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time-delay system; feedback; spectrum; right unimodular transformations; Invariant factors assignment; spectral controllability; Bezout ring; Volterra equation
It is well–known that every system with commensurable delays can be assigned a finite spectrum by feedback, provided that it is spectrally controllable. In general, the feedback involves distributed delays, and it is defined in terms of a Volterra equation. In the case of multivariable time–delay systems, one would be interested in assigning not only the location of the poles of the closed–loop system, but also their multiplicities, or, equivalently, the invariant factors of the closed–loop system. We answer this question. Our basic tool is the ring of operators that includes derivatives, localized and distributed delays. This ring is a Bezout ring. It is also an elementary divisor ring, and finally one can show that every matrix over this ring can be brought in column reduced form using right unimodular transformations. The formulation of the result we finally obtain in the case of time-delay systems differs from the well–known fundamental theorem of state feedback for finite dimensional systems, mainly because the reduced column degrees of a matrix of operators are not uniquely defined in general.
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