Biomedical Engineering Reference
In-Depth Information
solution, spatio-temporal system modeling and
space-distributed structures control as is shown
in (Fortuna
et al.
, 2001).
The purpose here is to give sufficient condi-
tions for the
absolute stability
(
robustness
) of
CNN with time delay feedback and zero control
templates defined by
restrictions being nondecreasing and globally
Lipschitzian verifying the inequalities (3) and
(4) with
L =
1.
Using the technique due to Kharitonov &
Zhabko (2003) one obtains a more complex
Lyapunov functional—the so-called “exact”
Krasovskii-Lyapunov functional, which gives
improved (less restrictive) conditions on system's
parameters. When applied to neural networks with
time delays this approach has to be completed with
some robustness conditions suggested by Malkin
(1952) for globally Lipschitz nonlinearities (as the
sigmoid functions of the RNN are).
Denoting
(
)
∈
x
=
−
a
x
(
t
)
+
c
f
x
(
t
−
)
+
I
,
i
=
1
m
i
i
i
ij
j
j
j
i
j
N
(34)
where
j
is the index for the cells of the nearest
neighborhood
N
of the
i
th
cell and t
j
are positive
delays. The nonlinearities for the cellular neural
networks are of the bipolar ramp type
(−=
, the Krasovskii-Lyapu-
nov functional is (see Danciu & Răsvan, 2005)
A
diag
a
)
m
0
1
( )
T
V
x
=
x
(
t
)
U
(
0
x
(
t
)
t
(
)
,
f
(
x
)
=
1
x
+
1
−
x
−
1
(35)
0
m
i
i
i
[
]
2
∑
∫
=
−
T
+
x
(
t
+
)
P
+
(
+
)
R
x
(
t
+
)
d
j
j
j
j
1
j
hence they are bounded, nondecreasing and glob-
ally Lipschitzian with
L
i
=
1 (Figure 4). Worth
mentioning that other sigmoidal nonlinear func-
tions, which are to be met in neural networks,
may also be considered.
We shall use the system in deviations
(37)
The sign condition on the derivative
W
of (37)
along the solutions of system (36)
m
( )
∑
T
T
W
x
=
x
(
t
)
P
x
(
t
)
+
x
(
t
−
)
P
x
(
t
−
)
t
0
j
j
j
(
)
∈
z
(
t
)
=
−
a
z
(
t
)
+
c
g
z
(
t
−
)
,
i
=
1
m
j
=
1
i
i
i
ij
j
j
j
j
N
0
m
∑
∫
T
+
x
(
t
+
)
R
x
(
t
+
)
d
(36)
j
j
=
−
1
j
where functions
g
j
are also subject to sector
(38)
Figure 4. The bipolar ramp function
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