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Optimal Fractional Order Proportional—Integral—Differential Controller for Inverted Pendulum with Reduced Order Linear Quadratic Regulator

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Fractional Order Control and Synchronization of Chaotic Systems

Part of the book series: Studies in Computational Intelligence ((SCI,volume 688))

Abstract

The objective of this chapter is to present an optimal Fractional Order Proportional—Integral-Differential (FOPID) controller based upon Reduced Linear Quadratic Regulator (RLQR) using Particle Swarm Optimization (PSO) algorithm and compared with PID controller. The controllers are applied to Inverted Pendulum (IP) system which is one of the most exciting problems in dynamics and control theory. The FOPID or PID controller with a feed-forward gain is responsible for stabilizing the cart position and the RLQR controller is responsible for swinging up the pendulum angle. FOPID controller is the recent advances improvement controller of a conventional classical PID controller. Fractional-order calculus deals with non-integer order systems. It is the same as the traditional calculus but with a much wider applicability. Fractional Calculus is used widely in the last two decades and applied in different fields in the control area. FOPID controller achieves great success because of its effectiveness on the dynamic of the systems. Designing FOPID controller is more flexible than the standard PID controller because they have five parameters with two parameters over the standard PID controller. The Linear Quadratic Regulator (LQR) is an important approach in the optimal control theory. The optimal LQR needs tedious tuning effort in the context of good results. Moreover, LQR has many coefficients matrices which are designer dependent. These difficulties are talked by introducing RLQR. RLQR has an advantage which allows for the optimization technique to tune fewer parameters than classical LQR controller. Moreover, all coefficients matrices that are designer dependent are reformulated to be included into the optimization process. Tuning the controllers’ gains is one of the most crucial challenges that face FOPID application. Thanks to the Metaheuristic Optimization Techniques (MOTs) which solves this dilemma. PSO technique is one of the most widely used MOTs. PSO is used for the optimal tuning of the FOPID controller and RLQR parameters. The control problem is formulated to attain the combined FOPID controllers’ gains with a feed forward gain and RLQR into a multi-dimensions control problem. The objective function is designed to be multi-objective by considering the minimum settling time, rise time, undershoot and overshoot for both the cart position and the pendulum angle. It is evident from the simulation results, the effectiveness of the proposed design approach. The obtained results are very promising. The design procedures are presented step by step. The robustness of the proposed controllers is tested for internal and external large and fast disturbances.

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References

  1. Nasir, A. N. K., Ahmad, M. A., & Rahmat, M. F. A. (2008). Performance comparison between LQR and PID controllers for an inverted pendulum system. In International conference on power control and optimization: Innovation in Power control for optimal industry (Vol. 1052, No. 1, pp. 124–128). AIP Publishing.

    Google Scholar 

  2. Chakraborty, S. (2014). An experimental study for stabilization of inverted pendulum. (Doctoral dissertation).

    Google Scholar 

  3. Li, Z., Yang, C., & Fan, L. (2012). Advanced control of wheeled inverted pendulum systems. Springer Science and Business Media.

    Google Scholar 

  4. Noh, K. K., Kim, J. G., & Huh, U. Y. (2004). Stability experiment of a biped walking robot with inverted pendulum. In 30th annual conference of IEEE industrial electronics society, 2004. IECON 2004 (Vol. 3, pp. 2475–2479). IEEE.

    Google Scholar 

  5. Li, Z., & Yang, C. (2012). Neural-adaptive output feedback control of a class of transportation vehicles based on wheeled inverted pendulum models. IEEE Transactions on Control Systems Technology, 20(6), 1583–1591.

    Article  Google Scholar 

  6. Azar, A. T., & Vaidyanathan, S. (Eds.). (2016). Advances in chaos theory and intelligent control (Vol. 337). Springer.

    Google Scholar 

  7. Azar, A. T., & Serrano, F. E. (2015a). Stabilizatoin and control of mechanical systems with backlash. Handbook of research on advanced intelligent control engineering and automation. Advances in computational intelligence and robotics (ACIR), IGI-Global, USA (pp. 1–60).

    Google Scholar 

  8. Altinoz, O. T., Yilmaz, A. E., & Weber, G. W. (2010). Chaos particle swarm optimized PID controller for the inverted pendulum system. In 2nd international conference on engineering optimization, Lisbon, Portugal.

    Google Scholar 

  9. Azar, A. T., & Serrano, F. E. (2015b). Adaptive sliding mode control of the Furuta pendulum. In Advances and Applications in Sliding Mode Control systems (pp. 1–42). Springer International Publishing.

