DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 151 
A table of minimal prototype response functions is given in Table 7.1. 
It is understood that, if a minimal prototype is used, the system will 
respond without error for any lower-order input function in the steady 
state. 
The third stated requirement for a minimal prototype is that the 
system settle in a finite time after the application of aninput. As used in 


TasBLe 7.1. Mintmau PrRotrotyPE REsPponseE FUNCTIONS 
Input r(t) R(z) K(z) 
1 il 
Step KO) 9) ares g 
Tam i a 
Ramp tu(t) Gd —22 22 = & 
: T2211 -+ 271) 
Acceleration (2201 (G) a |e ee ag > = Be 2 op ae 
(1 — 27})3 



this application, this means that the transient must disappear at sampling 
instants only. There is the very likely possibility that, between sampling 
instants, a continuous output will differ from the desired output and that 
the system will ripple about the steady-state value. However, in a 
minimal prototype, the output will be exact at sampling instants. This 
effect is shown in Fig. 7.4, where it is seen that the output has settled 
after one sampling interval in so far 
as sampling instants are concerned, 
but that there is a substantial ripple 
component during the sampling in- 
tervals. A system which is designed 
both to have finite settling time at 
sampling instants and to be ripple- 
free is one which can truly be called a 
finite-settling-time system. This re- 




quires special treatment and willbe ° zy ae Sree ee ioe 
considered later. Minimal proto- Wie 
: P Fic. 7.4. Ripple in finite-settling-time 
types are concerned only with sam- systems. 
pling instants. 
In the expression for an over-all response function as given in (7.9) and 
(7.15), K(z) may be the ratio of two polynomials in z~!. In (7.15) the 
unspecified function F(z) may be the ratio of two polynomials. In 
minimal prototypes, F(z) is taken as unity, resulting in a minimal expres- 
sion containing zeros anywhere in the z plane but having all its poles at 
