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Identification of contractive Lur'e-type systems via kernel-based Lipschitz design

Cesare Donati, Fabrizio Dabbene, Constantino Lagoa, Carlo Novara, Yoshio Ebihara

TL;DR

The paper tackles identifying contractive Lur’e-type systems by coupling a known linear structure with a kernel-based model of the static nonlinearity, while enforcing contractivity through Lipschitz design. It develops a kernel regression framework in which the nonlinear residual lives in an RKHS and can be tuned to satisfy a contraction bound jointly with the linear part. Two practical identification strategies are proposed: a post hoc contractivity check and a constrained optimization that embeds contractivity, both leveraging nonexpansive kernels to guarantee Lipschitz properties. Numerical results show that enforcing contractivity improves parameter estimation and yields physically consistent models, highlighting the value of incorporating dynamical properties into data-driven system identification.

Abstract

This paper addresses the problem of identifying contractive Lur'e-type systems. Specifically, it proposes an identification framework that integrates linear prior knowledge with a kernel representation of the nonlinear feedback while systematically enforcing contractivity via Lipschitz constant design. The resulting algorithms provide models that are accurate in prediction, interpretable, and faithful to the contractive nature of the true system. Numerical experiments demonstrate that enforcing contractivity significantly improves parameter estimation and yields models that are both accurate and physically meaningful.

Identification of contractive Lur'e-type systems via kernel-based Lipschitz design

TL;DR

The paper tackles identifying contractive Lur’e-type systems by coupling a known linear structure with a kernel-based model of the static nonlinearity, while enforcing contractivity through Lipschitz design. It develops a kernel regression framework in which the nonlinear residual lives in an RKHS and can be tuned to satisfy a contraction bound jointly with the linear part. Two practical identification strategies are proposed: a post hoc contractivity check and a constrained optimization that embeds contractivity, both leveraging nonexpansive kernels to guarantee Lipschitz properties. Numerical results show that enforcing contractivity improves parameter estimation and yields physically consistent models, highlighting the value of incorporating dynamical properties into data-driven system identification.

Abstract

This paper addresses the problem of identifying contractive Lur'e-type systems. Specifically, it proposes an identification framework that integrates linear prior knowledge with a kernel representation of the nonlinear feedback while systematically enforcing contractivity via Lipschitz constant design. The resulting algorithms provide models that are accurate in prediction, interpretable, and faithful to the contractive nature of the true system. Numerical experiments demonstrate that enforcing contractivity significantly improves parameter estimation and yields models that are both accurate and physically meaningful.

Paper Structure

This paper contains 13 sections, 2 theorems, 33 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Identification framework
  3. Contractive Lur'e-type systems
  4. Kernel-based modeling framework for Lur’e systems
  5. Kernel-based approximation
  6. Kernel methods with structural priors donati2025kernelTAC
  7. Extension to the general case
  8. Lipschitz constant design via kernel regularization
  9. The proposed approach
  10. Unconstrained identification with post-contractivity check
  11. Constrained identification with embedded contractivity check
  12. Numerical example
  13. Conclusions

Key Result

Lemma 1

Consider the Lur’e-type model eqn:lure.model. Assume $\delta$ is Lipschitz with Lipschitz constant $\ell_\delta$ and $u_t\equiv0$, $\forall t$. Then, the following inequality holds: i.e., eqn:lure.model is Lipschitz with Lipschitz constant $\ell\leq \|A(\theta)\| + \ell_\delta$.

Figures (2)

  • Figure 1: Parametric identification error and the contractive region with respect to $\gamma$.
  • Figure 2: Root mean square errors with respect to $\gamma$.

Theorems & Definitions (7)

  • Definition 1: Contractivity shima_contractivity_2025
  • Remark 1: Contractivity in the Euclidean norm
  • Definition 2: Nonexpansive kernel
  • Lemma 1: Lur’e model Lipschitz bound
  • proof
  • Theorem 1: Lur'e model contractivity
  • proof