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Cited by 35Control Systems and IdentificationAdvanced Adaptive Filtering TechniquesSparse and Compressive Sensing Techniques

Identification of nonlinear dynamical systems with time delay

Open access
Leylaz, Ghazaale, Wang, Shuo, Sun, Jian-Qiao·2022·International Journal of Dynamics and Control
DOI 10.1007/s40435-021-00783-7
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Abstract

This paper proposes a technique to identify nonlinear dynamical systems with time delay. The sparse optimization algorithm is extended to nonlinear systems with time delay. The proposed algorithm combines cross-validation techniques from machine learning for automatic model selection and an algebraic operation for preprocessing signals to filter the noise and for removing the dependence on initial conditions. We further integrate the bootstrapping resampling technique with the sparse regression to obtain the statistical properties of estimation. We use Taylor expansion to parameterize time delay. The proposed algorithm in this paper is computationally efficient and robust to noise. A nonlinear Duffing oscillator is simulated to demonstrate the efficiency and accuracy of the proposed technique. An experimental example of a nonlinear rotary flexible joint is presented to further validate the proposed method.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper extends sparse identification of nonlinear dynamics (SINDy) to nonlinear systems with time delay by combining algebraic signal preprocessing, Taylor expansion of the delay, cross-validation, and bootstrapping. It validates the approach on a simulated nonlinear mass-spring-damper and a Quanser rotary flexible joint, estimating time delays and sparse polynomial models from noisy data.

Key findings

  • In the simulated mass-spring-damper, the selected polynomial order was \(N=3\), and estimated parameters were close to exact values: \(m=0.9885\) vs. 1, \(k_1=5.9830\) vs. 6, \(k_3=2.0041\) vs. 2, \(c_1=1.0609\) vs. 1, \(c_2=1.0074\) vs. 1, \(c_3=0.9973\) vs. 1; time delay was estimated as \(\tau=0.1656\) vs. the selected 0.2.
  • In the Quanser rotary flexible joint experiment, the selected polynomial order was \(N=5\) for both \(\theta\) and \(\alpha\), and the introduced 0.2 s delay was estimated as \(\tau=0.1734\) s.
  • The method identified a non-negligible Coulomb friction term in the experiment: \(f_c=0.0516\) for the \(\theta\)-equation and \(f_c=-0.0477\) for the \(\alpha\)-equation, with standard deviations 0.0083 and 0.0084, respectively.
  • Cross-validation error dropped near \(N=3\) and increased from \(N=5\) in the simulation; in the experiment, the minimum cross-validation error occurred at \(N=5\) for both coordinates.

Methods

  • The proposed algorithm uses an algebraic operation to preprocess signals, filter noise, and remove dependence on initial conditions; Taylor expansion parameterizes the time delay; a nonlinear extended state observer estimates derivatives; sparse regression (LASSO/SINDy) selects terms; cross-validation selects the sparse regulator \(\lambda\); and bootstrapping estimates parameter statistics.
  • Simulation: a second-order nonlinear oscillator under proportional control was used with training/test forces, different initial conditions, and Gaussian noise with \(\sigma_f=0.01\) and \(\sigma_x=0.01\); polynomial orders from 1 to 6 were searched with \(N=M\), using third-order Taylor expansion, 30 bootstrap samples at 50% of the data, and 50 \(\lambda\) values from \(10^{-6}\) to \(10^0\).
  • Experiment: a Quanser rotary flexible joint SIMO system under proportional control with a 0.2 s feedback delay was used; data were one minute long with sample time 0.001 s; polynomial orders from 1 to 7 were searched, using second-order Taylor expansion, 10 bootstrap samples at 50% of the training data, and 100 \(\lambda\) values from \(10^{-10}\) to \(10^0\).

Limitations

  • The paper states that estimation error comes from Taylor truncation error and regression error; keeping more higher-order Taylor terms may cause instability, so third order was chosen as a compromise for accuracy, complexity, and stability.
  • The paper states assumptions that only the system response \(x(t)\) is measured, the system is second order, the restoring force structure is unknown, and measurements contain zero-mean Gaussian white noise.
  • Evident from the reported tables: time delay estimates are biased low in both examples (simulation 0.1656 vs. 0.2; experiment 0.1734 vs. 0.2), and the experimental tracking performance is poor because proportional control alone is inadequate and the delay further deteriorates it, which the authors say was chosen only to generate stable data.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1007/s40435-021-00783-7

Cites

38 of 38 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

2 of 12 cited sources checked · 2 supported · 0 flagged

  • Findingown data

    The algorithm demonstrates excellent performance in identification.

