48 SAMPLED-DATA CONTROL SYSTEMS 
of integrators to generate a third-order polynomial. The additional 
element which requires explanation is the first integrator which precedes 
the zero-order hold. It is noted that this integrator has samplers pre- 
ceding and following it. In effect, this causes the element to act as a 
summing device which adds all the past error samples in ef(t). This 
element is necessary when it is considered that the error at any particular 
sampling instant n7' is actually the result of the summation of effects 
of all previously generated correction polynomials and not just the 
difference between the most recent error polynomial contribution and 
the input. The zero-order hold is required to convert the pulse at any 
instant n7' into a constant signal for the following interval as required 
in the generation of the error polynomial. Thus, the signals applied to 
the various integrators of the chain are the summation of effects of the 
previous error polynomials in addition to the changes required for the 
most recent error polynomial. 
The system shown in Fig. 3.20 fulfills the objectives of the extrapolation 
by inserting weighted components of the error as required by (3.29) 
in the manner shown in Fig. 3.19. If the input were actually a third- 
order polynomial and if there were no noise or drift in the system, the 
output Q(t) would be an exact replica of the input polynomial after a 
fixed settling time has elapsed. This would mean that the error sequence 
er(t) is zero at all times and that the signals which would be applied 
to the various summing points in the integrator chain would be the 
constants required to generate the polynomial. On the other hand, if the 
input were not a polynomial, the system would generate an approximation 
to the input signal and at each sample time an error pulse e:1(m7’) would 
appear and alter the output polynomial accordingly. 
In some applications considered by Porter and Stoneman‘*® it was 
considered undesirable for the output to experience sudden changes or 
steps. The system illustrated in Fig. 3.20 would act in this manner. 
This can be seen when it is considered that the presence of an error 
pulse e:(n7’) causes a sudden change in the output of the zero-order hold 
and that this change is applied directly to the output at the last summing 
point in the chain. One way of avoiding this effect is to apply this 
component of the signal to the next-to-last summing point, that is, pre- 
ceding the final integrator. In this manner, the output of the system 
gradually changes to the required value and the complete correction 
takes place at the end of the sampling interval under consideration rather 
than at the beginning. The polynomial so generated does not close out 
the error instantly, but the output is considerably smoother and is often 
more desirable as a result. 
The feedback implementation of higher-order holds is only slightly 
more complex than the open-cycle implementation illustrated in Fig. 3.16. 
