Time delay systems are known to take an important place in many fields of application. We refer to [7] and the references therein for more details on the subject. We propose a method for parameters and delay identification, inspired by the work of Fliess and Sira-Ramìrez [3].
We write the system
, with
. The output is
where
stands for the noise
which is such that
goes to
when
goes
to infinity. The main idea is to use a family of functions
such that
for
. Let
. Integrating by
parts, we get
. So we
may estimate the values of the coefficients
and
by solving
the system
for
, by
the mean squares method. We can estimate
and its derivatives in
the same way, using functions
such that
for
and
for
, with
.
In practice we have used
, the integration being done
between
and the current time. A good approximation of the
integrals is obtained by integrating the system
if
, with
initial conditions
:
tends quickly to
for
great enough. Numerical simulations
are given at example 1.
In section 3, we consider a delay system
, with
. We use the notation
, where
is such that
, for
with
. Let
denote the equation
In example 2, we used and investigate the precision of
the evaluation depending on the size of the noise. See Tableau 1.
In section 4, we adapt the method of section 1 to the delay situation,
solving the system
Example 3 shows a simulation with . Greater delays could be
considered by changing the time scale. Example 4 considers the case of
a slowly varying delay. Scilab simulation files are available at url
[11].