MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 305 
Substituting k = n + 1 in the first and last sums, 
» YalkT)2-*z > ya(nT)z—" + T/2 » a(nT)2-" 
k=1 n=0 n=0 
i} 
+ 7/2 » a(kT)e-ke (11.52) 
k=1 
By adding and subtracting the initial terms in the first and last sums in 
(11.52), this equation can be written in terms of the z transforms of y(nT) 
and z(nT) 
aYa(z) — y(O)] = Ya(z) + T/2[X(z)] + T/2{a[X(z) — x(0)]} (11.58) 
Simplifying this expression, Y.(z) becomes 
_ RO see Wee) 40) eee) 
VA@) = SS oo Vee Se) (11.54) 
In this manner, the z transform is corrected for the initial values of the 
independent and dependent variables. 
The operations indicated by (11.54) are shown schematically in Fig. 
11.14. The input or independent variable x(t) is applied at the input, 
(0) z x(0) 
Fig. 11.14. Sampled model of integration process with initial conditions. 


while the initial conditions are inserted as inputs at the intermediate 
summing point, as shown. This equivalent system suggests the tech- 
nique which is used to insert initial conditions in problem solutions. The 
system equations are written in terms of Laplace transforms, and a sys- 
tem block diagram is drawn containing only elementary integration oper- 
ations. This type of reduction is much the same as that which would be 
applied if the problem were to be set up on an analogue computer contain- 
ing only summers and integrators. Each integrator is then replaced by 
the sampled equivalent circuit shown in Fig. 11.14 and the initial condi- 
tions inserted. ‘The sampled model which results can be solved by the 
usual z-transform inversion methods. This is best illustrated by means 
of a simple example. 
EXAMPLE 
The resistance-inductance circuit shown in Fig. 11.15 is subjected 
to a unit step voltage applied at time ¢ = 0. It is desired to set up a 
computational model which permits the insertion of an initial current 
