Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems

Authors

Abstract

After recalling the definitions of standard port-Hamiltonian systems and their algebraic constraints, called here Dirac algebraic constraints, an extended class of port-Hamiltonian systems is introduced. This is based on replacing the Hamiltonian function by a general Lagrangian submanifold of the cotangent bundle of the state space manifold, motivated by developments in (Barbero-Linan et al., J. Geom. Mech. 11, 487–510, 2019 ) and extending the linear theory as developed in (van der Schaft and Maschke, Syst. Control Lett. 121, 31–37, 2018 ) and (Beattie et al., Math. Control Signals Syst. 30, 17, 2018 ). The resulting new type of algebraic constraints equations are called Lagrange algebraic constraints. It is shown how Dirac algebraic constraints can be converted into Lagrange algebraic constraints by the introduction of extra state variables, and, conversely, how Lagrange algebraic constraints can be converted into Dirac algebraic constraints by the use of Morse families.

Keywords

Differential-algebraic equations; Nonlinear control; Hamiltonian systems; Dirac structures; Lagrangian submanifolds; 34A09; 65L80; 53D12; 70B45; 93C10

Citation

Journal: Vietnam Journal of Mathematics
Year: 2020
Volume: 48
Issue: 4
Pages: 929–939
Publisher: Springer Science and Business Media LLC
DOI: 10.1007/s10013-020-00419-x

BibTeX

@article{van_der_Schaft_2020,
  title={{Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems}},
  volume={48},
  ISSN={2305-2228},
  DOI={10.1007/s10013-020-00419-x},
  number={4},
  journal={Vietnam Journal of Mathematics},
  publisher={Springer Science and Business Media LLC},
  author={van der Schaft, Arjan and Maschke, Bernhard},
  year={2020},
  pages={929--939}
}

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References

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