Representation of Linear Sampled-Data Systems in the Continuous State Space

In sampled-data systems, informations are transmitted in continuous time as well as at discrete instants. Therefore, a strict and unified representation of the systems needs a new pair of state and output equations. Initially, the variable space of the systems is formulated to the sum set of real functions and Dirac's delta function series. And a sampling operation which satisfies the causality is defined in this space, in order to realize approximately sampled-data systems with or without holding elements. Then, a new system equation is set up for the systems obtained by introducing the above sampling to proper and linear continuous time systems. Finally, some basic properties of sampled-data systems are shown by analizing generally the state transition of linear and first-order sampled-data systems, and the new system equation is suggested to be a unified expression of continuous time systems, discrete time ones and sampled-data ones all of which are proper and linear.