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arxiv: 2605.02517 · v2 · submitted 2026-05-04 · 📡 eess.SY · cs.SY

Least Costly Space-Filling Experiment Design for the Identification of a Nonlinear System

M\'at\'e Kiss , Maarten Schoukens , Roland T\'oth This is my paper

Pith reviewed 2026-05-13 06:29 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords nonlinear system identificationspace-filling designGaussian processoptimal input designexperiment costMonte Carlo simulationfeature space
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The pith

Gaussian process optimization designs low-cost excitation signals that maintain space-fillingness for nonlinear system identification

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to generate space-filling datasets in the feature space of a nonlinear system model while minimizing a user-defined experimentation cost. It extends Gaussian process-based optimal input design by adding cost as an objective and enforcing a prescribed level of space-fillingness. The approach works for a broad class of signals and models and still includes information measures via optimality criteria. Monte Carlo simulations show that the resulting signals cut experimental cost substantially while the identified models retain adequate performance. This matters because high experiment costs frequently limit the amount and quality of data available for practical nonlinear system identification.

Core claim

The central claim is that an optimization problem can be solved to produce an excitation signal that minimizes user-specified experimentation cost subject to a minimum space-filling level in the feature space, using Gaussian processes to quantify space-fillingness and incorporating optimality criteria, thereby yielding cheaper yet effective data for nonlinear model identification across broad signal and model types.

What carries the argument

Gaussian process-based optimal input design that minimizes cost while enforcing a prescribed space-filling level in feature space

Load-bearing premise

A user-specified level of space-fillingness in the feature space can be maintained while minimizing cost, and the Gaussian process model accurately represents the relevant nonlinear dynamics for the optimality criteria.

What would settle it

A Monte Carlo simulation in which models identified from the cost-optimized low-cost signals exhibit substantially higher prediction errors or worse performance metrics than models from standard space-filling designs, even when the space-filling criterion is met.

Figures

Figures reproduced from arXiv: 2605.02517 by Maarten Schoukens, M\'at\'e Kiss, Roland T\'oth.

Figure 1
Figure 1. Figure 1: The concept of space-fillingness in the feature space: view at source ↗
Figure 2
Figure 2. Figure 2: Nonlinear mass-spring-damper system. Then, γ is chosen to be larger than γ◦ by a user-defined margin. The chosen γ value has to give sufficient room to the least costly design to make a trade-off between space￾fillingness and the cost of the experiment; e.g., we used a 5% margin in the simulation study below. 3. SIMULATION STUDY The proposed least costly input design approach is tested in a Monte Carlo stu… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Least costly u LC θ (red) and classical u Cl θ (green) space-filling signal, (b) frequency plot of u Cl θ (green) and u LC θ (red), (c) box plot of input powers with their mean value (yellow circle), (d,e) classical DCl and least costly DLC (red) space-filling dataset. sampling frequency fs=100 Hz and N=1024 data samples per period. The amplitudes {Al} lmax l=lmin are parametrized such that their value… view at source ↗
Figure 4
Figure 4. Figure 4: Results for different test sets: (a) multisine signal view at source ↗
read the original abstract

The quality of an estimated nonlinear model highly depends on the data quality that was used for the system identification. By using a Gaussian Process-based optimal input design approach, a so-called space-filling dataset can be generated in the feature space of the system model. The design method is applicable for a broad type of signals and models and also incorporates information measures through optimality criteria into the signal design. However, the resulting input design can be costly to apply to the real system. The goal of this paper is to propose a space-filling input design that can minimize the experimentation cost in terms of a user defined measure, while still guaranteeing a prescribed level of space-fillingness. Through a Monte Carlo simulation study we demonstrate that the proposed method can appropriately shape the excitation signal to significantly reduce the experimental cost while the identified model performance remains adequate.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a Gaussian process-based optimal input design method for nonlinear system identification that generates space-filling datasets in the model's feature space while minimizing a user-defined experimental cost. The approach incorporates optimality criteria and is validated through Monte Carlo simulations claiming significant cost reduction with adequate model performance.

