Weighted performance criteria for the synthesis of discrete systems

For the free and forced motion of an asymptotically stable linear stationary discrete control system of order
k
, Performance Indexes given by the general expressions
\Sigma\min{n=0}\max{\infty} c(n)x_{(l)}(n)
and
\Sigma\min{n=0}\max{\infty} c(n)\Delta^{m}x_{(l)}(n)
where
c(n)
is any
z
-transformable function,
x_{(l)}(n)
is a vector whose elements are products of the state variables of the system in the form
\frac{l!}{a!b!...h!}x_{1}^{a}(n)x_{2}^{b}(n) ... x_{k}^{h}(n);a + b + ... + h = !
and Δ
^m
is the
m
th forward difference, can be computed from the system's parameter without obtaining an explicit solution of the response. This is achieved by the introduction of a matrix
L
formed in a certain manner from the system's matrix
A
and the introduction of a "Matrix
z
-transform"
C[(L'+ I)]
of
c(n)
.