Introduction

Background

It is well known that discrete systems are suitable for computer realization and continuous systems are convenient for theoretical analysis. Usually, the discrete-time model is expressed in the form of the shift operator. However, the first disadvantage of using shift operator is an inconvenience not like the continuous-time operator. The second weakness of the normal shift operating systems are the problem of crowding poles within the boundary of stability circle at small sampling interval and the difficulties of truncation and round-off errors. Goodwin constructed delta operator instead of traditional shift operator for sampling continuous systems in [82] and [170]. The proposed method can unify some previous related results of the continuous and discrete systems into the framework of the DOSs framework. In contrast to the discrete shift operator, the discrete delta operator approximates the Euler derivative. The shorter the sampling period is, the better the system performances are for discrete time control systems. Therefore, the delta operator method is significantly less sensitive than the standard shift operator method at high sampling rate. This leads to a quasi-continuous time s-domain model for high frequencies and gives a better representation of the underlying physical model. That is the poles approach the system poles and the zeros approaches the system zeros for fast sampling. The zeros introduced by the sample and hold circuit tend to minus infinity.