Biomedical Engineering Reference
In-Depth Information
are reasonable taking into account that the artifi-
cial neurons operate in times about
the whole real axis. Given an initial condition (in
our example,
x
0
= [1,2]), the internal behavior of
the unforced system (the neural network without
stimuli) can be discuss as time evolution of the
state's components (Figure 9) and the evolution
of the trajectory within the state space (Figure
10). Figure 9 shows an exponential convergence
in time of the state toward the equilibrium in
origin, whereas Figure 10 shows that this very
good convergence is ensured by a stable focus
in the origin of the state space.
Relating to the two theoretical results, let us
remark that the system may be quite large from
the dimensional point of view and the transmis-
sion delays may introduce additional difficulties
in what concern the direct check of the conditions
in theorems. Within this context,
Theorem 5
leads
to computer tractable Linear Matrix Inequalities,
while the frequency domain inequality in
Theo-
rem 6
is difficult to verify.
10
−
seconds.
The simulation was made using the MATLAB-
Simulink tool.
We verify the theorem's conditions:
9
1.
a
= >
2
0,
a
= >
;
5
0
1
2
2.
the nonlinear activation functions
f
i
,
i =
1, 2
of the neurons are sigmoidal: the hyperbolic
tangent function from the MATLAB-Simu-
link library
t
−
t
2
e
−
e
f t
( )
=
tanh( )
t
=
− =
1
1
+
e
−
2
t
e
t
+
e
−
t
its values are within the interval
[ 1,1]
−
and
the Lipschitz constant is
L =
1;
3.
For second order systems, as in our example,
the fulfilling of (44) gives: a small gain
condition on system's parameters
16
a
2
2
+
9
2
i
ii
c
<
ii
2
2
(
4
a
+
3
)
i
ii
The Estimation of the Admissible
Delay for Preserving the
Gradient-Like behavior
and for the free parameters θ
1
, θ
2
the
condition
.
(
)
The purpose here is to give an estimate of the
admissible delays that preserves the asymptotic
stability for each asymptotically stable equilib-
rium of a RNN with multiple equilibria. Consider
the Hopfield-type neural networks (5) together
with its model affected by time-delays (24) and
theirs systems in deviations (10), respectively
(25). Obviously, both systems (5) and (24) have
the same equilibria and we assume the
gradient
like-behavior
for the delayless Hopfield-type
neural network described by (5).
The following result is based on the estimate
for the state of system in deviation (25) using a
consequence of rearrangement inequality No.
368 from the classical topic of Hardy, Littlewood
and Polya (1934) and a follows from a direct
consequence of a technical Lemma belonging to
Halanay (1963) (see the proof in Danciu & Răsvan,
2001); it reads as follows
1
∈
0
0267
,
2
03106
2
It is easy to check that in our case,
76
c
=
2
<
2
,
c
=
3
<
11
.
81
and taking
11
22
1
=
, the frequency domain condition
of Popov is fulfilled.
1
2
4.
the external stimuli are bounded functions
on the whole real axis as shown in Figure
5 and Figure 6:
=
—an almost periodic signal with
d
(
t
)
sin(
t
)
+
sin(
t
)
1
|
d
( ) |
t
<
2
1
and
d
2
( )
t
=
2 cos( )
t
—a periodic signal with
|
d
( ) |
t
<
.
3
2
Figure 7 and Figure 8 represent the time
evolution of the state for the neural network
with external stimuli; one can see that the two
components of the state are almost periodic (as
an effect of the first stimulus) and bounded on
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