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Practical Explicit Model Predictive Control for a Class of Noise-embedded Chaotic Hybrid Systems

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  • Control Theory and Applications
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Abstract

Controlling a class of chaotic hybrid systems in the presence of noise is investigated in this paper. To reach this goal, an explicit model predictive control (eMPC) in combination with nonlinear estimators is employed. Using the eMPC method, all the computations of the common MPC approach are moved off-line. Therefore, the off-line control law makes it easier to be implemented in comparison with the on-line approach, especially for complex systems like the chaotic ones. In order to verify the proposed control structure practically, an op-amp based Chua’s chaotic circuit is designed. The white Gaussian noise is considered in this circuit. Therefore, the nonlinear estimators –extended and unscented Kalman filter (EKF and UKF)– are utilized to estimate signals from the noise-embedded chaotic system. Performance of these estimators for this experimental setup is compared in both open-loop and closed-loop systems. The experimental results demonstrate the effectiveness of the eMPC approach as well as the nonlinear estimators for chaos control in the presence of noise.

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References

  1. K.-S. Park, J.-B. Park, Y.-H. Choi, T.-S. Yoon, and G. Chen, “Generalized predictive control of discrete-time chaotic systems,” International Journal of Bifurcation and Chaos, vol. 8, no. 07, pp. 1591–1597, 1998.

    Article  MATH  Google Scholar 

  2. K.-S. Park, J.-M. Joo, J.-B. Park, Y.-H. Choi, and T.-S. Yoon, “Control of discrete-time chaotic systems using generalized predictive control,” IEEE International Symposium on Circuits and Systems, vol. 2, pp. 789–792, IEEE, 1997.

    Google Scholar 

  3. Q. Qian, A. Swain, and N. Patel, “Nonlinear continuous time generalized predictive controller for chaotic systems,” Proc. of IEEE International Conference on Industrial Technology, pp. 1–6, IEEE, 2008.

    Google Scholar 

  4. S. Li, Y. Li, B. Liu, and T. Murray, “Model-free control of lorenz chaos using an approximate optimal control strategy,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4891–4900, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Senouci and A. Boukabou, “Predictive control and synchronization of chaotic and hyperchaotic systems based on a T-S fuzzy model,” Mathematics and Computers in Simulation, vol. 105, pp. 62–78, 2014.

    Article  MathSciNet  Google Scholar 

  6. Z. Longge and L. Xiangjie, “The synchronization between two discrete-time chaotic systems using active robust model predictive control,” Nonlinear Dynamics, vol. 74, no. 4, pp. 905–910, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Jiang, H. Wang, J. Lu, G. Cai, and W. Qin, “Synchronization for chaotic systems via mixed-objective dynamic output feedback robust model predictive control,” Journal of the Franklin Institute, vol. 354, no. 12, pp. 4838–4860, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Bemporad, F. Borrelli, and M. Morari, “Model predictive control based on linear programming˜ the explicit solution,” IEEE Transactions on Automatic Control, vol. 47, no. 12, pp. 1974–1985, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, “The explicit linear quadratic regulator for constrained systems,” Automatica, vol. 38, no. 1, pp. 3–20, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. TøNdel, T. A. Johansen, and A. Bemporad, “An algorithm for multi-parametric quadratic programming and explicit mpc solutions,” Automatica, vol. 39, no. 3, pp. 489–497, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  11. I. J. Wolf and W. Marquardt, “Fast NMPC schemes for regulatory and economic NMPC-a review,” Journal of Process Control, vol. 44, pp. 162–183, 2016.

    Article  Google Scholar 

  12. A. Alessio and A. Bemporad, “A survey on explicit model predictive control,” in Nonlinear Model Predictive Control, pp. 345–369, Springer, 2009.

    Chapter  Google Scholar 

  13. F. Bayat, T. A. Johansen, and A. A. Jalali, “Using hash tables to manage the time-storage complexity in a point location problem: Application to explicit model predictive control,” Automatica, vol. 47, no. 3, pp. 571–577, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Mariéthoz, S. Almér, M. Bâja, A. G. Beccuti, D. Patino, A. Wernrud, J. Buisson, H. Cormerais, T. Geyer, H. Fujioka, U. T. Jonsson, C.-Y. Kao, M. Morari, G. Papafotiou, A. Rantzer, and P. Riedingder, “Comparison of hybrid control techniques for buck and boost dc-dc converters,” IEEE Transactions on Control Systems Technology, vol. 18, no. 5, pp. 1126–1145, 2010.

    Article  Google Scholar 

  15. M. A. Mohammadkhani, F. Bayat, and A. A. Jalali, “Design of explicit model predictive control for constrained linear systems with disturbances,” International Journal of Control, Automation and Systems, vol. 12, no. 2, pp. 294–301, 2014.

    Article  Google Scholar 

  16. J. Zhang, X. Cheng, and J. Zhu, “Control of a laboratory 3-dof helicopter: Explicit model predictive approach,” International Journal of Control, Automation and Systems, vol. 14, no. 2, pp. 389–399, 2016.

