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Optimal LQR Controller Methods for Double Inverted Pendulum System on a Cart

Yıl 2023, Cilt: 14 Sayı: 2, 247 - 255, 20.06.2023
https://doi.org/10.24012/dumf.1253331

Öz

Most of the systems in our lives are inherently nonlinear and unstable. In control problems in the field of engineering, the aim is to define the control laws that maximize the operating efficiency of these systems under diverse security coefficients, and constraints and minimize error rates. This study aimed to model and optimally control a Double-Inverted Pendulum System on a Cart (DIPSC). A DIPSC was modeled using the Lagrange-Euler method, and classical and optimal Linear Quadratic Regulator (LQR) control methods were designed for the control of the system. The purpose of the designed controllers is to keep the arms of the double inverted pendulum on the moving cart vertically in balance and to bring the cart to the determined balance position. The critically important Q and R parameters of the LQR control technique that is one of the optimal control techniques were obtained using the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Grey Wolf Optimization (GWO) algorithms. The DIPSC system was checked using classical LQR and optimal LQR methods. All obtained results are given graphically. The proposed methods are presented and analyzed in tabular form using Settling time and Mean-Square-Error (MSE) performance criteria.

Kaynakça

  • [1] M. McGrath, D. Howard and R. Baker, "The strengths and weaknesses of inverted pendulum models of human walking." Gait & posture vol. 41, no. 2, pp. 389-394, 2015.
  • [2] Y. Fang, W. E. Dixon, D. M. Dawson and E. Zergeroglu, "Nonlinear coupling control laws for an underactuated overhead crane system." IEEE/ASME transactions on mechatronics vol. 8, no. 3, pp. 418-423, 2003.
  • [3] X. Huang and C. Chen "Research and Implementation of Self-Balancing Obstacle Vehicle Based On the Principle of Flywheel Inverted Pendulum." 2021 3rd International Conference on Artificial Intelligence and Advanced Manufacture. 2021, pp. 1070-1073.
  • [4] C. He, K. Huang, X. Chen, Y. Zhang and H. Zhao, "Transportation control of cooperative double-wheel inverted pendulum robots adopting Udwadia -control approach." Nonlinear Dynamics vol. 91, no. 4, pp. 2789-2802, 2018.
  • [5] H. Li, M. Zhihong and W. Jiayin, "Variable universe adaptive fuzzy control on the quadruple inverted pendulum." Science in China Series E: Technological Sciences vol. 45, no. 2, pp. 213-224, 2002.
  • [6] K. Furuta, T. Okutani and H. Sone, "Computer control of a double inverted pendulum." Computers & Electrical Engineering vol. 5. no.1, pp. 67-84, 1978.
  • [7] F. Cheng, G. Zhong, Y. Li and Z. Xu, “Fuzzy control of a double-inverted pendulum." Fuzzy sets and systems vol. 79, no. 3, pp. 315-321, 1996.
  • [8] W. Zhong and H. Rock, “Energy and passivity based control of the double inverted pendulum on a cart." Proceedings of the 2001 IEEE International Conference on Control Applications (CCA'01) (Cat. No. 01CH37204). IEEE, 2001, pp. 896-901.
  • [9] A. Bogdanov, "Optimal control of a double inverted pendulum on a cart." Oregon Health and Science University, Tech. Rep. CSE-04-006, OGI School of Science and Engineering, Beaverton, OR, 2004.
  • [10] D. Cheng-jun, D. Ping, Z. Ming-lu and Z. Yan-fang, "Double inverted pendulum system control strategy based on fuzzy genetic algorithm." 2009 IEEE International Conference on Automation and Logistics. IEEE, 2009, pp. 1318-1323.
  • [11] X. Xiong and Z. Wan, "The simulation of double inverted pendulum control based on particle swarm optimization LQR algorithm." 2010 IEEE International Conference on Software Engineering and Service Sciences. IEEE, 2010, pp. 253-256.
  • [12] C. W. Tao, J. Taur, J. H. Chang and S. F. Su, "Adaptive fuzzy switched swing-up and sliding control for the double-pendulum-and-cart system." IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) vol. 40, no.1, pp. 241-252, 2009.
  • [13] M. Adeli, H. Zarabadipour and M. A. Shoorehdeli, "Anti-swing control of a double-pendulum-type overhead crane using parallel distributed fuzzy LQR controller." The 2nd International Conference on Control, Instrumentation and Automation. IEEE, 2011, pp. 401-406.
  • [14] I. Hassanzadeh, A., Nejadfard and M. Zadi, “A multivariable adaptive control approach for stabilization of a cart-type double inverted pendulum." Mathematical Problems in Engineering 2011 (2011).
  • [15] N. Singh and S. K. Yadav, "Comparison of LQR and PD controller for stabilizing Double Inverted Pendulum System." International Journal of Engineering Research and Development vol. 1, no. 12, pp. 69-74, 2012.
