# State Estimation for Discrete-Time Stochastic Neural Networks with Mixed Delays

Authors: Liyuan Hou; Hong Zhu; Shouming Zhong; Yong Zeng; Lin Shi
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

This paper investigates the analysis problem for stability of  discrete-time neural networks (NNs) with discrete- and distribute-time delay. Stability theory and a linear matrix inequality (LMI) approach are developed to establish sufficient conditions for the NNs to be globally asymptotically stable and to design a state estimator for the discrete-time neural networks. Both the discrete delay and distribute delays employ decomposing the delay interval approach, and the Lyapunov-Krasovskii functionals (LKFs) are constructed on these intervals, such that a new stability criterion is proposed in terms of linear matrix inequalities (LMIs). Numerical examples are given to demonstrate the effectiveness of the proposed method and the applicability of the proposed method.


---

## Body

## 1. Introduction

In the past decades, recurrent neural networks (RNNs) have been widely studied due to their wide applications in some areas such as pattern recognition, associative memory, combinatorial optimization, and signal processing. Dynamical behaviors (e.g., stability, instability, periodic oscillatory, and chaos) of the neural networks are known to be crucial in applications. It is noted that the stability of neural networks is a prerequisite for some optimization problems. As is known to all, many biological and artificial neural networks contain inherent time delays in signal transmission due to the finite speed of information processing, which may cause oscillation, divergence, and instability. In recent years, a great number of papers have been published on various networks with time delays [1–10].On one hand, delay-dependent stability condition for continuous-time RNNs with time-varying delays was derived by defining a new Lyapunov functional, and the obtained condition could include some existing time delay-independent ones; see [11, 12]. Up to now, however, when we use computer to simulate, experimentalize or compute continuous-time RNNs, it is necessary to discretize the continuous-time networks to formulate a discrete-time system. So, the study on the dynamics of discrete-time neural networks is crucially needed. In particular, the stability of discrete-time neural networks (DNNs) has been studied in [13–18], since DNNs play a more important role than their continuous-time counterparts in today’s digital life.On the other hand, the neuron states are seldom fully available in the network outputs in many applications; the neuron state estimation problem becomes important to utilize the estimated neuron state through the available measurements. Recently, the state estimation problem for the neural networks has engaged lots of scholars' attention and interest. Therefore, delay-dependent state estimation problem has been studied widely for NNs; see [19–26].Stochastic disturbances are mostly inevitable owing to thermal noise in electronic implementations. It has also been revealed that certain stochastic inputs could make a neural network unstable.Summarizing the above discussion, in this paper, the stability problem is considered for discrete-time neural networks with discrete and distribute delays. Firstly, the mathematical models are established. Secondly, a less conservative and new stability criterion is derived by using a novel Lyapunov-Krasovskii functional. Thirdly, a numerical example is provided to show the effectiveness of the main result. The technical difficulties of our paper are the partition of the distributed time-varying delays. The novel contribution of this work with respect to existing literature is to construct a novel Lyapunov-Krasovskii functional according to the situation of the distributed time-varying delays' partition. In Corollary11, we use Lemma 7, which we have proved in Section 2, and we can get a new stability criterion.Notation. Throughout this paper, ℝ
n and ℝ
n
×
m denote, respectively, the n-dimensional Euclidean space and the set of all real matrices. The superscript T denotes matrix transposition and the notation X
≥
Y (resp., X
>
Y), where X and Y are symmetric matrices, which means that X
-
Y is positive semidefinite (resp., positive definite). In symmetric block matrices, the symbol * is used as an ellipsis for terms induced by symmetry. |
·
| stands for the Euclidean vector norm in ℝ
n. Sym
(
M
) is defined as Sym
(
M
)
=
M
+
M
T. Z
≥
0 denotes the set including zero and positive integers. 𝔼
{
x
} and 𝔼
{
x
∣
y
} denote the expectation of x and the expectation of x conditional on y. (
Ω
,
ℱ
,
𝒫
) is a probability space, where Ω is the sample space, ℱ is the σ-algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ.


## 2. Preliminaries

Consider the following discrete-time recurrent neural network with time-varying delays described by(1)
x
(
k
+
1
)
=
C
x
(
k
)
+
A
F
(
x
(
k
)
)
+
B
G
(
x
(
k
-
τ
(
k
)
)
)
+
D
∑
i
=
-
d
(
k
)
-
1
‍
H
(
x
(
k
+
i
)
)
+
δ
(
k
,
x
(
k
)
)
ω
(
k
)
+
J
,
where x
(
k
)
=
[
x
1
(
k
)
,
x
2
(
k
)
,
…
,
x
n
(
k
)
]
T is the neural state vector at time k; C
=
diag
⁡
[
c
1
,
c
2
,
…
,
c
n
] with |
c
i
|
<
1 is the state feedback coefficient matrix; the n
×
n matrices A
=
[
a
i
j
]
n
×
n, B
=
[
b
i
j
]
n
×
n and D
=
[
d
i
j
]
n
×
n are the connection weight matrix, the discretely delayed connection weight matrix and distributively delayed connection weight matrix, respectively; J
=
[
J
1
,
J
2
,
…
,
J
n
]
T  is the exogenous input; F
(
x
(
k
)
),   G
(
x
(
k
)
), and  H
(
x
(
k
)
)   are the neuron activation functions, which satisfy F
(
x
(
k
)
)
=
[
f
1
(
x
1
(
k
)
)
,
…
,
f
n
(
x
n
(
k
)
)
]
T,   G
(
x
(
k
)
)
=
[
g
1
(
x
1
(
k
)
)
,
g
2
(
x
2
(
k
)
)
,
…
,
g
n
(
x
n
(
k
)
)
]
T, and H
(
x
(
k
)
)
=
[
h
1
(
x
1
(
k
)
)
,
h
2
(
x
2
(
k
)
)
,
…
,
h
n
(
x
n
(
k
)
)
]
T; and τ
(
k
)
,
d
(
k
) respectively, denote the discrete and distributed time-varying delays. ω
(
k
) is a scalar Wiener process on a probability space (
Ω
,
ℱ
,
𝒫
) with 𝔼
{
ω
(
k
)
}
=
0, 𝔼
{
ω
2
(
k
)
}
=
1, and 𝔼
{
ω
(
i
)
ω
(
j
)
}
=
0
(
i
≠
j
).Assumption 1.
For anyx
,
y
∈
R, (
x
≠
y
), i
∈
{
1,2
,
…
,
n
}, the activation functions satisfy
(2)
f
i
-