    “The algorithm demonstrates excellent performances in identification.”
    Quote in the paperNot a substantial claim
  • Findingown data

    The accuracy of estimated parameters in the simulation is quite acceptable.

    “The accuracy of the estimated parameters is quite acceptable.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The third-order term is a good compromise for accuracy, complexity and stability.

    “We have found that the third order term is a good compromise for accuracy, complexity and stability.”
    Quote in the paperSubstantial claim
  • Interpretationnone given

    The estimation error of coefficients c, particularly time delay τ, can be attributed to truncation error of Taylor expansion and regression error.

    “The estimation error of the coefficients c, in particular, time delay τ , can be attributed to two sources: the truncation error of the Taylor expansion and the regression error.”
    Quote in the paperNot a substantial claim
  • Interpretationreasoning

    The proposed algorithm is noise resistant.

    “-Noise resistant The sparse regression is already robust.”
    • Also stated in: § 1 Introduction · plain
      “The algebraic operation offers additional filtering of the noise.”
    • Also stated in: § 1 Introduction · plain
      “Furthermore, the proposed algorithm is equipped with bootstrapping to study the statistics of estimation such as mean and standard deviations.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: Time delay caused by sensors and actuators is usually so small that it is difficult or even impossible to estimate correct values of time delay from noisy data.
  • Classified as probably not a substantial claim. 7
  • May combine two claims. 22
  • The quote only partly states this claim. 1

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Fliess and Ramirez proposed a robust and fast algebraic identification technique to estimate time invariant linear systems without time delay.

    “Fliess and Ramirez proposed a robust and fast algebraic identification technique to estimate time invariant linear systems without time delay [33].”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [33] Fliess M, SiraíRamírez H (2003) An algebraic framework for linear identification. ESAIM: Control. Optim Calculus Variat 9:151–168 source

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Robustness to noise and requirement for measurements of derivatives are concerns with the SINDy approach.

    “Robustness to noise and requirement for measurements of derivatives are concerns with the SINDy approach [28].”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [28] Champion K, Lusch B, Kutz JN, Brunton SL (2019) Data-driven discovery of coordinates and governing equations. Proc Natl Acad Sci 116(45):22445 source
    • Rests on an assumption: Traditional finite difference methods to estimate derivatives can amplify the noise.

  • May combine two claims.

    Findingown data

    In the experimental example, for both coordinates θ and α, polynomial order N=5 leads to minimum cross-validation error and is optimal.

    “The results suggest that for both coordinates θ and α, the polynomial order N = 5 leads to the minimum cross validation error.This is the optimal order for the polynomials.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The responses of the closed-loop system are not really tracking the references accurately, partly due to time delay and poor control design.

    “Figures 3 and 4 indicate that the responses of the closedloop system are not really tracking the references accurately.This is partly due to the effect of time delay and also the poor control design, as discussed earlier.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The values of parameters in Table 5 fall in a wide range from 0.002 to 2, so parameters can be several orders of magnitude apart, making it difficult to accurately estimate all parameters in presence of unwanted noises.

    “The values of the parameters in Table 5 fall in a wide range from 0.002 to 2. Hence, the parameters can be serval order of magnitudes apart, which makes it difficult to accurately estimate all the parameters in the presence of unwanted noises.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The estimated value of Coulomb friction f_c is not negligible and can play an important role in system dynamics.

    “The estimated value of Coulomb friction f c is not negligible.It can play an important role in the system dynamics.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The proposed algorithm successfully detected sparse terms and precisely estimated coefficients in the simulation.

    “The proposed algorithm has successfully detected the sparse terms and precisely estimated the coefficients.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the simulated example, the cross-validation error has a large drop approaching N=3, stays same for N=4, and starts increasing from N=5.

    “It is seen from the figure that the cross validation error has a large drop when approaching N = 3 and stays at the same range for N = 4 and starts increasing from N = 5.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Belkoura first investigated the identifiability conditions for a general class of systems described by convolution equations and introduced an algebraic formulation for online estimation of time delay and parameters of structured and arbitrary input-outputs.

