DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 191 
It is seen that these two conditions on Kpr(z) are satisfied by taking 
a. and 6; equal to zero and a; equal to unity. This would produce a 
system having a finite settling time of one sampling interval. The 
extra terms permit the arbitrary assignment of one coefficient. Taking 
a value of 0.567 for ai, it follows that a2 and b; must be 0.483 and 0.433, 
respectively. Thus, 
Kr(z) = 0.5672-1 + 0.4332-? 
This expression holds only for a unit step input since the form of K r(z) 
depends on the input. 
The output pulse sequence resulting from the application of a unit 
step input is obtained by inverting C(z), 
0.56724 + 0.4332 
U2) = 1 — 27! 
which upon inversion yields the output sequence c*(f), 
e*() = 0.5676@) + 1.06¢ — 7) + 1.00¢ — 27) +: --: 

The system settles at sampling instants in two sample times. In con- 
trast, the system without the digital controller would have a sequence 
of values measured at sampling instants given by 
e*() = 0.7426(¢) + 1.4305¢ — T) + 1.1936 — 27) 
L OLR08G = BM) se > 
The digitally compensated system settles in a shorter time and, as 
expected, can respond to a step input with zero steady-state error. It 
should be pointed out that there will be intersample ripple in the 
digitally compensated system which does not appear in the sequence 
representation. It should also be pointed out that the assumption is 
made that the input step function appears at the instant of the first 
sample. If this does not occur, there will be more overshoot since the 
digital compensator does not produce a command output until a sample 
time is reached. 
The illustrative example shows how the design procedure is applied to 
a simple problem. By properly designing the bypass digital controller, 
improved performance can be achieved. Naturally, the sampling period 
must be high enough to generate command outputs frequently. This is 
particularly important for those cases where the input occurs at instants 
of time not corresponding to sampling instants. During the period pre- 
ceding the first sample time, the system is controlled by the continuous 
elements and the desired response is not achieved. Experimental results 
obtained on a system like that of the illustrative example? show that 
systems may have overshoots which are many times greater than those 
obtained with a step function applied at or near a sampling instant. To 
