Kybernetika 53 no. 2, 296-330, 2017

Improving the performance of semiglobal output controllers for nonlinear systems

Abdallah Benabdallah and Walid HdidiDOI: 10.14736/kyb-2017-2-0296

Abstract:

For a large class of nonlinear control systems, the main drawback of a semiglobal stabilizing output feedback controllers $(\mathcal{U}_R)_{R>0}$ with increasing regions of attraction $(\Omega_R)_{R>0}$ is that, when the region of attraction $\Omega_R$ is large, the convergence of solutions of the closed-loop system to the origin becomes slow. To improve the performance of a semiglobal controller, we look for a new feedback control law that preserves the semiglobal stability of the nonlinear system under consideration and that is equal to some "fast" controller $\mathcal{U}_{R_0}$ on a neighborhood of the origin. Under an input-output-to-state stability (IOSS) assumption, we propose a new semiglobal stabilizing hybrid feedback controller that unifies a "slow" controller that has a large region of attraction with a "fast" controller having a small region of attraction. This unification is inspired from the elegant hybrid unification of a local controller with a global one given in \cite{Pri11}. Moreover, this unification is different from the recent result \cite{Pri13}, since in the cited paper the objective is just the stabilization; whereas in our study, the objective is the stabilization with \textit{high performance}. Finally, we illustrate our main result by means of two numerical examples.

Keywords:

nonlinear system, hybrid output feedback, semiglobal output stabilization, local performance

Classification:

93C10, 93D15

paper.pdf

References:

