DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 163 
expression for K(z) in order to meet all the minimum requirements. If 
additional coefficients had been included, then a number of arbitrary 
samples in response to a test function could be realized. It is important 
to note that this is done at the expense of additional storage requirements 
in the digital controller. As a matter of fact, if the number of storage 
positions in the digital controller were increased without limit, a complete 
specification of the output at all sampling instants could be made. 
7.6 Implementation of Systems with Staleness Factor 
It has been demonstrated in the previous section that systems which 
are designed for minimum and finite settling time often do not give good 
performance when subjected to an input other than that for which they 
are designed. In this sense, minimal systems may be regarded as highly 
“tuned.” In addition, in obtaining minimum finite settling time at 
sampling instants, the severe shocks which result in the plant cause sub- 
stantial ripple in the continuous output, even though the output is cor- 
rect at sampling instants. These effects were noted in the literature!” 
and led to the introduction of a term in the over-all response function 
known as the “‘staleness factor.” This factor led to a “‘softening”’ of the 
response, with the result that a system could be expected to respond ade- 
quately, though not perfectly, to a number of test inputs. The choice of 
staleness factor can be arrived at by optimizing procedures‘ or by obsery- 
ing the response to some most likely form of test input. 
The staleness factor is introduced in the over-all prototype pulse trans- 
fer function in the following manner: 
6) es en 
where c is a constant whose value for stable systems can range from —1.0 
to +1.0, N is an exponent which may assume any positive value, and 
K,,(z) 1s the minimal pulse transfer function. Investigations! have 
shown that not too much is to be gained by making the exponent N higher 
than unity, so that it generally is taken as unity in practical systems. 
The staleness factor is defined as the constant c. In some systems, a 
sequence of terms of the form (1 — c;z~—!) may be used in the denominator 
of (7.28), in which case a number of different staleness factors are used. 
To illustrate the effect of the staleness factor on the response of systems, 
the minimal prototype z~! is used for K,,(z). 
=i 
KO) = ; az 
(7.29) 
If a unit step function is applied to this system, application of the final- 
value theorem will show that the steady-state output is a/(1 — c), se 
