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Space-Filling Input Design for Nonlinear State-Space Identification

Máté Kiss, Roland Tóth, Maarten Schoukens

TL;DR

The paper tackles the challenge of designing informative experiments for nonlinear state-space identification by introducing a space-filling input design. It combines flexible input parametrizations (e.g., multisine signals) with a grid-based, squared-exponential membership cost to promote uniform coverage of a region of interest in the joint input-state space, optimized via gradient-based methods under amplitude/energy constraints. Demonstrated on a nonlinear 1-DOF mass-spring-damper, the approach achieves substantial improvements in data coverage compared to random-phase inputs and shows advantages over Schroeder multisine excitations in spreading data along the input axis. The work offers a practical, extendable framework for robust data collection in black-box nonlinear identification, with potential applicability to broader model classes and constraint settings.

Abstract

The quality of a model resulting from (black-box) system identification is highly dependent on the quality of the data that is used during the identification procedure. Designing experiments for linear time-invariant systems is well understood and mainly focuses on the power spectrum of the input signal. Performing experiment design for nonlinear system identification on the other hand remains an open challenge as informativity of the data depends both on the frequency-domain content and on the time-domain evolution of the input signal. Furthermore, as nonlinear system identification is much more sensitive to modeling and extrapolation errors, having experiments that explore the considered operation range of interest is of high importance. Hence, this paper focuses on designing space-filling experiments i.e., experiments that cover the full operation range of interest, for nonlinear dynamical systems that can be represented in a state-space form using a broad set of input signals. The presented experiment design approach can straightforwardly be extended to a wider range of system classes (e.g., NARMAX). The effectiveness of the proposed approach is illustrated on the experiment design for a nonlinear mass-spring-damper system, using a multisine input signal.

Space-Filling Input Design for Nonlinear State-Space Identification

TL;DR

The paper tackles the challenge of designing informative experiments for nonlinear state-space identification by introducing a space-filling input design. It combines flexible input parametrizations (e.g., multisine signals) with a grid-based, squared-exponential membership cost to promote uniform coverage of a region of interest in the joint input-state space, optimized via gradient-based methods under amplitude/energy constraints. Demonstrated on a nonlinear 1-DOF mass-spring-damper, the approach achieves substantial improvements in data coverage compared to random-phase inputs and shows advantages over Schroeder multisine excitations in spreading data along the input axis. The work offers a practical, extendable framework for robust data collection in black-box nonlinear identification, with potential applicability to broader model classes and constraint settings.

Abstract

The quality of a model resulting from (black-box) system identification is highly dependent on the quality of the data that is used during the identification procedure. Designing experiments for linear time-invariant systems is well understood and mainly focuses on the power spectrum of the input signal. Performing experiment design for nonlinear system identification on the other hand remains an open challenge as informativity of the data depends both on the frequency-domain content and on the time-domain evolution of the input signal. Furthermore, as nonlinear system identification is much more sensitive to modeling and extrapolation errors, having experiments that explore the considered operation range of interest is of high importance. Hence, this paper focuses on designing space-filling experiments i.e., experiments that cover the full operation range of interest, for nonlinear dynamical systems that can be represented in a state-space form using a broad set of input signals. The presented experiment design approach can straightforwardly be extended to a wider range of system classes (e.g., NARMAX). The effectiveness of the proposed approach is illustrated on the experiment design for a nonlinear mass-spring-damper system, using a multisine input signal.
Paper Structure (14 sections, 13 equations, 8 figures)

This paper contains 14 sections, 13 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Input design approach
  4. Input Parametrization
  5. Space-Filling Cost Function
  6. Optimization
  7. Simulation example
  8. The considered system
  9. Input parametrization
  10. Considered Domain and Cost Function
  11. Results
  12. Copmarison to Schroeder multisine
  13. Monte-Carlo Simulation
  14. Conclusion

Figures (8)

  • Figure 1: Region of interest ($\mathcal{D}_z$) in the joint input-state space with a center point ($\Tilde{z}_i$). The red circle defines the local region around the center point with some data points (blue dots) in it.
  • Figure 2: Nonlinear dynamical system
  • Figure 3: Initial (blue) and optimized (orange) input in time- and frequency domain.
  • Figure 4: Side view of the space-filling effect of the input before (blue) and after (orange) optimization and space-filling effect of the Schroeder multisine (purple). The black contour lines outline the borders of the domain of interest.
  • Figure 5: Optimized points in the 3D space. The grid points are shown by the yellow dots. The optimized input-state samples are shown by the black crosses.
  • ...and 3 more figures