Abstract
If R is a ring and X is an R-module, then denote ΛX as the R-module of truncated formal Laurent series in s -1 with coefficients in the module X. The truncation operator on ΛX is the operator which sets the polynomial part of an element in ΛX to zero. It is well known that ΛX is endowed with the structure of a R[s]-module by using the shift left operator as multiplication by s. This paper establishes a lemma which limits the size of the commutator of the truncation operator on ΛX and the system operator (sI - A)-1, defined on ΛX. A corollary to this lemma is the well known fact that the transmission blocking space, Z signal’ is a submodule of ΓU. This Lemma can also be used to show that the gamma zero module, Z Γ’ and Z signal are equal for reachable systems.
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References
G. Conte and A. M. Perdon, On Polynomial Matrices and Finitely Generated Torsion K[z]-Modules ,Lectures in Applied Mathematics: Algebraic and Geometric Methods in Linear Systems Theory, C. I. Byrnes and C. F. Martin, Editors, Volume 18. Providence: American Mathematical Society, Pages 27–36, 1980.
R. E. Kalman, Algebraic structure of linear dynamical systems. The module of ∑, in Proceedings National Academy of Science, Vol 54, 1965, pp. 1503–1508.
Michael K. Sain and Bostwick F. Wyman, The fixed zero constraint in dynamical system performance ,Proceedings Eighth International Symposium on the Mathematical Theory of Networks and Systems, C. I. Byrnes, C. F. Martin, and R. Saeks, Editors, North Holland, June 1987.
Bostwick F. Wyman and Michael K. Sain, On the Zeros of a Minimal Realization ,Linear Algebra and Its Applications, Volume 50, Pages 621–637, 1983.
Bostwick F. Wyman and Michael K. Sain, Module Theoretic Zero Structures for System Matrices ,SIAM Journal of Control and Optimization, Volume 25, Number 1, Pages 86–99, January 1987
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© 1991 Springer Science+Business Media New York
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Crown, G.D. (1991). A Computational Lemma in the Module Theory of Linear Systems. In: New Trends in Systems Theory. Progress in Systems and Control Theory, vol 7. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0439-8_26
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DOI: https://doi.org/10.1007/978-1-4612-0439-8_26
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6760-7
Online ISBN: 978-1-4612-0439-8
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