Minimal Representations for Delay Systems

There are many, nonequivalent notions of minimality in state space representations for delay systems. In this class, one can express the transfer function as a ratio of two exponential polynomials. Then one can introduce various notions of coprimeness in such a representation. For example, if there is no common zeros between the numerator and denominator, it corresponds to a spectrally minimal realization, i.e., all eigenspaces are reachable. Another fact is that if the numerator and denominator are approximately coprime in some sense, then it corresponds to approximate reachability. All these are nicely embraced in the class of pseudorational transfer functions introduced by the author. The central question here is to characterize the Bézout identity in this class. This is shown to correspond to a non-cancellation property in the extended complex plane, including infinity. This leads to a unified understanding of coprimeness conditions for commensurate and non-commensurable delay cases. Various examples are examined in the light of the general theorem obtained here.