    Article  Google Scholar 

  17. C.-S. Poon and M. Barahona, “Titration of chaos with added noise,” Proceedings of the National Academy of Sciences, vol. 98, no. 13, pp. 7107–7112, 2001.

    Article  MATH  Google Scholar 

  18. W.-w. Tung, J. Gao, J. Hu, and L. Yang, “Detecting chaos in heavy-noise environments,” Physical Review E, vol. 83, no. 4, p. 0462.0, 2011.

    Google Scholar 

  19. T. Carroll and F. Rachford, “Chaotic sequences for noisy environments,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 26, no. 10, p. 1031.4, 2016.

    Google Scholar 

  20. A. Leontitsis, J. Pange, and T. Bountis, “Large noise level estimation,” International Journal of Bifurcation and Chaos, vol. 13, no. 08, pp. 2309–2313, 2003.

    Article  MATH  Google Scholar 

  21. T.-L. Yao, H.-F. Liu, J.-L. Xu, and W.-F. Li, “Estimating the largest Lyapunov exponent and noise level from chaotic time series,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 22, no. 3, p. 0331.2, 2012.

    Google Scholar 

  22. G. Çoban, A. H. Büyüklü, and A. Das, “A linearization based non-iterative approach to measure the gaussian noise level for chaotic time series,” Chaos, Solitons & Fractals, vol. 45, no. 3, pp. 266–278, 2012.

    Article  MATH  Google Scholar 

  23. A. Serletis, A. Shahmoradi, and D. Serletis, “Effect of noise on the bifurcation behavior of nonlinear dynamical systems,” Chaos, Solitons & Fractals, vol. 33, no. 3, pp. 914–921, 2007.

    Article  MATH  Google Scholar 

  24. M. Nurujjaman, S. Shivamurthy, A. Apte, T. Singla, and P. Parmananda, “Effect of discrete time observations on synchronization in chua model and applications to data assimilation,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 22, no. 2, p. 0231.5, 2012.

    Google Scholar 

  25. V. Semenov, I. Korneev, P. Arinushkin, G. Strelkova, T. Vadivasova, and V. Anishchenko, “Numerical and experimental studies of attractors in memristor-based chua’s oscillator with a line of equilibria. noise-induced effects,” The European Physical Journal Special Topics, vol. 224, no. 8, pp. 1553–1561, 2015.

    Article  Google Scholar 

  26. D. S. Goldobin, “Noise can reduce disorder in chaotic dynamics,” The European Physical Journal Special Topics, vol. 223, no. 8, pp. 1699–1709, 2014.

    Article  Google Scholar 

  27. N. Sviridova and K. Nakamura, “Local noise sensitivity: Insight into the noise effect on chaotic dynamics,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 26, no. 12, p. 1231.2, 2016.

    Google Scholar 

  28. M. Kvasnica, P. Grieder, M. Baotic, and M. Morari, “Multiparametric toolbox (mpt), 2004.,” 2006.

    MATH  Google Scholar 

  29. M. Herceg, M. Kvasnica, C. N. Jones, and M. Morari, “Multi-parametric toolbox 3.0,” Proc. of European Control Conference (ECC),, pp. 502–510, IEEE, 2013.

    Google Scholar 

  30. K. Judd and L. Smith, “Indistinguishable states: I. perfect model scenario,” Physica D: Nonlinear Phenomena, vol. 151, no. 2, pp. 125–141, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  31. K. Judd, “Nonlinear state estimation, indistinguishable states, and the extended kalman filter,” Physica D: Nonlinear Phenomena, vol. 183, no. 3, pp. 273–281, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  32. S.-H. Fu and Q.-S. Lu, “Set stability of controlled Chua’s circuit under a non-smooth controller with the absolute value,” International Journal of Control, Automation and Systems, vol. 12, no. 3, pp. 507–517, 2014.

    Article  MathSciNet  Google Scholar 

  33. L. O. Chua, The Genesis of Chua’s Circuit, Electronics Research Laboratory, College of Engineering, University of California, 1992.

    Google Scholar 

  34. J. Wong, A Collection of Amp Applications, Analog Devices, Inc., 1992.

    Google Scholar 

  35. L. Oxley and D. A. George, “Economics on the edge of chaos: some pitfalls of linearizing complex systems,” Environmental Modelling & Software, vol. 22, no. 5, pp. 580–589, 2007.

    Article  Google Scholar 

  36. D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. Scokaert, “Constrained model predictive control: stability and optimality,” Automatica, vol. 36, no. 6, pp. 789–814, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Bemporad and C. Filippi, “An algorithm for approximate multiparametric convex programming,” Computational optimization and applications, vol. 35, no. 1, pp. 87–108, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Bemporad, F. Borrelli, and M. Morari, “Piecewise linear optimal controllers for hybrid systems,” Proceedings of the American Control Conference, vol. 2, pp. 1190–1194, IEEE, 2000.

