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t
m
of regulators as the vectors constituted by the right
members of Eqs. (3.17) evaluated along
t
steps (where the values of all the vari-
ables of the system are known), and analogously the
t
-expansions
G
t
1
,
t
1
t
2
t
-expansions
φ
,
φ
,...,
φ
G
t
2
,...,
G
t
d
,
,...,
of the regressors
g
1
g
d
along the same
t
steps. With this notation Eqs. 3.17
provide a linear system with
d
g
2
×
m
unknowns:
1
c
1
,
1
G
t
1
+
c
1
,
2
G
t
2
+
...
+
c
1
,
d
G
t
d
φ
=
(3.20)
2
c
2
,
1
G
t
1
+
c
2
,
2
G
t
2
+
...
+
c
2
,
d
G
t
d
φ
=
...
=
............
φ
t
m
c
m
,
1
G
t
1
+
c
m
,
2
G
t
2
+
...
+
c
m
,
d
G
t
d
.
=
Now, let
C
1
,
C
m
be the unknown column vectors of dimension
d
constituted
by the coefficients of the regressors providing the linear combinations of regulators
ϕ
1
,
ϕ
2
,...,
ϕ
m
we are searching for, and
C
2
,...,
C
1
,
C
2
,...,
C
m
)
C =(
the matrix having these vectors as columns.
Let also
F
be the following matrix constituted by
m
column vectors of
t
elements:
t
1
t
2
t
m
F =(
φ
,
φ
,...,
φ
)
.
Finally, let
G
t
1
,
G
t
2
,...,
G
t
d
)
G=(
be the matrix of dimension
t
d
having as columns the
t
-expansions of regressors.
With the notation above Eq. (3.20) becomes:
×
G
×
C= F
(3.21)
n
be the column vectors of dimension
t
constituted by
substance variations of substances, from step
i
to step
i
1
2
Moreover, let
Δ
,
Δ
,...,
Δ
+
1, for 0
≤
i
≤
t
, and:
t
1
t
2
t
n
D=(
Δ
,
Δ
,...,
Δ
)
the matrix having these vectors as columns.
By using matrix transposition (denoted by the exponent
T
), Eq. (3.16)
A
×
U
[
i
]=
Δ
[
i
]
becomes:
T
T
T
U
[
i
]
×
A
=
Δ
[
i
]
=(
Δ
[
i
]
,
Δ
[
i
]
,...,
Δ
[
i
])
n
1
2
which, expanded along the
t
time points, provides:
T
F
×
A
= D
(3.22)
