Sampling zeros and the Euler-Frobenius polynomials

• Published in: Proceedings of the 36th IEEE Conference on Decision and Control Vol. 2; pp. 1471 - 1476 vol.2
• Main Authors: Weller, S.R., Moran, W., Ninnes, B., Pollington, A.D.
• Format: Conference Proceeding
• Language: English
• Published: IEEE 1997
• Subjects: Control systems; Information science; Interpolation; Mathematics; Poles and zeros; Polynomials; Sampling methods; Spline
• ISBN: 0780341872; 9780780341876
• ISSN: 0191-2216
• DOI: 10.1109/CDC.1997.657672

In this paper, we show that the zeros of sampled-data systems resulting from rapid sampling of continuous-time systems preceded by a zero-order hold (ZOH) are the roots of the Euler-Frobenius polynomials, the properties of which have been studied in the context of cardinal spline interpolation and, more recently, wavelets. Using known properties of the Euler-Frobenius polynomials, we prove two conjectures of Hagiwara et al. (1993), the first of which concerns the simplicity, negative realness and interlacing properties of the sampling zeros of ZOH- and first-order hold (FOH)- sampled systems. To prove the second conjecture, we show that in the fast sampling limit, and as the continuous-time relative degree increases, the largest sampling zero for FOH-sampled systems approaches 1/e, where e is the base of the natural logarithm.