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On Space-Filling Input Design for Nonlinear Dynamic Model Learning: A Gaussian Process Approach

Yuhan Liu, Máté Kiss, Roland Tóth, Maarten Schoukens

TL;DR

The paper tackles nonlinear system identification by prioritizing robust generalization over parameter-variance minimization. It introduces an offline GP-based space-filling input design that places a GP prior on the joint input–state space and minimizes a posterior-variance–based cost over anchor points in a region of interest $\tilde{\mathcal{Z}}$, guaranteeing effective coverage. Theoretical results show that minimizing the design cost implies $\epsilon$-space filling within the ROI, irrespective of GP hyperparameters, while remaining applicable to flexible input parameterizations. Simulation studies on a linear system and a nonlinear mass–spring–damper illustrate enhanced space filling compared to unoptimized inputs and standard multisine designs, highlighting potential for efficient exploration of nonlinear dynamics. Overall, the approach provides a principled, gradient-friendly framework for informative data collection in nonlinear system identification with broad applicability and practical impact.

Abstract

While optimal input design for linear systems has been well-established, no systematic approach exists for nonlinear systems where robustness to extrapolation/interpolation errors is prioritized over minimizing estimated parameter variance. To address this issue, we develop a novel space-filling input design strategy for nonlinear system identification that ensures data coverage of a given region of interest. By placing a Gaussian Process (GP) prior on the joint input-state space, the proposed strategy leverages the GP posterior variance to construct a cost function that promotes space-filling input design. Consequently, this enables maximization of the coverage in the region of interest, thereby facilitating the generation of informative datasets. Furthermore, we theoretically prove that minimization of the cost function implies the space-filling property of the obtained data. Effectiveness of the proposed strategy is demonstrated on both an academic and a mass-spring-damper example, highlighting its potential practical impact on efficient exploration of the dynamics of nonlinear systems.

On Space-Filling Input Design for Nonlinear Dynamic Model Learning: A Gaussian Process Approach

TL;DR

The paper tackles nonlinear system identification by prioritizing robust generalization over parameter-variance minimization. It introduces an offline GP-based space-filling input design that places a GP prior on the joint input–state space and minimizes a posterior-variance–based cost over anchor points in a region of interest , guaranteeing effective coverage. Theoretical results show that minimizing the design cost implies -space filling within the ROI, irrespective of GP hyperparameters, while remaining applicable to flexible input parameterizations. Simulation studies on a linear system and a nonlinear mass–spring–damper illustrate enhanced space filling compared to unoptimized inputs and standard multisine designs, highlighting potential for efficient exploration of nonlinear dynamics. Overall, the approach provides a principled, gradient-friendly framework for informative data collection in nonlinear system identification with broad applicability and practical impact.

Abstract

While optimal input design for linear systems has been well-established, no systematic approach exists for nonlinear systems where robustness to extrapolation/interpolation errors is prioritized over minimizing estimated parameter variance. To address this issue, we develop a novel space-filling input design strategy for nonlinear system identification that ensures data coverage of a given region of interest. By placing a Gaussian Process (GP) prior on the joint input-state space, the proposed strategy leverages the GP posterior variance to construct a cost function that promotes space-filling input design. Consequently, this enables maximization of the coverage in the region of interest, thereby facilitating the generation of informative datasets. Furthermore, we theoretically prove that minimization of the cost function implies the space-filling property of the obtained data. Effectiveness of the proposed strategy is demonstrated on both an academic and a mass-spring-damper example, highlighting its potential practical impact on efficient exploration of the dynamics of nonlinear systems.

Paper Structure

This paper contains 13 sections, 2 theorems, 13 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Data-generating system
  4. Space-Filling Input Design Problem
  5. Gaussian Process Models
  6. GP-based Space-filling Input Design
  7. Classical Space-filling Design
  8. Space-filling Promoting Cost Function
  9. Theoretical Analysis
  10. Simulation Experiments
  11. Linear system example
  12. Nonlinear mass-spring-damper system example
  13. Conclusion

Key Result

Lemma 1

Consider a dataset $\mathcal{D}_N$ and a region of interest $\tilde{\mathcal{Z}}$ which is characterized by $M$ pre-defined anchor points $\mathcal{D}_\mathrm{I}\!=\!\{\tilde{z}_i\}_{i\!=\!1}^{M}$. The GP-based space-filling promoting cost function is appropriately defined as GPcost-data, with a ker then $\mathcal{D}_N$ is guaranteed to have the at least $1/\epsilon$ density w.r.t. $\tilde{\mathca

Figures (3)

  • Figure 1: Space-filling performance of $\mathcal{D}_N(\theta)$ () under different numbers of anchor points () $M=4,9,16$ with $N=40$ when the optimized $\mathcal{W}({\theta};\mathcal{D}_N(\theta))\to 0$, where represents the largest ball in $\tilde{\mathcal{Z}}$ that does not contain any point in $\mathcal{D}_N(\theta)$. The boxplot illustrates that $\mathcal{D}_N(\theta)$ always exhibits a lower $\rho$ than the theoretically guaranteed value ($\epsilon \!=\! 2.8, 1.4, 0.94)$ over 20 Monte Carlo realizations with $\theta_k^{(0)}\sim\mathcal{N}(0,1)$, $\forall k\in\mathbb{I}_1^{N}$.
  • Figure 2: An illustration of the nonlinear mass-spring-damper system.
  • Figure 3: Comparison of initial dataset $\mathcal{D}_N(\theta^{0})$(green,0.43;blue,0.75] (0,0) -- (0.08,0);) and optimized points $\mathcal{D}_N(\hat{\theta})$(green,0.35;blue,0.13] (0,0) -- (0.08,0);) in the 3D space. The anchor points are shown by the shaded cube.

Theorems & Definitions (11)

  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Definition 3
  • Remark 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • ...and 1 more