DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 181 
-- The first system for which all 6’s are zero is the minimal prototype. It 
shows that the unit step for which optimization was implemented has an 
integrated-square error of 2.0. The impulse-response and the ramp- 
response errors are 6.0 and 1.0, respectively. As the system becomes 
more complex, the integrated-square error sequences for the unit step 
input become smaller and smaller since this is the input for which minim- 
ization is implemented. The impulse input response likewise becomes 
better as more arbitrary constants are used. On the other hand, this is 
done at the expense of the ramp input, as the figures show. Such a state 
of affairs is typical since minimization with one form of input is generally 
achieved at the expense of the other form of input. Had the system been 
optimized for a ramp input, this performance would have been increas- 
ingly improved as the complexity of the system increased. 
The procedure outlined in this section is merely one way in which the 
arbitrary constants in the over-all response function can be evaluated. 
It is realized that the forms given here are for those systems where no 
complications such as those due to uncancellable poles and zeros arise. 
In these cases, the minimum number of constants is higher than for the 
class of systems described here, but the additional arbitrary constants 
over and above this minimum can be evaluated in much the same manner 
as for the uncomplicated case discussed here. It should be noted also 
that a compromise can be reached where the integrated-square error 
sequence is minimized not for just one input, such as the step chosen in 
this discussion, but that the response to a number of inputs can be 
reduced, though not to their respective minima. In all cases, the 
increased complication means that the digital-controller pulse transfer 
function D(z) will contain more terms in both numerator and denominator 
than with minimum systems. In the limit, if an infinite number of arbi- 
trary constants were permitted, the response to any input could be fully 
programmed, but, of course, this would require an infinite storage in the 
digital controller. 
7.10 Systems with Plant Saturation 
Systems in which a gain factor saturates are, of course, nonlinear and 
cannot be designed or analyzed by the procedures described in this chap- 
ter. Another view of such systems,* however, is that a controller be 
designed in such a manner that the system does not saturate when sub- 
jected to the most severe input expected. For smaller inputs than this 
one, the system certainly will not saturate. By adopting this approach, 
the design of the digital controller can be carried out as in the linear case. 
The requirement placed on the digital controller is that its output, which 
is the input to the plant, never exceed some upper limit under the worst 
