Ciencia Y Tecnologia

Tunable System Identification and
Convolution Integrals
Juan Tisza-Contreras
Decano de la Facultad de Ingeniería Electrónica y Mecatrónica
Universidad Tecnológica del Perú
Abstract
In this paper, I’ve developed a model of integrals of convolution where the main point is based on the
possibility of getting tunable system identification in a much more complete manner and which is based
onthe quantitative measuring of interactions in t ime. The way for that consists in implementing a set of
parameters which are solved in each step of computation of the convolution integrals without losing the
generality. As illustration of this method, I’ve investigated the sustainability of the systems which are
required to be stable along the control horizon and which are sensitive to collapseafter a long time.

I. Tunable System Identification
I’ve defined as tunable system identification to the
action of measuring and altering the output
variables from convolution integrals with several
parameters changing in time. When complex
systems variables are changing in their form and
properties, and it is rather fast, then the observer
has not chance to decide the next step to solvethe
integrals equations and preferably make the
choice as convenient as possible. Exceptions
occur when the number of free parameters is
small, and most of them are negligible when the
system goes over horizon and maintains its set
point for a longer time [1]. The case when the
system contains a huge amount of free parameters
is much more complicated because the measuring
needssophisticated tools to maintain the stability
and equilibrium in each time step. My proposal is
the reconstruction of an algorithm which should
guarantee the measuring in all time steps and
making the measure error much more smalls or at
least leaving the system invariant in a short future.
I’ve coined the term “tunable” because the
parameters are tuned one-to-one in time and their
values are knownprevious to the next step.
The solving of these parameters for a time t has
consequences for the new set of parameters in the
time t+T where is assumed the system is still
stable. For times where the system might fall
down by showing weakness and a possible
collapse, the measuring of parameters previous to
these actions, is very important because it gives us
valuable information ofsystems in ad vance, in
special their important parameters. As example I
review the case where the system contains transfer

and input function with up to 10 parameters and
which are solved simultaneously within a
systematic error whose presence is contemplated.
In the next I’m going to explain the main
ingredients of my proposal.

II. Tunable Convolution Integrals
A convolution integral isan operation where there
are two functions to be integrated together:

One can observe that only one is shifted or
delayed. It is more or less used in input/output
(I/O) relations and would serve to apply in control
theory. Now I will write the same but for a next
step in time

It is easy to note that the input function left
invariant because it does not depend on the
variable “t”. NowI’m going to compute the square
of difference expressed as

One sees that the third line contains the product of
the convolution in the step “t” times the one which
is function of “t+T”. It is the important term

because from here the tunable system
identification can be realized. The third line is then
rewritten

and we can see the product of two transfer
functions and by which can berealized as a
interaction instead a correlation. In effect,
correlation sometimes is called to the product of
two transfer functions when both differ from a
shift in time. Correlation is defined as the effect of
one function over the subsequent one when they
describe the same phenomenon [2]. In this case
I’m not interested in correlations but also in
interactions because my hypothesis...