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Cited by 10Control Systems and IdentificationAdvanced Control Systems DesignFractional Differential Equations Solutions

Application of Fractional Calculus for Parameter Estimation of Nonlinear Wiener Systems With Time Delay

Open access
Kothari, Kajal·2024·IEEE Access
DOI 10.1109/access.2024.3367441
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Abstract

The parameter identification technique for a fractional-order Wiener system with time delay is proposed using the Haar wavelet operational matrices. The method is applicable to lower and higher-order Wiener systems, where the order could be an integer or fractional value. The proposed method accomplishes identification in two stages, whereby a special input signal is utilized to separate the identification of the nonlinearity from the linear dynamics. The graphical method is applied to the step responses to compute the static nonlinearity. Subsequently, the sinusoidal response is utilized along with the obtained nonlinearity parameters to identify the linear subsystem. The lower-order models of both subsystems are utilized for identification to reduce complexity. This operational matrix-based algebraic approach is simple yet accurate, and it is applicable to noisy data. Moreover, the proposed approach does not demand prior knowledge about the fractional order or time delay. This method is simulated and validated in MATLAB for various Wiener systems. Numerical results and comparisons with the existing methods illustrate the efficiency of the proposed method.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proposes a two-stage Haar wavelet operational matrix method for parameter estimation of fractional-order Wiener systems with time delay, using fractional calculus and a special test signal. It reports accurate identification of linear and nonlinear subsystems for integer/fractional, commensurate/non-commensurate, and noisy cases without prior knowledge of fractional order or delay.

Key findings

  • The method identifies fractional-order Wiener systems with time delay without prior knowledge of the fractional order or time delay, and works for commensurate as well as non-commensurate orders, as demonstrated by Example 5 with \(G_5(s)=\frac{1}{s^{1.9}+s^{0.85}+1}e^{-2s}\).
  • It separates nonlinearity identification from linear dynamics: nonlinearity is estimated non-iteratively from unit-step and step responses, while the linear subsystem is estimated from sinusoidal response data using HWOM.
  • It can handle noise-contaminated data without preprocessing; Example 2 uses noisy data with SNR = 20, and the paper states results in all examples were obtained without filtering or pre-processing noisy data.
  • For Example 1, the estimated third-order nonlinearity is \(f_1(v)=1.3792v(t)-0.4880v^2(t)+0.1152v^3(t)\), and the fourth-order polynomial is reported as more accurate than the third-order polynomial for noise-free data.

Methods

  • Design: two-stage identification for a block-oriented Wiener system consisting of a static polynomial nonlinearity followed by a stable single-pole fractional-order linear subsystem with input time delay.
  • Data/sample: a special input test signal comprising a unit step, a step of amplitude \(w\), and a sinusoidal signal; \(N=256\) samples are used for each data segment, with \(w=0.8\) for the fourth-order polynomial and \(w=0.5\) for the third-order polynomial/noisy case.
  • Analysis: nonlinearity parameters \(c_1\)–\(c_4\) are obtained by a non-iterative graphical method and least-squares solution \(C=(W^TW)^{-1}W^TY\); linear parameters \(b_0,a_0,a_1,\alpha,\theta\) are estimated by minimizing Integral Time-Squared Error using MATLAB `fsolve`, with HWOM of fractional-order integration and delay. Numerical validation uses MATLAB simulations and comparisons with relay feedback methods RF1, RF2, and RF3.

Limitations

  • The paper states it is not applicable in its current form to Wiener systems with discontinuous-type (dead-zone) nonlinearity, nor to Hammerstein-Wiener or Wiener-Hammerstein systems; these could be addressed by modifying the nonlinearity equation or extending the method.
  • The paper states that for large-scale systems the linear part can be estimated accurately, but the graphical method needs improvement for nonlinearity estimation.
  • Evident from the text: the method assumes a monotonic static nonlinearity crossing the origin, a stable single-pole fractional-order linear subsystem with finite time delay, and a fourth-order polynomial nonlinearity; validation is simulation-based in MATLAB using examples from the literature, not real experimental data.

