424
The new form of equation (1) is now given by
( 1)
( )
( (
))
( )
x k
Ax k BS x k h Cu k
.
(2)
This is a basic requirement for controller design. Now, we are interested designing a feedback controller for the
system equation (2) as
( )
( ),
u k Kx k
where
K
is
n m
u
constant control gain matrix.
The new form of (2) is now given by
( 1)
( )
( (
))
( )
x k
Ax k BS x k h CKx k
.
(3)
Throughout this paper we assume the neuron activations
i
i
s x
( )
,
1, 2, ,
i
n
is bounded and monotonically
nondecreasing on
R
, and
i
i
s x
( )
is Lipschitz continuous, that is, there exist constant
0
i
l
!
for
1 2
i
n
, , ,
such that
1 2
1
2
1 2
,
, ( ) ( )
.
i
i
i
r r
s r s r l r r
d
(4)
By condition equation (4),
i
i
s x
( )
satisfy
( )
,
i
i
i
i
s x l x
d
1, 2,...,
i
n
.
(5)
Theorem 3.1
The zero solution of the delay-difference control system (3) is asymptotically stable if there exist
symmetric positive definite matrices
, ,
P G W
and
1
[ , , ] 0
n
L diag l
l
!
satisfying the following matrix
inequalities of the form
(1,1)
0 0
0 (2, 2) 0 0
0 0 (3, 3)
\
§
·
¨
¸
¨
¸
¨
¸
©
¹
,
(6)
where
