Networks of Hamiltonian systems and feedback

Research output: Contribution to journalArticlepeer-review

Abstract

Motivated by the design of complex networks transmitting pulse-like signals, we investigate and develop a network theory for Hamiltonian systems. The main topics of this paper are articulated around the following result: under a few natural assumptions, a network of Hamiltonian systems with energy-conserving interactions will itself be Hamiltonian for the symplectic form ωc l = ω - G* (K (I - C K)- 1) where ω is the symplectic form of the individual components of the network, K encodes the network structure and C, which we call the symplectic gain matrix, is related to the non-involutivity of the control vector fields. A further analysis of this form will reveal how certain interactions result in effectively putting constraints on the system, giving a new result in the flavor of the Dirac theory of constrained Hamiltonian systems but of a control-theoretic origin. Throughout the paper, we use the Toda lattice to illustrate the results and find a new approach to the description of the periodic Toda lattice as a Hamiltonian system on the space of Jacobi matrices.

Original languageEnglish (US)
Pages (from-to)217-224
Number of pages8
JournalSystems and Control Letters
Volume58
Issue number3
DOIs
StatePublished - Mar 1 2009
Externally publishedYes

Keywords

  • Dirac-Bergmann theory
  • Hamiltonian systems
  • Networks
  • Symplectic form
  • Toda flows

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science(all)
  • Mechanical Engineering
  • Electrical and Electronic Engineering

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