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

Integrity signals

Registry

Crossref · 10.1109/access.2024.3367441

Cites

43 of 43 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

3 of 31 cited sources checked · 3 supported · 0 flagged

  • Findingown data

    Numerical results and comparisons with existing methods illustrate the efficiency of the proposed method.

    “Numerical results and comparisons with the existing methods illustrate the efficiency of the proposed method.”
    Quote in the paperSubstantial claim
  • Findingown data

    The third-order polynomial obtained for the nonlinearity in Example 1 is f1(v)=1.3792v(t)-0.4880v^2(t)+0.1152v^3(t).

    “The obtained third order polynomial is f 1 (v) = 1.3792v(t) -0.4880v 2 (t) + 0.1152v 3 (t).”
    Quote in the paperNot a substantial claim
  • Findingnone given

    The proposed identification method gives satisfactory results in finite trials.

    “The proposed identification method gives satisfactory results in the finite trials.”
    Quote in the paperNot a substantial claim
  • Findingown data

    In Example 1, the estimated nonlinearity using the fourth-order polynomial is more accurate than using the third-order polynomial.

    “As shown in Fig. 6, the estimated nonlinearity using the fourth order polynomial is more accurate compared to the third order polynomial.”
    Quote in the paperNot a substantial claim
  • Findingown data

    The proposed method can accurately estimate both the linear and nonlinear subsystems in Example 1.

    “The results clearly display that the proposed method can estimate both subsystems accurately.”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 43
  • May combine two claims. 10

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

    Premisecitation

    The Wiener-type model can represent several nonlinear systems or processes, such as nonlinear lithium-ion battery, distillation column, polymerization reactor, pH neutralization process, heat exchanger, thermal process, robotic arm, and many more.

    “The Wiener-type model can represent several nonlinear systems or processes, such as nonlinear lithium-ion battery [1], distillation column [2], polymerization reactor [3], pH neutralization process [4], [5],”
    Quote in the paperNot a substantial claim1 of 3 sources checked
    • cites [1] source
    • cites [2] source
    • cites [3] source

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

    Premisecitation

    The parameter identification method for the fractional order Wiener system [32] is applicable to the Wiener system with commensurate orders and without time delay.

    “The parameter identification method for the fractional order Wiener system was developed and validated on a photovoltaic cell in [32], which is applicable to the Wiener system with commensurate orders and without time delay.”
    Quote in the paperNot a substantial claim1 of 1 sources checked

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

    Premisecitation

    Allafi et al. [24] demonstrated identification of the fractional-order Hammerstein-Wiener model using a simplified refined instrumental variable method, considering some prior knowledge.

    “Allafi et al. [24] demonstrated the identification of the fractional-order Hammerstein-Wiener model using a simplified refined instrumental variable method, considering some prior knowledge.”
    Quote in the paperNot a substantial claim1 of 1 sources checked

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

    Findingnone given

    Increasing the number of samples N increases computational overhead.

    “However, it increases computational overhead.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    The proposed method can estimate system parameters as well as time delays.

    “The proposed method can estimate system parameters as well as time delays.”
    Quote in the paperSubstantial claim

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

    Findingown data

    Different numbers of samples (N = 128, N = 256, and N = 512) do not give exactly the same accuracy.

    “The different number of samples (N = 128, N = 256, and N = 512) will not give exactly the same accuracy.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The proposed method works well with non-commensurate fractional orders and is suitable for commensurate as well as non-commensurate models.

