Authors

Arjan van der Schaft, Paolo Rapisarda

Abstract

We develop a new approach to the construction of state vectors for linear time-invariant systems described by higher-order differential equations. The basic observation is that the concatenation of two solutions of higher-order differential equations results in another (weak) solution once their remainder terms resulting from (repeated) integration by parts match. These remainder terms can be computed in a simple and efficient manner by making use of the calculus of bilinear differential forms and two-variable polynomial matrices. Factorization of the resulting two-variable polynomial matrix defines a state map, as well as a state map for the adjoint system. Minimality of these state maps is characterized. The theory is applied to three classes of systems with additional structure, namely self-adjoint Hamiltonian, conservative port-Hamiltonian, and time-reversible systems. For the first two classes it is shown how the factorization leading to a (minimal) state map is equivalent to the factorization of another two-variable polynomial matrix, which is immediately derived from the external system characterization, and defines a symplectic, respectively, symmetric, bilinear form on the minimal state space.

Citation

  • Journal: SIAM Journal on Control and Optimization
  • Year: 2011
  • Volume: 49
  • Issue: 6
  • Pages: 2415–2439
  • Publisher: Society for Industrial & Applied Mathematics (SIAM)
  • DOI: 10.1137/100806825

BibTeX

@article{van_der_Schaft_2011,
  title={{State Maps from Integration by Parts}},
  volume={49},
  ISSN={1095-7138},
  DOI={10.1137/100806825},
  number={6},
  journal={SIAM Journal on Control and Optimization},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={van der Schaft, Arjan and Rapisarda, Paolo},
  year={2011},
  pages={2415--2439}
}

Download the bib file

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