≤
f
i
(
x
)
-
f
i
(
y
)
x
-
y
≤
f
i
+
;
g
i
-

≤
g
i
(
x
)
-
g
i
(
y
)
x
-
y
≤
g
i
+
;
h
i
-
≤
h
i
(
x
)
-
h
i
(
y
)
x
-
y
≤
h
i
+
,
where f
i
-
,
f
i
+
,
g
i
-
,
g
i
+
;
h
i
-, and h
i
+ are constants.Remark 2.
The condition on the activation function in Assumption1 was originally employed in [27] and has been subsequently used in recent papers with the problem of stability of neural networks; see [5, 6, 11, 28, 29], for example.Assumption 3.
The noise intensity function vectorδ
(
·
,
·
)
:
ℤ
≥
0
×
ℝ
n
→
ℝ
n satisfies the Lipschitz condition; that is, there exists a constant ξ such that for any k
∈
ℤ
≥
0 the following inequality:
(3)
δ
(
k
,
x
(
k
)
)
T
δ
(
k
,
x
(
k
)
)
≤
ξ
x
T
(
k
)
x
(
k
)
.Assumption 4.
The time-varying delaysτ
(
k
) and d
(
k
) are bounded, 0
<
τ
m
≤
τ
(
k
)
≤
τ
M, 0
<
d
m
≤
d
(
k
)
≤
d
M, and its probability distribution can be observed. Assume that τ
(
k
) takes values in [
τ
0
,
τ
1
]
⋃
‍
⋯
⋃
‍
(
τ
n
1
-
1
,
τ
n
1
] and Prob
{
τ
(
k
)
∈
[
τ
i
-
1
,
τ
i
)
}
=
ρ
i
=
1
-
ρ
~
i, where 0
≤
ρ
i
≤
1, ∑
i
ρ
i
=
1, and τ
0
=
τ
m
,
τ
n
1
=
τ
M. Similarly, d
(
k
) takes values in [
d
0
,
d
1
]
⋃
‍
⋯
⋃
‍
(
d
n
2
-
1
,
d
n
2
], and Prob
{
d
(
k
)
∈
[
d
i
-
1
,
d
i
)
}
=
ξ
i
=
1
-
ξ
~
i, where 0
≤
ξ
i
≤
1, ∑
i
‍
ξ
i
=
1, and d
0
=
d
m
,
d
n
1
=
d
M.Remark 5.
It is noted that the introduction of binary stochastic variable was first introduced in [6].To describe the probability distribution of time-varying delays, we define the following sets𝒜
i
=
(
τ
i
-
1
,
τ
i
], i
=
1,2
,
…
,
n
1, and ℬ
i
=
(
d
i
-
1
,
d
i
], i
=
1,2
,
…
,
n
2. Define mapping functions
(4)
τ
i
(
k
)
=
{
τ
(
k
)
,
τ
(
k
)
∈
𝒜
i
,
τ
i
-
1
,
else
,
d
i
(
k
)
=
{
d
(
k
)
,
d
(
k
)
∈
ℬ
i
,
d
i
-
1
,
else
,
ρ
i
(
k
)
=
{
1
,
τ
(
k
)
∈
𝒜
i
,
0
,
else
,
ξ
i
(
k
)
=
{
1
,
d
(
k
)
∈
ℬ
i
,
0
,
else
.Remark 6.
ConsiderProb
{
ρ
i
(
k
)
=
1
}
=
𝔼
{
ρ
i
(
k
)
}
=
ρ
i, Prob
{
ρ
i
(
k
)
=
0
}
=
ρ
~
i,
(5)
𝔼
{
ρ
i
(
k
)
*
ρ
j
(
k
)
}
=
{
ρ
i
,
i
=
j
,
0
,
else
,
𝔼
{
(
ρ
i
(
k
)
-
ρ
i
)
(
ρ
j
(
k
)
-
ρ
j
)
}
=
{
ρ
i
ρ
~
i
,
i
=
j
,
-
ρ
i
ρ
j
,
else
.
Similarly,Prob
{
ξ
i
(
k
)
=
1
}
=
𝔼
{
ξ
i
(
k
)
}
=
ξ
i, Prob
{
ξ
i
(
k
)
=
0
}
=
ξ
~
i,
(6)
𝔼
{
ξ
i
(
k
)
*
ξ
j
(
k
)
}
=
{
ξ
i
,
i
=
j
,
0
,
else
,
𝔼
{
(
ξ
i
(
k
)
-
ξ
i
)
(
ξ
j
(
k
)
-
ξ
j
)
}
=
{
ξ
i
ξ
~
i
,
i
=
j
,
-
ξ
i
ξ
j
,
else
.Proof .
Wheni
≠
j, τ
(
k
)
∈
[
τ
i
-
1
,
τ
i
) and as Prob
{
τ
(
k
)
∈
[
τ
i
-
1
,
τ
i
)
}
=
ρ
i
=
1
-
ρ
~
i, we can easily deduce ρ
i
(
k
)
=
1
,
ρ
j
(
k
)
=
0, so
(7)
𝔼
{
ρ
i
(
k
)
*
ρ
j
(
k
)
}
=
{
ρ
i
,
i
=
j
,
0
,
else
,
𝔼
{
(
ρ
i
(
k
)
-
ρ
i
)
(
ρ
j
(
k
)
-
ρ
j
)
}
=
𝔼
{
ρ
i
(
k
)
ρ
j
(
k
)
-
ρ
i
ρ
j
(
k
)
-
ρ
i
(
k
)
ρ
j
+
ρ
i
ρ
j
}
=
0
-
ρ
i
ρ
j
-
ρ
i
ρ
j
+
ρ
i
ρ
j
=
-
ρ
i
ρ
j
.
Similarly, the result about ξ
i
(
k
) can be deduced. The proof is complete.The system (1) can be rewritten as
(8)
x
(
k
+
1
)
=
C
x
(
k
)
+
A
F
(
x
(
k
)
)
+
∑
i
=
1
n
1
‍
ρ
i
(
k
)
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
+
∑
i
=
1
n
2
‍
ξ
i
(
k
)
D
∑
j
=
-
d
i
(
k
)
-
1
‍
H
(
x
(
k
+
j
)
)
+
δ
(
k
,
x
(
k
)
)
ω
(
k
)
+
J
.As mentioned before, it is very difficult or even impossible to acquire the complete information of the neuron states in relatively large-scale neural networks. The main purpose of this study is to develop a novel approach to estimating the neuron states via the network outputs. As mentioned above, the objective of this study is to present an efficient algorithm to estimate the neuron states via available network outputs. It is assumed that the measured network outputs are of the form(9)
y
(
k
)
=
E
x
(
k
)
+
O
(
k
,
x
(
k
)
)
,
where y
(
t
)
∈
R
m is the measured output, E is known constant matrix with appropriate dimensions, and O
:
ℤ
≥
0
×
ℝ
n
→
ℝ
m is a nonlinear disturbance on the network outputs satisfying
(10)
|
O
(
k
,
x
)
-
O
(
k
,
y
)
|
≤
|
L
(
x
-
y
)
|
,
∀
k
∈
ℤ
≥
0
,
x
,
y
∈
ℝ
n
.As a matter of fact, the activation functionsF
(
·
) are known. In order to fully utilize the information of the activation function, the state estimator for the neural network is constructed as
(11)
x
^
(
k
+
1
)
=
C
x
^
(
k
)
+
A
F
(
x
^
(
k
)
)
+
∑
i
=
1
n
1
‍
ρ
i
B
G
(
x
^
(
k
-
τ
i
(
k
)
)
)
+
∑
i
=
1
n
2
‍
ξ
i
D
∑
j
=
k
-
d
i
(
k
)
k
-
1
‍
H
(
x
^
(
j
)
)
-
K
[
y
(
k
)
-
E
x
^
(
k
)
-
O
(
k
,
x
^
(
k
)
)
]
+
J
,
where x
^
(
k
+
1
) is the estimation of the neuron state and K
∈
ℝ
n
×