    “In addition to that, for the fractional higher-order linear part, the proposed method works well with the non-commensurate orders as described in Example 5.”
    • Also stated in: § Remark 6: · plain
      “Furthermore, the fractional differential orders in G 5 are non-commensurate orders; therefore, the proposed method is suitable for commensurate as well as noncommensurate model.”
    • Also stated in: § V. CONCLUSION · plain
      “It can estimate fractional order Wiener systems with commensurate as well as non-commensurate orders.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The numerical examples selected from the literature are representative of the class of Wiener-type nonlinear systems with time delay to which the method is claimed to apply.

  • Probably not a substantial claim: may not pass S4 (based).

    Findingnone given

    Conventional calculus-based methods cannot achieve the same accuracy as the proposed method when using lower-order models for the fractional linear subsystem.

    “The conventional calculus-based methods cannot achieve the same accuracy using the lower order models for the fractional linear subsystem.”
    Quote in the paperNot a substantial claim

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

    Premisecitation

    The method of Meher and Manjhi [19] uses data filtering for noisy data.

    “However, this method uses data filtering for noisy data.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [19] source

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

    Premisecitation

    Meher and Manjhi [19] illustrated a relay-based identification technique for Wiener systems applicable to stable and unstable systems.

    “Recently, Meher and Manjhi [19] illustrated another relay-based identification technique for Wiener systems, which is applicable to stable as well as unstable systems.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [19] source

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

    Premisecitation

    Park and Lee [5] used a special test signal to separate identification of nonlinearity from the linear part for Wiener systems.

    “Strategies for the identification of Wiener systems were represented in brief by Park and Lee in [5] using the special test signal whereby the identification of nonlinearity was separated from the linear part, and further extended for the Hammerstein-Wiener type process in [9].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [5] source
    • cites [9] source

  • Classified as probably not a substantial claim.

    Interpretationnone given

    The lower-order models of both subsystems are utilized for identification to reduce complexity.

    “The lower-order models of both subsystems are utilized for identification to reduce complexity.”
    Quote in the paperSubstantial claim

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

    Interpretationnone given

    For the nonlinear subsystem, if the order of the polynomial is increased, the graphical method needs to be improved; it might not give satisfactory results for the large-scale system in its current form.

    “Furthermore, for the nonlinear subsystem, if the order of the polynomial is increased, the graphical method needs to be improved.Therefore, it might not give satisfactory results for the large-scale system in its current form.”
    Quote in the paperNot a substantial claim

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

    Interpretationnone given

    The non-iterative manner of nonlinearity identification reduces computational effort.

    “The non-iterative manner of nonlinearity identification reduces computational effort.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based). May combine two claims.

    Interpretationnone given

    The lower-order model is highly desirable and adequate to eliminate memory constraint issues and ease its real-time implementation.

    “In the case of the fractional system, the lower-order model is highly desirable and adequate to eliminate memory constraint issues and ease its real-time implementation.”
    Quote in the paperNot a substantial claim

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

    Premisecitation

    Fuzzy-weighted differential evolution [31] was proposed for identification of time-delay fractional-order Wiener system, whereby linear subsystem was considered with commensurate fractional orders.

    “The fuzzy-weighted differential evolution [31] was proposed for the identification of time-delay fractional-order Wiener system, whereby linear subsystem was considered with the commensurate fractional orders.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [31] source

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

    Premisecitation

    The method of Meher and Manjhi [19] does not use lower-order models to estimate higher-order systems.

    “Moreover, it does not use lower-order models to estimate higher-order systems.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [19] source

  • May combine two claims.

    Interpretationreasoning

    The determinant of the identity matrix, as well as the orthogonal P matrix, is always non-zero, so the inverse matrix of a1I + a0P always exists.

    “The determinant of the identity matrix, as well as orthogonal P α N ×N matrix, is always non-zero.Therefore, the inverse matrix of a 1 I + a 0 P α N ×N always exists.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    The proposed method is applicable to Wiener systems with commensurate as well as non-commensurate fractional orders without prior knowledge.

    “The proposed method is applicable to Wiener systems with commensurate as well as non-commensurate fractional orders without prior knowledge.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    A very few methods are applicable to fractional models with noncommensurate orders.

