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Technical Report for Dissipativity Learning in Reproducing Kernel Hilbert Space

Xiuzhen Ye, Wentao Tang

TL;DR

This work tackles the problem of certifying stability and performance for unknown nonlinear systems without explicit models by learning dissipativity properties from data. It introduces a nonparametric RKHS framework that represents storage and supply via Hilbert–Schmidt operators acting on kernel features, formulated as an OC-SVM and reduced to finite kernel Gram matrices through the representer theorem. The method provides explicit generalization guarantees by combining kernel regression error bounds with OC-SVM margins, yielding confidence bounds on the dissipation rate and the $L_2$-gain. Numerical experiments on a nonlinear distillation column show effective identification of dissipative behavior from input–output data, with favorable numerical stability and robustness compared to parametric baselines, highlighting the method’s practical impact for model-free control analysis and synthesis.

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.

Technical Report for Dissipativity Learning in Reproducing Kernel Hilbert Space

TL;DR

This work tackles the problem of certifying stability and performance for unknown nonlinear systems without explicit models by learning dissipativity properties from data. It introduces a nonparametric RKHS framework that represents storage and supply via Hilbert–Schmidt operators acting on kernel features, formulated as an OC-SVM and reduced to finite kernel Gram matrices through the representer theorem. The method provides explicit generalization guarantees by combining kernel regression error bounds with OC-SVM margins, yielding confidence bounds on the dissipation rate and the -gain. Numerical experiments on a nonlinear distillation column show effective identification of dissipative behavior from input–output data, with favorable numerical stability and robustness compared to parametric baselines, highlighting the method’s practical impact for model-free control analysis and synthesis.

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.

Paper Structure

This paper contains 20 sections, 10 theorems, 71 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dissipative Systems
  3. Dissipativity parameters and its learning: quadratic case
  4. Dissipativity parameters
  5. OC-SVM for dissipativity parameters
  6. Dissipativity learning in RKHS
  7. Reproducing Kernel Hilbert Spaces
  8. OC-SVM for Nonparametric Dissipativity Learning
  9. The Finite-dim Optimization
  10. Operators in the cost
  11. Operators in the dissipativity inequality constraint
  12. Operators in the constraints for bounds of the storage and $L_2$ gain
  13. Error Analysis for Learning in RKHS
  14. Approximation Error from KRR
  15. $f_\theta\in \operatorname{Range}(L_K)$
  16. ...and 5 more sections

Key Result

Theorem 1

For a $(\Pi_{yy}, \Pi_{yu}, \Pi_{uu})$-dissipative system, if there exists positive real numbers $\alpha$ and $\beta$, such that then the system has an $L_2$-gain not greater than $\beta^{1/2}$.

Figures (3)

  • Figure 1: Comparison of the objectives of nonparametric dissipativity learning in RKHS (top) and parametric learning (bottom).
  • Figure 2: Comparison of the learned dissipativity margin $\rho$ and $\alpha$ from nonparametric dissipativity learning in RKHS (top) and parametric learning (bottom), with respect to weighting parameter $\lambda$.
  • Figure 3: Dissipativity verification for parametric learning.

Theorems & Definitions (29)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Theorem 2
  • proof
  • Definition 3
  • Definition 4
  • Remark 3
  • ...and 19 more