Significance. If validated under realistic model mismatch, the method could improve efficiency in nonlinear system identification experiments by trading off cost against data quality. The use of standard GP regression and optimality criteria is a strength, but the current Monte Carlo evidence under matched conditions provides only weak support for the central claim of maintained performance at reduced cost.

major comments (2)
  1. [Monte Carlo simulation study] Monte Carlo simulation study: the evaluation uses the same nominal GP both to design the input and to generate the data, so it does not probe robustness to mismatch between the nominal model and the true nonlinear dynamics; this directly weakens the claim that 'identified model performance remains adequate' when the space-filling guarantee is only enforced under the nominal feature space.
  2. [Abstract and results section] Abstract and results section: no quantitative metrics (e.g., specific cost reduction percentages, space-filling discrepancy values, or error bars on model performance) are reported, leaving the demonstration of 'significantly reduce the experimental cost' unsupported by concrete evidence.
minor comments (1)
  1. [Abstract] The abstract states that Monte Carlo simulations support the claim but provides no details on how space-fillingness is quantified or how the cost-space-filling trade-off is measured.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important aspects of the evaluation. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Monte Carlo simulation study] Monte Carlo simulation study: the evaluation uses the same nominal GP both to design the input and to generate the data, so it does not probe robustness to mismatch between the nominal model and the true nonlinear dynamics; this directly weakens the claim that 'identified model performance remains adequate' when the space-filling guarantee is only enforced under the nominal feature space.

    Authors: We agree that the Monte Carlo study assumes matched conditions between the nominal GP used for design and data generation, which is a common starting point but does not fully address robustness to model mismatch. This is a valid limitation for the robustness claim. In the revised version, we will add a new subsection presenting additional Monte Carlo results under controlled mismatch (e.g., using a different kernel length-scale or additive parameter perturbation in the true system) to quantify any degradation in space-filling quality and model performance. revision: yes

  2. Referee: [Abstract and results section] Abstract and results section: no quantitative metrics (e.g., specific cost reduction percentages, space-filling discrepancy values, or error bars on model performance) are reported, leaving the demonstration of 'significantly reduce the experimental cost' unsupported by concrete evidence.

    Authors: We acknowledge that the abstract and results section would benefit from explicit quantitative metrics to support the claims. In the revision, we will update the abstract with specific values (e.g., average cost reductions of X% across scenarios) and expand the results section to report space-filling discrepancy measures (such as the maximum minimum distance or star discrepancy) along with error bars (mean ± standard deviation) on model performance metrics like RMSE from the Monte Carlo runs. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper formulates input design as a constrained optimization problem that minimizes a user-specified cost while enforcing a prescribed space-filling level (via discrepancy or information measure) inside the feature space of a nominal Gaussian-process model. This is a standard optimal-design construction that does not reduce any claimed prediction or optimality criterion to a fitted quantity defined by the same data; the Monte Carlo study simply evaluates the resulting signals under matched nominal conditions, which is conventional method validation rather than a self-referential loop. No self-definitional steps, no load-bearing self-citations that collapse the argument, and no ansatz or uniqueness claim imported from prior author work appear in the derivation. The central result therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the approach appears to rest on standard assumptions of Gaussian process regression and optimal experiment design theory.

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Works this paper leans on

126 extracted references · 126 canonical work pages

  1. [1]

    Deep Learning , author=

    work page

  2. [2]

    Proceedings of the American Control Conference , pages=

    Least costly identification experiment for control: A solution based on a high-order model approximation , author=. Proceedings of the American Control Conference , pages=

    work page

  3. [3]

    Journal Europ

    Design of least costly identification experiments: The main philosophy accompanied by illustrative examples , author=. Journal Europ

    work page

  4. [4]

    arXiv preprint arXiv:2504.02653 , year=

    Online and Offline Space-Filling Input Design for Nonlinear System Identification: A Receding Horizon Control-Based Approach , author=. arXiv preprint arXiv:2504.02653 , year=

    work page arXiv

  5. [5]

    Advances in Neural Information Processing Systems , volume=

    Optimistic active exploration of dynamical systems , author=. Advances in Neural Information Processing Systems , volume=

    work page

  6. [6]

    Actively learning gaussian process dynamics , author=. In proc. of the Learning for dynamics and control conference , pages=

    work page

  7. [7]

    IEEE Control Systems Magazine , volume=

    Iterative model identification of nonlinear systems of unknown structure: Systematic data-based modeling utilizing design of experiments , author=. IEEE Control Systems Magazine , volume=. 2020 , publisher=

    work page 2020

  8. [8]

    2024 , booktitle =

    Optimized Excitation Signal Tailored to Pertinent Dynamic Process Characteristics , author=. 2024 , booktitle =

    work page 2024

  9. [9]

    Authorea , year=

    Differentiable Model Predictive Excitation: Generating Optimal Data Sets for Learning of Dynamical System Models , author=. Authorea , year=

    work page

  10. [10]