    “A very few methods are applicable to fractional models with noncommensurate orders.”
    Quote in the paperNot a substantial claim

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

    Premisecitation

    Zong et al. [34] demonstrated identification of fractional-order Wiener-Hammerstein-type nonlinear systems using fuzzy PSO, whereby a data filtering technique was applied prior to identification.

    “Recently, Zong et al. [34] demonstrated the identification of fractional-order Wiener-Hammerstein-type nonlinear systems using fuzzy PSO, whereby a data filtering technique was applied prior to identification.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [34] source

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    The conditions in Assumption 1 are necessary for convergence, use of HWOM, and safety of the system.

    “These conditions are necessary for the convergence, use of HWOM, and safety of the system.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The input signal u(t) is persistently exciting for the whole Wiener system.
    • The authors state this as an assumption: The input and output satisfy u(t) in L2[0, Tn] and y(t) in L2[0, Tn].
    • The authors state this as an assumption: The considered stable system generates bounded output for bounded input.

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    The proposed fractional calculus-based parameter estimation strategy for a block-oriented Wiener system aims at increased accuracy with reduced complexity.

    “In this paper, a fractional calculus-based parameter estimation strategy is presented for a block-oriented Wiener system that aims at increased accuracy with reduced complexity.”
    Quote in the paperNot a substantial claim

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

    Premisecitation

    A frequency domain strategy for the Wiener-Hammerstein system was devised and delivered in [21], which yielded favorable results.

    “A frequency domain strategy for the Wiener-Hammerstein system was devised and delivered in [21], which yielded favorable results.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [21] source

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

    Premisenone given

    In many of the above techniques, fractional calculus was used for parameter estimation of Wiener systems; however, this domain has not been well explored yet.

    “In many of the above techniques, fractional calculus was used for the parameter estimation of the Wiener systems; however, this domain has not been well explored yet.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    Not all methods were developed using lower-order models.

    “Not all methods were developed using lower-order models.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    Not all methods separate the identification of linear and nonlinear parts.

    “Moreover, not all methods separate the identification of linear and nonlinear parts.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Interpretationnone given

    The proposed strategy is developed using the Riemann-Liouville definition and the Haar Wavelet Operational Matrices of Fractional-Order Integration and delay.

    “The proposed strategy is developed using the Riemann-Liouville definition and the Haar Wavelet Operational Matrices (HWOM) of Fractional-Order Integration (FOI) and delay.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The proposed method uses a lower order fractional model with an algebraic operational matrix approach that is applicable to time-delayed Wiener systems with commensurate as well as non-commensurate orders, even in the presence of noise.

    “Therefore, these factors have motivated the author to develop a method using a lower order fractional model with an algebraic operational matrix approach that is applicable to time-delayed Wiener systems with commensurate as well as non-commensurate orders, even in the presence of noise.”
    Quote in the paperSubstantial claim

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

    Premisecitation

    Giordano et al. [23] developed a technique for identification of the fractional Wiener-Hammerstein type nonlinear system using Taylor series expansion that can avoid issues related to some configuration of poles and zeros of the linear part.

    “Later, Giordano et al. [23] developed a technique for identification of the fractional Wiener-Hammerstein type nonlinear system using Taylor series expansion that can avoid issues related to some configuration of poles and zeros of the linear part.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [23] source

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

    Premisenone given

    Very few techniques have been developed using fractional calculus, and none of them were developed using the operational matrix of fractional order integration for the Wiener type nonlinear model.

    “Very few techniques have been developed using fractional calculus, and none of them were developed using the operational matrix of fractional order integration for the Wiener type nonlinear model.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    Appropriate initial guess values make convergence faster with more accuracy.

    “For the optimization problem, appropriate initial guess values are applied, which makes convergence faster with more accuracy.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    Many of the above techniques cannot accurately estimate actual fractional-order systems.

