Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions

Yu-Gang Hyon; Hyon-Ju Sin; Myong-Hyok Sin; Chol Min Sin; Hun Kim

doi:doi:10.11648/j.engmath.20250902.12

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Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions

Yu-Gang Hyon, Hyon-Ju Sin, Myong-Hyok Sin^*

, Chol Min Sin

, Hun Kim

Published in Engineering Mathematics (Volume 9, Issue 2)

Received: 27 October 2025 Accepted: 22 November 2025 Published: 29 December 2025

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Abstract

In this paper, we propose a method for identification of continuous-time fractional-order systems with unknown states and input delays. In practice, many systems are modeled accurately with fractional differential equations. In particular, many systems are modeled as fractional differential equations with input delay and state delay. Since the geometric and physical meaning of fractional calculus is not clear, it is difficult to model the real system directly to fractional order systems based on mechanical analysis. Thus, the identification of fractional order systems is the main method for constructing fractional order models and is the subject of the main research by many scientists. To solve the identification problem of systems with input delay and state delay, we use the fact that the fractional integral operator matrix by the block pulse functions is an upper triangular Toeplitz matrix. We have presented an efficient method to identify the linear and nonlinear parameters separably by using the commutativity and nilpotent property for multiplication between upper triangular Toeplitz matrices. We also have presented an efficient algorithm to newly approximate the Jacobian of the variable projection functional to solve the least squares problem with nonlinear parameters. Several simulation examples have been used to verify the effectiveness of the proposed method. It is shown that the input delay and the state delay have a significant effect on the output characteristics of the system, especially the state delay has a larger effect than the input delay.

Published in	Engineering Mathematics (Volume 9, Issue 2)
DOI	10.11648/j.engmath.20250902.12
Page(s)	31-44
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Block Pulse Function, Delay, Fractional Order System, Operational Matrix, Parameter Identification

References

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APA Style

Hyon, Y., Sin, H., Sin, M., Sin, C. M., Kim, H. (2025). Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions. Engineering Mathematics, 9(2), 31-44. https://doi.org/10.11648/j.engmath.20250902.12

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ACS Style

Hyon, Y.; Sin, H.; Sin, M.; Sin, C. M.; Kim, H. Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions. Eng. Math. 2025, 9(2), 31-44. doi: 10.11648/j.engmath.20250902.12

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AMA Style

Hyon Y, Sin H, Sin M, Sin CM, Kim H. Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions. Eng Math. 2025;9(2):31-44. doi: 10.11648/j.engmath.20250902.12

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@article{10.11648/j.engmath.20250902.12,
  author = {Yu-Gang Hyon and Hyon-Ju Sin and Myong-Hyok Sin and Chol Min Sin and Hun Kim},
  title = {Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions},
  journal = {Engineering Mathematics},
  volume = {9},
  number = {2},
  pages = {31-44},
  doi = {10.11648/j.engmath.20250902.12},
  url = {https://doi.org/10.11648/j.engmath.20250902.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20250902.12},
  abstract = {In this paper, we propose a method for identification of continuous-time fractional-order systems with unknown states and input delays. In practice, many systems are modeled accurately with fractional differential equations. In particular, many systems are modeled as fractional differential equations with input delay and state delay. Since the geometric and physical meaning of fractional calculus is not clear, it is difficult to model the real system directly to fractional order systems based on mechanical analysis. Thus, the identification of fractional order systems is the main method for constructing fractional order models and is the subject of the main research by many scientists. To solve the identification problem of systems with input delay and state delay, we use the fact that the fractional integral operator matrix by the block pulse functions is an upper triangular Toeplitz matrix. We have presented an efficient method to identify the linear and nonlinear parameters separably by using the commutativity and nilpotent property for multiplication between upper triangular Toeplitz matrices. We also have presented an efficient algorithm to newly approximate the Jacobian of the variable projection functional to solve the least squares problem with nonlinear parameters. Several simulation examples have been used to verify the effectiveness of the proposed method. It is shown that the input delay and the state delay have a significant effect on the output characteristics of the system, especially the state delay has a larger effect than the input delay.},
 year = {2025}
}

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TY  - JOUR
T1  - Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions
AU  - Yu-Gang Hyon
AU  - Hyon-Ju Sin
AU  - Myong-Hyok Sin
AU  - Chol Min Sin
AU  - Hun Kim
Y1  - 2025/12/29
PY  - 2025
N1  - https://doi.org/10.11648/j.engmath.20250902.12
DO  - 10.11648/j.engmath.20250902.12
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 31
EP  - 44
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20250902.12
AB  - In this paper, we propose a method for identification of continuous-time fractional-order systems with unknown states and input delays. In practice, many systems are modeled accurately with fractional differential equations. In particular, many systems are modeled as fractional differential equations with input delay and state delay. Since the geometric and physical meaning of fractional calculus is not clear, it is difficult to model the real system directly to fractional order systems based on mechanical analysis. Thus, the identification of fractional order systems is the main method for constructing fractional order models and is the subject of the main research by many scientists. To solve the identification problem of systems with input delay and state delay, we use the fact that the fractional integral operator matrix by the block pulse functions is an upper triangular Toeplitz matrix. We have presented an efficient method to identify the linear and nonlinear parameters separably by using the commutativity and nilpotent property for multiplication between upper triangular Toeplitz matrices. We also have presented an efficient algorithm to newly approximate the Jacobian of the variable projection functional to solve the least squares problem with nonlinear parameters. Several simulation examples have been used to verify the effectiveness of the proposed method. It is shown that the input delay and the state delay have a significant effect on the output characteristics of the system, especially the state delay has a larger effect than the input delay.
VL  - 9
IS  - 2
ER  -

