APPLICATION OF CONVENTIONAL TECHNIQUES 119 
ing network will be of a realizable form and, furthermore, successive 
applications of the cut-and-try procedure are simply and quickly made. 
In the case of sampled-data systems, only the first of these reeommenda- 
tions can be made without reservation. Unfortunately, the cut-and-try 
procedure is neither simple nor quick in the sampled-data case, so that one 
is typically forced to adopt approximate schemes which have limited 
ranges of validity. The source of the difficulty may be readily seen from 
an analysis of Fig. 6.2. Using the methods of Chap. 5, one can write 
COP eLLNG®) 
R(z) 1+ HNG(z) 

(6.1) 
where HNG(z) is the pulse transfer function corresponding to the product 
H(s)N(s)G(s). The stability and general performance of the system 
depends upon the location of the roots of this pulse transfer function, and 
design depends upon the modification of these root locations by changes 
in N(s). The relation between the roots of 1 + HNG(z) and the param- 
eters of N(s) is very complicated, and it is this complication which is the 
source of the difficulty. In terms of the frequency domain the transfer 
function can be written as 
C(jw) _ HGw)N Gjo)G Ga) 
R* (jw) 1 + HNG*(jw) 
: H (jes)N (jeo)G (je) ee) 
me = > FG eee OO IN Gon nonG Gaeeanen) 


The function which must be plotted for a Nyquist- or Bode-type analysis 
and shaping is 
F(jw) = 7 ») A (jw + jnwo)N (jw + jnoo)GYo + jnwo) (6.3) 
The expression (6.3) shows immediately that no simple product of vectors 
or addition of logarithms will suffice, in general, to show the effect of a 
change in N(jw) on the over-all plot. In other words, each new com- 
pensating network parameter requires a completely new frequency 
plot to display its effect on relative stability and other performance 
characteristics. 
In preference to an exact replotting of the frequency transfer function, 
the design engineer may take advantage of several alternatives and 
approximations. Although it is difficult to give a quantitative measure 
of the range of validity of the approximate methods, it is clear that quali- 
tatively the determining factor is the ratio of sampling frequency fo to 
