Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System

Djamel Boucherma; Mohamed Lotfi Cherrad; Khaled Chettah; Toufik Achour; Mohamed Chaour; Sofiane Boulkroune; Billel Hamadi

doi:doi:10.11648/j.ajmme.20240802.12

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| Peer-Reviewed

Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System

Djamel Boucherma^*

, Mohamed Lotfi Cherrad

, Khaled Chettah

, Toufik Achour

, Mohamed Chaour

, Sofiane Boulkroune

, Billel Hamadi

Published in American Journal of Mechanical and Materials Engineering (Volume 8, Issue 2)

Received: 3 September 2024 Accepted: 24 September 2024 Published: 10 October 2024

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Abstract

The stabilization of the non-linear inverted pendulum system requires a robust control strategy, as this system is inherently unstable and sensitive to disturbances. This research utilizes Lagrangian mechanics, a powerful technique in analytical dynamics, to derive the mathematical representation of the system. By applying the principles of Lagrangian dynamics, we can accurately model the energies involved and derive the equations of motion that govern the pendulum’s behavior. Following this, state-space feedback is employed to determine the Proportional, Integral, and Derivative (PID) values essential for effective control. This control strategy is particularly useful due to its ability to minimize error over time and ensure stability. To further enhance the control process, a comprehensive mathematical model is developed to establish the transfer function that correlates the pendulum's angle with the displacement of the cart. This relationship is crucial for understanding how changes in the cart's position affect the pendulum's stability. To validate the proposed control law, extensive simulations are conducted, allowing for comparative analysis against an Integer Order Controller. These simulations not only highlight the effectiveness of the PID controller but also provide insights into the dynamic behavior of the system under various conditions. The results demonstrate significant improvements in settling time and overshoot, showcasing enhanced performance metrics for the selected objective functions. This research contributes to the broader field of control systems engineering, suggesting that advanced control strategies can effectively manage complex, non-linear systems.

Published in	American Journal of Mechanical and Materials Engineering (Volume 8, Issue 2)
DOI	10.11648/j.ajmme.20240802.12
Page(s)	33-38
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), 2024. Published by Science Publishing Group

Keywords

Inverted Pendulum Dynamics, Cart Position Control, Pendulum Angle Stabilization, Fractional PID Control, Integer PID Control

References

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APA Style

Boucherma, D., Cherrad, M. L., Chettah, K., Achour, T., Chaour, M., et al. (2024). Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System. American Journal of Mechanical and Materials Engineering, 8(2), 33-38. https://doi.org/10.11648/j.ajmme.20240802.12

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ACS Style

Boucherma, D.; Cherrad, M. L.; Chettah, K.; Achour, T.; Chaour, M., et al. Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System. Am. J. Mech. Mater. Eng. 2024, 8(2), 33-38. doi: 10.11648/j.ajmme.20240802.12

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AMA Style

Boucherma D, Cherrad ML, Chettah K, Achour T, Chaour M, et al. Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System. Am J Mech Mater Eng. 2024;8(2):33-38. doi: 10.11648/j.ajmme.20240802.12

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@article{10.11648/j.ajmme.20240802.12,
  author = {Djamel Boucherma and Mohamed Lotfi Cherrad and Khaled Chettah and Toufik Achour and Mohamed Chaour and Sofiane Boulkroune and Billel Hamadi},
  title = {Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System
},
  journal = {American Journal of Mechanical and Materials Engineering},
  volume = {8},
  number = {2},
  pages = {33-38},
  doi = {10.11648/j.ajmme.20240802.12},
  url = {https://doi.org/10.11648/j.ajmme.20240802.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmme.20240802.12},
  abstract = {The stabilization of the non-linear inverted pendulum system requires a robust control strategy, as this system is inherently unstable and sensitive to disturbances. This research utilizes Lagrangian mechanics, a powerful technique in analytical dynamics, to derive the mathematical representation of the system. By applying the principles of Lagrangian dynamics, we can accurately model the energies involved and derive the equations of motion that govern the pendulum’s behavior. Following this, state-space feedback is employed to determine the Proportional, Integral, and Derivative (PID) values essential for effective control. This control strategy is particularly useful due to its ability to minimize error over time and ensure stability. To further enhance the control process, a comprehensive mathematical model is developed to establish the transfer function that correlates the pendulum's angle with the displacement of the cart. This relationship is crucial for understanding how changes in the cart's position affect the pendulum's stability. To validate the proposed control law, extensive simulations are conducted, allowing for comparative analysis against an Integer Order Controller. These simulations not only highlight the effectiveness of the PID controller but also provide insights into the dynamic behavior of the system under various conditions. The results demonstrate significant improvements in settling time and overshoot, showcasing enhanced performance metrics for the selected objective functions. This research contributes to the broader field of control systems engineering, suggesting that advanced control strategies can effectively manage complex, non-linear systems.
},
 year = {2024}
}

