Load Matching, Grid Interaction, and Advanced Control - Modeling, Design, and Optimization of Net-Zero Energy Buildings - page 323

(6.18)
This effectively results in replacing the sequence with a polynomial in terms
of z − 1 . The definition presented in Eq. (6.18) implies that the z-transform of
a sequence of values can simply be found by inspection. In a z -transform,
the auxiliary variable z is defined as
(6.19)
This substitution illustrates that the z -transform corresponds to a special
case of the Laplace transform applied to a sequence of rectangular pulses
with period T . By definition, the Laplace transform of a function y ( t ) is
(6.20)
Let the function y ( t ) be a sequence of values at time intervals T .
(6.21)
Equation (6.20) can be then written in terms of z as
(6.22)
Transfer functions can also be defined in terms of z -transforms:
(6.23)
The advantage of a z -transfer function, such as the one in Eq. (6.23) , is that
it can be linked to a difference equation involving consecutive values of the
input and output (Moudgalya, 2007). For example, consider the z -transfer
function
(6.24)
With some algebraic manipulation