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Filtered Repetitive Control with Nonlinear Systems: An Actuator-Focused Design Method

  • Quan QuanEmail author
  • Kai-Yuan Cai
Chapter

Abstract

The internal model principle (IMP) was first proposed by Francis and Wonham [2, 3]. It states that if any exogenous signal can be regarded as the output of an autonomous system, then the inclusion of this signal model, namely, internal model, in a stable closed-loop system can assure asymptotic tracking or asymptotic rejection of the signal. Until now, to the best of the authors’ knowledge, there exist at least two viewpoints on IMP. In the early years, for linear time-invariant (LTI) systems, IMP implies that the internal model is to supply closed-loop transmission zeros which cancel the unstable poles of the disturbances and reference signals. This is called cancelation viewpoint here and only works for problems able to be formulated in terms of transfer functions. In the mid-1970s, Francis and Wonham proposed the geometric approach [4] to design an internal model controller [2, 3]. The purpose of internal models is to construct an invariant subspace for the closed-loop system and make the regulated output zero at each point of the invariant subspace. This is called geometrical viewpoint here.

References

  1. 1.
    Quan, Q., & Cai, K.-Y. (2019). Repetitive control for nonlinear systems: an actuator-focussed design method. International Journal of Control,.  https://doi.org/10.1080/00207179.2019.1639077.CrossRefGoogle Scholar
  2. 2.
    Francis, B. A., & Wonham, W. M. (1976). The internal model principle of control theory. Automatica, 12(5), 457–465.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Wonham, W. M. (1976). Towards an abstract internal model principle. IEEE Transaction on Systems, Man, and Cybernetics, 6(11), 735–740.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Wonham, W. M. (1979). Linear multivariable control: A geometric approach. New York: Springer.CrossRefGoogle Scholar
  5. 5.
    Isidori, A., & Byrnes, C. I. (1990). Output regulation of nonlinear systems. IEEE Transactions on Automatic Control, 35(2), 131–140.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Isidori, A., Marconi, L., & Serrani, A. (2003). Robust autonomous guidance: An internal model-based approach. London: Springer.CrossRefGoogle Scholar
  7. 7.
    Huang, J. (2004). Nonlinear output regulation: Theory and applications. Philadelphia: SIAM.CrossRefGoogle Scholar
  8. 8.
    Memon, A. Y., & Khalil, H. K. (2010). Output regulation of nonlinear systems using conditional servocompensators. Automatica, 46(7), 1119–1128.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Wieland, P., Sepulchre, R., & Allgower, F. (2011). An internal model principle is necessary and sufficient for linear output synchronization. Automatica, 47(5), 1068–1074.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Knobloch, H. W., Isidori, A., & Flockerzi, D. (2014). Disturbance attenuation for uncertain control systems. London: Springer.CrossRefGoogle Scholar
  11. 11.
    Chen, Z., & Huang, J. (2014). Stabilization and regulation of nonlinear systems: A robust and adaptive approach. London: Springer.Google Scholar
  12. 12.
    Trip, S., Burger, M., & De Persis, C. (2016). An internal model approach to (optimal) frequency regulation in power grids with time-varying voltages. Automatica, 64, 240–253.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hara, S., Yamamoto, Y., Omata, T., & Nakano, M. (1988). Repetitive control system: A new type servo system for periodic exogenous signals. IEEE Transactions on Automatic Control, 33(7), 659–668.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Weiss, G., & Häfele, M. (1999). Repetitive control of MIMO systems using H\(^{\infty }\) design. Automatica, 35(7), 1185–1199.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Byrnes, C., Laukó, I., Gilliam, D., & Shubov, V. (2000). Output regulation for linear distributed parameter systems. IEEE Transactions on Automatic Control, 45(12), 2236–2252.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hämäläinen, T. (2005). Robust low-gain regulation of stable infinite-dimensional systems. Doctoral dissertation, Tampere University of Technology, Tampere, Finland.Google Scholar
  17. 17.
    Immonen, E. (2007). Practical output regulation for bounded linear infinite-dimensional state space systems. Automatica, 43(5), 786–794.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Immonen, E. (2007). On the internal model structure for infinite-dimensional systems: two common controller types and repetitive control. SIAM Journal on Control and Optimization, 45(6), 2065–2093.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hämäläinen, T., & Pohjolainen, S. (2010). Robust regulation of distributed parameter systems with infinite-dimensional exosystems. SIAM Journal on Control and Optimization, 48(8), 4846–4873.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Paunonen, L., & Pohjolainen, S. (2010). Internal model theory for distributed parameter systems. SIAM Journal on Control and Optimization, 48(7), 4753–4775.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Natarajan, V., Gilliam, D. S., & Weiss, G. (2014). The state feedback regulator problem for regular linear systems. IEEE Transactions on Automatic Control, 59(10), 2708–2723.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Xu, X., & Dubljevic, S. (2017). Output and error feedback regulator designs for linear infinite-dimensional systems. Automatica, 83, 170–178.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Quan, Q., & Cai, K.-Y. (2010). A new viewpoint on the internal model principle and its application to periodic signal tracking. In The 8th World Congress on Intelligent Control and Automation, Jinan, Shandong (pp. 1162–1167).Google Scholar
  24. 24.
    Burton, T. A. (1985). Stability and periodic solutions of ordinary and functional differential equations. London: Academic Press.zbMATHGoogle Scholar
  25. 25.
    Hale, J. K., & Lunel, S. M. V. (1993). Introduction to functional differential equations. New York: Springer.CrossRefGoogle Scholar
  26. 26.
    Quan, Q., & Cai, K.-Y. (2011). A filtered repetitive controller for a class of nonlinear systems. IEEE Transaction on Automatic Control, 56, 399–405.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Spong, M. W., & Vidyasagar, M. (1989). Robot dynamics and control. New York: Wiley.Google Scholar
  28. 28.
    Lewis, F. L., Abdallah, C. T., & Dawson, D. M. (1993). Control of robot manipulators. New York: Macmillan.Google Scholar
  29. 29.
    Khalil, H. K. (2002). Nonlinear systems. Englewood Cliffs: Prentice-Hall.zbMATHGoogle Scholar
  30. 30.
    Shkolnikov, I. A., & Shtessel, Y. B. (2002). Tracking in a class of nonminimum-phase systems with nonlinear internal dynamics via sliding mode control using method of system center. Automatica, 38(5), 837–842.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Quan, Q., & Cai, K.-Y. (2017). A new generator of causal ideal internal dynamics for a class of unstable linear differential equations. International Journal of Robust and Nonlinear Control, 27(12), 2086–2101.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Quan, Q., & Cai, K.-Y. (2012). A new method to obtain ultimate bounds and convergence rates for perturbed time-delay systems. International Journal of Robust Nonlinear Control, 22, 1873–1880.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.School of Automation Science and Electrical EngineeringBeijing University of Aeronautics and AstronauticsBeijingChina

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