m is the estimator gain matrix to be determined. Define the error signal e
(
k
+
1
)
=
x
(
k
+
1
)
-
x
^
(
k
+
1
); thus, we obtain the error state system as follows:
(12)
e
(
k
+
1
)
=
(
C
+
K
E
)
e
(
k
)
+
A
(
F
(
x
(
k
)
)
-
F
(
x
^
(
k
)
)
)
+
∑
i
=
1
n
1
‍
ρ
i
B
(
G
(
x
(
k
-
τ
i
(
k
)
)
)
-
G
(
x
^
(
k
-
τ
i
(
k
)
)
)
)
+
∑
i
=
1
n
2
‍
∑
j
=
-
d
i
(
k
)
-
1
‍
ξ
i
D
[
H
(
x
(
k
+
j
)
)
-
H
(
x
^
(
k
+
j
)
)
]
+
∑
i
=
1
n
1
‍
(
ρ
i
(
k
)
-
ρ
i
)
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
+
∑
i
=
1
n
2
‍
∑
j
=
-
d
i
(
k
)
-
1
‍
(
ξ
i
(
k
)
-
ξ
i
)
D
H
(
x
(
k
+
j
)
)
+
K
[
O
(
k
,
x
(
k
)
)
-
O
(
k
,
x
^
(
k
)
)
]
+
δ
(
k
,
x
(
k
)
)
ω
(
k
)
.Denotef
(
k
)
=
F
(
x
(
k
)
)
-
F
(
x
^
(
k
)
),  g
(
k
-
τ
i
(
k
)
)
=
G
(
x
(
k
-
τ
i
(
k
)
)
)
-
G
(
x
^
(
k
-
τ
i
(
k
)
)
),  h
(
k
+
i
)
=
H
(
x
(
k
+
i
)
)
-
H
(
x
^
(
k
+
i
),  o
(
k
)
=
O
(
k
,
x
(
k
)
)
-
O
(
k
,
x
^
(
k
)
), then (12) can be rewritten as
(13)
e
(
k
+
1
)
=
(
C
+
K
E
)
e
(
k
)
+
A
f
(
k
)
+
∑
i
=
1
n
1
‍
ρ
i
B
g
(
k
-
τ
i
(
k
)
)
+
∑
i
=
1
n
2
‍
∑
j
=
-
d
i
(
k
)
-
1
‍
ξ
i
D
h
(
k
+
i
)
+
K
o
(
k
)
+
δ
(
k
,
x
(
k
)
)
ω
(
k
)
+
∑
i
=
1
n
2
‍
∑
j
=
-
d
i
(
k
)
-
1
‍
(
ξ
i
(
k
)
-
ξ
i
)
D
H
(
x
(
k
+
j
)
)
+
∑
i
=
1
n
1
‍
(
ρ
i
(
k
)
-
ρ
i
)
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
.The initial condition associated with the error system (13) is given as
(14)
e
(
k
)
=
ϕ
(
k
)
,
k
=
-
ς
M
,
-
ς
M
+
1
,
…
,
0
,
where ς
M
=
max
⁡
{
τ
M
,
d
M
} and ∥
ϕ
∥
=
sup
⁡
-
ς
M
≤
k
≤
0
|
ϕ
|
2
<
∞.By defininge
^
(
k
)
=
[
x
T
(
k
)
,
e
T
(
k
)
]
T and combing (8) and (13) with J
=
0, we can obtain the following system:
(15)
e
^
(
k
+
1
)
=
C
^
e
^
(
k
)
+
A
^
f
^
(
k
)
+
ρ
^
(
k
)
B
^
G
^
(
k
)
+
ρ
^
1
B
^
g
^
(
k
)
+
ξ
^
(
k
)
D
^
H
^
(
k
)
+
ξ
^
1
D
^
h
^
(
k
)
+
K
^
o
(
k
)
+
I
^
δ
(
k
,
x
(
k
)
)
ω
(
k
)
,
where
(16)
ρ
^
(
k
)
=
[
ρ
1
(
k
)
I
ρ
2
(
k
)
I
⋯
ρ
n
1
(
k
)
I
ρ
1
(
k
)
I
-
ρ
1
I
ρ
2
(
k
)
I
-
ρ
2
I
⋯
ρ
n
1
(
k
)
I
-
ρ
n
1
I
]
,
ρ
^
1
=
[
0
0
⋯
0
ρ
1
I
ρ
2
I
⋯
ρ
n
1
I
]
,
ξ
^
(
k
)
=
[
ξ
1
(
k
)
I
ξ
2
(
k
)
I
⋯
ξ
n
2
(
k
)
I
ξ
1
(
k
)
I
-
ξ
1
I
ξ
2
(
k
)
I
-
ξ
2
I
⋯
ξ
n
2
(
k
)
I
-
ξ
n
2
I
]
,
ξ
^
1
=
[
0
0
⋯
0
ξ
1
I
ξ
2
I
⋯
ξ
n
2
I
]
,
f
^
(
k
)
=
(
F
T
(
x
(
k
)
)
,
f
T
(
k
)
)
T
,
G
^
(
k
)
=
(
G
T
(
x
(
k
-
τ
1
(
k
)
)
)
,
C
C
I
G
T
(
x
(
k
-
τ
2
(
k
)
)
)
,
…
,
G
T
(
k
-
τ
n
1
(
k
)
)
)
T
,
g
^
(
k
)
=
(
(
k
-
τ
n
1
(
k
)
)
g
T
(
k
-
τ
1
(
k
)
)
,
c
c
c
i
g
T
(
k
-
τ
2
(
k
)
)
,
…
,
g
T
(
k
-
τ
n
1
(
k
)
)
)
T
,
H
^
(
k
)
=
(
∑
i
=
-
d
1
(
k
)
-
1
‍
H
T
(
k
+
i
)
,
c
c
c
v
∑
i
=
-
d
2
(
k
)
-
1
‍
H
T
(
k
+
i
)
,
…
,
∑
i
=
-
d
n
2
(
k
)
-
1
‍
H
T
(
k
+
i
)
)
T
,
h
^
(
k
)
=
(
∑
i
=
-
d
1
(
k
)
-
1
‍
h
T
(
k
+
i
)
T
,
c
c
i
c
i
∑
i
=
-
d
2
(
k
)
-
1
‍
h
T
(
k
+
i
)
T
,
…
,
∑
i
=
-
d
n
2
(
k
)
-
1
‍
h
T
(
k
+
i
)
)
T
,
C
^
=
[
C
0
0
C
+
K
E
]
T
,
A
^
=
[
A
0
0
A
]
,
B
^
=
[
B
⋱
B
]
n
1
×
n
1
,
D
^
=
[
D
⋱
D
]
n
2
×
n
2
,
K
^
=
[
0
K
]
,
I
^
=
[
I
I
]
.
Then, it is easy to show the following equations:
(17)
E
{
ρ
^
(
k
)
}
=
ρ
^
2
=
[
ρ
1
I
ρ
2
I
⋯
ρ
n
1
I
0
0
⋯
0
]
,
E
{
ξ
^
(
k
)
}
=
ξ
^
2
=
[
ξ
1
I
ξ
2
I
⋯
ξ
n
2
I
0
0
⋯
0
]
.Lemma 7.
For any constant matrixM
∈
R
n
×
n, any integers γ
2
≥
γ
1, and any vector function ω
:
[
γ
1
,
γ
1
+
1
,
…
,
γ
2
]
→
R
n where M staisfies M
=
M
T
>
0 such that the sums in the following are well defined, then
(18)
-
∑
i
=
γ
1
γ
2
‍
ω
T
(
i
)
M
ω
(
i
)
≤
(
γ
2
-
γ
1
+
1
)
ζ
T
(
k
)
F
T
M
-
1
F
ζ
(
k
)
+
2
ζ
T
(
k
)
F
T
∑
i
=
γ
1
γ
2
‍
ω
(
i
)
,
where, matrix F and vector ζ
(
k
) independent of γ
1 and γ
2 are appropriate dimensional arbitrary ones.Proof.
It's well known that(19)
-
2
a
T
b
≤
a
T
W
-
1
a
+
b
T
W
b
,
where where the vector a, b, W with appropriate dimensional and W
>
0. From this, we can get
(20)
-
2
ζ
T
(
k
)
F
T
∑
i
=
γ
1
γ
2
‍
ω
(
i
)
=
∑
i
=
γ
1
γ
2
‍
-
2
ζ
T
(
k
)
F
T
ω
(
i
)
≤
∑
i
=
γ
1
γ
2
‍
ω
T
(
i
)
M
ω
(
i
)
+
∑
i
=
γ
1
γ
2
‍
ζ
T
(
k
)
F
T
M
-
1
F
ζ
(
k
)
≤
∑
i
=
γ
1
γ
2
‍
ω
T
(
i
)
M
ω
(
i
)
+
(
γ
2
-
γ
1
+
1
)
ζ
T
(
k
)
F
T
M
-
1
F
ζ
(
k
)
,
which is equivalent to (18).Lemma 8 (Zhu and Yang [28]).
For any constant matrixM
∈
R
n
×
n, any integers γ
2
≥
γ
1, and any vector function ω
:
[
γ
1
,
γ
1
+
1
,
…
,
γ
2
]
→
R
n where M satisfies M
=
M
T
>
0, such that the sums in the following are well defined; then
(21)
-
(
γ
2
-
γ
1
+
1
)
∑
i
=
γ
1
γ
2
‍
ω
T
(
i
)
M
ω
(
i
)
≤
-
(
∑
i
=
γ
1
γ
2
‍
ω
(
i
)
)
T
M
(
∑
i
=
γ
1
γ
2
‍
ω
(
i
)
)
.


