302 SAMPLED-DATA CONTROL SYSTEMS 
approximation can be made that 
ACEI oe 
dt” Or 


(11.38) 
If this approximation is acceptable, then (11.36) can be expressed as 

n n—1 
d ee: ae et +--+ + gy = fi) (11.39) 
Equations of the form in (11.39) can be solved by numerical methods 
which are set up using z forms. The procedure is to take the Laplace 
transform of both sides of (11.39) 
8’Gn(s) + s™7G,ai(s) + - - + + G(s) = F(s) (11.40) 
where G(s) is L(gny). It is assumed here that all initial conditions on 
gny are zero. If this does not obtain, the initial conditions can be inserted 
in the usual manner. Dividing (11.40) through by s”, 
Gone Gn(8) Shi ext < G(s) zs + Fs) (11.41) 
This relation can be interpreted as being an equality between various 
integrals on the left side of the equation and integrals on the right side. 
The various 1/s* terms can be approximated by the z forms of Table 11.1. 
Clarification of the approximation of the various G;(s) is required. 
It is recalled that G;(s) is the Laplace transform of the time function 
gi(t)y(t), which has been abbreviated in this development as g,y. The 
Laplace transform is, by definition, 
Gia i * a(t)y(t)e-* dt (11.42) 
The integral can be approximated by the summation 
Gi(s) = > gi(nT)y(n Tye"? T (11.43) 
n=0 
This summation is recognized as 
and, similarly, 
F(s) = TF(z) (11.45) 
The procedure is to replace the various 1/s* terms by the z forms and the 
Laplace transforms by the approximate z transforms. Abbreviating the 
symbol for the z forms as ZF; to represent the z form corresponding to 
1/s*, (11.41) becomes, after dividing through by T, 
Gn(z) + ZPiGra(z) + + + * + ZF.Go(z) = ZF.F(z) (11.46) 
