Authors

Kenji Fujimoto, Jacquelien M.A. Scherpen, W.Steven Gray

Abstract

This paper addresses the issue of state-space realizations for nonlinear adjoint operators. In particular, the relationships between nonlinear Hilbert adjoint operators, Hamiltonian extensions and port-controlled Hamiltonian systems are established. Then, characterizations of the adjoints of controllability, observability and Hankel operators are derived from this analysis. The state-space realizations of such adjoint operators provide new insights on singular value analysis and duality issues in nonlinear control systems theory. Finally, a duality between the controllability and observability energy functions is proved.

Keywords

controllability, duality, nonlinear systems, observability, state-space realization

Citation

BibTeX

@article{Fujimoto_2002,
  title={{Hamiltonian realizations of nonlinear adjoint operators}},
  volume={38},
  ISSN={0005-1098},
  DOI={10.1016/s0005-1098(02)00079-1},
  number={10},
  journal={Automatica},
  publisher={Elsevier BV},
  author={Fujimoto, Kenji and Scherpen, Jacquelien M.A. and Gray, W.Steven},
  year={2002},
  pages={1769--1775}
}

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References

  • Arnold, (1989)
  • Ball, J. A. & Van der Schaft, A. J. J-inner-outer factorization, J-spectral factorization, and robust control for nonlinear systems. IEEE Trans. Automat. Contr. 41, 379–392 (1996) – 10.1109/9.486639
  • Crouch, Variational and Hamiltonian control systems. (1987)
  • Fujimoto, K. & Sugie, T. Canonical Transformation and Stabilization of Generalized Hamiltonian Systems. IFAC Proceedings Volumes 31, 523–528 (1998) – 10.1016/s1474-6670(17)40390-9
  • Gray, W. S. & Scherpen, J. M. A. Hankel operators and Gramians for nonlinear systems. Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171) vol. 2 1416–1421 – 10.1109/cdc.1998.758485
  • Gray, W. S. & Scherpen, J. M. A. Hankel singular value functions from Schmidt pairs for nonlinear input-output systems. Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301) 3540–3545 vol.5 (2002) doi:10.1109/acc.2002.1024477 – 10.1109/acc.2002.1024477
  • Milnor, Morse theory. (1963)
  • Scherpen, J. M. A. Balancing for nonlinear systems. Systems & Control Letters 21, 143–153 (1993) – 10.1016/0167-6911(93)90117-o
  • Scherpen, Minimality and local state decompositions of a nonlinear state-space realization using energy functions. IEEE Transactions on Automatic Control (2000)
  • SCHERPEN, J. M. A. & VAN DER SCHAFT, A. J. Normalized coprime factorizations and balancing for unstable nonlinear systems. International Journal of Control 60, 1193–1222 (1994) – 10.1080/00207179408921517
  • Van der Schaft, (2000)
  • Walsh, (1975)
  • Young, (1969)
  • Zhou, (1996)