## 3. New Stability Criteria

In this section, we will establish new stability criteria for system (1). Since the system in (8) involves a stochastic parameter, to investigate its stability, we need the following definition.Definition 9.
The system (11) is said to be globally asymptotically state estimator of the system (8), if the estimation error system (13) satisfies is globally asymptotical stable in mean square; that is,
(22)
lim
⁡
k
→
+
∞
𝔼
{
|
e
(
k
)
|
2
}
=
0
.Theorem 10.
Under Assumptions1, 3, and 4, the system (15) is globally asymptotically stable in mean square, if there exist matrices P
=
P
T
=
diag
⁡
{
P
1
,
P
2
}
>
0, R
i
=
diag
⁡
{
R
i
1
,
R
i
2
}
≥
0
(
i
=
1
,
…
,
n
2
-
1
), S
i
≥
0
(
i
=
1
,
…
,
n
1
), X
>
0, and T
i
≥
0
(
i
=
1
,
…
,
n
2
), and positive diagonal matrices U
=
diag
⁡
(
u
1
,
…
,
u
n
), V
=
diag
⁡
(
v
1
,
…
,
v
n
), and W
=
diag
⁡
(
w
1
,
…
,
w
n
), and scalars ϵ
>
0 and ϑ
>
0 such that the following LMI holds:(23)
Γ
=
[
Γ
1
Γ
1,2
0
Γ
1,4
0
0
Γ
1,7
Γ
1,8
0
Γ
1,10
Γ
1,11
Γ
1,12
0
X
^
-
I
X
^
*
Γ
2
0
0
0
0
Γ
2,7
0
0
0
0
0
0
0
0
*
*
Γ
3
0
0
0
0
0
0
0
0
0
0
0
0
*
*
*
Γ
4
0
0
Γ
4,7
Γ
4,8
0
0
0
0
0
A
^
T
P
A
^
T
P
*
*
*
*
Γ
5
0
0
0
0
0
0
0
0
B
^
T
ρ
^
1
T
P
B
^
T
ρ
^
1
T
P
*
*
*
*
*
Γ
6
0
0
0
0
0
0
0
D
^
T
ξ
^
1
T
P
D
^
T
ξ
^
1
T
P
*
*
*
*
*
*
Γ
7
Γ
7,8
0
0
0
0
0
0
0
*
*
*
*
*
*
*
Γ
8
0
0
0
0
0
0
0
*
*
*
*
*
*
*
*
Γ
9
0
0
0
0
X
T
X
T
*
*
*
*
*
*
*
*
*
Γ
10
0
0
0
0
0
*
*
*
*
*
*
*
*
*
*
Γ
11
0
0
0
0
*
*
*
*
*
*
*
*
*
*
*
Γ
12
0
0
0
*
*
*
*
*
*
*
*
*
*
*
*
Γ
13
0
0
*
*
*
*
*
*
*
*
*
*
*
*
*
-
P
h
·
τ
M
0
*
*
*
*
*
*
*
*
*
*
*
*
*
*
-
P
]
<
0
,
(24)
Γ
1
=
-
2
P
+
∑
i
=
1
n
2
-
1
‍
(
d
i
+
1
-
d
i
-
1
)
R
i
+
[
σ
ξ
I
0
0
ϑ
L
T
L
]
+
[
-
U
F
1
-
V
H
1
0
0
-
U
F
1
-
V
H
1
-
W
G
1
]
,
Γ
1,2
=
(
P
,
…
,
0
)
,
Γ
1,4
=
[
U
F
2
0
0
U
F
2
]
,
Γ
1,7
=
X
^
ρ
^
2
B
^
,
Γ
1,8
=
X
^
ξ
^
2
D
^
,
Γ
1,10
=
[
0
W
G
2
]
,
Γ
1,11
=
[
0
V
H
2
]
,
Γ
1,12
=
[
V
H
2
0
]
,
Γ
2
=
[
-
2
P
P
⋱
⋱
-
2
P
P
-
P
]
(
n
1
+
1
)
×
(
n
1
+
1
)
+
[
-
W
G
1
^
⋱
-
W
G
1
^
0
]
,
Γ
2,7
=
[
W
G
2
^
⋱
W
G
2
^
0
]
,
Γ
3
=
[
-
1
d
2
-
d
m
R
1
1
d
2
-
d
m
R
1
-
1
d
2
-
d
m
R
1
-
1
d
3
-
d
1
R
2
1
d
3
-
d
1
R
2
⋱
⋱
⋱
1
d
n
2
-
d
n
2
-
2
R
n
2
-
1
-
1
d
n
2
-
d
n
2
-
2
R
n
2
-
1
]
(
n
2
×
n
2
)
,
Γ
4
=
-
U
^
,
Γ
4,7
=
A
^
T
P
ρ
^
2
B
^
,
Γ
4,8
=
A
^
T
P
ξ
^
2
D
^
,
Γ
5
=
-
S
^
,
Γ
6
=
-
T
^
2
,
Γ
7
=
2
B
^
T
P
^
B
^
-
W
^
,
Γ
7,8
=
B
^
T
ρ
^
2
T
P
ξ
^
2
D
^
,
Γ
8
=
2
D
^
T
P
¯
D
^
-
T
^
1
,
Γ
8,9
=
D
^
T
ξ
^
2
T
[
0
X
]
,
Γ
9
=
-
ϑ
I
,
Γ
10
=
∑
i
=
1
n
1
‍
(
τ
i
-
τ
i
-
1
+
1
)
S
i
-
W
,
Γ
11
=
∑
i
=
1
n
2
‍
(
d
i
-
d
i
-
1
+
1
)
(
d
i
+
d
i
-
1
)
2
T
i
2
-
V
,
Γ
12
=
∑
i
=
1
n
2
‍
(
d
i
-
d
i
-
1
)
(
d
i
+
d
i
-
1
+
1
)
2
T
i
1
-
V
,
Γ
13
=
2
(
P
1
+
P
2
)
-
σ
I
,
X
^
=
[
C
T
P
1
0
0
C
T
P
2
+
E
T
X
]
,
h
=
max
⁡
i
=
2
,
…
,
n
1
{
τ
m
-
τ
1
(
k
)
,
τ
M
-
τ
n
1
(
k
)
,
τ
i
-
τ
i
-
1
(
k
)
}
,
F
1
=
diag
⁡
(
f
1
-
f
1
+
,
f
2
-
f
2
+
,
…
,
f
n
-
f
n
+
)
,
F
2
=
diag
⁡
(
f
1
-
+
f
1
+
2
,
…
,
f
n
-
+
f
n
+
2
)
,
G
1
=
diag
⁡
(
g
1
-
g
1
+
,
g
2
-
g
2
+
,
…
,
g
n
-
g
n
+
)
,
G
2
=
diag
⁡
(
g