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TY  - JOUR
T1  - Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System

AU  - Djamel Boucherma
AU  - Mohamed Lotfi Cherrad
AU  - Khaled Chettah
AU  - Toufik Achour
AU  - Mohamed Chaour
AU  - Sofiane Boulkroune
AU  - Billel Hamadi
Y1  - 2024/10/10
PY  - 2024
N1  - https://doi.org/10.11648/j.ajmme.20240802.12
DO  - 10.11648/j.ajmme.20240802.12
T2  - American Journal of Mechanical and Materials Engineering
JF  - American Journal of Mechanical and Materials Engineering
JO  - American Journal of Mechanical and Materials Engineering
SP  - 33
EP  - 38
PB  - Science Publishing Group
SN  - 2639-9652
UR  - https://doi.org/10.11648/j.ajmme.20240802.12
AB  - The stabilization of the non-linear inverted pendulum system requires a robust control strategy, as this system is inherently unstable and sensitive to disturbances. This research utilizes Lagrangian mechanics, a powerful technique in analytical dynamics, to derive the mathematical representation of the system. By applying the principles of Lagrangian dynamics, we can accurately model the energies involved and derive the equations of motion that govern the pendulum’s behavior. Following this, state-space feedback is employed to determine the Proportional, Integral, and Derivative (PID) values essential for effective control. This control strategy is particularly useful due to its ability to minimize error over time and ensure stability. To further enhance the control process, a comprehensive mathematical model is developed to establish the transfer function that correlates the pendulum's angle with the displacement of the cart. This relationship is crucial for understanding how changes in the cart's position affect the pendulum's stability. To validate the proposed control law, extensive simulations are conducted, allowing for comparative analysis against an Integer Order Controller. These simulations not only highlight the effectiveness of the PID controller but also provide insights into the dynamic behavior of the system under various conditions. The results demonstrate significant improvements in settling time and overshoot, showcasing enhanced performance metrics for the selected objective functions. This research contributes to the broader field of control systems engineering, suggesting that advanced control strategies can effectively manage complex, non-linear systems.

VL  - 8
IS  - 2
ER  -

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Author Information

Djamel Boucherma

Center of Research in Mechanics, Constantine, Algeria; Research Center in Industrial Technologies, Cheraga Algiers, Algeria

Contact Email

http://orcid.org/0000-0003-4373-6846
Mohamed Lotfi Cherrad

Center of Research in Mechanics, Constantine, Algeria; Research Center in Industrial Technologies, Cheraga Algiers, Algeria

Contact Email

http://orcid.org/0000-0001-6012-2243
Khaled Chettah

Center of Research in Mechanics, Constantine, Algeria; Research Center in Industrial Technologies, Cheraga Algiers, Algeria

Contact Email

http://orcid.org/0000-0002-0312-600X
Toufik Achour

Center of Research in Mechanics, Constantine, Algeria; Research Center in Industrial Technologies, Cheraga Algiers, Algeria

Contact Email

http://orcid.org/0000-0002-7822-9118
Mohamed Chaour

Center of Research in Mechanics, Constantine, Algeria; Research Center in Industrial Technologies, Cheraga Algiers, Algeria

Contact Email

http://orcid.org/0009-0000-6365-152X
Sofiane Boulkroune

Center of Research in Mechanics, Constantine, Algeria; Research Center in Industrial Technologies, Cheraga Algiers, Algeria

Contact Email

http://orcid.org/0009-0000-3446-1807
Billel Hamadi

Center of Research in Mechanics, Constantine, Algeria; Research Center in Industrial Technologies, Cheraga Algiers, Algeria

Contact Email

http://orcid.org/0000-0001-7437-0475

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Plain Text BibTeX RIS

APA Style

Boucherma, D., Cherrad, M. L., Chettah, K., Achour, T., Chaour, M., et al. (2024). Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System. American Journal of Mechanical and Materials Engineering, 8(2), 33-38. https://doi.org/10.11648/j.ajmme.20240802.12

