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

Xiuzhen Ye, Wentao Tang

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.

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

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.

Paper Structure

This paper contains 17 sections, 14 theorems, 42 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Dissipativity
  4. Koopman Operator on Sobolev-Type RKHS
  5. Linear--Radial Kernel and its RKHS
  6. DISSIPATIVITY AS A LINEAR OPERATOR INEQUALITY
  7. Kernel Quadratic Forms
  8. Dissipative Inequality
  9. Finite-Rank Approximation from Data
  10. DATA-DRIVEN ESTIMATION OF DISSIPATIVITY AND GENERALIZATION ERROR
  11. Estimation of $S$ by Kernel Ridge Regression
  12. Generalization Error Bound
  13. NUMERICAL EXPERIMENTS
  14. Case 1: Polynomial System with Analytical Storage
  15. Case 2: Inverted Pendulum
  16. ...and 2 more sections

Key Result

Lemma 1

Suppose that (i) $\mathbb{X}\subset \mathbb{R}^{d_x}$ has a Lipschitz boundary, and $\mathbb{U}\subset \mathbb{R}^{d_u}$ is bounded, (ii) $f\in C_\mathrm{b}^s(\mathbb{X}\times \mathbb{U}, \mathbb{X})$ for some $s\in \mathbb{N}$, and (iii) $\inf_{x\in \mathbb{X}, u\in \mathbb{U}} \lvert \mathrm{D}_xf

Figures (3)

  • Figure E1: Storage function estimates for Case 1.
  • Figure E2: Storage function estimate (left) and dissipativity satisfaction check (right) for Case 2 under $\beta = 6$.
  • Figure E3: Storage function estimate and distribution of dissipativity violations for Case 3.

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Definition 3
  • Lemma 2: Wendland wendland2004scattered
  • Corollary 1
  • Corollary 2
  • Lemma 3: tang-ye2025koopman
  • Corollary 3
  • ...and 19 more