Skip to main content
SpringerLink
Log in

Geometric structure of generalized controlled Hamiltonian systems and its application

  • Published:
Science in China Series E: Technological Sciences Aims and scope Submit manuscript

Abstract

The main purpose of this paper is to provide a systematic geometric frame for generalized controlled Hamiltonian systems. The pseudo-Poisson manifold and the ω-manifold are proposed as the statespace of the generalized controlled Hamiltonian systems. A Lie group, calledN-group, and its Lie algebra, calledN-algebra, are introduced for the structure analysis of the systems. Some properties, including spectrum, structure-preservation, etc. are investigated. As an example the theoretical results are applied to power systems. The stabilization of excitation systems is investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham, R. A., Marsden, J. E., Foundations of Mechanics, 2nd ed., Benjamin: Cummings Pub. Com. Inc., 1978.

    MATH  Google Scholar 

  2. Linermann, P., Marle, C. M., Symplectic Geometry and Analytic Mechanics, Reidel: Dordrecht, 1986.

    Google Scholar 

  3. Oliver, P. J., Applications of Lie Groups to Differential Equations, New York: Soringer-Verlag, 1980.

    Google Scholar 

  4. van der Schaft, A.,L 2 Gain and Passivity Techniques in Nonlinear Control, Berlin: Springer-Verlag, 1986.

    Google Scholar 

  5. Maschke, B. M. J., van der Schaft, A. J., Port controlled Hamiltonian systems: nodeling origins and system theoretic properties, Proc. 2nd IFAC Symp. on Nonlinear Control Systems Design, NOLCOS’92, Bordeaux, 1986, 282–288

  6. Sarlashkar, J. V., Hamilton/Lagrange formalisms in stability analysis of detailed power system models, Ph D Thesis, Univ. of Wisconsin, 1996.

  7. Ortega, R., Stankove A., Stefanov, P., A passivation approach to power systems stabilization, IFAC Symp. Nonlinear Control Systems Design, Enschede, NL, July 1–3, 1988.

  8. Maschke, B. M. J., Ortega, R., van der Schaft, A. J., Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation, Proc. of CDC98, 1998, 3599–3604.

  9. Cheng, D., Xi, Z., Hong, Y., et al., Energy-Based Stabilization of Forced Hamiltonian Systems with Its Application to Power Systems, Proc. IFAC’99, Beijing, 1999, 297–302.

  10. Brockett, R. W., Control theory and analytic mechanics,in Geometric Control Theory (eds. Martin, C., Herman, R.), Brookline: Math. Sci. Press., 1977, 1–46.

    Google Scholar 

  11. van der Schaft, A., System Theory and Mechanics” in Three Decades of Mathematical System Theory (eds. Nijmeier, H., Schumacher, J. M.), Lect. Notes Contr. Inf. Sci., 1989, 185: 426.

    Article  Google Scholar 

  12. Nijmeijer, H., Van der Schaft, A. J., Nonlinear Dynamical Control Systems, New York: Springer-Verlag, 1990.

    MATH  Google Scholar 

  13. Boothby, W. M., Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed., New York: Academic Press, 1986.

    MATH  Google Scholar 

  14. Cheng, D., On generalized Hamiltonian systems, Proc. of ICARCV’98, Singapore, World Scientifics, 1998, 185–189.

  15. Clarke, C., Elementary General Relativity, New York: John Wiley & Sons, Inc., 1979.

    Google Scholar 

  16. Feng, K., Wang, D., Dynamical systems and geometric construction of algorithms, in Computational Mathematics in China, Contemporary Mathematics, 1994, 163: 1.

    MathSciNet  Google Scholar 

  17. Wang, D., Some aspects of Hamiltonian systems and symplectic algorithms, Physics D, 1994, 73: 1.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Systems and Control, Institute of Systems Science, Chinese Academy of Sciences, 100080, Beijing, China

    Daizhan Cheng & Zairong Xi

  2. Department of Electrical Engineering, Tsinghua University, 100084, Beijing, China

    Qiang Lu & Shengwei Mei

Authors
  1. Daizhan Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zairong Xi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qiang Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shengwei Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daizhan Cheng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheng, D., Xi, Z., Lu, Q. et al. Geometric structure of generalized controlled Hamiltonian systems and its application. Sci. China Ser. E-Technol. Sci. 43, 365–379 (2000). https://doi.org/10.1007/BF02916984

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02916984

Keywords

Search

Navigation