DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 
153 
error, as expected. Analysis will show that the steady-state error 
should be equal to T?, which it is on the diagram. Had a higher-order 
time function been applied, the system would fail to respond with a 
finite system error, and the error would gradually grow to infinity. 
The example illustrates a typical characteristic of minimal prototypes 
when subjected to other than the test input for which they are designed. 
When subjected to inputs which are 
time functions of an order lower 
than the required one, the minimal 
prototype systems tend to have 
highly oscillatory performance. In 
the illustration, it was seen that 100 
per cent overshoot resulted when a 
step instead of a ramp was applied. 
Minimal prototype systems are 
“tuned”? to a particular form of 
input and not to a broad class of 
inputs. 
In order to produce an average 
acceptable performance to a number 
of input test functions, modifica- 
tions to the minimal prototype 
response function must be made. 
Typical of these is the use of the 
‘“‘staleness factor,’ to be discussed 
in a later section. When properly 
applied, the staleness factor can pro- 
duce an adequate compromise per- 
formance for a whole class of in- 
put functions rather than highly 
“tuned’’ performance with a par- 
ticular test function. The objec- 
Output c(nT) Output c(nT) 
Output c(nT) 

0 © OF” oF CP GE 
Time 
(c) 
Fic. 7.5. Response of minimal prototype 
system to step, ramp, and acceleration. 
tions arising from ripple oscillations caused by the use of minimal proto- 
types are overcome by using ripple-free over-all response functions, to be 
described later. 
While useful in many applications, the minimal proto- 
types serve more as a guide as to the minimum response which can be 
obtained, subject to the conditions outlined in this section. 
7.4 Compensation of Open-loop Systems 
While the main interest of the designer is to compensate closed-loop 
systems, several important points can be made by considering the com- 
