Adaptive compensators for perturbed positive real infinite-dimensional systems

Ruth Curtain; Michael Demetriou; Kazufumi Ito

International Journal of Applied Mathematics and Computer Science (2003)

  • Volume: 13, Issue: 4, page 441-452
  • ISSN: 1641-876X

Abstract

top
The aim of this investigation is to construct an adaptive observer and an adaptive compensator for a class of infinite-dimensional plants having a known exogenous input and a structured perturbation with an unknown constant parameter, such as the case of static output feedback with an unknown gain. The adaptive observer uses the nominal dynamics of the unperturbed plant and an adaptation law based on the Lyapunov redesign method. We obtain conditions on the system to ensure uniform boundedness of the estimator dynamics and the parameter estimates, and the convergence of the estimator error. For the case of a known periodic exogenous input we design an adaptive compensator which forces the system to converge to a unique periodic solution. We illustrate our approach with a delay example and a diffusion example for which we obtain convincing numerical results.

How to cite

top

Curtain, Ruth, Demetriou, Michael, and Ito, Kazufumi. "Adaptive compensators for perturbed positive real infinite-dimensional systems." International Journal of Applied Mathematics and Computer Science 13.4 (2003): 441-452. <http://eudml.org/doc/207656>.

@article{Curtain2003,
abstract = {The aim of this investigation is to construct an adaptive observer and an adaptive compensator for a class of infinite-dimensional plants having a known exogenous input and a structured perturbation with an unknown constant parameter, such as the case of static output feedback with an unknown gain. The adaptive observer uses the nominal dynamics of the unperturbed plant and an adaptation law based on the Lyapunov redesign method. We obtain conditions on the system to ensure uniform boundedness of the estimator dynamics and the parameter estimates, and the convergence of the estimator error. For the case of a known periodic exogenous input we design an adaptive compensator which forces the system to converge to a unique periodic solution. We illustrate our approach with a delay example and a diffusion example for which we obtain convincing numerical results.},
author = {Curtain, Ruth, Demetriou, Michael, Ito, Kazufumi},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {adaptive controllers; positive real systems; infinite dimensional systems; adaptive observer; adaptive compensator; infinite-dimensional systems; stability; Lyapunov redesign method},
language = {eng},
number = {4},
pages = {441-452},
title = {Adaptive compensators for perturbed positive real infinite-dimensional systems},
url = {http://eudml.org/doc/207656},
volume = {13},
year = {2003},
}

TY - JOUR
AU - Curtain, Ruth
AU - Demetriou, Michael
AU - Ito, Kazufumi
TI - Adaptive compensators for perturbed positive real infinite-dimensional systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2003
VL - 13
IS - 4
SP - 441
EP - 452
AB - The aim of this investigation is to construct an adaptive observer and an adaptive compensator for a class of infinite-dimensional plants having a known exogenous input and a structured perturbation with an unknown constant parameter, such as the case of static output feedback with an unknown gain. The adaptive observer uses the nominal dynamics of the unperturbed plant and an adaptation law based on the Lyapunov redesign method. We obtain conditions on the system to ensure uniform boundedness of the estimator dynamics and the parameter estimates, and the convergence of the estimator error. For the case of a known periodic exogenous input we design an adaptive compensator which forces the system to converge to a unique periodic solution. We illustrate our approach with a delay example and a diffusion example for which we obtain convincing numerical results.
LA - eng
KW - adaptive controllers; positive real systems; infinite dimensional systems; adaptive observer; adaptive compensator; infinite-dimensional systems; stability; Lyapunov redesign method
UR - http://eudml.org/doc/207656
ER -

