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A holistic optimization approach for inverted cart-pendulum control tuning

  • Maude J. BlondinEmail author
  • Panos M. Pardalos
Methodologies and Application
  • 25 Downloads

Abstract

The inverted cart-pendulum (ICP) is a nonlinear underactuated system, which dynamics are representative of many applications. Therefore, the development of ICP control laws is important since these laws are suitable to other systems. Indeed, many nonlinear control strategies have emerged from the control of the ICP. For these reasons, the ICP remains a canonical and fundamental benchmark problem in control theory and robotics that is of interest to the scientific community. Till now, the trial-and-error method is still widely applied for ICP controller tuning as well as the sequential tuning referring to tune the swing-up controller and thereafter, the stabilization controller. Therefore, the aim of this paper is to automate and facilitate the ICP control in one step. Thus, this paper proposes to holistically optimize ICP controllers. The holistic optimization is performed by a simplified Ant Colony Optimization method with a constrained Nelder–Mead algorithm (ACO-NM). Holistic optimization refers to a simultaneous tuning of the swing-up, stabilization and switching mode parameters. A new cost function is designed to minimize swing-up time, achieve high stabilization performance and consider system constraints. The holistic approach optimizes four controller structures, which include controllers that have never been tuned by a specific method besides by the trial-and-error method. Simulation results on a ICP nonlinear model show that ACO-NM in the holistic approach is effective compared to other algorithms. In addition, contrary to the majority of work on the subject, all the optimized controllers are validated experimentally. The simulation and experimental results obtained confirm that the holistic approach is an efficient optimization tool and specifically responds to the need of optimization technique for the potential-well controller structure and for the Q [diagonal of the matrix and the full matrix] in the linear–quadratic regulator (LQR) technique. Moreover, ICP experimental response analysis demonstrates that using the full Q provides greater experimental stabilization performance than using its diagonal terms in the LQR technique.

Keywords

Inverted cart-pendulum system Nonlinear control Optimization Holistic approach Swing-up Stabilization 

