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Section: New Results

Coprimeness of fractional representations

Participants : Catherine Bonnet, Le Ha Vy Nguyen, Yutaka Yamamoto [Kyoto Univ] .

Coprimeness of a fractional representation plays various crucial roles in many different contexts, for example, stabilization of a given plant, minimality of a state space representation, etc. It should be noted however that coprimeness depends crucially on the choice of a ring (or algebra) where such a representation is taken, which reflects the choice of a plant, and particular problems that one studies. Such relationships are particularly delicate and interesting when dealing with infinite-dimensional systems. We have disucssed various coprimeness issues for different rings, typically for $H_{\infty }$ and pseudorational transfer functions. The former is related to $H_{\infty }$-stabilizability, and the latter to controllability of behaviors. We have also given some intricate examples where a seemingly non-coprime factorization indeed turns out to be a coprime factorization over $H_{\infty }$ [28], [29].