DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 159 
consists of an integrator and a double simple time delay. The data 
hold is a simple zero-order hold. ‘The problem is to design a linear 
program for the digital controller which will produce an over-all 
response of minimum finite settling time. If possible, this should be a 
minimal prototype. In addition, the over-all system is to respond to a 
ramp with zero steady-state error. 
The continuous feedforward plant transfer function G(s) is 
pes Og 
s(s ++ 1)? 
The z transform corresponding to this transfer function is 
(1 + 2.342-1)(1 + 0.162-!)2z-! 
@=2 0 = 0/3682 )2 
The pulse transfer function G(z) contains zeros at —2.34, —0.16, and 
co, and poles at 1.0 and 0.368, the latter being a double pole. To 
obtain a finite settling time, the prototype response function K(z) con- 
G(s) = 10 
G(z) = 


Fia. 7.8. Illustrative example of minimal prototype design. 
tains only a numerator polynomial in z2~!. The lowest order in z~! in 
the numerator of the function G(z) is unity, so that A(z) must con- 
tain 2~! as its lowest order also. It is seen that G(z) contains a zero, 
2.34, which lies outside of the unit circle in the z plane and for this 
reason must be contained in K(z). The prototype response which 
satisfies these requirements is of the form 
K(zg) = (1 + 2.3427!) (arz7! + age? + azz? + + + -) 
where the various a’s are to be determined. To obtain the minimum 
settling time, only a minimum number of a’s required to satisfy the 
other requirements will be used. 
In order to respond with zero error to an input ramp function, the 
additional requirement is placed on K(z) as follows: 
1-K@) =(—2)0 + bet + bet + °°) 
It is seen that this requirement automatically satisfies the condition 
that 1 — K(z) contain the poles of G(z) which lie outside or on the unit 
circle. 
