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Module theoretic zero structures for system matrices. (English) Zbl 0617.93010
The authors extend their previous study [IEEE Trans. Circuits Syst. CAS- 28, 112-126 (1981; Zbl 0474.93023)] referring to a coordinate-free, module-theoretic treatment of transmission zeros for multi-input, multi- output transfer functions in order to include nonminimal systems (A,B,C,D(s)). After two introductory sections containing some required material about modules, a detailed analysis of the Rosenbrock system matrix $$\Sigma$$ is organized on three levels: rational, finitely generated free-modular and torsion-divisible.
A first result characterizes the rational vector spaces ker $$\Sigma$$ and coker $$\Sigma$$, establishing the isomorphisms ker $$\Sigma\cong \ker G(s)$$ and coker $$\Sigma\cong co\ker G(s)$$, G(s) denoting the transfer function.
By the introduction of an $$\Omega$$-zero module $$Z_{\Omega}$$ and then by its imbedding in an exact sequence of modules, one demonstrates that the input-decoupling zero module is contained as a factor module in $$Z_{\Omega}$$. A $$\Gamma$$-zero module $$Z_{\Gamma}$$ is also defined and its connection with the output-decoupling zero module derives from the imbedding in a short exact sequence and allows a discussion of the Desoer-Schulman blocked transmission philosophy.
The direct sums $$Z_{\Omega}$$ $$=$$ (finitely generated free-module) $$+$$ M2, respectively $$Z_{\Gamma}$$ $$=$$ (torsion-divisible module) $$+$$ M1, where M1 and M2 are isomorphic finitely generated torsion modules, point out, separately, the ”lumped” and the ”generic” zeros of the system.
The cases when the system transfer function has a right or a left inverse are specially examined, the former from the free-module view point, the latter from the torsion-divisible module one.
Reviewer: O.Pastravanu

##### MSC:
 93B25 Algebraic methods 13C10 Projective and free modules and ideals in commutative rings 93C05 Linear systems in control theory 93C35 Multivariable systems, multidimensional control systems
##### Keywords:
transmission zeros; nonminimal systems; zero module
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