Z. Gu et al. / Information Sciences 457–458 (2018) 99–112 101
It is noticed that a large number of statistical experiments is needed for a big F . For facilitate implementation, one can
choose an interval of the nonlinearity with a probability α, then the probability of the rest is 1 − α, i.e. F = 2 . Therefore, we
mainly discuss this simplified situation for the follow-on study. Then, it is true that P rob{ t ∈ F
1
} ∪ P rob{ t ∈ F
2
} = 1 , that is,
α
1
= 1 − α
2
= α.
2.2. The dynamic output feedback control law
In this study, suppose that the controller and the sensors are connected by a wireless network [7] . The sampling period
is set by h . For a convenient description, the following notations are given:
• t
k
: the instants of data releasing;
• a
k
: the instants of the data arriving at the actuator;
• L
k
: the intervals between a
k
and a
k +1
, i.e. L
k
= [ a
k
, a
k +1
) ;
The dynamic output feedback control law is developed by
˙
x
c
(t) = A
c1
x
c
(t) + A
c2
x
c
(t
k
h ) + B
c
y (t
k
h )
u (t ) = Kx
c
(t)
(3)
for t ∈ L
k
, where x
c
(t) ∈ R
n
is the state vector of the output feedback controller; and A
c 1
, A
c 2
, B
c
and K are constant real
matrices to be determined.
2.3. Improved ETM
Define a set
I
t
k
=
l| e (t)
T
e (t ) ≤ δ
1
y
T
(t
k
h )y (t
k
h ) +
δ
2
2
y
T
(t
k
h )e (t) + e
T
(t)y (t
k
h )
(4)
where e (t) = y (t
k
h ) −
¯
y (t
k
h ) ;
¯
y
(t
k
h ) is a mean value between the latest released data and the current sampled data, that
is
¯
y
(t
k
h ) =
y (t
k
h )+ y (t
k
h + lh )
2
for l ∈ L
k
{ 0 , 1 , 2 , ···, l
M
} ; δ
1
and δ
2
are positive scalars, and > 0 is a weighting matrix. Then
borrowing the idea from [17] , the next releasing instant is determined by
t
k +1
h = t
k
h + (l
M
+ 1) h. (5)
where l
M
= max
l∈ I
t
k
l.
Remark 2. If one sets δ
2
= 0 in (4) , the ETM is the same as that in [17] . However, the definition of e ( t ) is different from
the conventional one (see [10,17] , and the references therein). A mean value is introduced to get the error in this study, by
which an unexpected releasing event (we call it mal-releasing) due to some unknown abrupt disturbance can be avoided.
Moreover, the data-releasing rate can be further reduced, which will be verified in Section 4 .
Remark 3. ETMs in some existing literature mainly focus on reducing NDR for the system with suitable control performance.
To get better control performance, the item in (4) with δ
2
is introduced, which plays a crucial role in enhancing the data-
releasing rate during the period when the system is disturbed by external signals while it keeps a low level in the other
period.
The objective of this study is to design a dynamic output feedback controller with the form of (3) such that the system
with stochastic nonlinearity in (1) satisfies
• The instants of data-releasing are decided by (5) .
• For the given disturbance attenuation γ > 0, the system (1) is mean square stable and under zero initial state condition
it satisfies E {
∞
t
0
z
T
(s ) z(s ) ds } ≤ E {
∞
t
0
γ
2
ω
T
(s ) ω(s ) ds } .
3. Co-design of the controller and ETM
In this section, we are in a position to design the controller in (3) for the system (1) under the proposed new ETM. Firstly,
the closed-loop control system is modeled as a time-delay system with consideration of the ETM proposed in Section 2.3 .
Define a
l
k
= t
k
h + lh + d
l
t
k
and L
l
k
[ a
l
k
, a
l+1
k
) . Let L
k
= ∪
l
M
l=0
L
l
k
, and one can know that d
0
t
k
= d
t
k
and d
l
M
+1
t
k
= d
t
k +1
, which
are networked induced delays at instants t
k
h and t
k +1
h, respectively. d
l
t
k
for l ∈ L
k
\{ 0 } is an artificial delay to guarantee the
interval L
l
k
is meaningful.
Define
d(t) = t − t
k
h (6)