190 SAMPLED-DATA CONTROL SYSTEMS 
From the definition given in (7.75), A(s) is 
1 
BNO) s?+0.5s + 1 
lt is recalled that A(s) is the over-all continuous response transfer 
function with the digital controller not connected. In this respect, the 
system is stable in the absence of the controller, but it is underdamped 
and has an undamped resonant period of some 6.6 sec. 
The transfer function A(s)H(s) is given by 
1 
Als IEN(s 97 Geren t**) =e EaSa ea) 
Taking the z transform of this expression, 
0.7424" + Oo f2z3 
1 — 0.162-! + 0.472 
It is seen that AH(z) has no zeros of magnitude greater than unity. 
This means that the restriction set forth in rule 3 above does not apply. 
Thus, Kr(z) can be chosen arbitrarily and is taken as 
Kr(z) = ay “E Ag? 
where the a’s are constants to be determined. 
To satisfy the condition that the steady-state error in response to a 
unit step input be zero, the condition on Kpr(z) is that 
1 — Ke(z) = A — 27) $+ dye) 
The final condition is that given in rule 4. The required expression is 
AR(e) _ Z{1/[s(s? + 0.58 + 1)]} 
ALG) i= 



R(z) TAS 
which is 
Athy (@ uma ae ih 
none Caos s(s? + 0.5s + 1) 
It is seen that this expression is identical to AH(z) in this problem so 
that the previously derived numerical expression can be used. Sub- 
stituting this expression in the following, 
AR(z) _ 1 — 0.9027! — 0.102-2 
R(z) 1 — 0.162! + 0.472 
The zeros of this expression are at 1.0 and —0.10, which means that 
1 — Kr(z) must also contain a zero at 1.0 to produce a stable system. 
It so happens that this requirement is also that for having the system 
respond to a step input without steady-state error. Thus, 
Kr(z) = ayz71 + ogne 
and 1 — Kr(z) = 0 — 2) H+ bie) 
i 