1
-
+
g
1
+
2
,
…
,
g
n
-
+
g
n
+
2
)
,
H
1
=
diag
⁡
(
h
1
-
h
1
+
,
…
,
h
n
-
h
n
+
)
,
H
2
=
diag
⁡
(
h
1
-
+
h
1
+
2
,
…
,
h
n
-
+
h
n
+
2
)
,
S
^
=
diag
⁡
{
S
1
,
S
2
,
…
,
S
n
1
}
,
V
^
=
diag
⁡
{
V
,
V
,
…
,
V
}
,
W
G
1
^
=
diag
⁡
{
W
G
1
,
0
}
,
W
^
=
diag
⁡
{
W
,
W
,
…
,
W
}
,
U
^
=
diag
⁡
{
U
,
U
}
,
T
^
2
=
diag
⁡
{
T
12
d
1
,
T
22
d
2
,
…
,
T
n
2
2
d
n
2
}
,
T
^
1
=
diag
⁡
{
T
11
d
1
,
T
21
d
2
,
…
,
T
n
2
1
d
n
2
}
,
W
G
2
^
=
diag
⁡
{
W
G
2
,
0
}
,
P
^
=
[
ρ
1
P
1
+
ρ
1
ρ
~
1
P
2
-
ρ
1
ρ
2
P
2
-
ρ
1
ρ
3
P
2
⋯
-
ρ
1
ρ
n
1
P
2
*
ρ
2
P
1
+
ρ
2
ρ
~
2
P
2
-
ρ
2
ρ
3
P
2
⋯
-
ρ
2
ρ
n
1
P
2
*
*
ρ
3
P
1
+
ρ
3
ρ
~
3
P
2
⋯
-
ρ
3
ρ
n
1
P
2
⋮
⋮
⋮
⋱
⋮
*
*
*
⋯
ρ
n
1
P
1
+
ρ
n
1
ρ
~
n
1
P
2
]
;
P
¯
=
[
ξ
1
P
1
+
ξ
1
ξ
~
1
P
2
-
ξ
1
ξ
2
P
2
-
ξ
1
ξ
3
P
2
⋯
-
ξ
1
ξ
n
2
P
2
*
ξ
2
P
1
+
ξ
2
ξ
~
2
P
2
-
ξ
2
ξ
3
P
2
⋯
-
ξ
2
ξ
n
2
P
2
*
*
ξ
3
P
1
+
ξ
3
ξ
~
3
P
2
⋯
-
ξ
3
ξ
n
2
P
2
⋮
⋮
⋮
⋱
⋮
*
*
*
⋯
ξ
n
2
P
1
+
ξ
n
2
ξ
~
n
2
P
2
]
.
And the estimator gain can be designed asK
=
P
2
-
1
X.Proof.
We construct a new Lyapunov-Krasovskii functional as(25)
V
(
e
^
k
,
k
)
=
∑
i
=
1
5
‍
V
i
(
e
^
k
,
k
)
,
where
(26)
V
1
(
e
^
k
,
k
)
=
e
^
T
(
k
)
P
e
^
(
k
)
,
V
2
(
e
^
k
,
k
)
=
h
∑
i
=
-
τ
M
-
1
‍
∑
j
=
k
+
i
k
-
1
‍
η
T
(
j
)
P
η
(
j
)
,
V
3
(
e
^
k
,
k
)
=
∑
i
=
1
n
2
-
1
‍
∑
j
=
-
d
i
+
1
-
d
i
-
1
-
1
‍
∑
l
=
k
+
j
k
-
1
‍
e
^
T
(
l
)
R
i
e
^
(
l
)
,
V
4
(
e
^
k
,
k
)
=
∑
i
=
1
n
1
‍
∑
j
=
k
-
τ
i
(
k
)
k
-
1
‍
g
T
(
j
)
S
i
g
(
j
)
+
∑
i
=
1
n
1
‍
∑
j
=
τ
i
-
1
τ
i
-
1
‍
∑
i
=
k
-
j
k
-
1
‍
g
T
(
i
)
S
i
g
(
i
)
,
V
5
(
e
^
k
,
k
)
=
∑
i
=
1
n
2
‍
{
∑
j
=
-
d
i
(
k
)
-
1
‍
∑
l
=
k
+
j
k
-
1
‍
h
¯
T
(
l
)
T
i
h
¯
(
l
)
ccccccc
+
∑
j
=
-
d
i
-
d
i
-
1
-
1
‍
∑
l
=
j
+
1
-
1
‍
∑
r
=
k
+
l
k
-
1
‍
h
¯
T
(
r
)
T
i
h
¯
(
r
)
}
,
η
(
k
)
=
e
^
(
k
+
1
)
-
e
^
(
k
)
,
h
¯
(
k
)
=
(
H
T
(
x
(
k
)
)
,
h
T
(
k
)
)
T
,
T
i
=
diag
⁡
{
T
i
1
,
T
i
2
}
.
Taking the difference of the functional along the solution of the system, we obtain(27)
𝔼
{
Δ
V
(
e
^
k
,
k
)
}
=
∑
i
=
1
4
‍
𝔼
{
Δ
V
i
(
e
^
k
,
k
)
}
,
𝔼
{
Δ
V
1
(
e
^
k
,
k
)
}
=
𝔼
{
𝔼
{
V
1
(
e
^
k
+
1
,
k
+
1
)
∣
e
^
k
}
-
V
1
(
e
^
k
,
k
)
}
=
𝔼
{
e
^
T
(
k
+
1
)
P
e
^
(
k
+
1
)
-
e
^
T
(
k
)
P
e
^
(
k
)
}
=
e
^
T
(
k
)
(
C
^
T
P
C
^
-
P
)
e
^
(
k
)
+
2
e
^
T
(
k
)
C
^
T
P
A
^
f
^
(
k
)
+
2
e
^
T
(
k
)
C
^
T
P
K
^
o
(
k
)
+
2
e
^
T
(
k
)
C
^
T
P
ρ
^
1
B
^
g
^
(
k
)
+
2
e
^
T
(
k
)
C
^
T
P
ξ
^
1
D
^
h
^
(
k
)
+
2
e
^
T
(
k
)
C
^
T
P
ρ
^
2
B
^
G
^
(
k
)
+
2
e
^
T
(
k
)
C
^
T
P
ξ
^
2
D
^
H
^
(
k
)
+
f
^
T
(
k
)
A
^
T
P
A
^
f
^
(
k
)
+
2
f
^
T
(
k
)
A
^
T
P
ρ
^
1
B
^
g
^
(
k
)
+
2
f
^
T
(
k
)
A
^
T
P
ξ
^
1
D
^
h
^
(
k
)
+
2
f
^
T
(
k
)
A
^
T
P
ρ
^
2
B
^
G
^
(
k
)
+
2
f
^
T
(
k
)
A
^
T
P
ξ
^
2
D
^
H
^
(
k
)
+
2
f
^
T
(
k
)
A
^
T
P
K
^
o
(
k
)
+
o
T
(
k
)
K
^
T
P
K
^
o
(
k
)
+
2
o
T
(
k
)
K
^
T
P
ρ
^
1
B
^
g
^
(
k
)
+
2
o
T
(
k
)
K
^
T
P
ξ
^
1
D
^
h
^
(
k
)
+
2
o
T
(
k
)
K
^
T
P
ρ
^
2
B
^
G
^
(
k
)
+
2
o
T
(
k
)
K
^
T
P
ξ
^
2
D
^
H
^
(
k
)
+
g
^
T
(
k
)
B
^
T
ρ
^
1
T
P
ρ
^
1
B
^
g
^
(
k
)
+
2
g
^
T
(
k
)
B
^
T
ρ
^
1
T
P
ξ
^
1
D
^
h
^
(
k
)
+