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Author Information

Yu-Gang Hyon

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

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Hyon-Ju Sin

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

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Myong-Hyok Sin

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

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http://orcid.org/0009-0009-0994-5469
Chol Min Sin

Institute of Mathematics, State Academy of Sciences, Pyongyang, Democratic People's Republic of Korea

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http://orcid.org/0000-0002-6442-7105
Hun Kim

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

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APA Style

Hyon, Y., Sin, H., Sin, M., Sin, C. M., Kim, H. (2025). Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions. Engineering Mathematics, 9(2), 31-44. https://doi.org/10.11648/j.engmath.20250902.12

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ACS Style

Hyon, Y.; Sin, H.; Sin, M.; Sin, C. M.; Kim, H. Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions. Eng. Math. 2025, 9(2), 31-44. doi: 10.11648/j.engmath.20250902.12

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AMA Style

Hyon Y, Sin H, Sin M, Sin CM, Kim H. Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions. Eng Math. 2025;9(2):31-44. doi: 10.11648/j.engmath.20250902.12

Copy | Download

@article{10.11648/j.engmath.20250902.12,
  author = {Yu-Gang Hyon and Hyon-Ju Sin and Myong-Hyok Sin and Chol Min Sin and Hun Kim},
  title = {Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions},
  journal = {Engineering Mathematics},
  volume = {9},
  number = {2},
  pages = {31-44},
  doi = {10.11648/j.engmath.20250902.12},
  url = {https://doi.org/10.11648/j.engmath.20250902.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20250902.12},
  abstract = {In this paper, we propose a method for identification of continuous-time fractional-order systems with unknown states and input delays. In practice, many systems are modeled accurately with fractional differential equations. In particular, many systems are modeled as fractional differential equations with input delay and state delay. Since the geometric and physical meaning of fractional calculus is not clear, it is difficult to model the real system directly to fractional order systems based on mechanical analysis. Thus, the identification of fractional order systems is the main method for constructing fractional order models and is the subject of the main research by many scientists. To solve the identification problem of systems with input delay and state delay, we use the fact that the fractional integral operator matrix by the block pulse functions is an upper triangular Toeplitz matrix. We have presented an efficient method to identify the linear and nonlinear parameters separably by using the commutativity and nilpotent property for multiplication between upper triangular Toeplitz matrices. We also have presented an efficient algorithm to newly approximate the Jacobian of the variable projection functional to solve the least squares problem with nonlinear parameters. Several simulation examples have been used to verify the effectiveness of the proposed method. It is shown that the input delay and the state delay have a significant effect on the output characteristics of the system, especially the state delay has a larger effect than the input delay.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions
AU  - Yu-Gang Hyon
AU  - Hyon-Ju Sin
AU  - Myong-Hyok Sin
AU  - Chol Min Sin
AU  - Hun Kim
Y1  - 2025/12/29
PY  - 2025
N1  - https://doi.org/10.11648/j.engmath.20250902.12
DO  - 10.11648/j.engmath.20250902.12
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 31
EP  - 44
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20250902.12
AB  - In this paper, we propose a method for identification of continuous-time fractional-order systems with unknown states and input delays. In practice, many systems are modeled accurately with fractional differential equations. In particular, many systems are modeled as fractional differential equations with input delay and state delay. Since the geometric and physical meaning of fractional calculus is not clear, it is difficult to model the real system directly to fractional order systems based on mechanical analysis. Thus, the identification of fractional order systems is the main method for constructing fractional order models and is the subject of the main research by many scientists. To solve the identification problem of systems with input delay and state delay, we use the fact that the fractional integral operator matrix by the block pulse functions is an upper triangular Toeplitz matrix. We have presented an efficient method to identify the linear and nonlinear parameters separably by using the commutativity and nilpotent property for multiplication between upper triangular Toeplitz matrices. We also have presented an efficient algorithm to newly approximate the Jacobian of the variable projection functional to solve the least squares problem with nonlinear parameters. Several simulation examples have been used to verify the effectiveness of the proposed method. It is shown that the input delay and the state delay have a significant effect on the output characteristics of the system, especially the state delay has a larger effect than the input delay.
VL  - 9
IS  - 2
ER  -

Copy | Download