![](bg8.png)
Theorem 4. Consider the system of (2a-c) and (3)
where the system matrices lie within the polytope
¯
Ω
of (7). For a prescribed scalar γ > 0 and a positive
tuning scalar ε
f
, there exists a filter of the structure
(4) that achieves J
F
< 0, where J
F
is given in (6), for
all nonzero w ∈
˜
l
2
([0, ∞);R
q
), n ∈
˜
l
2
([0, ∞);R
p
), if
there exist n × n matrices
¯
X > 0, Y > 0, 2n × 2n ma-
trix
¯
R
p
> 0, n × n matrices K
0
and U, 2n × 2n matrix
˜
P
M
and a n × l matrix Z, and matrices H and G that
satisfy (31). In the latter case the filter parameters can
be extracted using (26) as explained in Theorem 3.
6 CONCLUSIONS
In this paper the theory of linear H
∞
filtering of state-
multiplicative noisy discrete-time delayed systems,
is extended to the robust polytopic vertex-dependant
case. Delay dependent analysis and synthesis meth-
ods are developed for the robust case which are based
on the input-output approach. Sufficient conditions
are thus derived for the robust stability of the system
and the existence of a solution to the corresponding
robust BRL. Based on the robust vertex-dependant
BRL derivation, the robust filtering problem is for-
mulated and solved.
An inherent overdesign is admitted to our solu-
tion due to the use of the bounded operators which
enable us to transform the retarded system to a norm-
bounded one. Some additional overdesign is also ad-
mitted in our solution due to the special structure im-
posed on R
2
.
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