The zero-order hold equivalent transfer function for a time-delayed process

An explicit expression is derived for the zero-order hold equivalent z transfer function associated with the Laplace transfer function Gp(s)exp(−(r + α)Ts), where T is the sampling period, r is a positive integer or zero and 0α1 Vector-matrix generating rules are given for the numerator polynomial N(z, α) and structural relations between its coefficients are derived by invoking a ‘correspondence principle’. Approximate root locus analysis is given for N(z, α) under the assumption that T is chosen so that, for all i = 1,..., n the relation | pi T | ≪ 1 is satisfied, where p I, are the poles of G p(s).