    “Many of the above techniques are efficient; however, these techniques cannot accurately estimate the actual fractional-order systems.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    Independent identification of nonlinearity permits freedom to use any suitable algorithm to identify the linear subsystem.

    “Furthermore, independent identification of nonlinearity permits freedom to use any suitable algorithm to identify the linear subsystem.”
    Quote in the paperNot a substantial claim

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

    Interpretationreasoning

    The consideration of time delay makes the method applicable to an additional subset of Wiener systems, as many industrial systems are time-delayed.

    “The consideration of time delay makes the method applicable to an additional subset of Wiener systems, as many industrial systems are time-delayed.”
    Quote in the paperNot a substantial claim

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

    Interpretationnone given

    w = 0.5 with a third order polynomial can also be used for noise-free data with slightly reduced accuracy compared to a fourth order polynomial.

    “It is to be noted that w = 0.5 with a third order polynomial can also be used for noise-free data with slightly reduced accuracy compared to a fourth order polynomial.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    For large and complex systems, increasing the order leads to better accuracy of identification.

    “For large and complex systems, one can increase the order, which leads to better accuracy of identification.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    The derived expression increases accuracy with reduced complexity.

    “This is the main contribution of the presented work, which increases accuracy with reduced complexity.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    The piece-wise constant Haar wavelets are chosen for the proposed technique due to their unique characteristics, such as high accuracy, mathematical simplicity, faster computation, ease of implementation with other standard algorithms, and better noise immunity.

    “The piece-wise constant Haar wavelets are chosen for the proposed technique due to their unique characteristics, such as high accuracy, mathematical simplicity, faster computation, ease of implementation with other standard algorithms, and better noise immunity.”
    Quote in the paperNot a substantial claim

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

    Interpretationreasoning

    The increased datalength from linear interpolation does not affect computational complexity much because the graphical method does not deal with differential equations or heavy calculations.

    “Nevertheless, this increased datalength does not affect the computational complexity much because the graphical method does not deal with differential equations or heavy calculations.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    Not all existing methods are applicable to time delayed Wiener models.

    “Not all existing methods are applicable to time delayed Wiener models.”
    Quote in the paperNot a substantial claim

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

    Premisenone given

    The Legendre wavelet based method [30] is not applicable to block-oriented nonlinear models.

    “This method is not applicable to block-oriented nonlinear models.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [30] source

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

    Interpretationnone given

    Increasing the number of poles and zeros increases the complexity of identification.

    “However, this also increases the complexity of the identification.”
    Quote in the paperNot a substantial claim

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

    Premisecitation

    The method in [21] was developed with an iterative minimization algorithm that could be caught in local minima.

    “However, it was developed with an iterative minimization algorithm that could be caught in local minima.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [21] source

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

    Premisecitation

    Giordano and Sjöberg [22] reviewed and extended the method in [21] using a time-domain approach to avoid the local minima problem.

    “Giordano and Sjöberg [22] reviewed and extended this method using the time-domain approach to avoid the local minima problem.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [22] source

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

    Premisecitation

    The Haar wavelet delay matrix was derived in [35] using the generalized delay matrix.

    “The Haar wavelet delay matrix was derived in [35] using the generalized delay matrix.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [35] source

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

    Premisecitation

    Compared to other functions of the wavelet family, the Haar wavelet exhibits faster computation capabilities.

    “In comparison to other functions of the wavelet family, the Haar wavelet exhibits faster computation capabilities and can be defined for time interval T n as [41],”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [41] source

  • May combine two claims.

    Interpretationnone given

    The operational matrix approach transforms the identification problem into algebraic equations, which reduces complexity.

    “Although it is based on fractional calculus, the operational matrix approach transforms the identification problem into algebraic equations, which reduces complexity.”
    Quote in the paperSubstantial claim
48 of 48

  • The authors state this as an assumption.