  • [16] J. L. Zhang and W. Zhang, "LQR self-adjusting based control for the planar double inverted pendulum." Physics Procedia vol. 24, pp. 1669-1676, 2012.
  • [17] N. S. Bhangal, "Design and performance of LQR and LQR based fuzzy controller for double inverted pendulum system." Journal of Image and Graphics vol. 1. no. 3, pp. 143-146, 2013.
  • [18] L. Wang, H. Ni, W. Zhou, P. M. Pardalos, J. Fang and M. Fei, “MBPOA-based LQR controller and its application to the double-parallel inverted pendulum system." Engineering Applications of Artificial Intelligence vol. 36, pp. 262-268, 2014.
  • [19] Z. Sun, N. Wang, and Y. Bi, "Type-1/type-2 fuzzy logic systems optimization with RNA genetic algorithm for double inverted pendulum. " Applied Mathematical Modelling, vol. 39, no. (1), pp. 70-85, (2015).
  • [20] G. A. Sultan and Z. K. Farej, "Design and Performance Analysis of LQR Controller for Stabilizing Double Inverted Pendulum System." Circ. Comput. Sci. vol. 2, no. 9, pp.1-5, 2017.
  • [21] N. Bandari, A. Hooshiar, M. Razdan, J, Dargahi and C. Su, "Stabilization of double inverted pendulum on a cart: LQR approach. Su, International Journal of Mechanical and Production Engineering, vol. 5, no. 2, 2017.
  • [22] R. Banerjee, N. Dey, U. Mondal and B. Hazra, "Stabilization of double link inverted pendulum using LQR." 2018 International Conference on Current Trends towards Converging Technologies (ICCTCT). IEEE, 2018, pp. 1-6.
  • [23] Ü. Önen, A. Çakan and I. Ilhan, “Performance comparison of optimization algorithms in LQR controller design for a nonlinear system." Turkish Journal of Electrical Engineering and Computer Sciences vol. 27, no. 3, pp. 1938-1953, 2019.
  • [24] B. He, S. Wang and Y. Liu, "Underactuated robotics: a review." International Journal of Advanced Robotic Systems vol.16, no.4, 1729881419862164, 2019.
  • [25] T. M. Tijani and I. A. Jimoh, "Optimal control of the double inverted pendulum on a cart: A comparative study of explicit MPC and LQR." Applications of Modelling and Simulation vol.5, pp. 74-87. 2021.
  • [26] G. S. Maraslidis, T. L. Kottas, M. G. Tsipouras and G. F. Fragulis, "Design of a Fuzzy Logic Controller for the Double Pendulum Inverted on a Cart." Information vol. 13, no. 8, pp. 379, 2022.
  • [27] G. P. Gil, W. Yu and H. Sossa, "Reinforcement learning compensation based PD control for a double inverted pendulum" IEEE Latin America Transactions, vol. 17, no. 2, pp. 323-329, 2019.
  • [28] N. Bu and X. Wang, "Swing-up design of double inverted pendulum by using passive control method based on operator theory" International Journal of Advanced Mechatronic Systems, vol. 10, no. 1, pp. 1-7, 2023.
  • [29] J. He, L. Cui, J. Sun, P. Huang and Y. Huang, "Chaotic dynamics analysis of double inverted pendulum with large swing angle based on Hamiltonian function" Nonlinear Dynamics, vol. 108, no. 4, pp. 4373-4384,2022.
  • [30] L. Fan, A. Zhang, G. Pan, Y. Du, and J. Qiu, "Swing‐up and fixed‐time stabilization control of underactuated cart‐double pendulum system" IET Control Theory & Applications, vol. 17, no. 6, pp. 662-671, 2023.
  • [31] B. D. Anderson and J. B. Moore, Optimal control: linear quadratic methods. Courier Corporation, 2007.
  • [32] T. Abut, "Modeling and optimal control of a DC motor." Int. J. Eng. Trends Technol vol. 32, no. 3, pp.146-150, 2016.
  • [33] L. M. Argentim, W. C. Rezende, P. E. Santos and R. A. Aguiar, “PID, LQR and LQR-PID on a quadcopter platform." 2013 International Conference on Informatics, Electronics and Vision (ICIEV). IEEE, 2013, pp. 1-6.
  • [34] S. N. Sivanandam and S. N. Deepa, Genetic algorithms. Springer Berlin Heidelberg, 2008, pp. 15-37.
  • [35] T. Abut, "Dynamic Model and Optimal Control of A Snake Robot: TAROBOT–1." International Journal of Scientific & Technology Research vol. 4, no.11, pp. 4, 2015.
  • [36] J. Kennedy and R. Eberhart, "Particle swarm optimization." Proceedings of ICNN'95-international conference on neural networks. vol. 4. IEEE, 1995, pp. 1942-1948.
  • [37] Y. Shi, "Particle swarm optimization: developments, applications and resources." Proceedings of the 2001 congress on evolutionary computation (IEEE Cat. No. 01TH8546). Vol. 1. IEEE, 2001, vol. 1, pp. 81-86.
  • [38] S. Mirjalili, S. M. Mirjalili and A. Lewis, “Grey wolf optimizer." Advances in engineering software vol. 69 pp. 46-61, 2014.
  • [39] T. Abut and S. Soyguder, "Optimal adaptive computed torque control for haptic-teleoperation system with uncertain dynamics." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering vol. 236, no. 4, pp. 800-817, 2022.
  • [40] H. Faris, I. Aljarah, M. A. Al-Betar and S. Mirjalili, "Grey wolf optimizer: a review of recent variants and applications." Neural computing and applications vol. 30, 413-435, 2018.

