Limiting properties of the zeros of sampled-data systems with zero and first-order holds

The zeros of sampled-data systems resulting from continuous-time systems preceded by a hold and followed by a sampler are discussed. The holds considered are a zero-order hold and a first-order hold. For a sufficiently small or large sampling period, such zeros are called limiting zeros. For a sufficiently small sampling period, 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. Basic properties of the zeros of sampled-data systems are shown for a sufficiently small sampling period. The correspondence between the former type of zeros of the sampled-data systems and the zeros of the original continuous-time system are clarified in more detail, including the stability property of these zeros. Stability properties of the latter type of zeros are also studied. In addition, limiting zeros are studied for a sufficiently large sampling period from the viewpoint of the stability of the zeros. The results are combined to derive the conditions which assure the stability of all limiting zeros. The conditions lead to the consequence that first-order hold provides no advantage over zero-order hold as far as the stability of the zeros of the resulting sampled-data systems is concerned