    Google Scholar 

  10. Azar, A. T., & Vaidyanathan, S. (2014). Handbook of research on advanced intelligent control engineering and automation. IGI Global.

    Google Scholar 

  11. Pengpeng, Z., Lei, Z., & Yanhai, H. (2013). BP neural network control of single inverted pendulum. In 2013 3rd international conference on computer science and network technology (ICCSNT) (pp. 1259–1262). IEEE.

    Google Scholar 

  12. Jaleel, J. A., & Francis, R. M. (2013). Simulated annealing based control of an inverted pendulum system. In 2013 International conference on control communication and computing (ICCC) (pp. 204–209). IEEE.

    Google Scholar 

  13. Hanafy, T. O. (2012). Stabilization of inverted pendulum system using particle swarm optimization. In 8th international conference on informatics and systems (INFOS), 2012 (pp. BIO-207). IEEE.

    Google Scholar 

  14. Zhao, J., & Spong, M. W. (2001). Hybrid control for global stabilization of the cart–pendulum system. Automatica, 37(12), 1941–1951.

    Article  MathSciNet  MATH  Google Scholar 

  15. Park, M. S., & Chwa, D. (2009). Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-mode control method. IEEE Transactions on Industrial Electronics, 56(9), 3541–3555.

    Article  Google Scholar 

  16. Mason, P., Broucke, M., & Piccoli, B. (2008). Time optimal swing-up of the planar pendulum. IEEE Transactions on Automatic Control, 53(8), 1876–1886.

    Article  MathSciNet  Google Scholar 

  17. Mills, A., Wills, A., & Ninness, B. (2009). Nonlinear model predictive control of an inverted pendulum. In 2009 American control conference (pp. 2335–2340). IEEE.

    Google Scholar 

  18. Wang, H. O., Tanaka, K., & Griffin, M. F. (1996). An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Transactions on Fuzzy Systems, 4(1), 14–23.

    Article  Google Scholar 

  19. Chang, W. D., Hwang, R. C., & Hsieh, J. G. (2002). A self-tuning PID control for a class of nonlinear systems based on the Lyapunov approach. Journal of Process Control, 12(2), 233–242.

    Article  Google Scholar 

  20. Åström, K. J., & Furuta, K. (2000). Swinging up a pendulum by energy control. Automatica, 36(2), 287–295.

    Article  MathSciNet  MATH  Google Scholar 

  21. Åström, K. J. (1999). Hybrid control of inverted pendulums. In Learning, control and hybrid systems (pp. 150–163). London: Springer.

    Google Scholar 

  22. Mazenc, F., & Bowong, S. (2003). Tracking trajectories of the cart-pendulum system. Automatica, 39(4), 677–684.

    Article  MathSciNet  MATH  Google Scholar 

  23. Azar, A. T., & Zhu, Q. (Eds.). (2015). Advances and applications in sliding mode control systems. Berlin: Springer.

    MATH  Google Scholar 

  24. Holzhüter, T. (2004). Optimal regulator for the inverted pendulum via Euler-Lagrange backward integration. Automatica, 40(9), 1613–1620.

    Article  MathSciNet  MATH  Google Scholar 

  25. Chernousko, F. L., & Reshmin, S. A. (2007). Time-optimal swing-up feedback control of a pendulum. Nonlinear Dynamics, 47(1–3), 65–73.

    MathSciNet  MATH  Google Scholar 

  26. Srinivasan, B., Huguenin, P., & Bonvin, D. (2009). Global stabilization of an inverted pendulum–control strategy and experimental verification. Automatica, 45(1), 265–269.

    Article  MathSciNet  MATH  Google Scholar 

  27. Azar, A. T., & Vaidyanathan, S. (Eds.). (2015). Chaos modeling and control systems design. Germany: Springer.

    MATH  Google Scholar 

  28. Karnopp, D. C., Margolis, D. L., & Rosenberg, R. C. (2012). System dynamics: modeling, simulation, and control of mechatronic systems. Wiley.

    Google Scholar 

  29. Ogata, K. (2010). Modern control engineering. книгa.

    Google Scholar 

  30. Box, G. E. (1957). Evolutionary operation: A method for increasing industrial productivity. Applied statistics (pp. 81–101).

    Google Scholar 

  31. Friedman, G. J. (1959). Digital simulation of an evolutionary process. General systems yearbook (Vol. 4, pp. 171–184).

    Google Scholar 

  32. Dianati, M., Song, I., & Treiber, M. (2002). An introduction to genetic algorithms and evolution strategies. Technical report, University of Waterloo, Ontario, N2L 3G1, Canada.

