Stability of the limiting zeros of sampled-data systems with zero-and first-order holds

This paper is concerned with the zeros of sampled-data systems resulting from continuous-time systems preceded by a hold and followed by a sampler. The holds we shall consider are a zero-order hold and a first-order hold. For sufficiently small or large sampling periods, such zeros are called limiting zeros. For sufficiently small sampling periods, they are known to consist of two different types of zeros: the zeros of the first type correspond to the zeros of the original continuous-time system, while those of the second type have no continuous-time counterparts. We first show basic properties of the zeros of sample-data systems for sufficiently small sampling periods. Next, we clarify, in more detail, the correspondence between the former-type zeros of the sampled-data systems and the zeros of the original continuous-time system, including the stability property of these zeros. We also study stability properties of the latter-type zeros. In addition, we study limiting zeros for sufficiently large sampling periods from the viewpoint of stability of the zeros. Finally, the above results are combined to derive the conditions which assure stability of all limiting zeros. The conditions lead to a consequence that a first-order hold provides no advantage over a zero-order hold as far as stability of the zeros of the resulting sampled-data systems is concerned.