Abstract
The important concept of the strict system equivalence of polynomial realisations has been extended to realisations over an arbitrary principal ideal domainR. We use the algebraic concept of the localisation of a ring at a prime ideal to study the local properties, as opposed to the global properties, of realisations over arbitrary principal ideal domains. This allows us to gain a new understanding of strict system equivalence. We show that twoR-realisations of aK-matrix, whereK is the field of fractions of the principal ideal domainR, are strictly system equivalent if and only if they are locally system equivalent at every prime ideal (p) ofR.
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References
B. D. O. Anderson, W. A. Coppel and D. J. Cullen, Strong system equivalence (I),J. Austral. Math. Soc. B. 27 (1985), 194–222.
N. Bourbaki, Commutative Algebra (Addison-Wesley, Reading, Mass., 1972).
W. A. Coppel, Matrices of rational functions,Bull. Austral. Math. Soc. 11 (1974), 89–113.
W. A. Coppel, Some remarks on strict system equivalence,Proc. 5th Internat. Symp. Math. Theory of Networks and Systems, Santa Monica (1981), 33–34.
W. A. Coppel and D. J. Cullen, Strong system equivalence (II),J. Austral. Math. Soc. B. 27 (1985), 223–237.
D. J. Cullen, The Equivalence of Linear Systems, Ph.D. Thesis, Australian National University, 1984.
P. A. Fuhrmann, On strict system equivalence and similarity,Int. J. Control 25 (1977), 5–10.
D. Hindrichsen and D. Prätzel-Wolters, Solution modules and system equivalence,Int. J. Control 32 (1980), 777–802.
S. Lang, Algebra (Addison-Wesley, Reading, Mass., 1965).
A. S. Morse, System Invariants under Feedback and Cascade Control, in Mathematical Systems Theory, Udine, 1975 (Springer-Verlag, Berlin, 1976).
L. Pernebo, Algebraic Control Theory for Linear Multivariable Systems, Ph.D. Dissertation, Lund Institute of Technology, 1978.
L. Pernebo, Notes on strict system equivalence,Internat. J. Control 25 (1977), 21–38.
H. H. Rosebrock, State-space and multivariable theory (Nelson, London, 1970).
H. H. Rosenbrock, The transformation of strict system equivalence,Internat. J. Control 25 (1977), 11–19.
O. Zariski and P. Samuel, Commutative Algebra Vols I and II (Springer-Verlag, New York, 1975).
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1980 Mathematics Subject Classification Number: 93B99
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Cullen, D.J. Local system equivalence. Math. Systems Theory 19, 67–78 (1986). https://doi.org/10.1007/BF01704906
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DOI: https://doi.org/10.1007/BF01704906