Sampling zeros and the Euler-Frobenius polynomials

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. Using known properties of these polynomials, we prove two conjectures of Hagiwara and co-workers, the first of which concerns the simplicity, negative realness and interlacing properties of the sampling zeros of ZOHand 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. Dept. of Electrical and Computer Engineering, and Centre for Integrated Dynamics and Control, University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected]. Work supported by the Australian Research Council yAuthor to whom correspondence should be addressed zDept. of Mathematics and Statistics, Flinders University, GPO 2100, SA 5001, Australia. Email: [email protected] xDept. of Electrical and Computer Engineering, and Centre for Integrated Dynamics and Control, University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected]. Work supported by the Australian Research Council {Dept. of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A. Email: [email protected]