4 Wei-Wei Che et al.
where e(k) = x(k) −ˆx(k) is the estimation error, and
¯
A
¯
β
= A + (1 −
¯
β)B
2
K,
¯
B
2
¯
β
=
(1 −
¯
β)B
2
¯
βB
2
,
¯
B
2
=
B
2
−B
2
,
¯
A = A − LC, ¯η
1
=
η
1
η
2
,
¯
C
1
¯
δ
=
(1 −
¯
δ)I
¯
δI
,
¯
C
2
=
−II
, ¯η
2
=
η
3
η
4
,
(12)
with
η
1
= µ
k
q
1
u(k)
µ
k
−
u(k)
µ
k
,
η
2
= µ
k−h
k
q
1
u(k − h
k
)
µ
k−h
k
−
u(k − h
k
)
µ
k−h
k
. (13)
η
3
= ν
k
q
2
y(k)
ν
k
−
y(k)
ν
k
,
η
4
= ν
k−l
k
q
2
y(k − l
k
)
ν
k−l
k
−
y(k − l
k
)
ν
k−l
k
.
Also, (11) can be rewritten in a compact form as:
x
e
(k + 1) =
¯
A
δ
x
e
(k) +
¯
A
1
x
e
(k − h
k
)
+
¯
A
2
x
e
(k − l
k
) +
¯
A
3
ω(k) +
¯
A
4
¯η (14)
where x
T
e
(k) =
x
T
(k) e
T
(k)
, ¯η
T
=
¯η
T
1
¯η
T
2
, and
¯
A
δ
=
¯
A
¯
β
+ (
¯
β − β
k
)B
2
K −(1 −
¯
β)B
2
K − (
¯
β − β
k
)B
2
K
¯
δLC − (
¯
δ − δ
k
)LC
¯
A
¯
A
1
=
¯
βB
2
K − (
¯
β − β
k
)B
2
K −
¯
βB
2
K + (
¯
β − β
k
)B
2
K
00
¯
A
2
=
00
(
¯
δ − δ
k
)LC −
¯
δLC 0
,
¯
A
3
=
B
1
B
1
¯
A
4
=
¯
B
2
¯
β
+ (
¯
β − β
k
)
¯
B
2
0
0 −L
¯
C
1
¯
δ
+ (
¯
δ − δ
k
)L
¯
C
2
Due to the fact that the closed-loop system (14) is a
stochastic parameter system with stochastic variables
β(k) and δ(k), similar to [31], we introduce the notion
of stochastic stability in the mean-square sense for our
control problem formulation.
Definition 2.1: The quantized closed-loop system (14) is
said to be exponentially mean-square stable if with ω(k) =
0, there exist constants α>0 and τ ∈ (0, 1), such that
E{x
e
(k)
2
}≤ατ
k
E{x
e
(0)
2
}
By Definition 2.1, due to the effect of quantization errors
and random packet losses, the problem addressed in this
paper is as follows:
Quantized H
∞
Control Problem (QCPH
∞
): Design a
quantized H
∞
control strategy with the minimized quan-
tizer ranges M
1
and M
2
such that the closed-loop system
(14) satisfies the following two requirements simultane-
ously.
R1) The closed-loop system (14) is exponentially mean-
square stable.
R2) For a given scalar γ>0 and all nonzero ω(k), under
the zero-initial condition, the control output z(k) satisfies
∞
k=0
E{z(k)
2
}≤γ
2
∞
k=0
ω(k)
2
.
(15)
Remark 2.2: The performance criterion (15) with deter-
ministic signal ω(k) can be easily modified to deal with
the case where the disturbance signals may be random
ones.
2.3. Useful lemmas
Before giving the main results, the following lemmas are
given first, which will be used in the sequel.
Lemma 2.1: [31] Let V (η
k
) be a Lyapunov functional. If
there exist real scalars λ ≥ 0, µ>0, ν>0, and 0 <ψ<1
such that
µη
k
2
≤ V (η
k
) ≤ νη
k
2
(16)
and
E{V (η
k+1
)|η
k
}−V (η
k
) ≤ λ − ψ V (η
k
)
(17)
then the sequence η
k
satisfies
E{η
k
2
}≤
ν
µ
η
0
2
(1 − ψ)
k
+
λ
µψ
(18)
Lemma 2.2: [31] Let B
2
∈ R
n×m
be of full-column rank.
Suppose there exist U
1
∈ R
m×n
,U
2
∈ R
(n−m)×n
and V ∈
R
m×m
satisfying
UB
2
V
=
U
1
U
2
B
2
V =
0
, (19)
where U , V are two orthogonal matrices, and =
diag{σ
1
,σ
2
,..., σ
m
} with σ
i
∈{1,2, ..., m} being nonzero
singular values of B
2
. Then there exists a nonsingular
matrix P ∈ R
m×m
such that B
2
P = P
1
B
2
, where P
1
has the
following structure
P
1
= U
T
P
11
0
0 P
22
U = U
T
1
P
11
U
1
+ U
T
2
P
22
U
2
(20)
with P
11
∈ R
m×m
> 0 and P
22
∈ R
(n−m)×(n−m)
> 0.In
particular, P is with the structure P = V
−1
P
11
V
T
.