    “Inspired by the algebraic operation, Belkoura in [12] and [13] first investigated the identifiability conditions for a general class of systems described by convolution equations.Then, an algebraic formulation is introduced for online estimation of time delay and parameters of structured and arbitrary input-outputs.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [12] Belkoura L, Richard JP, Fliess M (2009) Parameters estimation of systems with delayed and structured entries. Automatica 45(5):1117–1125 source
    • cites [13] Belkoura L (2010) Identifiability and algebraic identification of time delay systems. IFAC Proc 43(2):1–8 source

  • May combine two claims.

    Interpretationown data

    The proposed system identification algorithm can estimate time delay and identify nonlinearities when linear models are no longer adequate.

    “This highlights the ability of the proposed system identification algorithm to estimate the time delay and to identify the nonlinearities in the system when the linear models are no longer adequate.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    The nonlinear least square-based algorithm in [20] may have several local minima in its proposed cost function, making it highly dependent on the initial guess for parameters and especially time delay.

    “The main problem with this algorithm is that the proposed cost function may have several local minima.Therefore, it highly depends on the initial guess for the parameters and especially the time delay.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [20] Chen F, Garnier H, Gilson M (2015) Robust identification of continuous-time models with arbitrary time-delay from irregularly sampled data. J Process Control 25:19–27 source

  • May combine two claims.

    Premisecitation

    Sparse regression is a convex optimization problem with a global minimum.

    “The sparse regression is a convex optimization problem with a global minimum [34].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [34] Lai Z, Nagarajaiah S (2019) Sparse structural system identification method for nonlinear dynamic systems with hysteresis/inelastic behavior. Mech Syst Signal Process 117:813–842 source

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Interpretationreasoning

    In real world, time delay is usually small and appears in high order terms of the Taylor expansion; therefore, the LASSO will penalize these terms, such as the coefficient of term P1(t) in Eq. (11).

    “In real world, time delay is usually small.Moreover, time delay appears in high order terms of the Taylor expansion.Therefore, the LASSO will penalize these terms, such as the coefficient of term P 1 (t) in Eq. (11).”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: Time delay caused by sensors and actuators is usually so small that it is difficult or even impossible to estimate correct values of time delay from noisy data.

  • May combine two claims.

    Interpretationnone given

    The proposed algorithm relies on data to make selection and is not limited to linear and SISO systems.

    “The proposed algorithm relies on the data to make selection and is not limited to linear and SISO systems.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationown data

    The small standard deviations in Table 1 indicate that randomness of data is not significant so bootstrapping samples are not sufficiently different.

    “This fact is an indication that the randomness of the data is not significant so that the bootstrapping samples are not sufficiently different.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given

    Keeping more higher-order terms of Taylor expansion helps reduce truncation error but may risk instability of the truncated model.

    “While keeping more higher order terms of the Taylor expansion helps to reduce truncation error, we may run the risk of instability of the truncated model.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    The algebraic signal pre-processing and nonlinear state observer are essential to compute needed derivatives of measured time series and other signals without knowing initial conditions, and to filter noises due to random excitations and measurements.

    “These operations are essential to compute the needed derivatives of measured time series and other signals without the need to know initial conditions, and to filter noises due to random excitations and measurements.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    Higher order expansions can reduce truncation error, but the system can become unstable even when the system is linear with a constant time delay.

    “Even though higher order expansions can reduce the truncation error, the system can become unstable even when the system is linear with a constant time delay [1].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [1] Richard JP (2003) Time-delay systems: an overview of some recent advances and open problems. Automatica 39(10):1667–1694 source

  • The quote only partly states this claim.

    Interpretationreasoning

    The proposed algorithm overcomes the multiple local minima limitation of modified least square methods with recursive approach.

    “The simulation and experimental results illustrate that the proposed algorithm can overcome the following limitations of the existing techniques:”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationnone given

    Keeping too many higher order terms in the Taylor expansion may not be beneficial to the identification process; only a sufficient number of terms is needed to generate enough equations to determine the unknown parameters including time delay.

    “We should point out that keeping too many higher order terms may not be beneficial to the identification process.We only need to keep a sufficient number of terms to generate enough equations to determine the unknown parameters including time delay.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: Time delay caused by sensors and actuators is usually so small that it is difficult or even impossible to estimate correct values of time delay from noisy data.

  • May combine two claims.

    Interpretationnone given

    The linear model in Eq. (21) does not include the friction term, although the actual device always has friction.

