INTRODUCTION a 
There are many possible configurations possible in addition to those 
shown in Figs. 1.4 and 1.5. Sampled signals may exist at several points 
in the system as well as in the error line. There may be dynamical 
elements in the feedback line, and there may be multiple loops. The 
transform methods for sampled-data systems must be applicable to all 
possible configurations. 
1.5 The Z Transformation 
Continuous linear dynamical systems are described mathematically 
by a set of linear differential equations. While their solution can be 
carried out by classical methods, the use of the Laplace transformation 
organizes and simplifies the process. What is even more important, 
inversion of the transform of the variable of interest is rarely necessary 
in order to deduce the important characteristics of the system and their 
relation to the system constants. Mapping techniques on the complex 
plane in the form of transfer loci or root loci further clarify the properties 
of the system. Certainly, the value of the Laplace transform as a tool 
for the analysis and synthesis of linear continuous systems is indisputable. 
Linear sampled-data dynamical systems are shown to be described 
by a set of linear difference equations, provided that all the samplers 
in the system are synchronous, that is, their sampling periods are equal 
or related by integers. Some of this earlier work, as reported by Olden- 
bourg and Sartorius,4® was motivated by the use of intermittent error- 
sensing devices such as the chopper-bar galvanometer, shown sche- 
matically in Fig. 1.6. In this type of device, a small error voltage or 
current is applied to the galvanometer coil. While the chopper bar is 
raised, the sensitive galvanometer movement is free and the coil responds 
with a large displacement in response to the weak signal. Periodically, 
the chopper bar is lowered and the projecting galvanometer needle causes 
a bell crank to be rotated more or less proportionately to the deflection 
angle 6. The bell crank causes the output shaft to rotate with a torque 
capacity determined by the chopper-bar drive rather than the galva- 
nometer-coil drive. 
The main point of interest here is that a datum is stored in the output 
shaft just once per cycle of the chopper-bar drive. In a sense, the inter- 
mittency of the output signal has been accepted in return for a high 
sensitivity of the system. The early work by Oldenbourg and Sartorius 
generalized systems of this type into the form of the sampled-data block 
diagrams of Figs. 1.4 and 1.5. It was shown that these systems could be 
described by a set of linear difference equations whose solution could be 
obtained by classical methods. The linear sampled-data system was 
therefore placed in the same status as the continuous system, using 
classical methods to solve the differential equations. 