Notes

Acknowledgements

This work was supported by the Vanier Canada Graduate Scholarship, the Michael Smith Foreign Study Supplements Program from the Natural Sciences. It was also supported by the Bourse Mobilité Étudiante from Ministère de l’Éducation du Québec and the CEMF Claudette MacKay-Lassonde Graduate Engineering Ambassador Award. This work was partially supported by Postdoctoral research scholarship from Fonds de recherche nature et Technologies du Québec and by the Paul and Heidi Brown preeminent professorship of ISE, University of Florida.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Abualigah LMQ (2019) Feature selection and enhanced krill herd algorithm for text document clustering. Studies in computational intelligence, Springer, BerlinGoogle Scholar
  2. Abualigah LMQ, Hanandeh ES (2015) Applying genetic algorithms to information retrieval using vector space model. Int J Comput Sci Eng Appl 5(1):19Google Scholar
  3. Abualigah LM, Khader AT (2017a) Unsupervised text feature selection technique based on hybrid particle swarm optimization algorithm with genetic operators for the text clustering. J Supercomput 73(11):4773–4795Google Scholar
  4. Abualigah LM, Khader AT, Hanandeh ES (2017b) A new feature selection method to improve the document clustering using particle swarm optimization algorithm. J Comput Sci 25:456–466Google Scholar
  5. Abualigah LM, Khader AT, Hanandeh ES, Gandomi AH (2017c) A novel hybridization strategy for krill herd algorithm applied to clustering techniques. Appl Soft Comput 60:423–435Google Scholar
  6. Abualigah LM, Khader AT, Hanandeh ES (2018a) Hybrid clustering analysis using improved krill herd algorithm. Appl Intell 48:4047–4071Google Scholar
  7. Abualigah LM, Khader AT, Hanandeh ES (2018b) A combination of objective functions and hybrid krill herd algorithm for text document clustering analysis. Eng Appl Artif Intell 73:111–125Google Scholar
  8. Adhikary N, Mahanta C (2013) Integral backstepping sliding mode control for underactuated systems: swing-up and stabilization of the Cart-Pendulum System. ISA Trans 52:870–880Google Scholar
  9. Aguilar Ibanez C, Martinez Garcia JC, Soria Lopez A, Rubio JdJ, Suarez Castanon MS (2018) Stabilization of the inverted cart-pendulum system with linear friction. IEEE Latin Am Trans 16(6):1650–1657Google Scholar
  10. Agustinah T, Jazidie A, Nuh M (2011) Hybrid fuzzy control for swinging up and stabilizing of the pendulum-cart system. IEEE Int Conf Comput Sci Autom Eng 4:109–113Google Scholar
  11. Almobaied M, Eksin I, Guzelkaya M (2016) Design of LQR controller with big bang-big crunch optimization algorithm based on time domain criteria. In: Mediterranean conference on control and automation, pp 1192–1197Google Scholar
  12. Anderson CW (1989) Learning to control an inverted pendulum using neural networks. IEEE Control Syst Mag 9(3):31–37Google Scholar
  13. Apkarian J, Lacheray H, Martin P (2012a) Linear pendulum gantry experiment for MATLAB/simulink users. Instructor workbook. Quanser Inc., Markham, pp 1–40Google Scholar
  14. Apkarian J, Lacheray H, Martin P (2012b) Linear inverted pendulum gantry experiment for MATLAB/simulink users. Instructor workbook. Quanser Inc., Markham, pp 1–33Google Scholar
  15. Aström KJ, Furuta K (2000) Swinging up a pendulum by energy control. Automatica 36(2):287–295MathSciNetzbMATHGoogle Scholar
  16. Ata B, Coban R (2015) Artificial bee colony algorithm based linear quadratic optimal controller design for a nonlinear inverted pendulum. Int J Intell Syst Appl Eng 3(1):1–6Google Scholar
  17. Barr RS, Golden BL, Kelly JP, Resende MG, Stewart WR (1995) Designing and reporting on computational experiments with heuristic methods. J Heuristics 1(1):9–32zbMATHGoogle Scholar
  18. Bettayeb M, Boussalem C, Mansouri R, Al-Saggaf UM (2014) Stabilization of an inverted pendulum-cart system by fractional PI-state feedback. ISA Trans 53(2):508–516Google Scholar
  19. Blasco Ferragud FX (1999) Control predictivo basado en modelos mediante técnica de optimización heurística. PhD Tesis (en espagnol) editorial UPV. ISBN: 84-699-5429-6Google Scholar
  20. Blondin MJ, Sicard P, Pardalos PM (2018a) The ACO-NM algorithm for controller tuning for an inverted cart-pendulum. In: International symposium on power electronics, electrical drives, automation and motion, pp 1370–1375Google Scholar
