Upper bound on the dimension of minimal realizations of linear time-invariant dynamical systems

A linear time-invariant dynamical system described by means of its transfer function matrix is considered, and an upper bound on the dimension of a minimal internal description of the system is computed. This upper bound is also the minimal dimension of a realization having an invariant structure with respect to the coefficients of the transfer-function matrix for a given location of its distinct poles. Finally, a simple algorithm is proposed for constructing a realization having dimension equal to that upper bound.