284 SAMPLED-DATA CONTROL SYSTEMS 
Laplace transform is R(s). In this case, the problem can be solved by 
computing the Laplace transform of the output, C(s), from the simple: 
relation 
C(s) = G,(s) R(s) (11.1) 
The time-domain expression for the output c(t) is obtained exactly from 
the inverse Laplace transformation, 
c(t) = £-(C(s)] (11.2) 
The output c(t) is then sketched by substituting a sequence of values of 
time ¢ into the expression and plotting the result point by point. 
nares Giz) 7 
po? ----08als) 
\ cqlt) 
Ris) Cis) Rls) * R,(s) ! 
r({t) elt) r(t) rt) Lhold | r,t) alt) 
(a) (b) 
Fic. 11.1. Open-cycle approximate sampled model. 
The other approach is that of constructing the equivalent approximate 
sampled model shown in Fig. 11.16. Here the continuous input r(¢) is 
applied to the system through a sampler which closes every T sec. A 
data hold reconstructs the function as 7,(t), which is an acceptably 
accurate reproduction of the continuous function r(t), and applies it to 
the system whose transfer function is G,(s). The output of this element 
is Ca(#), which is an approximation of c(¢). The errors in c,(¢) are caused 
by the imperfect reproduction of the function r(t), in other words, by the 
difference between r(t) and r,(t). The approximate output function ca(¢) 
is then sampled synchronously, as shown in Fig. 11.16, to produce a pulse 
sequence c*(t). 
The advantage of the simulation of the continuous system by this 
equivalent sampled-data system is that the relation between C*(s) and 
R*(s) can be obtained by the application of sampled-data theory. The 
variable s is replaced by z to give the generating function for the output 
C(z) as follows: 
C(z) = G(z) R(z) (1a) 
where G(z) is the z transform corresponding to the Laplace transform of 
the process G,(s) and the data hold in cascade. The pulse sequence 
obtained by the inversion of C(z) will give a sequence of numbers repre- 
senting the output values at sampling instants. If the sampling interval 
T and the form of the data hold are judiciously chosen, the output will be 
acceptably accurate. In mathematical terms, what has been done is the 
