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then we can read the equation
EMA
by reversing the known values with the un-
known ones. In fact, by writing the substance variation vector:
s
[
i
+
1
]
−
s
[
i
]=
Δ
[
i
]
and, assuming
n
substances and
m
reactions, we get the following system, which we
call
ADA
(Avogadro-Dalton-Action, see [94]), for remarking that here, differently
from
EMA
of Eqs. (3.5), the unknown values are fluxes
U
[
i
]
:
A
×
U
[
i
]=
Δ
[
i
]
.
(3.16)
For the determination of the regulators which provide the best approximate solution
of Eq. (3.16), we apply a procedure which we call
stoichiometric expansion
(see
[102, 103, 104]).
Given a positive integer
d
, let us assume that the regulators we are searching for
can be expressed as linear combinations of some basic regressors
g
1
,
g
2
,...,
g
d
which usually include constants, powers, and products of substances, plus some ba-
sic functions which are considered suitable in the specific cases under investigation:
ϕ
1
=
c
1
,
1
g
1
+
c
1
,
2
g
2
+
...
+
c
1
,
d
g
d
(3.17)
ϕ
2
=
c
2
,
1
g
1
+
c
2
,
2
g
2
+
...
+
c
2
,
d
g
d
...
=
............
ϕ
c
m
,
d
g
d
.
Equation (3.17) can be written, in matrix notation, in the following way, where
U
=
+
+
...
+
c
m
,
1
g
1
c
m
,
2
g
2
m
[
i
]
is the column vector of regulators evaluated at state
s
i
,
G[
i
]
the column vector of
T
regressors evaluated at the same state, and
d
of the unknown
coefficients of regressors (exponent T denotes matrix transposition):
C
is the matrix
m
×
T
U
[
i
]=C
×
G[
i
]
.
(3.18)
Substituting the right member of Eq. (3.18) into Eq. (3.16), we obtain the following
system of equations (
A
is the stoichiometric matrix):
T
A
×
C
×
G[
i
]=
Δ
[
i
]
.
(3.19)
Now, if we consider
t
systems of type (3.19), for 1
t
,andif
n
is the number
of variables, we obtain
nt
equations with
md
unknown coefficients of
≤
i
≤
md
and the system has maximum rank, then we can apply a Least Square Evaluation
which provides the coefficients that minimize the errors between the left and right
sides of the equations. These coefficients provide the regulator representations that
we are searching for.
Now we provide a compact representation of the unknown coefficients con-
stituting the matrix
C
.If
nt
>
C
, by using suitable matrix operations. Let us consider the
