152 SAMPLED-DATA CONTROL SYSTEMS 
the origin. It is this characteristic which produces a finite-settling- 
time response. The reason for this is found in the fact that the impulsive 
response of a system containing only a numerator polynomial in K(z) is of 
finite duration. If properly designed, a system of this type will continue 
to generate the correct number after all the necessary past samples are 
weighted and added. Thus, minimal systems and, for that matter, all 
finite-settling-time systems are characterized by the fact that they have 
only a numerator polynomial in z~! in their pulse transfer function. 
EXAMPLE 
A sampled-data system is to be designed with a minimal prototype 
response K(z) such that it responds to a ramp input without error. 
From Table 7.1, the minimal prototype for this condition is 
TZ) ee 
The response to an input whose z transform is R(z) is given by 
Cz) = @z4 — 2) RE) 
Taking three inputs, a unit step, a unit ramp (for which the system is 
specifically designed), and a unit acceleration, the following outputs 
result: 
For a unit step, 
Dem a a 
Dee 
For a unit ramp, f, 
ill a Se 
CO = TaN 22 ga) 
Ga iem) 
For a unit acceleration, ¢?/2, 
Pz aes cee) 
2(1 — 27})3 
The three transforms are inverted, resulting in the pulse sequences 
which are plotted in Fig. 7.5. 
The response of the system to a ramp input for which the minimal 
prototype is designed is relatively docile. The system settles to zero 
error in two sampling intervals, with no overshoot. It is understood, 
of course, that the continuous output of the system probably does have 
overshoot between sampling instants but this component is referred to 
as ripple and is not considered in the minimal prototype. The response 
of the system to a step input is seen to have a severe overshoot of 100 
per cent. The reason for this is that upon application of the first 
sample at t = 0, the system responds as though it were being subjected 
to a unit ramp since the sample at ¢ = —T is zero. Finally, the 
response to a unit acceleration is seen to have finite steady-state 


Ce) — 

