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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

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
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