2
g
^
T
(
k
)
B
^
T
ρ
^
1
T
P
ρ
^
2
B
^
G
^
(
k
)

+
2
g
^
T
(
k
)
B
^
T
ρ
^
1
T
P
ξ
^
2
D
^
H
^
(
k
)
+
h
^
T
(
k
)
D
^
T
ξ
^
1
T
P
ξ
^
1
D
^
h
^
(
k
)
+
2
h
^
T
(
k
)
D
^
T
ξ
^
1
T
P
ρ
^
2
B
^
G
^
(
k
)
+
2
h
^
T
(
k
)
D
^
T
ξ
^
1
T
P
ξ
^
2
D
^
H
^
(
k
)
+
𝔼
{
G
^
T
(
k
)
B
^
T
ρ
^
T
(
k
)
P
ρ
^
(
k
)
B
^
G
^
(
k
)
ccci
+
H
^
T
(
k
)
D
^
T
ξ
^
T
(
k
)
P
ξ
^
(
k
)
D
^
H
^
(
k
)
}
+
2
G
^
T
(
k
)
B
^
T
ρ
^
2
T
P
ξ
^
2
D
^
H
^
(
k
)
+
δ
T
(
k
,
x
(
k
)
)
(
P
1
+
P
2
)
δ
(
k
,
x
(
k
)
)
.
From Remark 6, we can get
(28)
𝔼
{
G
^
T
(
k
)
B
^
T
ρ
^
(
k
)
T
P
ρ
^
(
k
)
B
^
G
^
(
k
)
}
=
𝔼
{
(
∑
i
=
1
n
1
‍
ρ
i
(
k
)
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
)
T
×
P
1
(
∑
i
=
1
n
1
‍
ρ
i
(
k
)
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
)
+
(
∑
i
=
1
n
1
‍
(
ρ
i
(
k
)
-
ρ
i
)
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
)
T
×
P
2
(
∑
i
=
1
n
1
‍
(
ρ
i
(
k
)
-
ρ
i
)
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
)
(
∑
i
=
1
n
1
‍
ρ
i
(
k
)
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
)
T
}
=
∑
i
=
1
n
1
‍
ρ
i
G
T
(
x
(
k
-
τ
i
(
k
)
)
)
B
T
P
1
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
+
∑
i
=
1
n
1
‍
ρ
i
ρ
~
i
G
T
(
x
(
k
-
τ
i
(
k
)
)
)
B
T
P
2
B
G
(
x
(
k
-
τ
i
(
k
)
)
)
-
∑
i
=
1
n
1
‍
∑
j
=
1
,
i
≠
j
n
1
‍
ρ
i
ρ
j
G
T
(
x
(
k
-
τ
i
(
k
)
)
)
×
B
T
P
2
B
G
(
x
(
k
-
τ
j
(
k
)
)
)
=
G
^
T
(
k
)
B
^
T
P
^
B
^
G
^
(
k
)
.
Similarly, the following equation can be deduced:(29)
𝔼
{
H
^
T
(
k
)
D
^
T
ξ
^
T
(
k
)
P
ξ
^
(
k
)
D
^
H
^
(
k
)
}
=
H
^
T
(
k
)
D
^
T
P
¯
D
^
H
^
(
k
)
.
And it is easy to deduce that B
^
T
ρ
^
1
T
P
ρ
^
2
B
^
=
0, B
^
T
ρ
^
1
T
P
ξ
^
2
D
^
=
0, and D
^
T
ξ
^
1
T
P
ρ
^
2
B
^
=
0. Consider
(30)
𝔼
{
Δ
V
2
(
e
k
,
k
)
}
=
𝔼
{
𝔼
{
V
2
(
e
k
+
1
,
k
+
1
)
∣
e
k
}
-
V
2
(
e
k
,
k
)
}
=

𝔼
{

h
∑
i
=
-
τ
M
-
1
‍
∑
j
=
k
+
i
+
1
k
‍
η
T

(
j
)
P
η
(
j
)
-
h
∑
i
=
-
τ
M
-
1
‍
∑
j
=
k
+
i
k
-
1
‍

η
T

(
j
)
P
η
(
j
)