Optimal LQR Controller Methods for Double Inverted Pendulum System on a Cart

Yıl 2023, Cilt: 14 Sayı: 2, 247 - 255, 20.06.2023
https://doi.org/10.24012/dumf.1253331

Öz

Most of the systems in our lives are inherently nonlinear and unstable. In control problems in the field of engineering, the aim is to define the control laws that maximize the operating efficiency of these systems under diverse security coefficients, and constraints and minimize error rates. This study aimed to model and optimally control a Double-Inverted Pendulum System on a Cart (DIPSC). A DIPSC was modeled using the Lagrange-Euler method, and classical and optimal Linear Quadratic Regulator (LQR) control methods were designed for the control of the system. The purpose of the designed controllers is to keep the arms of the double inverted pendulum on the moving cart vertically in balance and to bring the cart to the determined balance position. The critically important Q and R parameters of the LQR control technique that is one of the optimal control techniques were obtained using the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Grey Wolf Optimization (GWO) algorithms. The DIPSC system was checked using classical LQR and optimal LQR methods. All obtained results are given graphically. The proposed methods are presented and analyzed in tabular form using Settling time and Mean-Square-Error (MSE) performance criteria.

Kaynakça

  • [1] M. McGrath, D. Howard and R. Baker, "The strengths and weaknesses of inverted pendulum models of human walking." Gait & posture vol. 41, no. 2, pp. 389-394, 2015.
  • [2] Y. Fang, W. E. Dixon, D. M. Dawson and E. Zergeroglu, "Nonlinear coupling control laws for an underactuated overhead crane system." IEEE/ASME transactions on mechatronics vol. 8, no. 3, pp. 418-423, 2003.
  • [3] X. Huang and C. Chen "Research and Implementation of Self-Balancing Obstacle Vehicle Based On the Principle of Flywheel Inverted Pendulum." 2021 3rd International Conference on Artificial Intelligence and Advanced Manufacture. 2021, pp. 1070-1073.
  • [4] C. He, K. Huang, X. Chen, Y. Zhang and H. Zhao, "Transportation control of cooperative double-wheel inverted pendulum robots adopting Udwadia -control approach." Nonlinear Dynamics vol. 91, no. 4, pp. 2789-2802, 2018.
  • [5] H. Li, M. Zhihong and W. Jiayin, "Variable universe adaptive fuzzy control on the quadruple inverted pendulum." Science in China Series E: Technological Sciences vol. 45, no. 2, pp. 213-224, 2002.
  • [6] K. Furuta, T. Okutani and H. Sone, "Computer control of a double inverted pendulum." Computers & Electrical Engineering vol. 5. no.1, pp. 67-84, 1978.
  • [7] F. Cheng, G. Zhong, Y. Li and Z. Xu, “Fuzzy control of a double-inverted pendulum." Fuzzy sets and systems vol. 79, no. 3, pp. 315-321, 1996.
  • [8] W. Zhong and H. Rock, “Energy and passivity based control of the double inverted pendulum on a cart." Proceedings of the 2001 IEEE International Conference on Control Applications (CCA'01) (Cat. No. 01CH37204). IEEE, 2001, pp. 896-901.
  • [9] A. Bogdanov, "Optimal control of a double inverted pendulum on a cart." Oregon Health and Science University, Tech. Rep. CSE-04-006, OGI School of Science and Engineering, Beaverton, OR, 2004.
  • [10] D. Cheng-jun, D. Ping, Z. Ming-lu and Z. Yan-fang, "Double inverted pendulum system control strategy based on fuzzy genetic algorithm." 2009 IEEE International Conference on Automation and Logistics. IEEE, 2009, pp. 1318-1323.