    2024 , booktitle =

    Space-filling input design for nonlinear state-space identification , author=. 2024 , booktitle =

    work page 2024

  11. [11]

    Statistics and Computing , volume=

    Design of computer experiments: space filling and beyond , author=. Statistics and Computing , volume=. 2012 , publisher=

    work page 2012

  12. [12]

    2025 , journal=

    On Space-Filling Input Design for Nonlinear Dynamic Model Learning: A Gaussian Process Approach , author=. 2025 , journal=

    work page 2025

  13. [13]

    Journal of statistical planning and inference , volume=

    Minimax and maximin distance designs , author=. Journal of statistical planning and inference , volume=. 1990 , publisher=

    work page 1990

  14. [14]

    IEEE Control Systems Magazine , volume=

    Linear system identification in a nonlinear setting: Nonparametric analysis of the nonlinear distortions and their impact on the best linear approximation , author=. IEEE Control Systems Magazine , volume=. 2016 , publisher=

    work page 2016

  15. [15]

    2021 European Control Conference (ECC) , pages=

    Improved initialization of state-space artificial neural networks , author=. 2021 European Control Conference (ECC) , pages=. 2021 , organization=

    work page 2021

  16. [16]

    The Bell system technical journal , volume=

    A mathematical theory of communication , author=. The Bell system technical journal , volume=. 1948 , publisher=

    work page 1948

  17. [17]

    Advanced Lectures on Machine Learning , pages=

    Gaussian Processes in Machine Learning , author=. Advanced Lectures on Machine Learning , pages=. 2004 , organization=

    work page 2004

  18. [18]

    1992 , publisher=

    Optimum experimental designs , author=. 1992 , publisher=

    work page 1992

  19. [19]

    Technicla report , year=

    A review of optimal Bayesian designs , author=. Technicla report , year=

    work page

  20. [20]

    Statistical Science , volume=

    Modern Bayesian experimental design , author=. Statistical Science , volume=. 2024 , publisher=

    work page 2024

  21. [21]

    Automatica , volume=

    Informative input design for kernel-based system identification , author=. Automatica , volume=. 2018 , publisher=

    work page 2018

  22. [22]

    Annals of the Institute of Statistical Mathematics , volume=

    Some Bayesian considerations of the choice of design for ranking, selection and estimation , author=. Annals of the Institute of Statistical Mathematics , volume=. 1976 , publisher=

    work page 1976

  23. [23]

    Statistical science , pages=

    Bayesian experimental design: A review , author=. Statistical science , pages=. 1995 , publisher=

    work page 1995

  24. [24]

    The Annals of Mathematical Statistics , volume=

    On a measure of the information provided by an experiment , author=. The Annals of Mathematical Statistics , volume=. 1956 , publisher=

    work page 1956

  25. [25]

    Handbook of Statistics , volume=

    29 Review of optimal bayes designs , author=. Handbook of Statistics , volume=. 1996 , publisher=

    work page 1996

  26. [26]

    IFAC Proceedings Volumes , volume=

    Input signal generation for constrained multiple-input multple-output systems , author=. IFAC Proceedings Volumes , volume=. 2014 , publisher=

    work page 2014

  27. [27]

    , volume=

    On Space-Filling Input Design for Nonlinear Dynamic Model Learning: \\ A Gaussian Process Approach , author=. , volume=. 2024 , publisher=

    work page 2024

  28. [28]

    IEEE Transactions on Automatic Control , volume=

    Input design for kernel-based system identification from the viewpoint of frequency response , author=. IEEE Transactions on Automatic Control , volume=. 2018 , publisher=

    work page 2018

  29. [29]

    Automatica , volume=

    On the informativity of direct identification experiments in dynamical networks , author=. Automatica , volume=. 2023 , publisher=

    work page 2023

  30. [30]

    2019 18th European Control Conference (ECC) , pages=

    Informativity: how to get just sufficiently rich for the Identification of MISO FIR Systems with Multisine Excitation? , author=. 2019 18th European Control Conference (ECC) , pages=. 2019 , organization=

    work page 2019

  31. [31]

    Automatica , volume=

    Data informativity for the open-loop identification of MIMO systems in the prediction error framework , author=. Automatica , volume=. 2020 , publisher=

    work page 2020

  32. [32]

    Control Engineering Practice , volume=

    Space-filling optimized excitation signals for nonlinear system identification of dynamic processes of a diesel engine , author=. Control Engineering Practice , volume=. 2024 , publisher=

    work page 2024

  33. [33]

    Proceedings of the 2021 3rd International Conference on Management Science and Industrial Engineering , pages=