    The static nonlinearity f(v) is monotonic and crossing at the origin.

    “Therefore, it is assumed that the static nonlinearity f (v) is monotonic and crossing at the origin, and G(s) represents the transfer function of a stable linear subsystem, which is a single-pole fractional-order model with time delay.”

  • The authors state this as an assumption.

    The linear subsystem G(s) is a stable single-pole fractional-order model with time delay.

    “Therefore, it is assumed that the static nonlinearity f (v) is monotonic and crossing at the origin, and G(s) represents the transfer function of a stable linear subsystem, which is a single-pole fractional-order model with time delay.”

  • The authors state this as an assumption.

    The input time delay θ is finite.

    “The parameter θ is assumed to be a finite value.”

  • The authors state this as an assumption.

    The static nonlinearity can be adequately represented by a fourth-order polynomial for identification purposes.

    “In the presented work, a fourthorder nonlinear polynomial function is utilized, which is sufficient for identification purposes.”

  • The authors state this as an assumption.

    The input signal u(t) is persistently exciting for the whole Wiener system.

    “The input signal u(t) is persistently exciting for whole Wiener system.”

  • The authors state this as an assumption.

    The input and output satisfy u(t) in L2[0, Tn] and y(t) in L2[0, Tn].

    “The input and output satisfy the condition of u(t) ∈ L 2 [0, T n ] and y(t) ∈ L 2 [0, T n ].”

  • The authors state this as an assumption.

    The considered stable system generates bounded output for bounded input.

    “Moreover, the considered stable system generates bounded output for the bounded input.”

  • The paper relies on this without support.

    Complex nonlinear systems with significant nonlinearity cannot be accurately identified using linear models.

    “Complex nonlinear systems with significant nonlinearity cannot be accurately identified using linear models.”

  • The paper relies on this without support.

    Estimating the complete nonlinear system efficiently with minimal computational effort is an open problem.

    “Therefore, estimating the complete nonlinear system efficiently with minimal computational effort is an open problem.”

  • The paper relies on this without support.

    Various types of static memoryless nonlinearities can be approximated efficiently with reduced complexity using the polynomial form.

    “Various types of static memoryless nonlinearities can be approximated efficiently with reduced complexity using the polynomial form.”

  • The paper relies on this without support.

    The numerical examples selected from the literature are representative of the class of Wiener-type nonlinear systems with time delay to which the method is claimed to apply.

    “Wienertype nonlinear systems with significant nonlinearity have been selected from the literatures [17], [19], and [39] and simulated to evaluate the performance of the proposed method compared to existing relay feedback methods RF1 method [19], RF2 method [39], and RF3 method [17].”
11 of 11

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Authors
Kothari, K.
Affiliations
Fiji National University (FJ)
Corresponding author
Kothari, Kajal
ORCID
Kothari, Kajal
Venue
IEEE Access
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Published
2024
Volume
12
Pages
26281-26294
ISSN
2169-3536 (ISSN-L)
DOI
10.1109/access.2024.3367441
OpenAlex
W4391929619
Semantic Scholar
c8fcd72339ca1bde5b3f686c5d1e29a0b5f298e8
Cited by
10
References
41
Citations by year
2026: 4 · 2025: 4 · 2024: 2
FWCI
1.60
Type
journal article
Language
en
Open access
Licence
Locations
  • IEEE Access (publishedVersion, cc-by-nc-nd)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
Funders
Fiji National University
Topics
  • Physical Sciences › Engineering › Control and Systems Engineering › Control Systems and Identification
  • Physical Sciences › Engineering › Control and Systems Engineering › Advanced Control Systems Design
  • Physical Sciences › Mathematics › Modeling and Simulation › Fractional Differential Equations Solutions
Keywords
fractional calculus, Wiener systems, time-delay systems, parameter estimation, fractional-order systems, nonlinear system identification, sinusoidal response, step response, noisy data

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