DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 169 
numerator. This can be readily ascertained from (7.5), where the ratio of 
K(z) and 1 — K(z) produces a polynomial ratio in z which is of no higher 
order than the numerator of K(z) if this condition obtains. Since most 
practical designs use a denominator in K(z) which is of first order, no 
increase in complexity of the controller is expected. 
7.7 The Design of Ripple-free Systems 
In general, there is ripple in the output of systems which implement a 
minimal prototype or staleness factor. This ripple may be objectionable 
under certain circumstances. For instance, if mechanical elements are 
involved, excessive oscillatory stresses will be superimposed on those 
required to cause the system to respond smoothly to an input. A desir- 
able objective is that the system contain no ripple after a suitably short 
transient period has elapsed. In this case, once the system has come to 
the steady state, no excess command input to the plant other than that 
required to maintain steady state is applied. 
It is evident that, to obtain this type of performance, the feedforward 
components of the control system must be capable of generating a smooth 
output which is the same as the input. For instance, if the input is a 
ramp function, the output of the plant must also bea ramp function. To 
generate such a ramp, an integration must be present in the plant transfer 
function so that a constant command to the plant will, in the steady state 
produce the desired output. If the integration were not present in the 
plant, it would have to be supplied by some other cascaded element. It 
will be assumed in the discussions which follow that the feedforward ele- 
ment satisfies the condition that it contain the necessary integrations to 
produce ripple-free outputs as required. The problem then is to design 
the digital controller such that it will drive the system to this steady-state 
condition after some finite transient period. 
The philosophy underlying such a design is to arrange the controller 
pulse transfer function in such a manner that the error-sequence response 
is of finite length, that is, the pulse transfer function H.(z)/R(z) contains 
a finite number of terms in 2z~!. Referring to Fig. 7.3, the pulse sequence 
E2(z) which is applied to the plant and hold system typified by the pulse 
transfer function G(z) is given by 
ju) = ae (7.46) 
aid COL KORG) (7.47) 
Substituting C(z) from (7.47) in (7.46), there results 
Ex(z) _ K(z) 
R(z) Gz) 

(7.48) 