  21. Blondin MJ, Saez JS, Sicard P, Herrero JM (2018b) New optimal controller tuning method for an AVR system using a simplified Ant Colony Optimization with a new constrained Nelder–Mead algorithm. Appl Soft Comput 62:216–229Google Scholar
  22. Boubaker O (2012) The inverted pendulum: a fundamental benchmark in control theory and robotics. In: IEEE international conference on education and e-learning innovations, pp 1–6Google Scholar
  23. Boubaker O (2013) The inverted pendulum benchmark in nonlinear control theory: a survey. Int J Adv Robot Syst 10:233Google Scholar
  24. Chakraborty A, Dey J (2015) Global stabilization of cart-pendulum system with sliding mode controller: experimental results. In: IEEE international conference on industrial technology, pp 277–282Google Scholar
  25. Chatterjee D, Patra A, Joglekar KH (2002) Swing-up and stabilization of a cart-pendulum system under restricted cart track length. Syst Control Lett 47(4):355–364MathSciNetzbMATHGoogle Scholar
  26. Chiha I, Liouane N, Borne P (2012) Tuning PID controller using multiobjective ant colony optimization. Appl Comput Intell Soft Comput.  https://doi.org/10.1155/2012/536326 Google Scholar
  27. Das RR, Elumalai VK, Subramanian RG, Kumar KVA (2018) Adaptive predator-prey optimization for tuning of infinite horizon LQR applied to vehicle suspension system. Appl Soft Comput 72:1568–4946Google Scholar
  28. Duan M, Okwudire CE (2018) Proxy-based optimal control allocation for dual-input over-actuated systems. IEEE/ASME Trans Mechatron 23:895–905Google Scholar
  29. Ghommam J, Mnif F (2017) Predictor-based control for an inverted pendulum subject to networked time delay. ISA Trans 67:306–316Google Scholar
  30. Ghosh A, Krishnan TR, Subudhi B (2012) Robust proportional-integral-derivative compensation of an inverted cart-pendulum system: an experimental study. IET Control Theory Appl 6(8):1145–1152MathSciNetGoogle Scholar
  31. Gordillo F, Aracil J (2008) A new controller for the inverted pendulum on a cart. Int J Robust Nonlinear Control 18(17):1607–1621MathSciNetzbMATHGoogle Scholar
  32. Hanwate SD, Budhraja A, Hote YV (2015) Improved performance of cart inverted pendulum system using LQR based PID controller and ANN. In: IEEE UP section conference on electrical computer and electronicsGoogle Scholar
  33. Hassani K, Lee WS (2016) Multi-objective design of state feedback controllers using reinforced quantum-behaved particle swarm optimization. Appl Soft Comput 41:66–76Google Scholar
  34. Henmi T, Park Y, Deng M, Inoue A (2010) Stabilization controller for a cart-type inverted pendulum via a partial linearization method. In: Proceedings of the 2010 international conference on modelling, identification and controlGoogle Scholar
  35. Heris SMK (2015) Implementation of artificial bee colony in MATLAB. Yarpiz, Project Code: YPEA114Google Scholar
  36. Howimanporn S, Thanok S, Chookaew S, Sootkaneung W (2016) Design and implementation of PSO based LQR control for inverted pendulum through PLC. In: IEEE/SICE international symposium on system integration, pp 664–669Google Scholar
  37. Huang SJ, Huang CL (2000) Control of an inverted pendulum using grey prediction model. IEEE Trans Ind Appl 36(2):452–458MathSciNetGoogle Scholar
  38. Jacknoon A, Abido MA (2017) Ant Colony based LQR and PID tuned parameters for controlling inverted pendulum. In: International conference on communication, control, computing and electronics engineering (ICCCCEE)Google Scholar
  39. Kawaji S, Ogasawara K, Honda H (1994) Swing up control of a pendulum using genetic algorithms. In: 33rd IEEE conference on decision and control vol 4, pp 3530–3532Google Scholar
  40. Kumar EV, Jerome J (2013) Robust LQR controller design for stabilizing and trajectory tracking of inverted pendulum. Procedia Eng 64:169–178Google Scholar
  41. Kumar EV, Raaja GS, Jerome J (2016) Adaptive PSO for optimal LQR tracking control of 2 DoF laboratory helicopter. Appl Soft Comput 41:77–90Google Scholar
  42. Lagarias JC, Reeds JA, Wright MH, Wright PE (1998) Convergence properties of the Nelder–Mead simplex method in low dimensions. SIAM J Optim 9(1):112–147MathSciNetzbMATHGoogle Scholar
  43. Lewis FL, Vrabie D, Syrmos VL (2012) Optimal control. Wiley, HobokenzbMATHGoogle Scholar
  44. Li W, Ding H, Cheng K. (2012) An investigation on the design and performance assessment of double-PID and LQR controllers for the inverted pendulum. In: Proceedings of 2012 UKACC international conference on control, pp 190–196Google Scholar
