304 SAMPLED-DATA CONTROL SYSTEMS 
terms are 
y(1) =4 
y(2) = 73 
ete. 
11.5 Problems Involving Initial Conditions 
In the developments of the previous sections, the systems were gen- 
erally considered to be initially relaxed so that initial values and deriva- 
tives of the variables were assumed to be zero. Techniques for inserting 
initial conditions in the problem will now be considered. In the first 
place, the use of z forms described in Sec. 11.3 permits the insertion of 
initial conditions readily because the technique consists of approximating 
the Laplace transform of the variable of interest with a z transform. 
This means that, if initial conditions are nonzero, they may be inserted 
directly into the Laplace transform by the standard methods and then 
conversion into a z transform carried out. In the approach which 
replaces the continuous system with a sampled model a general technique 
will be developed which permits the application of initial conditions 
directly into the model rather than into the Laplace transform. 
A typical element of the system represents a process given by the 
exact integral 
y(t) = [s. a(t) dt (11.47) 
This integral is approximated by considering values at every T sec, as 
described in Chap. 4. Thus, (11.47) can be written 
‘ace b= ina x(t) dt + aay x(t) dt (11.48) 
This simplifies to 
ln + 1) T] = yt) + [PP 2 at (11.49) 
The integral in (11.49) is approximated by means of the polygonal 
approximation to produce an approximate value of y[(n + 1)T] as 
follows: 
yal(n + 1)T] = ya(nT) + T/2{x(nT) + 2[(n + 1)T]} (11.50) 
To obtain the z transform of (11,50), both sides are multiplied by z~ and 
summed over all positive values of n 
» yal(n + 1)T]z7 = 2, yo(nT)e-* + T/2 y a(nT)z-" 
0 n= n=0 
--) 
+ 7/2 > al(n + 1)T]e (11.51) 
n=O 