    Google Scholar 

  33. James, K., & Russell, E. (1995). Particle swarm optimization. In Proceedings of 1995 IEEE international conference on neural networks (pp. 1942–1948).

    Google Scholar 

  34. Mehdi, H., & Boubaker, O. (2011). Position/force control optimized by particle swarm intelligence for constrained robotic manipulators. In 11th international conference on intelligent systems design and applications (ISDA) (pp. 190–195). IEEE.

    Google Scholar 

  35. Ebrahim, M. A., El-Metwally, A., Bendary, F. M., Mansour, W. M., Ramadan, H. S., Ortega, R., & Romero, J. (2011). Optimization of proportional-integral-differential controller for wind power plant using particle swarm optimization technique. International Journal of Emerging Technologies in Science and Engineering.

    Google Scholar 

  36. Ebrahim, M. A., Mostafa, H. E., Gawish, S. A., & Bendary, F. M. (2009). Design of decentralized load frequency based-PID controller using stochastic particle swarm optimization technique. In International conference on electric power and energy conversion systems, 2009. EPECS’09 (pp. 1–6). IEEE.

    Google Scholar 

  37. Jagatheesan, K., Anand, B., & Ebrahim, M. A. (2014). Stochastic particle swarm optimization for tuning of PID controller in load frequency control of single area reheat thermal power system. International Journal of Electrical Power and Energy, 8(2), 33–40.

    Google Scholar 

  38. Mousa, M. E., Ebrahim, M. A., & Hassan, M. M. (2015). Stabilizing and swinging-up the inverted pendulum using pi and pid controllers based on reduced linear quadratic regulator tuned by PSO. International Journal of System Dynamics Applications (IJSDA), 4(4), 52–69.

    Article  Google Scholar 

  39. Ali, A. M., Ebrahim, M. A., & Hassan, M. M. (2016). Automatic voltage generation control for two area power system based on particle swarm optimization. Indonesian Journal of Electrical Engineering and Computer Science, 2(1), 132–144.

    Article  Google Scholar 

  40. Kennedy, J. (1997). The particle swarm: social adaptation of knowledge. In IEEE international conference on evolutionary computation (pp. 303–308). IEEE.

    Google Scholar 

  41. Wang, J. J. (2011). Simulation studies of inverted pendulum based on PID controllers. Simulation Modelling Practice and Theory, 19(1), 440–449.

    Article  Google Scholar 

  42. Ciprian, P. P., Luminita, D., & Lucia, P. (2011). Control optimization using MATLAB. INTECH Open Access Publisher.

    Google Scholar 

  43. Branch, S. T. (2011). Optimal design of LQR weighting matrices based on intelligent optimization methods.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Ministry of Civil Aviation, Cairo, Egypt

    M. E. Mousa

  2. Faculty of Engineering at Shoubra, Department of Electrical Engineering, Benha University, Cairo, Egypt

    M. A. Ebrahim

  3. FCLab FR CNRS 3539, FEMTO-ST UMR CNRS 6174, UTBM, 90010, Belfort Cedex, France

    M. A. Ebrahim

  4. Faculty of Engineering, Department of Electrical Engineering, Cairo University, Giza, Egypt

    M. A. Moustafa Hassan

Authors
  1. M. E. Mousa
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  2. M. A. Ebrahim
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  3. M. A. Moustafa Hassan
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Correspondence to M. E. Mousa .

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Editors and Affiliations

  1. Faculty of Computers and Information, Benha University Faculty of Computers and Information, AND Nanoelectronics Integrated Systems Center (NISC), Nile University, Egypt. , Benha, Egypt

    Ahmad Taher Azar

  2. Research and Development Centre, Vel Tech University Research and Development Centre, Chennai, Tamil Nadu, India

    Sundarapandian Vaidyanathan

  3. Department of Mathematics and CS, University of Tebessa Department of Mathematics and CS, Tebessa, Algeria

    Adel Ouannas

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Mousa, M.E., Ebrahim, M.A., Moustafa Hassan, M.A. (2017). Optimal Fractional Order Proportional—Integral—Differential Controller for Inverted Pendulum with Reduced Order Linear Quadratic Regulator. In: Azar, A., Vaidyanathan, S., Ouannas, A. (eds) Fractional Order Control and Synchronization of Chaotic Systems. Studies in Computational Intelligence, vol 688. Springer, Cham. https://doi.org/10.1007/978-3-319-50249-6_8

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  • DOI: https://doi.org/10.1007/978-3-319-50249-6_8

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