MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 283 
the uses of sampled-data theory as applied to continuous systems which 
do not include a sampler but which can be so approximated. 
11.1 Approximation of Open-loop Continuous Systems by a Sampled 
Model 
Continuous dynamical systems are characterized by a set of differ- 
ential equations containing derivatives with respect to time. The 
response in the time domain to a test input is studied by integrating the 
differential equations and plotting the resultant solution. An alternative 
is the experimental approach, using an analogue computer which is so 
programmed that its describing equations are identical to those of the 
actual system, except for scale factors. Still another alternative is to 
program a digital computer or to use manual computation, employing 
desk calculators, which solves an acceptably accurate difference equation 
derived from the differential equation. This section deals with tech- 
niques adapted from the theory of sampled-data systems which can be 
used to set up numerical routines for solving the continuous differential 
equations of linear systems. 
If the system is linear, the solution of the differential equations describ- 
ing it can be carried out in a straightforward manner. The Laplace 
transform of the output variable is obtained analytically and is inverted 
into the time domain by the use of tables or by contour integration. The 
resultant analytical expression is not readily interpreted and is generally 
plotted in order to obtain significant information required by the designer. 
The plotting is done by computing the value of the output variable at 
selected instants of time, plotting the points on a graph, and fitting a 
smooth curve connecting these points. It is noted that, in spite of the 
analytic nature of the solution, the final graph is obtained by employing 
some form of numerical computation at the selected time instants. 
The alternate approach to the problem is to convert the differential 
equations into difference equations that can be solved numerically by the 
application of a recursion formula. There are well-known techniques for 
accomplishing this, but it so happens that, for linear systems, sampled- 
data theory can be applied directly.°°?!7 The technique is to replace the 
continuous system with an acceptably accurate sampled-data model. 
The z transform of the output sequence is readily obtained by the usual 
methods and its inversion accomplished either by contour integration or 
by the process of long division, easily implemented by digital-computer 
programming or by means of desk calculators. 
To describe the technique, a simple open-cycle system will be used. 
Referring to Fig. 11.1a, it is desired to obtain the response of the linear 
system whose transfer function is G,(s) to an input function whose 
