CONTROL BY (STATE–MODULATED) INTERCONNECTION OF PORT–HAMILTONIAN SYSTEMS

Authors

Romeo Ortega, Arjan van der Schaft, Fernando Castaños, Alessandro Astolfi

Abstract

It is well known that the dynamics of many physical processes can be suitably described by Port–Hamiltonian (PH) models. In this paper we consider the Passivity–Based Control (PBC) technique of Control by Interconnection (CbI), where the controller is another PH system connected to the plant to add up their energy functions. We propose two extensions to this method, first, we exploit the non–uniqueness of the PH representation of the system to generate new cyclo–passive outputs. Applying CbI through these new port variables overcomes the so–called dissipation obstacle. Second, when the plant state variables are measurable, we show that the conditions for applicability of the method can be relaxed replacing the simple unitary feedback by a state–modulated interconnection. A central contribution of the paper is the proof that the conditions for energy shaping via CbI are equivalent to those imposed in Interconnection and Damping Assignment PBC, providing in this way a nice geometric interpretation to this successful controller design technique.

Keywords

Passivity; nonlinear systems; stabilization; interconnection; passivity–based control; Hamiltonian systems

Citation

• Journal: IFAC Proceedings Volumes
• Year: 2007
• Volume: 40
• Issue: 12
• Pages: 28–35
• Publisher: Elsevier BV
• DOI: 10.3182/20070822-3-za-2920.00005
• Note: 7th IFAC Symposium on Nonlinear Control Systems

BibTeX

@article{Ortega_2007,
  title={{CONTROL BY (STATE–MODULATED) INTERCONNECTION OF PORT–HAMILTONIAN SYSTEMS}},
  volume={40},
  ISSN={1474-6670},
  DOI={10.3182/20070822-3-za-2920.00005},
  number={12},
  journal={IFAC Proceedings Volumes},
  publisher={Elsevier BV},
  author={Ortega, Romeo and van der Schaft, Arjan and Castaños, Fernando and Astolfi, Alessandro},
  year={2007},
  pages={28--35}
}

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References

• Bernhard, Energy-balancing and ida pbc of nonlinear systems. Journal of the Franklin Institute (1992)
• Bloch, A. M., Leonard, N. E. & Marsden, J. E. Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem. IEEE Trans. Automat. Contr. 45, 2253–2270 (2000) – 10.1109/9.895562
• Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM J. Control Optim. 37, 54–91 (1998) – 10.1137/s0363012996312039
• Fujimoto, K. & Sugie, T. Canonical transformation and stabilization of generalized Hamiltonian systems. Systems & Control Letters 42, 217–227 (2001) – 10.1016/s0167-6911(00)00091-8
• García-Canseco, Power shaping control of nonlinear systems: a benchmark example. (2006)
• Hill, D. J. & Moylan, P. J. Dissipative Dynamical Systems: Basic Input-Output and State Properties. Journal of the Franklin Institute 309, 327–357 (1980) – 10.1016/0016-0032(80)90026-5
• Jeltsema, D., Ortega, R. & M.A. Scherpen, J. An energy-balancing perspective of interconnection and damping assignment control of nonlinear systems. Automatica 40, 1643–1646 (2004) – 10.1016/j.automatica.2004.04.007
• Krstic, (1995)
• Marsden, (1999)
• Ortega, R. & García-Canseco, E. Interconnection and Damping Assignment Passivity-Based Control: A Survey. European Journal of Control 10, 432–450 (2004) – 10.3166/ejc.10.432-450
• Ortega, R. & Spong, M. W. Adaptive motion control of rigid robots: A tutorial. Automatica 25, 877–888 (1989) – 10.1016/0005-1098(89)90054-x
• Ortega, (1998)
• Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38, 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
• Ortega, Putting energy back in control. (2001)
• Ortega, R., Jeltsema, D. & Scherpen, J. M. A. Power shaping: A new paradigm for stabilization of nonlinear RLC circuits. IEEE Trans. Automat. Contr. 48, 1762–1767 (2003) – 10.1109/tac.2003.817918
• Ortega, R., Spong, M. W., Gomez-Estern, F. & Blankenstein, G. Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Automat. Contr. 47, 1218–1233 (2002) – 10.1109/tac.2002.800770
• van der Schaft, (2000)
• Willems, J. C. Dissipative dynamical systems part I: General theory. Arch. Rational Mech. Anal. 45, 321–351 (1972) – 10.1007/bf00276493