Limiting zeros of sampled systems with time delay

We consider the problem of asymptotic analysis of the zeros of a sampled system of a linear time-invariant continuous system as the sampling period decreases. We show that for a continuous prototype system with delay the limits of a part of the model’s zeros are roots of certain polynomials whose coefficients are determined by the relative order of the prototype system and delay. In the special case of zero delay these polynomials coincide with Euler polynomials. Zeros of these generalized Euler polynomials are localized: we show that they are all simple and negative, and that they move monotonically between the zeros of classical Euler polynomials as the fractional part of the delay divided by the sampling period grows. Our results lead to sufficient and “almost necessary” conditions for the stable invertible of the discrete model for all sufficiently small values of the sampling period.