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ACS Style

Boucherma, D.; Cherrad, M. L.; Chettah, K.; Achour, T.; Chaour, M., et al. Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System. Am. J. Mech. Mater. Eng. 2024, 8(2), 33-38. doi: 10.11648/j.ajmme.20240802.12

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AMA Style

Boucherma D, Cherrad ML, Chettah K, Achour T, Chaour M, et al. Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System. Am J Mech Mater Eng. 2024;8(2):33-38. doi: 10.11648/j.ajmme.20240802.12

Copy | Download

@article{10.11648/j.ajmme.20240802.12,
  author = {Djamel Boucherma and Mohamed Lotfi Cherrad and Khaled Chettah and Toufik Achour and Mohamed Chaour and Sofiane Boulkroune and Billel Hamadi},
  title = {Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System
},
  journal = {American Journal of Mechanical and Materials Engineering},
  volume = {8},
  number = {2},
  pages = {33-38},
  doi = {10.11648/j.ajmme.20240802.12},
  url = {https://doi.org/10.11648/j.ajmme.20240802.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmme.20240802.12},
  abstract = {The stabilization of the non-linear inverted pendulum system requires a robust control strategy, as this system is inherently unstable and sensitive to disturbances. This research utilizes Lagrangian mechanics, a powerful technique in analytical dynamics, to derive the mathematical representation of the system. By applying the principles of Lagrangian dynamics, we can accurately model the energies involved and derive the equations of motion that govern the pendulum’s behavior. Following this, state-space feedback is employed to determine the Proportional, Integral, and Derivative (PID) values essential for effective control. This control strategy is particularly useful due to its ability to minimize error over time and ensure stability. To further enhance the control process, a comprehensive mathematical model is developed to establish the transfer function that correlates the pendulum's angle with the displacement of the cart. This relationship is crucial for understanding how changes in the cart's position affect the pendulum's stability. To validate the proposed control law, extensive simulations are conducted, allowing for comparative analysis against an Integer Order Controller. These simulations not only highlight the effectiveness of the PID controller but also provide insights into the dynamic behavior of the system under various conditions. The results demonstrate significant improvements in settling time and overshoot, showcasing enhanced performance metrics for the selected objective functions. This research contributes to the broader field of control systems engineering, suggesting that advanced control strategies can effectively manage complex, non-linear systems.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System

AU  - Djamel Boucherma
AU  - Mohamed Lotfi Cherrad
AU  - Khaled Chettah
AU  - Toufik Achour
AU  - Mohamed Chaour
AU  - Sofiane Boulkroune
AU  - Billel Hamadi
Y1  - 2024/10/10
PY  - 2024
N1  - https://doi.org/10.11648/j.ajmme.20240802.12
DO  - 10.11648/j.ajmme.20240802.12
T2  - American Journal of Mechanical and Materials Engineering
JF  - American Journal of Mechanical and Materials Engineering
JO  - American Journal of Mechanical and Materials Engineering
SP  - 33
EP  - 38
PB  - Science Publishing Group
SN  - 2639-9652
UR  - https://doi.org/10.11648/j.ajmme.20240802.12
AB  - The stabilization of the non-linear inverted pendulum system requires a robust control strategy, as this system is inherently unstable and sensitive to disturbances. This research utilizes Lagrangian mechanics, a powerful technique in analytical dynamics, to derive the mathematical representation of the system. By applying the principles of Lagrangian dynamics, we can accurately model the energies involved and derive the equations of motion that govern the pendulum’s behavior. Following this, state-space feedback is employed to determine the Proportional, Integral, and Derivative (PID) values essential for effective control. This control strategy is particularly useful due to its ability to minimize error over time and ensure stability. To further enhance the control process, a comprehensive mathematical model is developed to establish the transfer function that correlates the pendulum's angle with the displacement of the cart. This relationship is crucial for understanding how changes in the cart's position affect the pendulum's stability. To validate the proposed control law, extensive simulations are conducted, allowing for comparative analysis against an Integer Order Controller. These simulations not only highlight the effectiveness of the PID controller but also provide insights into the dynamic behavior of the system under various conditions. The results demonstrate significant improvements in settling time and overshoot, showcasing enhanced performance metrics for the selected objective functions. This research contributes to the broader field of control systems engineering, suggesting that advanced control strategies can effectively manage complex, non-linear systems.

VL  - 8
IS  - 2
ER  -

Copy | Download