DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 183 
_.The step-by-step procedure which is applied is to consider the first 
relation of (7.67). Here ro is the first pulse of the specified input pulse 
sequence, k» is the first coefficient of the plant sequence, and é is the first 
pulse in the command sequence whose pulse transfer function is H(z). 
From this relation, k» is obtained arbitrarily, so long as vt 1s not required for 
€o to be greater than its wpper bound. If the fastest possible rise time is 
desired, éo is set at this limit and k,, computed. The second coefficient of 
K(@), km+1, is obtained in a similar manner from the second equation in 
(7.67) by attempting to set k»+1 at the value which satisfies the require- 
ments for responding to a unit step or ramp without error. If so setting 
the value of k»41 causes the error coefficient e; to exceed the limit, e1 is set 
at this limit and the coefficient k»1 computed from this relation. The 
procedure is repeated until such time as the limitation on the coefficient 
eé, is no longer set by the equations of (7.67), after which time the other 
necessary conditions on K(z) can be imposed. The simultaneous satisfy- 
ing of the relations of (7.67) and those ordinarily imposed on K(z) will 
complete the design. 
It is recognized that while K(z) will represent a finite settling time 
using this procedure, this finite settling time will not be minimal or even 
minimum. Additional k’s are required to simultaneously satisfy the 
requirements of (7.67) with regard to the upper limit of the e’s. In 
effect, this procedure takes the controller design step by step during the 
transient period and limits the command magnitudes which the controller 
produces. At the same time, the general conditions on K(z) concerning 
its structure to follow input test functions and the requirement of non- 
cancellation of poles and zeros on or outside the unit circle in the z plane 
must be observed also. 
EXAMPLE 
An error-sampled system has unity feedback and a feedforward pulse 
transfer function G(z), including the data-hold circuit, given by the 
following: 
(0.368 + 0.2642-!)271 
SO) = Gee ca 

This pulse transfer function can be expanded into a power series in z~! 
by the simple process of long division, yielding the following series: 
G(z) = 0.3682-1 + 0.76722 + 0.9152-3 + 0.9682-4 + ++ - 
The coefficients of the series in z~! are the various values of g; given in 
(7.67). The input is assumed to be a unit step function whose pulse 
transfer function is, in series form, the following: 
IWM(®) = se ee ee ae ee ae 6 e - 