    “We should point out that the linear model in Eq. ( 21) does not include the friction term, although the actual device always has friction.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    The optimization problem in the framework of the LASSO is convex, which implies the existence of a unique optimal solution.

    “It should be noted that the optimization problem in the framework of the LASSO is convex [34], which implies the existence of a unique optimal solution.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [34] Lai Z, Nagarajaiah S (2019) Sparse structural system identification method for nonlinear dynamic systems with hysteresis/inelastic behavior. Mech Syst Signal Process 117:813–842 source

  • May combine two claims.

    Premisecitation

    The LASSO penalizes terms with small coefficients by regularization and keeps the important terms in the system model.

    “The LASSO penalizes the terms with small coefficients by regularization and keeps the important terms in the system model.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [35] Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning: data mining, inference, and prediction. Springer, Berlin source

  • Classified as probably not a substantial claim.

    Premisecitation

    Xu demonstrates that the low order Taylor expansion gives promising estimation of small time delay.

    “Xu demonstrates that the low order Taylor expansion gives promising estimation of small time delay [19].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [19] Xu L (2014) A proportional differential control method for a time-delay system using the Taylor expansion approximation. Appl Math Comput 236:391–399 source

  • May combine two claims.

    Interpretationnone given

    The proposed algorithm is computationally efficient and robust to noise.

    “The proposed algorithm in this paper is computationally efficient and robust to noise.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The only assumption of the proposed algorithm about the structure of the system is that the system is sparse in the space of base functions.
25 of 25

  • The authors state this as an assumption.

    Only the system response x(t) is measured.

    “1.Only the system response x(t) is measured.”

  • The authors state this as an assumption.

    The system is exposed to random excitations included in f(t).

    “2. The system is exposed to random excitations included in f (t).”

  • The authors state this as an assumption.

    The measurement contains Gaussian white noise with zero mean denoted as x.

    “3. The measurement contains Gaussian white noise with zero mean denoted as x .”

  • The authors state this as an assumption.

    The system structure in terms of the function g(x, ẋ) is unknown.

    “4. The system structure in terms of the function g(x, ẋ) is unknown.”

  • The authors state this as an assumption.

    The system is second order.

    “5.The system is second order.”

  • The authors state this as an assumption.

    The only assumption of the proposed algorithm about the structure of the system is that the system is sparse in the space of base functions.

    “The only assumption of the proposed algorithm about the structure of the system is that the system is sparse in the space of base functions.”

  • The authors state this as an assumption.

    The dataset consists of two long time series of the system response, one used for training and another for cross-validation.

    “In this work, we assume that the dataset consists of two long time series of the system response.One is used for training and another for cross-validation.”

  • The paper relies on this without support.

    The initial conditions (x0, v0) are generally unknown.

    “The system starts from initial conditions (x 0 , v 0 ) which are generally unknown.”

  • The paper relies on this without support.

    Time delay caused by sensors and actuators is usually so small that it is difficult or even impossible to estimate correct values of time delay from noisy data.

    “Time delay caused by sensors and actuators is usually so small that it is difficult or even impossible to estimate correct values of time delay from noisy data.”

  • The paper relies on this without support.

    Traditional finite difference methods to estimate derivatives can amplify the noise.

    “Traditional finite difference methods to estimate derivatives can amplify the noise.”

  • The paper relies on this without support.

    The elbow of the Pareto front is often ambiguous due to existence of a cluster of candidate models near the elbow.

    “Unfortunately, in most of cases, the elbow of the Pareto front is ambiguous due to existence of a cluster of candidate models near the elbow.”

  • The paper relies on this without support.

    A big data matrix P(x) may not always be available for real-world applications.

    “However, a big data matrix P(x) may not always be available for real-world applications.”

  • The paper relies on this without support.

    Real measurements often contain noises, and can have outliers and missing data.

    “Real measurements often contain noises, and can have outliers and missing data.”

  • The paper relies on this without support.

    Most existing methods for time delay estimation rely on the knowledge of the system model.

    “Most existing methods for time delay estimation rely on the knowledge of the system model.”

  • The paper relies on this without support.

    Another general assumption in time delay estimation is that the initial conditions are zero, while in most practical applications this is not true.

    “Another general assumption in time delay estimation is that the initial conditions are zero while in most of practical applications, it is not true.”

  • The authors state this as an assumption.

    Sensors have measurement noises such that the system output is x(t)+x where x is normally distributed random noise with zero mean and standard deviation σ_x=0.01.

