The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation

IEEE Transactions on Automatic Control

Volumn 52, Issue 2, 2007, Pages 371-377

  Guo, Bao Zhu ^a,b   Xu, Cheng Zhong ^c,d  

^a CHINESE ACADEMY OF SCIENCES   (China)

^b UNIVERSITY OF THE WITWATERSRAND   (South Africa)

^c UNIVERSITÉ LYON 1   (France)

^d LUDONG UNIVERSITY   (China)

Author keywords

Exact controllability; Exponential stability; Feedback control; Observer; Wave equation

Indexed keywords

ASYMPTOTIC STABILITY; CLOSED LOOP CONTROL SYSTEMS; CONTROL SYSTEM STABILITY; WAVE EQUATIONS;

BOUNDARY OUTPUT FEEDBACK; EXACT CONTROLLABILITY; RIESZ BASIS;

FEEDBACK CONTROL;

EID: 33947377930     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2006.890385     Document Type: Article

References (27)

1. R. H. Cannon and E. Schmitz, "Initial experiments on the end-point control of a flexible one-link robot," Int. J. Robot. Res., vol. 3, pp. 62-75, 1984.
2. P. A. Chodavarapu and M. W. Spong, "On Noncollocated control of a single flexible link," in Proc. IEEE Int. Conf. Robotics Automation, Minneapolis, MN, 1996, pp. 1101-1106.
3. R. F. Curtain, "The Salamon-Weiss class of well-posed infinite dimensional linear systems: A survey," IMA J. Math. Control Inform., vol. 14, pp. 207-223, 1997.
4. R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear System Theory. New York: Springer-Verlag, 1995.
5. A. J. Deguenon, G. Sallet, and C. Z. Xu, "A Kalman observer for infinite-dimensional skew-symmetric systems with application to an elastic beam," in Proc. 2nd Int. Symp. Communications, Control, Signal Processing, Marrakech, Morocco, Mar. 2006.
6. E. A. Guillemin, Synthesis of Passive Networks. New York: Wiley, 1957.
7. B. Z. Guo and J. M. Wang, "Riesz basis generation of an abstract second-order partial differential equation system with general non-separated boundary conditions," Numer. Funct. Anal. Optim., vol. 27, no. 3-4, pp. 291-328, 2006.
8. B. Z. Guo and X. Zhang, "The regularity of the wave equation with partial Dirichlet control and colocated observation," SIAM J. Control Optim., vol. 44, no. 5, pp. 1598-1613, 2005.
9. A. Guesmia, "A new approach of stabilization of nondissipative distributed systems," SIAM J. Control Optim., vol. 42, pp. 24-52, 2003.
10. L. Y. Liu and K. Yuan, "Noncollocated passivity-based control of a single-link flexible manipulator," Robotica, vol. 21, pp. 117-135, 2003.
11. I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, ser. Trans. Math. Monographs. Providence, RI: AMS, 1969, vol. 18.
12. W. Lacarbonara and H. Yabuno, "Closed-loop non-linear control of an initially imperfect beam with non-collocated input," J. Sound Vibrat., vol. 273, pp. 695-711, 2004.
13. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge, U.K.: Cambridge Univ. Press, 2000, vol. 1.
14. V. Mazja, Sobolev Spaces. New York: Springer-Verlag, 1985.
15. M. S. de Queiroz and C. D. Rahn, "Boundary control and noise in distributed parameter systems: An overview," Mech. Sys. Signal Process., vol. 16, pp. 19-38, 2002.
16. D. L. Russell and G. Weiss, "A general necessary condition for exact observability," SIAM J. Control Optim., vol. 32, pp. 1-23, 1994.
17. P. Rideau, Contrôle d'un assemblage de poutres flexibles par des capteurs-actionneurs ponctuels: Étude du spectre du système, Thèse de l'École Nationale Supérieure des Mines de Paris. Sophia-Antipolis, France, Nov. 1985.
18. J. H. Ryu, D. S. Kwon, and B. Hannaford, "Control of a flexible manipulator with nonconllocated feedback: Time-domain passivity approach," in Control Problems in Robotics, A. Bicchi, H. I. Christensen, and D. Prattichizzo, Eds. Berlin, Germany: Springer-Verlag, 2003, vol. 4, pp. 121-134, Springer Tracts in Advanced Robotics Series.
19. A. Smyshlyaev and M. Krstic, "Backstepping observers for a class of parabolic PDEs," Syst. Control Lett., vol. 54, pp. 613-625, 2005.
20. V. A. Spector and H. Flashner, "Modelling and design implecations of non-collocated control in flexible systems," ASME J. Dyna. Syst., Measure., Control, vol. 112, pp. 186-193, 1990.
21. F. E. Udwadia, "Noncollocated point control of nondispersive distributed-parameter systems using time delays," Appl. Math. Comput., vol. 42, pp. 23-63, 1991.
22. J. M. Wang, B. Z. Guo, and B. Chentouf, "Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach," ESAIM: Control, Optim. Calc. Variat., vol. 12, pp. 12-34, 2006.
23. S. T. Wu, "Virtual passive control of flexible arms with collocated and noncollocated feedback," J. Robot. Syst., vol. 18, pp. 645-655, 2001.
24. G. Weiss, "Admissible observation operators for linear semigroups," Israel J. Math., vol. 65, pp. 17-43, 1989.
25. B. Yang and C. D. Mote, Jr., "On time delay in noncolocated control of flexible mechanical systems," ASME J. Dyna. Syst., Measure., Control, vol. 114, pp. 409-415, 1992.
26. R. M. Young, An Introduction to Nonharmonic Fourier Series. London, U.K.: Academic, 1980.
27. K. Yuan and L. Y. Liu, "On the stability of zero dynamics of a single-link flexible manipulator for a class of parametrized outputs," J. Robot. Syst., vol. 20, pp. 581-586, 2003.