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State maps from integration by parts

State maps from integration by parts
State maps from integration by parts
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
adjoint systems, Hamiltonian systems, state, integration by parts, factorization
2415-2439
van der Schaft, Arjan
d7e3477b-ce6d-4443-98a0-c65360437f03
Rapisarda, Paolo
79efc3b0-a7c6-4ca7-a7f8-de5770a4281b
van der Schaft, Arjan
d7e3477b-ce6d-4443-98a0-c65360437f03
Rapisarda, Paolo
79efc3b0-a7c6-4ca7-a7f8-de5770a4281b

van der Schaft, Arjan and Rapisarda, Paolo (2011) State maps from integration by parts. SIAM Journal on Control and Optimization, 49 (6), 2415-2439. (doi:10.1137/100806825).

Record type: Article

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

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Published date: November 2011
Keywords: adjoint systems, Hamiltonian systems, state, integration by parts, factorization
Organisations: Southampton Wireless Group

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Local EPrints ID: 273012
URI: http://eprints.soton.ac.uk/id/eprint/273012
PURE UUID: a2bfc3e9-4a04-47ee-992f-e8ae7d69b753

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Date deposited: 17 Nov 2011 17:32
Last modified: 14 Mar 2024 10:16

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Contributors

Author: Arjan van der Schaft
Author: Paolo Rapisarda

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