    “We assume that the sensors have measurement noises such that the system output is given by x(t) + x where x is the normally distributed random noise with zero mean and standard deviation σ x = 0.01.”

  • The authors state this as an assumption.

    Motor parameters are known.

    “We assume that the motor parameters are known.”

  • The authors state this as an assumption.

    For each coordinate θ and α, the restoring force can be represented by a sum of four polynomials of a single variable, i.e. θ^i, θ̇^i, α^j, and α̇^j, together with a friction term related to θ, without considering cross-product terms.

    “For simplicity, we assume that for each coordinate θ and α, the restoring force can be represented by a sum of four polynomials of a single variable, i.e. θ i , θi , α j , and α j , together with a friction term related to θ , without considering the cross-product terms.”

  • The paper relies on this without support.

    No prior work has extended sparse regression to nonlinear dynamical systems with time delay.

    “For the first time, this study extends the sparse regression to nonlinear dynamical systems with time delay.”

  • The paper relies on this without support.

    The actual device always has friction.

    “although the actual device always has friction.”
20 of 20

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Parameter vector reverses three coefficient signs

    Minor

    Equation (9) gives the coefficients of P2, P3, and P6 as m − kpτ2/2, c1 + kpτ, and k1 − kp. The parameter vector instead uses plus, minus, and plus, respectively, so it does not represent the stated signal equation.

  • Test trajectory amplitude differs from the stated matching amplitude

    Minor

    The text says the square-wave test trajectory has the same amplitude as the training trajectory, whose stated amplitude is 2. The Figure 4 caption instead gives the square wave an amplitude of 1 radian.

  • Algorithm assigns the minimum error rather than its minimizing regulator

    Minor

    The stated selection rule is to choose the λ value that minimizes cross-validation error. Algorithm 1 instead assigns the minimum error value itself to λ min and then uses that value as the regulator.

  • Training bootstrap samples use the test force data

    Minor

    Algorithm 1 labels the bootstrapped force vector as training data but constructs it from p test. This conflicts with the paper's separation of training data from data used for cross-validation.

  • Two tables give different nominal equivalent inertias

    Minor

    Table 3 lists the equivalent moment of inertia as 9.76 × 10−5 kg m2, while Table 5 lists the nominal J eq as 0.00208. Both are presented as reference values for the rotary flexible joint.

5 of 5

Cites

Flagged first, then by citations

Research works printed but not linked: 4

  1. Gawthrop PJ, Nihtila MT, Rad AB (1989) Recursive parameter estimation of continuous systems with unknown time delay. Control Theory Adv Technol 5:227–248
  2. Quanser Inc (2011) Rotary flexible joint workbook. Markhan, Ontario
  3. Apkarian J, Lévis M, Gurocak H (2011) User manual SRV02 Rotary servo base unit set up and configuration. Quanser Inc., Markham
  4. Han J (1995) A class of extended state observers for uncertain systems. Control Decis 10:85–88

Cited by

Sorted by citations

Related

Authors
Leylaz, G., Wang, S., & Sun, J.-Q.
Affiliations
  • University of California, Merced (US)
  • PLA Information Engineering University (CN)
Corresponding author
Sun, Jian-Qiao
ORCID
Venue
International Journal of Dynamics and Control
Publisher
Springer Science and Business Media LLC
Published
2021-04-01
Volume
10
Issue
1
Pages
13-24
ISSN
  • 2195-268X (ISSN-L)
  • 2195-2698
DOI
10.1007/s40435-021-00783-7
OpenAlex
W3141930961
Cited by
35
References
37
Citations by year
2026: 2 · 2025: 13 · 2024: 8 · 2023: 6 · 2022: 3 · 2021: 3
FWCI
2.24
Type
journal article
Language
en
Open access
Licence
Locations
International Journal of Dynamics and Control (publishedVersion, cc-by)
Funders
Zhengzhou University
Topics
  • Physical Sciences › Engineering › Control and Systems Engineering › Control Systems and Identification
  • Physical Sciences › Engineering › Computational Mechanics › Advanced Adaptive Filtering Techniques
  • Physical Sciences › Engineering › Computational Mechanics › Sparse and Compressive Sensing Techniques
Keywords
nonlinear system identification, time-delay systems, sparse optimization, sparse regression, cross-validation, model selection, bootstrapping, noise reduction, Duffing oscillator, sensitivity to initial conditions

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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

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