Notes on n-Dimensional System Theory
This paper makes three observations with regard to several issues of a fundamental nature that apparently must arise in any general theory of linear n-dimensional systems. It is shown, by means of three-specific interrelated counterexamples, that certain decomposition techniques which have proven to...
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todo:paper_00984094_v26_n2_p105_Youla2023-10-03T14:56:57Z Notes on n-Dimensional System Theory Youla, D.C. Gnavi, G. SYSTEMS SCIENCE AND CYBERNETICS - Multivariable Systems CONTROL SYSTEMS This paper makes three observations with regard to several issues of a fundamental nature that apparently must arise in any general theory of linear n-dimensional systems. It is shown, by means of three-specific interrelated counterexamples, that certain decomposition techniques which have proven to be basic for n = 1 and 2 are no longer applicable for n » 3. In fact, for n » 3, at least three equally meaningful but inequivalent notions of polynomial coprimeness emerge, namely, zero-coprimeness (ZC), minor-coprimeness (MC), and factor-coprimeness (FC). Theorems 1 and 3 clarify the differences (and similarities) between these concepts, and Theorem 2 gives the ZC and MC properties a useful system formulation. (Unfortunately, FC, which in our opinion is destined to play a major role, has thus far eluded the same kind of characterization.) Theorem 4 reveals that the structure of 2-variable elementary polynomial matrices is completely captured by the ZC concept. However, there is reason to believe that ZC is insufficient for n » 3 but a counterexample is not at hand. The matter is therefore unresolved. © 1979 IEEE Fil:Gnavi, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00984094_v26_n2_p105_Youla |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
SYSTEMS SCIENCE AND CYBERNETICS - Multivariable Systems CONTROL SYSTEMS |
| spellingShingle |
SYSTEMS SCIENCE AND CYBERNETICS - Multivariable Systems CONTROL SYSTEMS Youla, D.C. Gnavi, G. Notes on n-Dimensional System Theory |
| topic_facet |
SYSTEMS SCIENCE AND CYBERNETICS - Multivariable Systems CONTROL SYSTEMS |
| description |
This paper makes three observations with regard to several issues of a fundamental nature that apparently must arise in any general theory of linear n-dimensional systems. It is shown, by means of three-specific interrelated counterexamples, that certain decomposition techniques which have proven to be basic for n = 1 and 2 are no longer applicable for n » 3. In fact, for n » 3, at least three equally meaningful but inequivalent notions of polynomial coprimeness emerge, namely, zero-coprimeness (ZC), minor-coprimeness (MC), and factor-coprimeness (FC). Theorems 1 and 3 clarify the differences (and similarities) between these concepts, and Theorem 2 gives the ZC and MC properties a useful system formulation. (Unfortunately, FC, which in our opinion is destined to play a major role, has thus far eluded the same kind of characterization.) Theorem 4 reveals that the structure of 2-variable elementary polynomial matrices is completely captured by the ZC concept. However, there is reason to believe that ZC is insufficient for n » 3 but a counterexample is not at hand. The matter is therefore unresolved. © 1979 IEEE |
| format |
JOUR |
| author |
Youla, D.C. Gnavi, G. |
| author_facet |
Youla, D.C. Gnavi, G. |
| author_sort |
Youla, D.C. |
| title |
Notes on n-Dimensional System Theory |
| title_short |
Notes on n-Dimensional System Theory |
| title_full |
Notes on n-Dimensional System Theory |
| title_fullStr |
Notes on n-Dimensional System Theory |
| title_full_unstemmed |
Notes on n-Dimensional System Theory |
| title_sort |
notes on n-dimensional system theory |
| url |
http://hdl.handle.net/20.500.12110/paper_00984094_v26_n2_p105_Youla |
| work_keys_str_mv |
AT youladc notesonndimensionalsystemtheory AT gnavig notesonndimensionalsystemtheory |
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1807316979510411264 |