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A constrained optimization approach to nonlinear system identification through simulation error minimization

Vito Cerone, Sophie M. Fosson, Simone Pirrera, Diego Regruto

TL;DR

This work tackles nonlinear system identification by replacing prediction-error minimization with simulation-error minimization (SEM) and addressing vanishing-gradient issues via a constrained optimization formulation. It introduces FL-CMO, a feedback-linearization controlled multiplier optimization algorithm, and proves local convergence and favorable computational properties using sparse QR. Theoretical results connect the constrained SEM to the traditional unconstrained formulation, while extensions cover errors-in-variables and state-space models. Numerical experiments on fluid damper, Bouc-Wen, MIMO Wiener-Hammerstein, and magnetic levitation benchmarks demonstrate improved accuracy and reduced training time relative to gradient-based methods and recurrent networks.

Abstract

This paper introduces a novel approach to system identification for nonlinear input-output models that minimizes the simulation error and frames the problem as a constrained optimization task. The proposed method addresses vanishing gradient issues, enabling faster convergence than traditional gradient-based techniques. We present an algorithm based on feedback linearization control of Lagrange multipliers and conduct a theoretical analysis of its performance. We prove that the algorithm converges to a local minimum, and it enhances computational efficiency by exploiting the problem's structure. Numerical experiments demonstrate that our approach outperforms gradient-based methods in both computational effort and estimation accuracy.

A constrained optimization approach to nonlinear system identification through simulation error minimization

TL;DR

This work tackles nonlinear system identification by replacing prediction-error minimization with simulation-error minimization (SEM) and addressing vanishing-gradient issues via a constrained optimization formulation. It introduces FL-CMO, a feedback-linearization controlled multiplier optimization algorithm, and proves local convergence and favorable computational properties using sparse QR. Theoretical results connect the constrained SEM to the traditional unconstrained formulation, while extensions cover errors-in-variables and state-space models. Numerical experiments on fluid damper, Bouc-Wen, MIMO Wiener-Hammerstein, and magnetic levitation benchmarks demonstrate improved accuracy and reduced training time relative to gradient-based methods and recurrent networks.

Abstract

This paper introduces a novel approach to system identification for nonlinear input-output models that minimizes the simulation error and frames the problem as a constrained optimization task. The proposed method addresses vanishing gradient issues, enabling faster convergence than traditional gradient-based techniques. We present an algorithm based on feedback linearization control of Lagrange multipliers and conduct a theoretical analysis of its performance. We prove that the algorithm converges to a local minimum, and it enhances computational efficiency by exploiting the problem's structure. Numerical experiments demonstrate that our approach outperforms gradient-based methods in both computational effort and estimation accuracy.

Paper Structure

This paper contains 21 sections, 3 theorems, 50 equations, 3 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Proposed System Identification Framework
  3. Problem Setup and Modeling
  4. Background and Related Work
  5. Identification using FL-CMO algorithm
  6. Constrained optimization formulation
  7. A motivating example
  8. Identification algorithm based on FL-CMO
  9. Theoretical Analysis
  10. Relation with the unconstrained formulation
  11. Analysis of local convergence
  12. Analysis of the computational complexity
  13. Extensions to error-in-variables and state-space models
  14. Errors-in-variables models
  15. State-space models
  16. ...and 6 more sections

Key Result

Theorem 1

If a point $[\theta^\top, y_1^\top, \dots, y_N^\top]^\top \in \mathbb{R}^{n_\theta+pN}$ satisfies the first-order optimality conditions of Problem cns_opt_nio, the point $[\theta^\top, y_1^\top, \dots, y_n^\top]^\top \in \mathbb{R}^{n_\theta+pn}$ is a stationary point of Problem eq:uncons_formulatio

Figures (3)

  • Figure 1: Block diagram of the data generation process
  • Figure 2: MIMO Wiener-Hammerstein system diagram.
  • Figure 3: Example 3. Validation output of the top-performing models in each class for output $y_1$ (left) and $y_2$ (right).

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1
  • Theorem 2: Convergence of Algorithm \ref{['alg:cap']}
  • Proposition 1
  • Remark 2