  • [11] X. Xiong and Z. Wan, "The simulation of double inverted pendulum control based on particle swarm optimization LQR algorithm." 2010 IEEE International Conference on Software Engineering and Service Sciences. IEEE, 2010, pp. 253-256.
  • [12] C. W. Tao, J. Taur, J. H. Chang and S. F. Su, "Adaptive fuzzy switched swing-up and sliding control for the double-pendulum-and-cart system." IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) vol. 40, no.1, pp. 241-252, 2009.
  • [13] M. Adeli, H. Zarabadipour and M. A. Shoorehdeli, "Anti-swing control of a double-pendulum-type overhead crane using parallel distributed fuzzy LQR controller." The 2nd International Conference on Control, Instrumentation and Automation. IEEE, 2011, pp. 401-406.
  • [14] I. Hassanzadeh, A., Nejadfard and M. Zadi, “A multivariable adaptive control approach for stabilization of a cart-type double inverted pendulum." Mathematical Problems in Engineering 2011 (2011).
  • [15] N. Singh and S. K. Yadav, "Comparison of LQR and PD controller for stabilizing Double Inverted Pendulum System." International Journal of Engineering Research and Development vol. 1, no. 12, pp. 69-74, 2012.
  • [16] J. L. Zhang and W. Zhang, "LQR self-adjusting based control for the planar double inverted pendulum." Physics Procedia vol. 24, pp. 1669-1676, 2012.
  • [17] N. S. Bhangal, "Design and performance of LQR and LQR based fuzzy controller for double inverted pendulum system." Journal of Image and Graphics vol. 1. no. 3, pp. 143-146, 2013.
  • [18] L. Wang, H. Ni, W. Zhou, P. M. Pardalos, J. Fang and M. Fei, “MBPOA-based LQR controller and its application to the double-parallel inverted pendulum system." Engineering Applications of Artificial Intelligence vol. 36, pp. 262-268, 2014.
  • [19] Z. Sun, N. Wang, and Y. Bi, "Type-1/type-2 fuzzy logic systems optimization with RNA genetic algorithm for double inverted pendulum. " Applied Mathematical Modelling, vol. 39, no. (1), pp. 70-85, (2015).
  • [20] G. A. Sultan and Z. K. Farej, "Design and Performance Analysis of LQR Controller for Stabilizing Double Inverted Pendulum System." Circ. Comput. Sci. vol. 2, no. 9, pp.1-5, 2017.
  • [21] N. Bandari, A. Hooshiar, M. Razdan, J, Dargahi and C. Su, "Stabilization of double inverted pendulum on a cart: LQR approach. Su, International Journal of Mechanical and Production Engineering, vol. 5, no. 2, 2017.
  • [22] R. Banerjee, N. Dey, U. Mondal and B. Hazra, "Stabilization of double link inverted pendulum using LQR." 2018 International Conference on Current Trends towards Converging Technologies (ICCTCT). IEEE, 2018, pp. 1-6.
  • [23] Ü. Önen, A. Çakan and I. Ilhan, “Performance comparison of optimization algorithms in LQR controller design for a nonlinear system." Turkish Journal of Electrical Engineering and Computer Sciences vol. 27, no. 3, pp. 1938-1953, 2019.
  • [24] B. He, S. Wang and Y. Liu, "Underactuated robotics: a review." International Journal of Advanced Robotic Systems vol.16, no.4, 1729881419862164, 2019.
  • [25] T. M. Tijani and I. A. Jimoh, "Optimal control of the double inverted pendulum on a cart: A comparative study of explicit MPC and LQR." Applications of Modelling and Simulation vol.5, pp. 74-87. 2021.