  1. T. Ahmed-Ali, V. Van Assche, J. Massieu and P. Dorleans: Continuous-discrete observer for state affine systems with sampled and delayed measurements. IEEE Trans. Automat. Control 58 (2013), 4, 1085-1091.   DOI:10.1109/tac.2012.2225555
  2. T. Ahmed-Ali, I. Karafyllis and F. Lamnabhi-Lagarrigue: Global exponential sampled-data observers for nonlinear systems with delayed measurements. System Control Lett. 62 (2013), 7, 539-549.   DOI:10.1016/j.sysconle.2013.03.008
  3. V. Andrieu and L. Praly: A unifying point of view on output feedback designs for global asymptotic stabilization. Automatica 45 (2009), 8, 1789-1798.   DOI:10.1016/j.automatica.2009.04.015
  4. D. V. Efimov: Uniting global and local controllers under acting disturbances. Automatica 42 (2006), 489-495.   DOI:10.1016/j.automatica.2005.11.003
  5. D. V. Efimov, A. Loria and E. Panteley: Robust output stabilization: improving performance via supervisory control. Int. J. Robust Nonlinear Control 21 (2011), 10, 1219-1236.   DOI:10.1002/rnc.1660
  6. R. Freeman and P. Kokotovic: Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. Birkhauser, Boston 1996.   DOI:10.1007/978-0-8176-4759-9
  7. J. Gauthier, H. Hammouri and S. Othman: A simple observer for nonlinear systems: Application to bioreactor. IEEE Trans. Automat. Control 37 (1992), 875-880.   DOI:10.1109/9.256352
  8. R. Goebel, R. G. Sanfelice and A. R. Teel: Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press 2012.   CrossRef
  9. R. Geobel and A. R. Teel: Solutions to hybrid inclusions via set and graphical convergence with stability theory applications. Automatica 42 (2006), 573-587.   DOI:10.1016/j.automatica.2005.12.019
  10. A. Isidori: Nonlinear Control Systems. (Third edition) Springer Verlag, London 1995.   DOI:10.1007/978-1-84628-615-5
  11. Z. P. Jiang: Discussion on the paper "Global asymptotic output feedback stabilization of feedforward systems", by F. Mazenc and J. C. Vivalda. Europ. J. Control 8 (2002), 6, 531-534.   DOI:10.3166/ejc.8.531-534
  12. P. Jouan and J. Gauthier: Finite singularities of nonlinear systems, output stabilization, observability and observers. J. Dynamical Control Systems 2 (1996), 2, 255-288.   DOI:10.1007/bf02259528
  13. H. Khalil and F. Esfandiari: Semiglobal stabilization of a class of nonlinear systems using output feedback. IEEE Trans. Automat. Control 38 (1993), 2, 1412-1415.   DOI:10.1109/9.237658
  14. M. Krichman, E. Sontag and Y. Wang: Input-output-to-state stability. SIAM J. Control Optim. 39 (2001), 1874-1928.   DOI:10.1137/s0363012999365352
  15. R. Marino and P. Tomei: A class of globally output feedback stabilizable nonlinear nonminimum phase systems. IEEE Trans. Automat. Control 50 (2005), 2097-2101.   DOI:10.1109/tac.2005.858652
  16. F. Mazenc and J. C. Vivalda: Global asymptotic output feedback stabilization of feedforward systems. Europ. J. Control 8 (2002), 6, 519-530.   DOI:10.3166/ejc.8.519-530
  17. Z. Pan, K. Ezal, A. J. Krener and P. V. Kokotovic: Backstepping design with local optimality matching. IEEE Trans. Automat. Control 46 (2001), 7, 1014-1027.   DOI:10.1109/9.935055
  18. L. Praly: Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate. IEEE Trans. Automat. Control 48 (2003), 12, 1103-1108.   DOI:10.1109/tac.2003.812819
  19. L. Praly and A. R. Teel: Tools for semiglobal stabilization by partial state and output feedback. SIAM J. Control Optim. 33 (1995), 5, 1443-1488.   DOI:10.1137/s0363012992241430
  20. C. Prieur, R. Goebel and A. R. Teel: Hybrid feedback control and robust stabilization of nonlinear systems. IEEE Trans. Automat. Control 52 (2007), 11, 2103-2117.   DOI:10.1109/tac.2007.908320
  21. C. Prieur and A. R. Teel: Uniting local and global output feedback controllers. IEEE Trans. Automat. Control 56 (2011), 1636-1649.   DOI:10.1109/tac.2010.2091436
  22. C. Qian and W. Lin: Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm. IEEE Trans. Automat. Control 47 (2002), 10, 1710-1715.   DOI:10.1109/tac.2002.803542
  23. R. G. Sanfelice and R. Goebel: Generalized solutions to hybrid dynamical systems. ESAIM: Control, Optimisation and Calculus of Variations 14 (2008), 699-724.   DOI:10.1051/cocv:2008008
  24. R. G. Sanfelice and C. Prieur: Robust supervisory control for uniting two output-feedback hybrid controllers with different objectives. Automatica 49 (2013), 1958-1969.   DOI:10.1016/j.automatica.2013.03.009
  25. Y. Shen, D. Zhang, Y. Huang and Y. Liu: Global $\mathcal K$-exponential stabilization of a class of nonlinear networked control systems. Int. J. Systems Sci. 47 (2016), 15, 3545-3553.   DOI:10.1080/00207721.2015.1091899
  26. Y. Shen, D. Zhang and X. Xia: Continuous output feedback stabilization for nonlinear systems based on sampled and delayed output measurements. Int. J. Robust. Nonlinear Control 26 (2016), 14, 3075-3087.   DOI:10.1080/00207721.2015.1091899
  27. E. D. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems. (Second edition) Springer, New York 1998.   DOI:10.1007/978-1-4612-0577-7
  28. E. D. Sontag and Y. Wang: Output-to-state stability and detectability of nonlinear systems. Systems Control Lett. 29 (1997), 5, 279-290.   DOI:10.1016/s0167-6911(97)90013-x
  29. S. Tarbouriech, G. Garcia, J. M. G. da Silva Jr. and I. Queinnec: Stability and Stabilization of Linear Systems with Saturating Actuators. Springer-Verlag, London 2011.   DOI:10.1007/978-0-85729-941-3
  30. A. R. Teel and N. Kapoor: Uniting global and local controllers. In: European Control Conference, Brussels 1997.   CrossRef