}
=
τ
M
h
η
T
(
k
)
P
η
(
k
)
-
∑
i
=
-
τ
M
-
1
‍
h
η
T
(
k
+
i
)
P
η
(
k
+
i
)
,
𝔼
{
Δ
V
3
(
e
k
,
k
)
}
=
𝔼
{
𝔼
{
V
3
(
e
k
+
1
,
k
+
1
)
∣
e
k
}
-
V
3
(
e
k
,
k
)
}
=
∑
i
=
1
n
2
-
1
‍
∑
j
=
-
d
i
+
1
-
d
i
-
1
-
1
‍
∑
l
=
k
+
j
+
1
k
‍
e
^
T
(
l
)
R
i
e
^
(
l
)
-
∑
i
=
1
n
2
-
1
‍
∑
j
=
-
d
i
+
1
-
d
i
-
1
-
1
‍
∑
l
=
k
+
j
k
-
1
‍
e
^
T
(
l
)
R
i
e
^
(
l
)
=
∑
i
=
1
n
2
-
1
‍
∑
j
=
-
d
i
+
1
-
d
i
-
1
-
1
[
(
e
^
T
(
k
)
R
i
e
^
(
k
)
+
∑
l
=
k
+
j
+
1
k
-
1
‍
e
^
T
(
l
)
R
i
e
^
(
l
)
)
‍
cccccccccccccccc
-
(
∑
l
=
k
+
j
+
1
k
-
1
‍
e
^
T
(
l
)
R
i
e
^
(
l
)
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
+
e
^
T
(
k
+
j
)
R
i
e
^
(
k
+
j
)
∑
l
=
k
+
j
+
1
k
-
1
‍
)
(
e
^
T
(
k
)
R
i
e
^
(
k
)
+
∑
l
=
k
+
j
+
1
k
-
1
‍
e
^
T
(
l
)
R
i
e
^
(
l
)
)
]
=
∑
i
=
1
n
2
-
1
‍
∑
j
=
-
d
i
+
1
-
d
i
-
1
-
1
‍
(
e
^
T
(
k
)
R
i
e
^
(
k
)
-
e
^
T
(
k
+
j
)
R
i
e
^
(
k
+
j
)
)
.
Then, by using Lemma 7 and d
i
(
k
)
∈
(
d
i
-
1
,
d
i
], we have
(31)
-
∑
j
=
-
d
i
+
1
-
d
i
-
1
-
1
e
^
T
(
k
+
j
)
R
i
e
^
(
k
+
j
)
)
‍
≤
-
1
d
i
+
1
-
d
i
-
1
(
∑
j
=
-
d
i
+
1
-
d
i
-
1
-
1
‍
e
^
(
k
+
j
)
)
T
×
R
i
(
∑
j
=
-
d
i
+
1
-
d
i
-
1
-
1
‍
e
^
T
(
k
+
j
)
)
≤
-
1
d
i
+
1
-
d
i
-
1
(
∑
j
=
k
-
d
i
+
1
(
k
)
k
-
d
i
(
k
)
‍
e
^
(
j
)
)
T
R
i
(
∑
j
=
k
-
d
i
+
1
(
k
)
k
-
d
i
(
k
)
‍
e
^
(
j
)
)
=
-
1
d
i
+
1
-
d
i
-
1
(
∑
j
=
k
-
d
i
+
1
(
k
)
k
-
1
‍
e
^
(
j
)
-
∑
j
=
k
-
d
i
(
k
)
k
-
1
‍
e
^
(
j
)
)
T
×
R
i
(
∑
j
=
k
-
d
i
+
1
(
k
)
k
-
1
‍
e
^
(
j
)
-
∑
j
=
k
-
d
i
(
k
)
k
-
1
‍
e
^
(
j
)
)
,
𝔼
{
Δ
V
4
(
e
k
,
k
)
}
=
𝔼
{
𝔼
{
V
4
(
e
k
+
1
,
k
+
1
)
∣
e
k
}
-
V
4
(
e
k
,
k
)
}
=
𝔼
{
∑
i
=
1
n
1
‍
∑
j
=
k
+
1
-
τ
i
(
k
+
1
)
k
‍
g
T
(
j
)
S
i
g
(
j
)
ccc
+
∑
i
=
1
n
1
‍
∑
j
=
τ
i
-
1
τ
i
-
1
‍
∑
i
=
k
-
j
+
1
k
‍
g
T
(
i
)
S
i
g
(
i
)
ccc
-
[
∑
i
=
1
n
1
‍
∑
j
=
k
-
τ
i
(
k
)
k
-
1
‍
g
T
(
j
)
S
i
g
(
j
)
cccccci
+
∑
i
=
1
n
1
‍
∑
j
=
τ
i
-
1
τ
i
-
1
‍
∑
i
=
k
-
j
k
-
1
‍
g
T
(
i
)
S
i
g
(
i
)
]
}
=
∑
i
=
1
n
1
‍
{
∑
j
=
τ
i
-
1
τ
i
-
1
‍
g
T
(
k
)
S
i
g
(
k
)
-
g
T
(
k
-
τ
i
(
k
)
)
S
i
g
(
k
-
τ
i
(
k
)
)
ccccci
+
∑
j
=
k
-
τ
i
(
k
+
1
)
+
1
k
-
1
‍
g
T
(
j
)
S
i
g
(
j
)
ccccci
-
∑
j
=
k
-
τ
i
(
k
)
+
1
k
-
1
‍
g
T
(
j
)
S
i
g
(
j
)
ccccci
+
∑
j
=
τ
i
-
1
τ
i
-
1
‍
(
g
T
(
k
)
S
i
g
(
k
)
-
g
T
(
k
-
j
)
S
i
g
(
k
-
j
)
)
}
≤
∑
i
=
1
n
1
‍
{
(
k
-
τ
i
(
k
)
)
(
τ
i
-
τ
i
-
1
+
1
)
g
T
(
k
)
S
i
g
(
k
)
ciiici
-
g
T
(
k
-
τ
i
(
k
)
)
S
i
g
(
k
-
τ
i
(
k
)
)
}
,
𝔼
{
Δ
V
5
(
e
k
,
k
)
}
=
𝔼
{
𝔼
{
V
5
(
e
k
+
1
,
k
+
1
)
∣
e
k
}
-
V
5
(
e
k
,
k
)
}
=
∑
i
=
1
n
2
{
∑
j
=
-
d
i
(
k
+
1
)
-
1
‍
∑
l
=
k
+
j
+
1
k
‍
h
¯
T
(
l
)
T
i
h
¯
(
l
)
‍
cccci
+
∑
j
=
-
d
i
-
d
i
-
1
-
1
‍
∑
l
=
j
+
1
-
1
‍
∑
r
=
k
+
l
+
1
k
‍
h
¯
T
(
r
)
T
i
h
¯
(
r
)
cccci
-
∑
j
=
-
d
i
(
k
)
-
1
‍
∑
l
=
k
+
j
k
-
1
‍
h
¯
T
(
l
)
T
i
h
¯
(
l
)
cccci
-
∑
j
=
-
d
i
-
d
i
-
1
-
1
‍
∑
l
=
j
+
1
-
1
‍
∑
r
=
k
+
l
k
-
1
‍
h
¯
T
(
r
)
T
i
h
¯
(
r
)
}
=
∑
i
=
1
n
2
‍
{
∑
j
=
-
d
i
(
k
+
1
)
-
1
‍
∑
l
=
k
+
j
+
1
k
-
1
‍
h
¯
T
(
l
)
T
i
h
¯
(
l
)
ccccc
+
∑
j
=
-
d
i
(
k
+
1
)
-
1
‍
h
¯
T
(
l
)
T
i
h
¯
(
l
)
ccccc
-
∑
j
=
-
d
i
(
k
)
-
1
‍
∑
l
=
k
+
j
+
1
k
-
1
‍
h
¯
T
(
l
)
T
i
h
¯
(
l
)
ccccc
-
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
T
(
k
+
j
)
T
i
h
¯
(
k
+
j
)
ccccc
+
∑
j
=
-
d
i
-
d
i
-
1
-
1
‍
∑
l
=
j
+
1
-
1
‍
(
h
¯
T
(
k
)
T
i
h
¯
(
k
)
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
i
-
h
¯
T
(
k
+
l
)
T
i
h
¯
(
k
+
l
)
)
∑
j
=
-
d
i
(
k
+
1
)
-
1
‍
}
≤
∑
i
=
1
n
2
‍
{
(
∑
j
=
-
d
i
-
d
i
-
1
-
1
‍
+
∑
j
=
-
d
i
-
1
-
1
‍
)
∑
l
=
k
+
j
+
1
k
-
1
‍
h
¯
T
(
l
)
T
i
h
¯
(
l
)
cccccccccccccccccccccccccccccccccc
+
d
i
h
¯
T
(
k
)
T
i
h
¯
(
k
)
ccccc
-
∑
j
=
-
d
i
-
1
-
1
‍
∑
l
=
k
+
j
+
1
k
-
1
‍
h
¯
T
(
l
)
T
i
h
¯
(