  • [26] G. S. Maraslidis, T. L. Kottas, M. G. Tsipouras and G. F. Fragulis, "Design of a Fuzzy Logic Controller for the Double Pendulum Inverted on a Cart." Information vol. 13, no. 8, pp. 379, 2022.
  • [27] G. P. Gil, W. Yu and H. Sossa, "Reinforcement learning compensation based PD control for a double inverted pendulum" IEEE Latin America Transactions, vol. 17, no. 2, pp. 323-329, 2019.
  • [28] N. Bu and X. Wang, "Swing-up design of double inverted pendulum by using passive control method based on operator theory" International Journal of Advanced Mechatronic Systems, vol. 10, no. 1, pp. 1-7, 2023.
  • [29] J. He, L. Cui, J. Sun, P. Huang and Y. Huang, "Chaotic dynamics analysis of double inverted pendulum with large swing angle based on Hamiltonian function" Nonlinear Dynamics, vol. 108, no. 4, pp. 4373-4384,2022.
  • [30] L. Fan, A. Zhang, G. Pan, Y. Du, and J. Qiu, "Swing‐up and fixed‐time stabilization control of underactuated cart‐double pendulum system" IET Control Theory & Applications, vol. 17, no. 6, pp. 662-671, 2023.
  • [31] B. D. Anderson and J. B. Moore, Optimal control: linear quadratic methods. Courier Corporation, 2007.
  • [32] T. Abut, "Modeling and optimal control of a DC motor." Int. J. Eng. Trends Technol vol. 32, no. 3, pp.146-150, 2016.
  • [33] L. M. Argentim, W. C. Rezende, P. E. Santos and R. A. Aguiar, “PID, LQR and LQR-PID on a quadcopter platform." 2013 International Conference on Informatics, Electronics and Vision (ICIEV). IEEE, 2013, pp. 1-6.
  • [34] S. N. Sivanandam and S. N. Deepa, Genetic algorithms. Springer Berlin Heidelberg, 2008, pp. 15-37.
  • [35] T. Abut, "Dynamic Model and Optimal Control of A Snake Robot: TAROBOT–1." International Journal of Scientific & Technology Research vol. 4, no.11, pp. 4, 2015.
  • [36] J. Kennedy and R. Eberhart, "Particle swarm optimization." Proceedings of ICNN'95-international conference on neural networks. vol. 4. IEEE, 1995, pp. 1942-1948.
  • [37] Y. Shi, "Particle swarm optimization: developments, applications and resources." Proceedings of the 2001 congress on evolutionary computation (IEEE Cat. No. 01TH8546). Vol. 1. IEEE, 2001, vol. 1, pp. 81-86.
  • [38] S. Mirjalili, S. M. Mirjalili and A. Lewis, “Grey wolf optimizer." Advances in engineering software vol. 69 pp. 46-61, 2014.
  • [39] T. Abut and S. Soyguder, "Optimal adaptive computed torque control for haptic-teleoperation system with uncertain dynamics." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering vol. 236, no. 4, pp. 800-817, 2022.
  • [40] H. Faris, I. Aljarah, M. A. Al-Betar and S. Mirjalili, "Grey wolf optimizer: a review of recent variants and applications." Neural computing and applications vol. 30, 413-435, 2018.
Toplam 40 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Makaleler
Yazarlar

Tayfun Abut 0000-0003-4646-3345

Erken Görünüm Tarihi 19 Haziran 2023
Yayımlanma Tarihi 20 Haziran 2023
Gönderilme Tarihi 20 Şubat 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 14 Sayı: 2

Kaynak Göster

IEEE T. Abut, “Optimal LQR Controller Methods for Double Inverted Pendulum System on a Cart”, DÜMF MD, c. 14, sy. 2, ss. 247–255, 2023, doi: 10.24012/dumf.1253331.
DUJE tarafından yayınlanan tüm makaleler, Creative Commons Atıf 4.0 Uluslararası Lisansı ile lisanslanmıştır. Bu, orijinal eser ve kaynağın uygun şekilde belirtilmesi koşuluyla, herkesin eseri kopyalamasına, yeniden dağıtmasına, yeniden düzenlemesine, iletmesine ve uyarlamasına izin verir. 24456