168. SAMPLED-DATA CONTROL SYSTEMS 
To illustrate further how minimizing techniques can be used to obtain 
the staleness factor, a system similar to that of the preceding illustrative 
example is used. Here 
(1 + 27})(ay27! + aye?) 

K(2) = —— (7.40) 
and 1 — K(z) = es ee ba) (7.41) 
By substituting (7.40) into (7.41) and solving for b; in terms of c, it is 
found that 

ee (7.42) 
Thus, the z transform of the error sequence is given by (7.34), resulting in 
E(z) = ao pylat Ee R(z) Gas. 
1 — cz} 

The sum of the squared errors is found by evaluation of (4.67), which 
results in the following for a ramp input: 
SIG == 2 
acu) oe 
and for a step input 
_ Stace 240 
S= aa ae (7.45) 
These functions are plotted in Fig. 7.14 over the permissible range of the 
staleness factor c. In this situation it is seen that no minimum can be 
found which will result in optimum 
performance for both step and ramp 
inputs, and it depends on which 
type of response is to be favored as 
to where the staleness factor is set. 
For the illustrative problem in this 
section, the choice of 0.5 favored 
the step response at the expense of 
the ramp response. 
Generally speaking, the use of the 
staleness factor is a means of obtain- 

-10 -05 0 0.5 1.0 ing a compromise performance of a 
Staleness factor, c system to more than one input. 
Fic. 7.14. Integrated-square error as Its use does not require additional 
function of staleness factor. complexity in the digital controller 
so long as the order of the denominator of K (z) does not exceed that of the 
