DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 171 
which lie outside the unit circle in the z plane. Thus, 
K(z) = 271(1 + 2.342-1)(1 + 0.16271) (@o + az!) 
By containing no denominator polynomial in 2~1, this form of K(z) 
assures finite settling time, as well as ripple-free operation. 
The second specification on K(z) is that the system be capable of 
following a ramp input with zero steady-state error, in accordance with 
the requirement given by (7.15). This results in the equation for 
1 — K(z), given by 
1 — K(z) = 0 — 21)? + biz! + Doe?) 
Solving for K(z) from the two relationships which contain it, there 
result the following numerical values for the constants: 
ao = 0.73 
Cn —().47 
which produce an over-all prototype response K(z) given by 
IK(@) = 00825" qp Leis — OMe = Ole 
This solution should be compared to that of the minimal prototype 
design given in Sec. 7.5, where it is seen that K(z) contains terms in 
z—! only up to the third power, while the ripple-free design contains 
terms up to the fourth power. This means that in so far as sampling 
instants are concerned, this design has a longer settling time by one 
sampling interval. 
The response of the system to a unit ramp and a unit step are 
obtained by substitution of A(z) in 
CO) — a @@) 
where the appropriate expression for F(z) is used. The resultant out- 
put sample sequences obtained from the inversion of C(z) are plotted in 
Figs. 7.15 and 7.16, respectively. The intersample behavior of the 
system is sketched in solid lines, and it is seen that after the fourth 
sampling instant, the output ripple is reduced to zero. In the case of 
the step response plotted in Fig. 7.16, the overshoot is about 200 per 
cent, a figure which is comparable with that obtained in minimal proto- 
type design and higher than that obtained with a system designed with 
a staleness factor. 
The pulse transfer function of the digital compensator required to 
implement this over-all pulse transfer function is found by substitution 
in (7.5), resulting in 
Os OU Tegy > OAL See O06 3525° 
PO = Orem. — l O9deqa Onl iG2n* 

D(z) = 