  45. Lozano R, Fantoni I, Block DJ (2000) Stabilization of the inverted pendulum around its homoclinic orbit. Syst Control Lett 40(3):197–204MathSciNetzbMATHGoogle Scholar
  46. MathWorks (2018) Global optimization toolbox user’s guide. Revised for version 4.0 (release 2018b)Google Scholar
  47. Mihara K, Yokoyama J, Suemitsu H (2012) Swing-up and stabilizing control of an inverted pendulum by two step control method. In: International conference on advanced mechatronic systems, pp 323–328Google Scholar
  48. Mua’zu MB, Salawudeen AT, Sikiru TH, Abdu AI, Mohammad A (2015) Weighted artificial fish swarm algorithm with adaptive behaviour based linear controller design for nonlinear inverted pendulum. J Eng Res 20(1):1–12Google Scholar
  49. Muskinja N, Tovornik B (2006) Swinging up and stabilization of a real inverted pendulum. IEEE Trans Ind Electron 53(2):631–639Google Scholar
  50. Mathew NJ, Rao KK, Sivakumaran N (2013) Swing up and stabilization control of a rotary inverted pendulum. In: IFAC international symposium on dynamics and control of process systems, international federation of automatic control, pp 654–659Google Scholar
  51. Oróstica R, Duarte-Mermoud MA, Jáuregui C (2016) Stabilization of inverted pendulum using LQR, PID and fractional order PID controllers: a simulated study. In: IEEE international conference on automatica, pp 1–7Google Scholar
  52. Pandey SK, Vijaya L (2015) Optimal control of twin rotor MIMO system using LQR technique. Comput Intell Data Min 1:11–21Google Scholar
  53. Park MS, Chwa D (2009) Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-mode control method. IEEE Trans Ind Electron 56:3541–3555Google Scholar
  54. Prasad LB, Tyagi B, Gupta HO (2014) Optimal control of nonlinear inverted pendulum system using PID controller and LQR: performance analysis without and with disturbance input. Int J Autom Comput 11(6):661–670Google Scholar
  55. Quanser Inc. (2009) Quanser Inc. SIP and SPG user manual, 2009Google Scholar
  56. Rabah M, Rohan A, Kim SH (2018) Comparison of position control of a gyroscopic inverted pendulum using PID, fuzzy logic and fuzzy PID controllers. Int J Fuzzy Logic Intell Syst 18(2):103–110Google Scholar
  57. Reddy NS, Saketh MS, Pal P, Dey R (2016) Optimal PID controller design of an inverted pendulum dynamics: a hybrid pole-placement firefly algorithm approach. In: IEEE international conference on control, measurement and instrumentation, pp 305–310Google Scholar
  58. Rohan A, Rabah M, Nam K-H, Kim SH (2018) Design of fuzzy logic based controller for gyroscopic inverted pendulum system. Int J Fuzzy Logic Intell Syst 18(1):58–64Google Scholar
  59. Rybovic A, Priecinsky M, Paskala M (2012) Control of the inverted pendulum using state feedback control. ELEKTROGoogle Scholar
  60. Sakurama K, Hara S, Nakano K (2007) Swing-up and stabilization control of a cart- pendulum system via energy control and controlled Lagrangian methods. Electr Eng Jpn 160(4):24–31Google Scholar
  61. Srikanth K, Kumar GNV (2017) Novel fuzzy preview controller for rotary inverted pendulum under time delays. Int J Fuzzy Logic Intell Syst 17(4):257–263Google Scholar
  62. Wang JJ (2011) Simulation studies of inverted pendulum based on PID controllers. Simul Model Pract Theory 19(1):440–449Google Scholar
  63. Wang Z, Chen Y, Fang N (2004) Minimum-time swing-up of a rotary inverted pendulum by iterative impulsive control. In: American control conference, vol 2, pp 1335–1340Google Scholar
  64. Wang L, Ni H, Zhou W, Pardalos PM, Fang J, Fei M (2014) MBPOA-based LQR controller and its application to the double-parallel inverted pendulum system. Eng Appl Artif Intell 36:262–268Google Scholar
  65. Wongsathan C, Sirima C (2008) Application of GA to design LQR controller for an inverted pendulum system. In: IEEE international conference on robotics and biomimetics, pp 951–954Google Scholar
  66. Yang JH, Shim SY, Seo JH, Lee YS (2009) Swing-up control for an inverted pendulum with restricted cart rail length. Int J Control Autom Syst 7(4):674–680Google Scholar
  67. Yusuf LA, Magaji N (2014) GA-PID controller for position control of inverted pendulum. In: IEEE 6th international conference on adaptive science and technology (ICAST), pp 1–5Google Scholar
  68. Zhou H, Deng H, Duan J (2018) Hybrid fuzzy decoupling control for a precision maglev motion system. IEEE/ASME Trans Mechatron 23:389–401Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of FloridaGainesvilleUSA
  2. 2.University of FloridaGainesvilleUSA

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