# Polynomial systems theory applied to the analysis and design of multidimensional systems

• Volume: 13, Issue: 1, page 15-27
• ISSN: 1641-876X

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## Abstract

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The use of a principal ideal domain structure for the analysis and design of multidimensional systems is discussed. As a first step it is shown that a lattice structure can be introduced for IO-relations generated by polynomial matrices in a signal space X (an Abelian group). It is assumed that the matrices take values in a polynomial ring F[p] where F is a field such that F[p] is a commutative subring of the ring of endomorphisms of X. After that it is analysed when a given F[p] acting on X can be extended to its field of fractions F(p). The conditions on the pair (F[p],X) are quite restrictive, i.e. each non-zero a(p)∊F[p] has to be an automorphism on X before the extension is possible. However, when this condition is met, say for operators {p1, p2, . . . , pn−1}, a polynomial ring F[p1, p2, . . . , pn] acting on X can be extended to F(p1, p2, . . . , pn−1)[pn], resulting in a principal ideal domain structure. Hence in this case all the rigorous principles of ‘ordinary’ polynomial systems theory for the analysis and design of systems is applicable. As an example, both an observer for estimating non-measurable outputs and a stabilizing controller for a distributed parameter system are designed.

## How to cite

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Hatonen, Jari, and Ylinen, Raimo. "Polynomial systems theory applied to the analysis and design of multidimensional systems." International Journal of Applied Mathematics and Computer Science 13.1 (2003): 15-27. <http://eudml.org/doc/207620>.

@article{Hatonen2003,
abstract = {The use of a principal ideal domain structure for the analysis and design of multidimensional systems is discussed. As a first step it is shown that a lattice structure can be introduced for IO-relations generated by polynomial matrices in a signal space X (an Abelian group). It is assumed that the matrices take values in a polynomial ring F[p] where F is a field such that F[p] is a commutative subring of the ring of endomorphisms of X. After that it is analysed when a given F[p] acting on X can be extended to its field of fractions F(p). The conditions on the pair (F[p],X) are quite restrictive, i.e. each non-zero a(p)∊F[p] has to be an automorphism on X before the extension is possible. However, when this condition is met, say for operators \{p1, p2, . . . , pn−1\}, a polynomial ring F[p1, p2, . . . , pn] acting on X can be extended to F(p1, p2, . . . , pn−1)[pn], resulting in a principal ideal domain structure. Hence in this case all the rigorous principles of ‘ordinary’ polynomial systems theory for the analysis and design of systems is applicable. As an example, both an observer for estimating non-measurable outputs and a stabilizing controller for a distributed parameter system are designed.},
author = {Hatonen, Jari, Ylinen, Raimo},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {partial differential equations; nD systems; polynomial systems theory; module of fractions; D systems; PDF; polynomials; observer; feedback},
language = {eng},
number = {1},
pages = {15-27},
title = {Polynomial systems theory applied to the analysis and design of multidimensional systems},
url = {http://eudml.org/doc/207620},
volume = {13},
year = {2003},
}

TY - JOUR
AU - Hatonen, Jari
AU - Ylinen, Raimo
TI - Polynomial systems theory applied to the analysis and design of multidimensional systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2003
VL - 13
IS - 1
SP - 15
EP - 27
AB - The use of a principal ideal domain structure for the analysis and design of multidimensional systems is discussed. As a first step it is shown that a lattice structure can be introduced for IO-relations generated by polynomial matrices in a signal space X (an Abelian group). It is assumed that the matrices take values in a polynomial ring F[p] where F is a field such that F[p] is a commutative subring of the ring of endomorphisms of X. After that it is analysed when a given F[p] acting on X can be extended to its field of fractions F(p). The conditions on the pair (F[p],X) are quite restrictive, i.e. each non-zero a(p)∊F[p] has to be an automorphism on X before the extension is possible. However, when this condition is met, say for operators {p1, p2, . . . , pn−1}, a polynomial ring F[p1, p2, . . . , pn] acting on X can be extended to F(p1, p2, . . . , pn−1)[pn], resulting in a principal ideal domain structure. Hence in this case all the rigorous principles of ‘ordinary’ polynomial systems theory for the analysis and design of systems is applicable. As an example, both an observer for estimating non-measurable outputs and a stabilizing controller for a distributed parameter system are designed.
LA - eng
KW - partial differential equations; nD systems; polynomial systems theory; module of fractions; D systems; PDF; polynomials; observer; feedback
UR - http://eudml.org/doc/207620
ER -

## References

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15. Ylinen R. (1975): On the algebraic theory of linear differential and difference systems with time-varying or operator coefficients. - Tech. Rep., Helsinki University of Technology, Systems Theory Laboratory, No. B23.
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18. Zampieri S. (1998): Causal input/output representation of 2D systemsin the behavioral approach. - SIAM J. Contr. Opt., Vol. 36, No. 5, pp. 1133-1146. Zbl0913.93002

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