122 SAMPLED-DATA CONTROL SYSTEMS 
a fictitious sampler to perform the separation of transfer functions neces- 
sary to apply conventional shaping techniques directly. The mathe- 
matical sampling represented by 72 in Fig. 6.3 is usually taken as a higher 
rate than the basic sampling rate 7’, in which case the system is a multi- 
rate problem, which will be treated in a later chapter. The hold opera- 
tion indicated by His part of the numerical or mathematical approxima- 
tion to a continuous connection between plant and shaping network and 
R(s) | C(s) 


[pane Compensating 
Fictitious sample network 
and hold 

Fic. 6.3. Block diagram for approximate design procedure, using mathematical 
sampling between plant and network. 
consequently need not be physically realizable. The use of nonphysical 
extrapolators permits considerably greater freedom in the choice of hold 
and consequently greater accuracy in the approximation. This method 
is comparable to method 2 (Linvill’s method) in range of applicability and 
accuracy. 
Method 4. Time-domain solution for continuous network compensa- 
tion. 
Design step 4.1. From considerations of the over-all transfer function 
from input to output specify the pulse transfer function HNG(z). 
Design step 4.2. Find a continuous transfer function F(s) correspond- 
ing to the pulse transfer function obtained in step 4.1. 
Design step 4.3. Solve for N(s) as 
This method, originally proposed by Truxal® as an extension of Guil- 
lemin’s control-system synthesis procedure to sampled-data systems, has 
great intuitive appeal but has not yet been satisfactorily applied in many 
practical cases. Because of the complicated relation between N(s) and 
the pulse transfer function of the system, no simple criterion exists to 
guide the initial choice of pulse transfer function in such a manner that 
the final network will be practical. The cut-and-try process is rather like 
a random search because failure at one step does not point toward success 
at later steps. Cut and try does not converge in this method. This 
peculiarity will be demonstrated with examples later in the chapter. 
Method 5. Pulsed-network compensation. 