References

top
  1. Balakrishnan A.V. (1995): On a generalization of the Kalman-Yakubovic lemma. - Appl. Math. Optim., Vol. 31, No. 2, pp. 177-187. Zbl0821.47031
  2. Baumeister J., Scondo W., Demetriou M. and Rosen I. (1997): On-line parameter estimation for infinite dimensional dynamical systems. - SIAM J. Contr. Optim., Vol. 3, No. 2, pp. 678-713. Zbl0873.93026
  3. Curtain R.F. (1996a): Corrections to the Kalman-Yakubovich-Popov Lemma for Pritchard-Salamon systems. - Syst. Contr. Lett., Vol. 28, No. 4, pp. 237-238. Zbl0883.93032
  4. Curtain R.F. (1996b): The Kalman-Yakubovich-Popov Lemma for Pritchard-Salamon systems. - Syst. Contr. Lett., Vol. 27, No. 1, pp. 67-72. Zbl0877.93065
  5. Curtain R.F. (2001): Linear operator inequalities for stable weakly regular linear systems. - Math. Contr.Sign. Syst., Vol. 14, No. 4, pp. 299-337. Zbl1114.93029
  6. Curtain R.F., Demetriou M.A. and Ito K. (1997): Adaptive observers for structurally perturbed infinite dimensional systems. - Proc. IEEE Conf. Decision and Control, San Diego, California, pp. 509-514. 
  7. Curtain R.F. and Pritchard A.J. (1978): Infinite Dimensional Linear Systems Theory. - Berlin: Springer. Zbl0389.93001
  8. Curtain R.F. and Zwart H.J. (1995): An Introduction to Infinite Dimensional Linear Systems Theory. - Berlin: Springer. Zbl0839.93001
  9. Demetriou M.A., Curtain R.F. and Ito K. (1998): Adaptive observers for structurally perturbed positive real delay systems. - Proc. 4th Int. Conf. Optimization: Techniques and Applications, Curtin University of Technology, Perth, Australia, pp. 345-351. 
  10. Demetriou M.A. and Ito K. (1996): Adaptive observers for a class of infinite dimensional systems. - Proc. 13th World Congress, IFAC, San Francisco, CA, pp. 409-413. 
  11. Hansen S. and Weiss G. (1997): New results on the operator Carleson measure criterion. - IMA J. Math. Contr. Inf., Vol. 14, No. 1, pp. 3-32. Zbl0874.93031
  12. Infante E. and Walker J. (1977): A Lyapunov functional for scalar differential difference equations. - Proc. Royal Soc. Edinburgh, Vol. 79A, Nos. 3-4, pp. 307-316. Zbl0417.34111
  13. Ito K. and Kappel F. (1991): A uniformly differentiable approximation scheme for delay systems using splines. - Appl. Math. Optim., Vol. 23, No. 3, pp. 217-262. Zbl0738.65055
  14. Khalil H.K. (1992): Nonlinear Systems. - New York: Macmillan. Zbl0969.34001
  15. Narendra K.S. and Annaswamy A.M. (1989): Stable Adaptive Systems. - Englewood Cliffs, NJ: Prentice Hall. Zbl0758.93039
  16. Oostveen J.C. and Curtain R.F. (1998): Riccati equations for strongly stabilizable bounded linear systems. - Automatica, Vol. 34, No. 8, pp. 953-967. Zbl0979.93092
  17. Pandolfi L. (1997): The Kalman-Yacubovich theorem: An overview and new results for hyperbolic boundary control systems. - Nonlin. Anal. Theory Meth. Applic., Vol. 30, No. 30, pp. 735-745. Zbl0896.93026
  18. Pandolfi L. (1998): Dissipativity and Lur'e problem for parabolic boundary control systems. - SIAM J. Contr. Optim., Vol. 36, No. 6, pp. 2061-2081. Zbl0913.43001
  19. Pazy A. (1983): Semigroups of Linear Operators and Applications to Partial Differential Equations. - New York: Springer. Zbl0516.47023
  20. Prato G.D. and Ichikawa A. (1988): Quadratic control for linear periodic systems. - Appl. Math. Optim., Vol. 18, No. 1, pp. 39-66. Zbl0647.93057
  21. Salamon D. (1984): Control and Observation of Neutral Systems.- Research Notes in Mathematics, 91, Boston: Pitman Advanced Publishing Program. Zbl0546.93041
  22. Staffans O.J. (1995): Quadratic optimal control of stable abstract linear systems. - Proc. IFIP Conf. Modelling and Optimization of DPS with Applications to Engineering, Warsaw, Poland, pp. 167-174. Zbl0878.49023
  23. Staffans O.J. (1997): Quadratic optimal control of stable well-posed linear systems. - Trans. Amer. Math. Soc., Vol. 349, No. 9, pp. 3679-3715. Zbl0889.49023
  24. Staffans O.J. (1998): Coprime factorizations and well-posed linear systems. - SIAM J. Contr. Optim., Vol. 36, No. 4, pp. 1268-1292. Zbl0919.93040
  25. Staffans O.J. (1999): Quadratic optimal control of well-posed linear systems. - SIAM J. Contr. Optim., Vol. 37, No. 1, pp. 131-164. Zbl0955.49018
  26. Titchmarsh E.C. (1962): Introduction to the Theory of Fourier Integrals. - New York: Oxford University Press. Zbl0017.40404
  27. Weiss M. (1994): Riccati Equations in Hilbert Spaces: A Popov Function Approach. - Ph.D. Thesis, Rijksuniversiteit Groningen, The Netherlands. 
  28. Weiss M. (1997): Riccati equation theory for Pritchard-Salamon systems: a Popov function approach. - IMA J. Math. Contr. Inf., Vol. 14, No. 1, pp. 45-83. Zbl0876.93023
  29. Weiss M. and Weiss G. (1997): Optimal control of stable weakly regular linear systems. - Math. Contr. Signals Syst., Vol. 10, No. 4, pp. 287-330. Zbl0884.49021

NotesEmbed ?

top

You must be logged in to post comments.