180 SAMPLED-DATA CONTROL SYSTEMS 
It is seen that the sum of the squared errors is a function of the constant 
bi, which can, by ordinary methods, be evaluated to minimize (7.61). 
Differentiating and setting the derivative equal to zero, the value of b; is 
found to be 0.5. 
With this value of bi, the integrated-square error as obtained by substi- 
tution in (7.61) is 1.5. Had the minimal prototype been used instead, the 
integrated-square error sequence would have been 2.0, showing how a 
judicious choice of constant lowered the error in the sense shown. An 
important corollary is that by increasing the number of terms in K(z) 
beyond the absolute minimum, the digital controller is complicated by 
the fact that it must have additional storage. The controller pulse 
transfer function D(z) is of higher order the greater the number of terms 
included in K(z). 
To illustrate what can be done by these minimization procedures, a 
number of systems having the capability of responding to a ramp input 
with zero steady-state error will be considered. In general such systems 
have the error z transform given by 
IKE) (ON ee ACU oe ies a ae Se (7.62) 
If the minimal prototype is assumed, then all b’s will be zero. On the 
other hand, increasing numbers of arbitrary constants are available for 
evaluation as more terms of (7.62) are used. A’tabulation of the per- 
formance of this type of system is given in Table 7.2 for increasingly com- 
plex systems. The integrated-square error sequence is given for a unit 
impulse, a unit step, and a unit ramp input to the system. 
TABLE 7.2. INTEGRATED-SQUARE ERROR SEQUENCES FOR OPTIMIZED RESPONSE 
to Unit Step Input with Unit Imputss, Step, anD Ramp INPUTS 
» [e(mT)]? 





Type 
Concent Impulse Step input Ramp 
input (optimum) input 
Minimal b, = 0 6.0 2.0 1.0 
b, = 0.5 3.5 1D 1.25 
b, = 0.667 2.889 1.33 1.556 
b; = 0.75 2.625 1.25 1.875 
be = 0.50 
bs = 0.25 