    Space-filling designs for experiments with assembled products , author=. Proceedings of the 2021 3rd International Conference on Management Science and Industrial Engineering , pages=

    work page 2021

  34. [34]

    IFAC Proceedings Volumes , volume=

    Input design for nonlinear stochastic dynamic systems--A particle filter approach , author=. IFAC Proceedings Volumes , volume=. 2011 , publisher=

    work page 2011

  35. [35]

    The Annals of Mathematical Statistics , volume=

    The sequential generation of D -optimum experimental designs , author=. The Annals of Mathematical Statistics , volume=. 1970 , publisher=

    work page 1970

  36. [36]

    Automatica , volume=

    Control-relevant experiment design for multivariable systems described by expansions in orthonormal bases , author=. Automatica , volume=. 2001 , publisher=

    work page 2001

  37. [37]

    Automatica , volume=

    A graph theoretical approach to input design for identification of nonlinear dynamical models , author=. Automatica , volume=. 2015 , publisher=

    work page 2015

  38. [38]

    49th IEEE Conference on Decision and Control (CDC) , pages=

    On optimal input design for nonlinear FIR-type systems , author=. 49th IEEE Conference on Decision and Control (CDC) , pages=. 2010 , organization=

    work page 2010

  39. [39]

    Chemometrics and Intelligent Laboratory Systems , volume=

    Optimal model-based experimental design in batch crystallization , author=. Chemometrics and Intelligent Laboratory Systems , volume=. 2000 , publisher=

    work page 2000

  40. [40]

    1997 , publisher=

    Model identification and control of batch crystallization for an industrial chemical system , author=. 1997 , publisher=

    work page 1997

  41. [41]

    IFAC Proceedings Volumes , volume=

    Optimal input design using linear matrix inequalities , author=. IFAC Proceedings Volumes , volume=. 2000 , publisher=

    work page 2000

  42. [42]

    Automatica , volume=

    Least costly identification experiment for control , author=. Automatica , volume=. 2006 , publisher=

    work page 2006

  43. [43]

    SIAM Journal on Control and Optimization , volume =

    Hildebrand, Roland and Gevers, Michel , title =. SIAM Journal on Control and Optimization , volume =

    work page

  44. [44]

    1975 , publisher=

    Notes on the design of optimal identification experiments , author=. 1975 , publisher=

    work page 1975

  45. [45]

    Automatica , volume=

    Optimal experiment design for linear systems with input-output constraints , author=. Automatica , volume=. 1977 , publisher=

    work page 1977

  46. [46]

    SIAM Journal on Control and optimization , volume=

    Identification for control: optimal input design with respect to a worst-case -gap cost function , author=. SIAM Journal on Control and optimization , volume=. 2002 , publisher=

    work page 2002

  47. [47]

    IFAC Proceedings Volumes , volume=

    Optimal experiment design in closed loop , author=. IFAC Proceedings Volumes , volume=. 2005 , publisher=

    work page 2005

  48. [48]

    IEEE Transactions on Instrumentation and Measurement , volume=

    Peak factor minimization using a time-frequency domain swapping algorithm , author=. IEEE Transactions on Instrumentation and Measurement , volume=. 1988 , publisher=

    work page 1988

  49. [49]

    IEEE transactions on Information Theory , volume=

    Synthesis of low-peak-factor signals and binary sequences with low autocorrelation , author=. IEEE transactions on Information Theory , volume=. 1970 , publisher=

    work page 1970

  50. [50]

    The annals of mathematical statistics , volume=

    Optimum designs in regression problems , author=. The annals of mathematical statistics , volume=. 1959 , publisher=

    work page 1959

  51. [51]

    The Annals of Mathematical Statistics , pages=

    Locally optimal designs for estimating parameters , author=. The Annals of Mathematical Statistics , pages=. 1953 , publisher=

    work page 1953

  52. [52]

    The annals of mathematical statistics , volume=

    On the efficiency of experimental designs , author=. The annals of mathematical statistics , volume=. 1955 , publisher=

    work page 1955

  53. [53]

    The Annals of Statistics , volume=

    D -optimum weighing designs , author=. The Annals of Statistics , volume=. 1980 , publisher=

    work page 1980

  54. [54]

    Automatica , volume=

    Optimal experiment designs with respect to the intended model application , author=. Automatica , volume=. 1986 , publisher=

    work page 1986

  55. [55]

    Mathematics in science and engineering , volume=

    Dynamic system identification: experiment design and data analysis , author=. Mathematics in science and engineering , volume=. 1977 , publisher=

    work page 1977

  56. [56]

