DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 179 
state error, 
l= ik@ = (l= eel sp Ore Se 9 8 8 Se Op EO] (7D) 
Since (7.53) and (7.54) must be satisfied simultaneously, the order of the 
polynomials obtained by multiplying out must be the same; hence the 
final term in (7.54) must be of order m — n, as shown. The minimal 
prototype is the one which uses the minimum number of terms required to 
satisfy both (7.53) and (7.54). Any additional coefficients can be 
assigned values on the basis of optimization of the system, using the 
integrated-square error criterion. For a unity-feedback error-sampled 
system, the error to be minimized has a pulse transfer function given by 
E(z) = [1 — K(2)|R(@) (7.55) 
To demonstrate the technique, a system which has been designed to 
respond to a unit ramp input without steady-state error will be used. 
For this case, the error z transform becomes 
IH) = (Ql = BY se Ore VE) (7.56) 
The constant 6; is to be assigned a value which optimizes the response of 
the system to a unit step input. It is recalled that finite-settling-time 
systems of the minimal type have considerable overshoot with step inputs, 
and the reduction of this effect is sought by choice of bi. The integrated- 
square error is 
» le(mT)]2 = sy i. E(@) Ee) dz (7.57) 
m=0 
For a unit step input, H(z) is 
B@) =< - 10 + bee) (7.58) 
which simplifies to 
Ei) =1— (1 — bi)2z7! — bie (7.59) 
Substituting (7.59) back in (7.57) and evaluating the contour integral by 
the method of residues, there results for the integrated-square error the 
following: 
Y lel) = 1 + Gl — bi)? + by? (7.60) 
m=0 
Simplifying this expression, 
Y le(mT)]? = 201 — b1 + 5x?) (7.61) 
m= 
