178 SAMPLED-DATA CONTROL SYSTEMS 
Solving these equations for the gain constants, 
Ko = 1.58 
K, = 0.786 
By adjusting the gains to these values, the servomechanism which is 
designed will have the ripple-free finite settling time indicated by K(z), 
which is 
K(zg) = 0.5812! + 0.4182—? 
The gain constant K, is seen to be a tachometer gain constant relating 
the signal which is fed back to the servoamplifier from the output 
shaft. 
The foregoing example illustrates how ‘‘early’’ feedback connections 
can implement the equivalent of a digital controller, provided that all the 
early outputs are available and can be instrumented. These “early” 
feedback connections are identified with the “states” of the system in 
that their specification fully describes the performance of the linear sys- 
tem. If all the ‘‘states’’ of the system are not available for instrumenta- 
tion, it is found that the full equivalence between the actual and desired 
feedforward pulse transfer function cannot be achieved and only partial 
compensation can be realized. In this case, a compromise in perform- 
ance is sought rather than the full realization, as was the case in the 
illustrative example. 
7.9 Finite-settling-time Systems Which Are Nonminimal 
Except for those systems employing a staleness factor, finite settling 
time has characterized the systems considered in previous sections. 
Even with the staleness factor, a minimum prototype is used in the basic 
over-all pulse transfer function. These systems contained only the mini- 
mum number of terms consistent with the requirement that they be 
stable and that they respond to a prescribed test input without steady- 
state error. If additional constants are used in the over-all pulse transfer 
function, it was stated that these constants could be arbitrarily assigned. 
This opens up the possibility of selecting the extra system constants with 
a view of optimizing the system under some criterion. An approach‘ to 
this problem is to minimize the integrated-square error sequence in 
response to a test input. 
In general, the over-all pulse transfer function for a system having 
finite settling time is given by 
K(@) =a! + aoe? + + > +s ane (7.53) 
and to respond to a test input polynomial of order n + 1 without steady- 
