nonlinear systems in robotics, power grids, and industrial processes where stability analysis is critical but accurate models are difficult to obtain.