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cribed, and in fourth section the application
to simulated data is given. From experimental
data, some ensembles are created and the
generation of data is thus achieved. It is revealed
a small discrepancy between formalism
and data. The results are in fact an interesting
indication that this formalism can be used in
advanced control strategies in order to guarantee
a certain efficiency of actuators towards
the set-point stability over the control horizon.
We also display curves by emphasizing the
incorporation of non-diagonal contribution on
the I/O relations in order to reduce the errors
and discrepancies. Finally, in fifth section conclusions
are sketched.
II. THE FORMALISM
A. Notation Used in this Scheme
In the Laplace space it is well-known the correspondence
between I/O functions linked
through the transfer function as Y(s) = H(s)
X(s) where it can be recognized as a purely
linear relation without affecting the basic
functionality of a continuous system [3]. In
time space sometimes one speak about convolution
where the input function appears to be
delayed as follows
For the sake of the simplicity this convolution
operation shall be denoted as a bracket as
which is recognized as the first order approximation.
To note that the brackets encloses
only lower indexes of and
functions, respectively. For second and third
order is possible write down as
(2)
It should be noted the case when
M=N=K=L=P=Q=1, and (1), (2) and (3) are put
all together, the full third order diagonal nonlinear
series is recovered. So, by gathering (1),
(2) and (3) as a sum of diagonal terms, then
and, in terms of the brackets definitions equation
(4) can also be written as
y(t) = [11,1] + [11,1 | 11,1] + [11,1 | 11,1 | 11,1]
(5)
and so on (of course, it s possible to introduce
high order terms but it goes beyond the scope
of this paper). In these studies, only third order
are taken into account inside variable interactions
expansions.
B. Second Order Interactions
Consider now a 2x2 MIMO systems whose I/O
dynamic variables can be related by the following
equations
Let us consider the most general case in which
the output y1(t), for instance, gets interactions
in all situations as follows
and the output finally can be defined under these
circumstances as the sum of the free terms
plus the one containing interactions,
where the second line gives account of the free
case as expressed in equation (6) and the third
and fourth ones take into account the interaction
through the product by the full expressions
(6) and (7). In terms of brackets nomenclature
as given in (2), the complete expression for (9)
reads
y1(t) = [11,1] + [12,2] +
[11,1|11,1] + [11,1|12,1] +
[11,1|21,1] + [11,1|22,2] +
[12,2|11,1] + [12,2|12,2] + [12,2|21,1] +
[12,2|22,2] (10)
Clearly, the [11,1] + [11,1|11,1] and [12,2] +
[12,2|12,2] are recognized as the diagonal terms
whereas the other ones are fully non-diagonal
interactions. In principle, all these interactions
might be necessary to adjust data to the “plant
curve”. Under these circumstances it is possible
rewrite (10) in a most friendship manner,
y1(t)= Diag2 [11,1] + Diag2 [12,2]
[11,1|12,1] + [11,1|21,1] + [11,1|22,2] +
[12,2|11,1] + [12,2|21,1] +
[12,2|22,2] (11)
with Diag2 [11,1] = [11,1] + [11,1|11,1] and
Diag2 [12,2] = [12,2] + [12,2|12,2] . It is interesting
to note that the diagonal expressions play
the role of self-interactions by the same input.
C. Third Order Interactions
Let us now to use the expansion for third order
in where is assumed a full interaction of up to
three I/O products. Based on it the complete expression
pushes us to write down as
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