160 SAMPLED-DATA CONTROL SYSTEMS 
The constants in the above expressions are obtained by substituting 
K(z) into the expression for 1 — K(z) and equating the coefficients of 
terms of the same power in z~!. This results in the following simul- 
taneous equations relating the coefficients: 
=< i— by —2 
—2.34a, —- a2 = 1- 2b1 
—2.34a. = by 
Three simultaneous equations define the values for three unknowns. 
Had more terms been included in K(z), there would have been more 
unknowns than equations, and arbitrary values could be assigned to 
the excess coefficients. This would not result in a minimum finite 
settling time, however, as specified in the problem. 
Solving the equations, the coefficients are 
a= 0.81 
ag = —0.51 
By substituting back in the expression for K(z), the over-all prototype 
which results is 
Ke@\i— OS leet AgS8ee> — 1 Oza 
The response of this system to a unit ramp input is given by 
il 
G@) S (Oss ss sips NI) me ee = (cee 
which upon inversion yields the output sequence plotted in Fig. 7.9. 
The dashed lines are for guidance to indicate the location of the output 
samples. Plotted with a solid line is an estimate of the continuous 
output c(t), which is seen to ripple about the final value after the system 
has settled at sampling instants. 
To show the disadvantage of a minimum response prototype, the 
response to a unit step input is computed. The output sequence for 
this input is defined by 
Cle) = O8le + 1.38? — 1.1974) ~— 

which upon inversion yields a pulse sequence plotted in Fig. 7.10. The 
overshoot is over 200 per cent and again, even though the system settles 
in three sample times, there will be a ripple in the output, as estimated 
by the solid-line curve. While disadvantages have been pointed out, 
it should also be emphasized that this system settles in the minumum 
number of sample times possible and that if the ripple and overshoot 
