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

Jari Hatonen; Raimo Ylinen

International Journal of Applied Mathematics and Computer Science (2003)

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

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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  1. Blomberg H. and Ylinen R. (1983): Algebraic Theory for Multivariable Linear Systems. - London: Academic Press. Zbl0556.93016
  2. Hinrichsen D. and Pratzel-Wolters D. (1980): Solution modules and system equivalence. - Int. J. Contr., Vol. 32, No. 5, pp. 777-802. Zbl0469.93036
  3. Hatonen J. and Ylinen R. (2000): Synthesis of nD systems using polynomial approach. - Proc. 2nd Int. Workshop Multidimensional (nD) Systems, Czocha Castle, Poland, pp. 159-164. Zbl0979.93038
  4. Kučera V. (1979): Discrete Linear Control: The Polynomial Equation Approach. - New York: Wiley. Zbl0432.93001
  5. Morf M., Levy B., Kung S.Y. and Kailath T. (1977): New results in 2-D systems theory, Part I and II. - Proc. IEEE, Vol. 65, No. 6, pp. 861-872; 945-961. 
  6. Napoli M. and Zampieri S. (1999): Two-dimensional proper rational matrices and causal inputoutput representations of two-dimensional behavioral systems. - SIAM J. Contr. Opt., Vol. 37, No. 5, pp. 1538-1552. Zbl0928.93006
  7. Northcott D.G. (1968): Lessons on Rings, Modules and Multiplicities. - Cambridge: Cambridge University Press. Zbl0159.33001
  8. Oberst U. (1990): Multidimensional constant linear systems. - Acta Applicande Mathematicae, Vol. 20, pp. 1-175. Zbl0715.93014
  9. Rosenbrock H.H. (1970): State-Space and Multivariable Theory. -London: Nelson. Zbl0246.93010
  10. Valcher M.E. and Willems J.C. (1999): Observer synthesis in the behavioral approach. - IEEE Trans. Automat. Contr., Vol. AC-44, No. 12, pp. 2297-2307. Zbl1136.93340
  11. Willems J.C. (1991): Paradigms and puzzles in the theory of dynamic systems. - IEEE Trans. Automat. Contr., Vol. AC-36, No. 3, pp. 259-294. Zbl0737.93004
  12. Willems J.C. (1997): On interconnections, control and feedback. - IEEE Trans. Automat. Contr., Vol. AC-42, No. 3, pp. 326-339. Zbl0872.93034
  13. Wolovich W.A. (1974): Linear Multivariable Systems. - New York: Springer. Zbl0291.93002
  14. Wood J. (2000): Modules and behaviours in nD systems theory. - Multidimensional Syst. Signal Proc., Vol. 11, No. 1-2, pp. 11-48. Zbl0963.93015
  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. 
  16. Ylinen R. (1980): An algebraic theory for analysis and synthesis of time-varying lineal differential systems. - Acta Polytechnica Scandinavica, No. Ma32. 
  17. Ylinen R. and Blomberg H. (1989): Order and equivalence relations on descriptions of finite dimensional linear system, In: Computer Aided Systems Theory-EUROCAST'89 (F. Pichler and R. Moreno Diaz, Eds.). - Berlin: Springer. 
  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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