    2007 , publisher=

    Parameter estimation for scientists and engineers , author=. 2007 , publisher=

    work page 2007

  57. [57]

    1993 , publisher=

    Fundamentals of statistical signal processing: estimation theory , author=. 1993 , publisher=

    work page 1993

  58. [58]

    Design of experiments in nonlinear models , author=

    work page

  59. [59]

    Encyclopedia of Systems and Control , pages=

    Experiment design and identification for control , author=. Encyclopedia of Systems and Control , pages=. 2021 , publisher=

    work page 2021

  60. [60]

    2014 , publisher=

    Identification of linear systems: a practical guideline to accurate modeling , author=. 2014 , publisher=

    work page 2014

  61. [61]

    IFAC Proceedings Volumes , volume=

    Experiment design , author=. IFAC Proceedings Volumes , volume=. 1982 , publisher=

    work page 1982

  62. [62]

    Automatica , volume=

    Computationally efficient identification of continuous-time Lur’e-type systems with stability guarantees , author=. Automatica , volume=. 2022 , publisher=

    work page 2022

  63. [63]

    IFAC-PapersOnLine , volume=

    On the initialization of nonlinear LFR model identification with the best linear approximation , author=. IFAC-PapersOnLine , volume=. 2020 , publisher=

    work page 2020

  64. [64]

    IFAC Symposium on System Identification , volume=

    LPVcore: MATLAB toolbox for LPV modelling, identification and control , author=. IFAC Symposium on System Identification , volume=. 2021 , publisher=

    work page 2021

  65. [65]

    Proceedings of 35th IEEE Conference on Decision and Control , volume=

    Robust and optimal control , author=. Proceedings of 35th IEEE Conference on Decision and Control , volume=. 1996 , organization=

    work page 1996

  66. [66]

    Journal of Mathematics and Physics , volume=

    On a certain linear fractional transformation , author=. Journal of Mathematics and Physics , volume=. 1960 , publisher=

    work page 1960

  67. [67]

    Journal of Physics: Conference Series , volume=

    Combining experiments for linear dynamic network identification in the presence of nonlinearities , author=. Journal of Physics: Conference Series , volume=. 2018 , organization=

    work page 2018

  68. [68]

    IFAC-PapersOnLine , volume=

    Detecting nonlinear modules in a dynamic network: A step-by-step procedure , author=. IFAC-PapersOnLine , volume=. 2018 , publisher=

    work page 2018

  69. [69]

    IEEE Transactions on Instrumentation and Measurement , volume=

    Best linear approximation of nonlinear continuous-time systems subject to process noise and operating in feedback , author=. IEEE Transactions on Instrumentation and Measurement , volume=. 2020 , publisher=

    work page 2020

  70. [70]

    IEEE Transactions on Automatic Control , volume=

    Extending the best linear approximation framework to the process noise case , author=. IEEE Transactions on Automatic Control , volume=. 2019 , publisher=

    work page 2019

  71. [71]

    IEEE Transactions on Automatic Control , year=

    Identification of physical networks through structured polynomial models , author=. IEEE Transactions on Automatic Control , year=

    work page

  72. [72]

    Computers & Chemical Engineering , volume=

    Identification in dynamic networks , author=. Computers & Chemical Engineering , volume=. 2018 , publisher=

    work page 2018

  73. [73]

    2012 , publisher=

    Mastering system identification in 100 exercises , author=. 2012 , publisher=

    work page 2012

  74. [74]

    2007 , institution=

    Nonlinear system identification for damage detection , author=. 2007 , institution=

    work page 2007

  75. [75]

    D-optimal

    An algorithm for the construction of “D-optimal” experimental designs , author=. Technometrics , volume=. 2000 , publisher=

    work page 2000

  76. [76]

    1972 , publisher=

    Theory of optimal experiments , author=. 1972 , publisher=

    work page 1972

  77. [77]

    Some aspects of the sequential design of experiments , author=

    work page

  78. [78]

    IFAC Proceedings Volumes , volume=

    Robust input design using sum of squares constraints , author=. IFAC Proceedings Volumes , volume=. 2006 , publisher=

    work page 2006

  79. [79]

    Mathematical Biosciences , volume=

    Robust experiment design via maximin optimization , author=. Mathematical Biosciences , volume=. 1988 , publisher=

    work page 1988

  80. [80]

    2019 , publisher=

    Industrial process identification: Perturbation Signal Design and Applications , author=. 2019 , publisher=

    work page 2019

Showing first 80 references.