Kybernetika 55 no. 4, 727-739, 2019

Inverse optimal control for linearizable nonlinear systems with input delays

Xiushan Cai, Jie Wu, Xisheng Zhan and Xianhe ZhangDOI: 10.14736/kyb-2019-4-0727

Abstract:

We consider inverse optimal control for linearizable nonlinear systems with input delays based on predictor control. Under a continuously reversible change of variable, a nonlinear system is transferred to a linear system. A predictor control law is designed such that the closed-loop system is asymptotically stable. We show that the basic predictor control is inverse optimal with respect to a differential game. A mechanical system is provided to illustrate the effectiveness of the proposed method.

Keywords:

nonlinear systems, inverse optimality, predictor control, input delays

Classification:

93Cxx, 93Dxx

paper.pdf

References:

  1. N. Bekiaris-Liberis and M. Krstic: Compensation of time-varying input and state delays for nonlinear systems. J. Dynamic Systems, Measurement Control 134 (2012), 1-14.   DOI:10.1115/1.4005278
  2. N. Bekiaris-Liberis and M. Krstic: Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations. Automatica 49 (2013), 1576-1590.   DOI:10.1016/j.automatica.2013.02.050
  3. N. Bekiaris-Liberis and M. Krstic: Compensation of wave actuator dynamics for nonlinear systems. IEEE Trans. Automat. Control 59 (2014), 1555-1570.   DOI:10.1109/tac.2014.2309057
  4. F. Bullo and A. Lewis: Reduction, linearization, and stability of relative equilibria for mechanical systems on riemannian manifolds. Acta Applicandae Mathematicae 99 (2007), 53-95.   DOI:10.1007/s10440-007-9155-5
  5. X. Cai, N. Bekiaris-Liberis and M. Krstic: Input-to-state stability and inverse optimality of linear time-varying-delay predictor feedbacks. IEEE Trans. Automat. Control 63 (2018), 233-240.   DOI:10.1109/tac.2017.2722104
  6. X. Cai, N. Bekiaris-Liberis and M. Krstic: Input-to-state stability and inverse optimality of predictor feedback for multi-input linear systems. Automatica 103 (2019), 549-557.   DOI:10.1016/j.automatica.2019.02.038
  7. X. Cai and M. Krstic: Nonlinear control under wave actuator dynamics with time- and state-dependent moving boundary. Int. J. Robust. Nonlinear Control 25 (2015), 222-253.   DOI:10.1002/rnc.3083
  8. X. Cai and M. Krstic: Nonlinear stabilization through wave PDE dynamics with a moving uncontrolled boundary. Automatica 68 (2016), 27-38.   DOI:10.1016/j.automatica.2016.01.043
  9. X. Cai, L. Liao, J. Zhang and W. Zhang: Oberver design for a class of nonlinear system in cascade with counter-convecting transport dynamics. Kybernetika 52 (2016), 76-88.   DOI:10.14736/kyb-2016-1-0076
  10. X. Cai, Y. Lin and L. Liu: Universal stabilisation design for a class of non-linear systems with time-varying input delays. IET Control Theory Appl. 9 (2015), 1481-1490.   DOI:10.1049/iet-cta.2014.1085
  11. X. Cai, C. Lin, L. Liu and X. Zhan: Inverse optimal control for strict-feedforward nonlinear systems with input delays. Int. J. Robust. Nonlinear Control 28 (2018), 2976-2995.   DOI:10.1002/rnc.4062
  12. M. Jankovic: Control Lyapunov Razumikhin functions and robust stabilization of time delay systems. IEEE Trans. Automat. Control 46 (2001), 1048-1060.   DOI:10.1109/9.935057
  13. M. Jankovic: Control of nonlinear systems with time delay. In: Proc. 42nd IEEE conference on Decision and Control, Maui 2003, pp. 4545-4550.   DOI:10.1109/cdc.2003.1272267
  14. I. Karafyllis and M. Krstic: Predictor Feedback for Delay Systems: Implementations and Approximations. Springer, 2016.   DOI:10.1007/978-3-319-42378-4
  15. M. Krstic: Feedback linearizability and explicit integrator forwarding controllers for classes of feedforward systems. IEEE Trans. Automat. Control 49 (2004), 1668-1681.   DOI:10.1109/tac.2004.835361
  16. M. Krstic: Integrator forwarding control laws for some classes of linearizable feedforward systems. In: Proc. American Control Conference, Boston, Massachusetts 2004, 4360-4365.   DOI:10.23919/acc.2004.1383994
  17. M. Krstic: Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch. Automatica 44 (2008), 2930-2935.   DOI:10.1016/j.automatica.2008.04.010
  18. M. Krstic: Input delay compensation for forward complete and feed forward nonlinear systems. IEEE Trans. Automat. Control 55 (2010), 287-303.   DOI:10.1109/tac.2009.2034923
  19. M. Krstic and Z. Li: Inverse optimal design of input-to-state stabilizing nonlinear controllers. IEEE Trans. Automat. Control 43 (1998), 336-350.   DOI:10.1109/9.661589
  20. W. Respondek and S. Ricardo: On linearization of mechanical control systems. In: Proc. 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Bertinoro 2012.   DOI:10.3182/20120829-3-it-4022.00053
  21. I. Tall: Linearizable feedforward systems: A special class. In: Proc. IEEE Multi-conference on Systems and Control, Texas 2008.   DOI:10.1109/cca.2008.4629662
  22. A. van der Schaft: Linearization of Hamiltonian and gradient systems. IMA J. Math. Control Inform. 1 (1994), 185-198.   DOI:10.1093/imamci/1.2.185
  23. X. Zhan, L. Cheng, J. Wu and Q. Yang: Optimal modified performance of MIMO networked control systems with multi-parameter constraints. ISA Trans. 84 (2019), 111-117.   DOI:10.1016/j.isatra.2018.09.018
  24. X. Zhan, Z.Guan, X. Zhang and F. Yuan: Optimal tracking performance and design of networked control systems with packet dropout. J. the Franklin Inst. 350 (2013), 3205-3216.   DOI:10.1016/j.jfranklin.2013.06.019
  25. X. Zhan, J. Wu, T. Jiang and X. Jiang: Optimal performance of networked control systems under the packet dropouts and channel noise. ISA Trans. 58 (2015), 214-221.   DOI:10.1016/j.isatra.2015.05.012
  26. Z. Zhang, S. Xu and B. Zhang: Asymptotic tracking control of uncertain nonlinear systems with unknown actuator nonlinearity. IEEE Trans. Automatic Control 59 (2014), 1336-1341.   DOI:10.1109/tac.2013.2289704
  27. Z. Zhang, S. Xu and B. Zhang: Exact tracking control of nonlinear systems with time delays and dead-zone input. Automatica 52 (2015), 272-276.   DOI:10.1016/j.automatica.2014.11.013