170 SAMPLED-DATA CONTROL SYSTEMS 
The over-all response function K(z) must be so chosen that H(z) /R(z) 
contain only a numerator polynomial in z~!. Recalling that G(z) is usu- 
ally the ratio of polynomials in z~!, it is immediately evident that K(z) 
must contain all the zeros of G(z), regardless of their location on the 
zplane. It is noted that this condition automatically includes the lesser 
condition applying to all error-sampled stabilized systems that K(z) con- 
tain zeros of G(z) which lie outside the unit circle in the z plane. In gen- 
eral, because K(z) must contain as its zeros the additional zeros, the 
settling time of ripple-free systems exceeds that of minimal systems in so 
far as sample times are concerned. 
The rules which apply to ripple-free design will be summarized: 
1. All the rules for minimal prototype response systems apply to ripple- 
free systems. 
2. To produce a ripple-free system response, it is necessary that the 
feedforward transfer function be capable of generating a continuous out- 
put function which is the same as the input function. 
3. The over-all pulse transfer function K(z) must contain as its zeros 
all the zeros of the plant pulse transfer function G(z) and not just the 
zeros of G(z) which lie outside the unit circle in the z plane. 
The application of these rules can best be illustrated by means of an 
example. The system used to illustrate the minimal prototype and stale- 
ness factor in Secs. 7.5 and 7.6 will be used. 
EXAMPLE 
It is desired to design the digital-controller program D(z) for the 
system of Fig. 7.8. The criterion is that the system respond to a step 
and ramp input with no ripple in the steady state and that the transient 
be of the shortest possible finite duration. First it is noted that the 
feedforward transfer function, including the zero-order hold, is capable 
of generating a continuous step or ramp function since a constant input 
E* into the hold system will cause a continuous ramp at the output. 
This being the case, the system design can be carried out in the manner 
outlined in this section. 
The feedforward pulse transfer function G(z) was found in the 
illustrative example in Sec. 7.5 to be 
@ 234257) =e OnlGzet ea 
60) = (= 2) oes 
It is seen that there are zeros of G(z) located at —2.34, —0.16, and o. 
To implement a ripple-free system, it is necessary that the over-all 
pulse transfer function K(z) contain all these zeros and not just those 
