Kybernetika 49 no. 6, 911-934, 2013

A sample-time adjusted feedback for robust bounded output stabilization

Patricio Ordaz, Hussain Alazki and Alexander Poznyak

Abstract:

This paper deals with a bounded control design for a class of nonlinear systems where the mathematical model may be not explicitly given. This class of uncertain nonlinear systems governed by a system of ODE with quasi-Lipschitz right-hand side and containing external perturbations as well. The Attractive Ellipsoid Method (AEM) application permits to describe the class of nonlinear feedbacks (containing a nonlinear projection operator, a linear state estimator and a feedback matrix-gain) guaranteeing a boundedness of all possible trajectories around the origin. To fulfill this property some modification of AEM are introduced: basically, some sort of sample-time corrections of the feedback parameters are required. The optimization of feedback within this class of controllers is associated with the selection of the feedback parameters which provide the trajectory converges within an ellipsoid of a "minimal size". The effectiveness of the suggested approach is illustrated by its application to a flexible arm system).

Keywords:

state estimation, sample-time data, attractive ellipsoid, saturated control process, flexible arm system

Classification:

93E12, 62A10

References:

  1. T. Ahmed-Ali and F. Lamnabhi-Lagarrigue: High gain observer design for some networked control systems. IEEE Trans. Automat. Control 57 (2012), 4, 995-1000.   CrossRef
  2. F. Blanchini: Set invariance in control, a survey. Automatica 35 (1999), 11, 1747-1767.   CrossRef
  3. F. Blanchini and F. Miani: Set-Theoretic Methods in Control. Birkhauser, Boston 2008.   CrossRef
  4. S.\thinspace A. Bortoff and A.\thinspace F. Lynch: Synthesis of Optimal Nonlinear Observersr. 34th IEEE Conference on Decision and Control 1 (1995), 95-100.   CrossRef
  5. M.\thinspace A. Dahleh, J.\thinspace B. Pearson and Jr.: Optimal rejection of persistent disturbances, robust stability, and mixed sensitivity minimization. IEEE Trans. on Automat. Control {\mi 33} (1988), 8, 722-731.   CrossRef
  6. J. Davila and A. Poznyak: Sliding mode parameter adjustment for perturbed linear systems with actuators via invariant ellipsoid method. Internat. J. Robust and Nonlinear Control 21 (2011), 473-487.   CrossRef
  7. G. J. Duncan and F. C. Schweppe: Control of linear dynamic systems with set constrained disturbances. IEEE Trans. Automat. Control 16 (1971), 5, 411-423.   CrossRef
  8. S. Gonzalez, A. Polyakov and A. Poznyak: Using the method of invariant ellipsoids for linear robust output stabilization of spacecraft. Automat. Remote Control 72 (2011), 3, 540-555.   CrossRef
  9. P. Ioannou and J. Sun: Robust Adaptive Control. Prentice Hall, Inc, 1996.   CrossRef
  10. M. L. Jong and H. L. Jay: Approximate dynamic programming-based approaches for input-output data-driven control of nonlinear processes. Automatica 41 (2005), 1281-1288.   CrossRef
  11. P. T. Kabamba and S. Hara: Worst-case analysis and design of sampled-data control systems. IEEE Trans. Automat. Control 38 (1993), 9, 1337-1358.   CrossRef
  12. A. B. Kurzhanski and V. M. Veliov: Modeling Techniques and Uncertain Systems. Birkhauser, New York 1994.   CrossRef
  13. S. R. Kou, D. L. Elliott and T. J. Tarn: Exponential observers for non-linear dynamic systems. Inform. Control 29 (1975), 393-428.   CrossRef
  14. W. Min, Z. Lan Zhou and S. Jinhua: Design of observer-based $H_{\infty }$ robust repetitive-control system. IEEE Trans. Automat. Control 56 (2011), 6, 1452-1457.   CrossRef
  15. K. S. Narendra and A. M. Annaswamy: Stable Adaptive Systems. Dover Publications Inc., 2005.   CrossRef
  16. A. Nazin, B. Polyak and M. Topunov: Rejection of bounded exogenous disturbances by the method of invariant ellipsoids. Automat. Remote Control 68 (2007), 3, 467-486.   CrossRef
  17. J. O'Reilly: Observers for Linear Systems. Academic Press, 1983.   CrossRef
  18. P. Ordaz and A. Poznyak: Stabilizaton of furuta's pendulum with out model: Attractive ellipsoid method. In: 51th IEEE Conference of Decision and Control, Hawaii 2012, pp. 7285-7290.   CrossRef
  19. A. Poliakov and A. Poznyak: Invariant ellipsoid method for minimization of unmatched disturbances effects in sliding mode control. Automatica 47 (2011), 1450-1454.   CrossRef
  20. A. Poznyak: Advanced Mathematical Tools for Automatic Control Engineers: Deterministic techniques. Vol. 1. Elsevier 2008.   CrossRef
  21. A. Poznyak, V. Azhmyakov and M. Mera: Practical output feedback stabilization for a class of continuous-time dynamic systems under sample-data outputs. Internat. J. Control 8 (2011), 4, 1408-1416.   CrossRef
  22. T. P. Proychev and R. L. Mishkov: Transformation of nonlinear systems in observer canonical from with reduced dependency on derivatives of the input. Automatica 29 (1993), 2, 495-498.   CrossRef
  23. W. Rudin: Functional Analysis. Second edition. MacGraw-Hill, Inc. 1991.   CrossRef
  24. H. J. Sussmann, E. D. Sontag and Y. Yang: A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Automat. Control 39 (1994), 12, 2411-2425.   CrossRef
  25. H. Tingshu and L. Zongli: Control Systems with Actuator Saturation: Analyze and Design. Birkhauser, Boston 2001.   CrossRef
  26. A. R. Teel: A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans. Automat. Control 41 (1996), 9, 1256-1270.   CrossRef
  27. V. Utkin: Sliding mode control design principles and applications to electric drives. IEEE Trans. on Industrial Electronics 40 (1993), 1, 23-36.   CrossRef
  28. B. Walcott, M. Corless and S. Zak: Comparative study of non-linear state-observation techniques. Internat. J. Control 45 (1987), 6, 2109-2132.   CrossRef
  29. M. Zeitz: The extended Luenberger observer for nonlinear systems. Systems Control Lett. 9 (1987), 149-156.   CrossRef