DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 167 
the system error in the integrated-square sense, that is, 
io} 
g= » [E(mT)]? > min (7.33) 
m=0 
Using this criterion, the value of ¢ which minimizes S can be arrived at in 
logical fashion. 
The sum S is obtained in the z domain by use of (4.67). To illustrate 
the technique, a minimal prototype with a single staleness factor as given 
in (7.30) is used. It is assumed that the feedback transfer function is 
unity, so that the error pulse transform is 
E(z) = {1 — K(z)]R(z) (7.34) 
Substituting (7.30) in (7.34), there results the expression for the error 
pulse transform 
jo) = : ae) (7.35) 
Now, if R(z) is a unit step function z transform, H(z) is simply 
1 
DNS = 
(7.36) 
Using the integral for the summation S as given in (4.67), 

ag if 1 ei 
S= Sa te La hee dz (7.37) 
where it is recalled that [ is the unit circle. The residue is obtained 
resulting in a value for S given by 
I 
Ss = coe (7.38) 
It is seen that S is minimized for ¢ set equal to zero, that is, the minimal 
prototype is best for this input using this criterion. Looking further, 
however, when a unit ramp is applied, the prototype given by (7.35) will 
produce a steady-state error e() in response to a unit ramp with a unit 
sampling period, which is given by 
1 
Il = @ 
In this instance, if c is made equal to —0.5, the integrated-square error 
given by (7.38) rises by one-third, but the steady-state error to a ramp is 
reduced by one-third. Thus, if the inputs to the system were mainly 
step functions and ramp functions, a compromise choice in staleness factor 
would be about —0.5. Considerations like these can be used to determine 
the optimum value of staleness factor in different situations. 
Qe) = 

(7.39) 
