Root
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Conference Proceedings
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5th International Conference on Physics and Control (PhysCon 2011)
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Feedback classification of single-input systems over von Neumann regular
rings
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This work deals with linear systems with scalars in a

commutative ring R with the property of being “von

Neumann regular”, i.e. R is zero-dimensional and has

no nonzero nilpotents. We prove that every single-input,

n-dimensional system over R is feedback equivalent to

a special normal form, whose existence actually characterizes

the class of von Neumann regular rings. This

normal form, which captures completely the structure

of the reachable submodule of the system, is associated

to a collection of n principal ideals generated by idempotent

elements f1, . . . , fn, each dividing the following

one. The normal form can be obtained by an explicit algorithm,

which is implemented in PARI-GP in the case

R = Z/(dZ), where d is a squarefree integer.

commutative ring R with the property of being “von

Neumann regular”, i.e. R is zero-dimensional and has

no nonzero nilpotents. We prove that every single-input,

n-dimensional system over R is feedback equivalent to

a special normal form, whose existence actually characterizes

the class of von Neumann regular rings. This

normal form, which captures completely the structure

of the reachable submodule of the system, is associated

to a collection of n principal ideals generated by idempotent

elements f1, . . . , fn, each dividing the following

one. The normal form can be obtained by an explicit algorithm,

which is implemented in PARI-GP in the case

R = Z/(dZ), where d is a squarefree integer.