    Google Scholar 

  39. A. Bemporad and M. Morari, “Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, no. 3, pp. 407–427, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  40. A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Elsevier, 1983.

    MATH  Google Scholar 

  41. E. Pistikopoulos, M. Georgiadis, and V. Dua, Multiparametric Programming: Theory, Algorithms and Applications, Volume, WileyVCH, Weinheim, 2007.

    Book  Google Scholar 

  42. J. Acevedo and E. N. Pistikopoulos, “A multiparametric programming approach for linear process engineering problems under uncertainty,” Industrial & Engineering Chemistry Research, vol. 36, no. 3, pp. 717–728, 1997.

    Article  Google Scholar 

  43. V. Dua and E. N. Pistikopoulos, “An algorithm for the solution of multiparametric mixed integer linear programming problems,” Annals of Operations Research, vol. 99, no. 1, pp. 123–139, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Lines, Nonlinear Dynamical Systems in Economics, vol. 476, Springer Science & Business Media, 2007.

    Google Scholar 

  45. M. S. Ghasemi and A. A. Afzalian, “Robust tube-based mpc of constrained piecewise affine systems with bounded additive disturbances,” Nonlinear Analysis: Hybrid Systems, vol. 26, pp. 86–100, 2017.

    MathSciNet  MATH  Google Scholar 

  46. M. Lazar, “Model predictive control of hybrid systems: Stability and robustness,” 2006.

    Google Scholar 

  47. E. F. Camacho, D. R. Ramírez, D. Limón, D. M. De La Peña, and T. Alamo, “Model predictive control techniques for hybrid systems,” Annual Reviews in Control, vol. 34, no. 1, pp. 21–31, 2010.

    Article  Google Scholar 

  48. J. Rodriguez and P. Cortes, Predictive Control of Power Converters and Electrical Drives, vol. 40, John Wiley & Sons, 2012.

    Book  Google Scholar 

  49. H. Nagashima and Y. Baba, Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena, CRC Press, 1998.

    MATH  Google Scholar 

  50. D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches, John Wiley & Sons, 2006.

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. Complex Nonlinear Systems laboratory, Electrical engineering faculty, Iran University of Science and Technology, 1684613114, Tehran, Iran

    Seyyed Mostafa Tabatabaei, Sara Kamali & Mohammad Reza Jahed-Motlagh

  2. Department of Electrical Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, 7616913439, Kerman, Iran

    Mojtaba Barkhordari Yazdi

Authors
  1. Seyyed Mostafa Tabatabaei
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  2. Sara Kamali
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  3. Mohammad Reza Jahed-Motlagh
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Correspondence to Mojtaba Barkhordari Yazdi.

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Recommended by Associate Editor Joseph Kwon under the direction of Editor Jay H. Lee.

Seyyed Mostafa Tabatabaei received the B.Sc. degree in electrical engineering from Urmia University, Urmia, Iran, in 2007, and his M.Sc. degree in control engineering from Iran University of Science and Technology, Tehran, Iran, in 2011. His research interests include nonlinear systems, switched systems, predictive control, adaptive control, chaos phenomena, and industrial automation control systems.

Sara Kamali received her B.Sc. degree in electrical engineering from Iran University of Science and Technology, Tehran, Iran, in 2014 and her M.Sc. degree in control engineering from Iran University of Science and Technology, Tehran, Iran, in 2017. Her research interests include nonlinear control, adaptive control, predictive control, chaos control, and switched systems.

Mohammad Reza Jahed-Motlagh was born in Tehran, Iran, in 1955. He received the B.Sc. degree in electrical engineering from the Sharif University of Technology, Tehran, Iran, in 1978, and the M.Sc. and Ph.D. degrees both in control theory and control engineering from the University of Bradford, Bradford, U.K., in 1986 and 1990, respectively. He is currently a Professor with the Department of Computer Engineering, Iran University of Science and Technology, Tehran. His current research interests include nonlinear control, hybrid control systems, multivariable control systems, chaos computing, and chaos control.

Mojtaba Barkhordari Yazdi received his B.Sc. degree in electrical engineering from the K. N. Toosi University of Technology, Tehran, Iran in 2001, and his M.Sc. and Ph.D. degrees in control engineering, in 2003 and 2010, both from Iran University of Science and Technology, Tehran, Iran. From September 2008 to March 2009, he was with the Department of Control Systems, Technische Universitat Berlin, Germany, as a visiting researcher. Since 2010, he has been an Assistant Professor of electrical engineering at Shahid Bahonar University of Kerman. His research interests include hybrid systems, multi-agent systems, robotics, and power system dynamics.

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Tabatabaei, S.M., Kamali, S., Jahed-Motlagh, M.R. et al. Practical Explicit Model Predictive Control for a Class of Noise-embedded Chaotic Hybrid Systems. Int. J. Control Autom. Syst. 17, 857–866 (2019). https://doi.org/10.1007/s12555-018-0384-3

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  • DOI: https://doi.org/10.1007/s12555-018-0384-3

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