Advertisement

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

  • Seyyed Mostafa Tabatabaei
  • Sara Kamali
  • Mohammad Reza Jahed-Motlagh
  • Mojtaba Barkhordari YazdiEmail author
Regular Papers Control Theory and Applications
  • 16 Downloads

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.

Keywords

Chaos control Chua’s circuit explicit model predictive control (eMPC) noise unscented Kalman filter 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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.CrossRefzbMATHGoogle Scholar
  2. [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. [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. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [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.MathSciNetCrossRefGoogle Scholar
  6. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [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.CrossRefGoogle Scholar
  12. [12]
    A. Alessio and A. Bemporad, “A survey on explicit model predictive control,” in Nonlinear Model Predictive Control, pp. 345–369, Springer, 2009.CrossRefGoogle Scholar
  13. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [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.CrossRefGoogle Scholar
  15. [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.CrossRefGoogle Scholar
  16. [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.CrossRefGoogle Scholar
  17. [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.CrossRefzbMATHGoogle Scholar
  18. [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. [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. [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.CrossRefzbMATHGoogle Scholar
  21. [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. [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.CrossRefzbMATHGoogle Scholar
  23. [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.CrossRefzbMATHGoogle Scholar
  24. [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. [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.CrossRefGoogle Scholar
  26. [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.CrossRefGoogle Scholar
  27. [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. [28]
    M. Kvasnica, P. Grieder, M. Baotic, and M. Morari, “Multiparametric toolbox (mpt), 2004.,” 2006.zbMATHGoogle Scholar
  29. [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. [30]
    K. Judd and L. Smith, “Indistinguishable states: I. perfect model scenario,” Physica D: Nonlinear Phenomena, vol. 151, no. 2, pp. 125–141, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [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.MathSciNetCrossRefGoogle Scholar
  33. [33]
    L. O. Chua, The Genesis of Chua’s Circuit, Electronics Research Laboratory, College of Engineering, University of California, 1992.Google Scholar
  34. [34]
    J. Wong, A Collection of Amp Applications, Analog Devices, Inc., 1992.Google Scholar
  35. [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.CrossRefGoogle Scholar
  36. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [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. [39]
    A. Bemporad and M. Morari, “Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, no. 3, pp. 407–427, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Elsevier, 1983.zbMATHGoogle Scholar
  41. [41]
    E. Pistikopoulos, M. Georgiadis, and V. Dua, Multiparametric Programming: Theory, Algorithms and Applications, Volume, WileyVCH, Weinheim, 2007.CrossRefGoogle Scholar
  42. [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.CrossRefGoogle Scholar
  43. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    M. Lines, Nonlinear Dynamical Systems in Economics, vol. 476, Springer Science & Business Media, 2007.Google Scholar
  45. [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.MathSciNetzbMATHGoogle Scholar
  46. [46]
    M. Lazar, “Model predictive control of hybrid systems: Stability and robustness,” 2006.Google Scholar
  47. [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.CrossRefGoogle Scholar
  48. [48]
    J. Rodriguez and P. Cortes, Predictive Control of Power Converters and Electrical Drives, vol. 40, John Wiley & Sons, 2012.CrossRefGoogle Scholar
  49. [49]
    H. Nagashima and Y. Baba, Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena, CRC Press, 1998.zbMATHGoogle Scholar
  50. [50]
    D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches, John Wiley & Sons, 2006.CrossRefGoogle Scholar

Copyright information

© ICROS, KIEE and Springer 2019

Authors and Affiliations

  1. 1.Complex Nonlinear Systems laboratory, Electrical engineering facultyIran University of Science and TechnologyTehranIran
  2. 2.Department of Electrical Engineering, Faculty of EngineeringShahid Bahonar University of KermanKermanIran

Personalised recommendations