DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 197 
where a digital computer forms part of the data-processing system, a 
control program could be used which would compute commands to the 
plant at sampling instants. The exact form of the digital controller 
depends on the application. 
7.14 Summary 
Feedback control systems can be compensated by means of active 
devices known as digital controllers. These devices accept a sequence 
of input samples and process them linearly to generate a sequence of 
commands which are applied to the plant. The linear program is 
designed to compensate for deficiencies in the plant pulse transfer func- 
tion and to produce a system which has an over-all pulse transfer function 
which is acceptable and desirable. 
There are a number of possible over-all prototype response functions. 
The minimal prototype is one which produces the shortest finite settling 
time consistent with steady-state characteristics in response to some test 
function applied to the input. In this, as well as other prototypes, there 
are certain limitations imposed by the poles and zeros of the plant pulse 
transfer function which lie outside of the unit circle in the z plane. In 
such cases, the minimal prototype response cannot be realized, but, 
instead, a longer minimum finite settling time results. One of the dis- 
advantages of minimal prototypes is that the system emerges from the 
transient in a nonneutral state and continues to ripple between sampling 
instants long after the system has settled at sampling instants. Another 
disadvantage is that the system response to inputs other than that for 
which it is designed is apt to be poor. 
A compromise prototype response function is the one employing a 
staleness factor, which essentially smooths the response to various inputs 
and which produces acceptable but not minimum response. This system 
also has ripple at its output, and the settling time even at sampling 
instants is infinite. Ripple-free prototypes having finite settling time 
can be devised. In this prototype, the system emerges from the transient 
in a neutral state after the transient period at sampling instants has 
passed. The settling time at sampling instants is always longer than 
that of the minimal prototype but the improvement obtained in ripple- 
free operation is generally desired. In all cases, the desired over-all 
prototypes can be realized by programming a linear relationship between 
the input and output samples of the digital controller. 
An equivalent effect can be achieved by the use of multiple early feed- 
back from the feedforward system. In this form, continuous elements 
are used, and by adjusting a set of gain constants, an equivalent effect to 
that produced by a digital controller is achieved. Still another form of 