l
)
ccccc
-
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
T
(
k
+
j
)
T
i
h
¯
(
k
+
j
)
ccccc
+
∑
j
=
-
d
i
-
d
i
-
1
-
1
‍
∑
l
=
j
+
1
-
1
‍
(
h
¯
T
(
k
)
T
i
h
¯
(
k
)
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
-
h
¯
T
(
k
+
l
)
T
i
h
¯
(
k
+
l
)
)
∑
j
=
-
d
i
-
d
i
-
1
-
1
‍
}
=
∑
i
=
1
n
2
‍
{
d
i
h
¯
T
(
k
)
T
i
h
¯
(
k
)
-
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
T
(
k
+
j
)
T
i
h
¯
(
k
+
j
)
ccccc
+
(
d
i
-
d
i
-
1
)
(
d
i
+
d
i
-
1
-
1
)
2
h
¯
T
(
k
)
T
i
h
¯
(
k
)
∑
j
=
-
d
i
(
k
)
-
1
‍
}
≤
∑
i
=
1
n
2
‍
{
-
(
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
(
k
+
j
)
)
T
T
i
d
i
(
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
(
k
+
j
)
)
ccccc
+
(
d
i
-
d
i
-
1
+
1
)
(
d
i
+
d
i
-
1
)
2
h
¯
T
(
k
)
T
i
h
¯
(
k
)
(
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
(
k
+
j
)
)
}
.
Then, by using Lemma 7, we have
(32)
-
h
∑
i
=
-
τ
M
-
1
‍
η
T
(
k
+
i
)
P
η
(
k
+
i
)
=
-
h
∑
i
=
k
-
τ
1
(
k
)
k
-
1
‍
η
T
(
i
)
P
η
(
i
)
-
h
∑
i
=
k
-
τ
2
(
k
)
k
-
τ
1
(
k
)
-
1
‍
η
T
(
i
)
P
η
(
i
)
-
⋯
-
h
∑
i
=
k
-
τ
M
k
-
τ
n
1
(
k
)
-
1
‍
η
T
(
i
)
P
η
(
i
)
≤
-
(
∑
i
=
k
-
τ
1
(
k
)
k
-
1
‍
η
(
i
)
)
T
P
(
∑
i
=
k
-
τ
1
(
k
)
k
-
1
‍
η
(
i
)
)
-
(
∑
i
=
k
-
τ
2
(
k
)
k
-
τ
1
(
k
)
-
1
‍
η
(
i
)
)
T
P
(
∑
i
=
k
-
τ
2
(
k
)
k
-
τ
1
(
k
)
-
1
‍
)
-
⋯
-
(
∑
i
=
k
-
τ
M
k
-
τ
n
1
(
k
)
-
1
‍
η
(
i
)
)
T
P
(
∑
i
=
k
-
τ
M
k
-
τ
n
1
(
k
)
-
1
‍
η
(
i
)
)
=
-
(
e
^
(
k
)
-
e
^
(
k
-
τ
1
(
k
)
)
)
T
P
(
e
^
(
k
)
-
e
^
(
k
-
τ
1
(
k
)
)
)
-
(
e
^
(
k
-
τ
1
(
k
)
)
-
e
^
(
k
-
τ
2
(
k
)
)
)
T
×
P
(
e
^
(
k
-
τ
1
(
k
)
)
-
e
^
(
k
-
τ
2
(
k
)
)
)
-
⋯
-
(
e
^
(
k
-
τ
n
1
(
k
)
)
-
e
^
(
k
-
τ
M
)
)
T
×
P
(
e
^
(
k
-
τ
n
1
(
k
)
)
-
e
^
(
k
-
τ
M
)
)
.
Let α
1
(
k
)
=
(
e
^
T
(
k
-
τ
1
(
k
)
), e
^
T
(
k
-
τ
2
(
k
)
)
,
…
,
e
^
T
(
k
-
τ
n
1
(
k
)
), e
^
T
(
k
-
τ
M
)
)
T; we have
(33)
-
h
2
∑
i
=
k
-
τ
M
k
-
τ
m
-
1
‍
η
T
(
i
)
Q
η
(
i
)
≤
(
e
^
T
(
k
)
,
α
1
T
(
k
)
)
×
[
-
P
P
0
⋯
0
*
-
2
P
P
⋯
0
⋮
⋮
⋱
⋮
⋮
*
*
*
-
2
P
P
*
*
*
*
-
P
]
(
e
^
(
k
)
α
1
(
k
)
)
.
From Assumption1, f
i
-
≤
(
f
i
(
x
)
-
f
i
(
y
)
)
/
(
x
-
y
)
≤
f
i
+ and g
i
-
≤
(
g
i
(
x
)
-
g
i
(
y
)
)
/
(
x
-
y
)
≤
g
i
+, we have (
f
(
k
)
-
f
i
-
y
(
k
)
)
(
f
(
k
)
-
f
i
+
y
(
k
)
)
≤
0 and (
g
(
k
-
τ
i
(
k
)
)
-
g
i
-
y
(
k
-
τ
i
(
k
)
)
)
(
g
(
k
-
τ
i
(
k
)
)
-
g
i
+
e
(
k
-
τ
i
(
k
)
)
)
≤
0.
It can be deduced that there existU
=
diag
⁡
[
u
1
,
u
2
,
…
u
n
]
>
0, W
=
diag
⁡
[
w
1
,
w
2
,
…
,
w
n
]
>
0, and V
=
diag
⁡
[
v
1
,
v
2
,
…
,
v
n
]
>
0, such that
(34)
∑
i
=
1
n
‍
u
i
[
e
(
k
)
f
(
k
)
]
T
[
f
i
-
f
i
+
e
i
e
i
T
-
f
i
-
+
f
i
+
2
e
i
e
i
T
-
f
i
-
+
f
i
+
2
e
i
e
i
T
e
i
e
i
T
]
[
e
(
k
)
f
(
k
)
]
=
[
e
(
k
)
f
(
k
)
]
T
[
U
F
1
-
U
F
2
-
U
F
2
U
]
[
e
(
k
)
f
(
k
)
]
≤
0
,
∑
i
=
1
n
‍
u
i
[
x
(
k
)
F
(
x
(
k
)
)
]
T
×
[
f
i
-
f
i
+
e
i
e
i
T
-
f
i
-
+
f
i
+
2
e
i
e
i
T
-
f
i
-
+
f
i
+
2
e
i
e
i
T
e
i
e
i
T
]
[
x
(
k
)
F
(
x
(
k
)
)
]
=
[
x
(
k
)
F
(
x
(
k
)
)
]
T
[
U
F
1
-
U
F
2
-
U
F
2
U
]
[
x
(
k
)
F
(
x
(
k
)
)
]
≤
0
,
∑
i
=
1
n
‍
w
i
[
x
(
k
-
τ
j
(
k
)
)
G
(
x
(
k
-
τ
i
(
k
)
)
)
]
T
[
g
i
-
g
i
+
e
i
e
i
T
-
g
i
-
+
g
i
+
2
e
i
e
i
T
-
g
i
-
+
g
i
+
2
e
i
e
i
T
e
i
e
i
T
]
×
[
x
(
k
-
τ
j
(
k
)
)
G
(
x
(
k
-
τ
i
(
k
)
)
)
]
=
[
x
(
k
-
τ
i
(
k
)
)
G
(
x
(
k
-
τ
i
(
k
)
)
)
]
T
[
W
G
1
-
W
G
2
-
W
G
2
W
]
×
[
x
(
k
-
τ
i
(
k
)
)
G
(
x
(
k
-
τ
i
(
k
)
)
)
]
≤
0
,
∑
i
=
1
n
‍
w
i
[
e
(
k
)
g
(
k
)
]
T
[
g
i
-
g
i
+
e
i
e
i
T
-
g
i
-
+
g
i
+
2
e
i
e
i
T
-
g
i
-
+
g
i
+
2
e
i
e
i
T
e
i
e
i
T
]
[
e
(
k
)
g
(
k
)
]
=
[
e
(
k
)
g
(
k
)
]
T
[
W
G
1
-
W
G
2
-
W
G
2
W
]
[
e
(
k
)
g
(
k
)
]
≤
0
,
∑
i
=
1
n
‍
v
i
[
e
(
k
)
h
(
k
)
]
T
[
h
i
-
h
i
+
e
i
e
i
T
-
h
i
-
+
h
i
+
2
e
i
e
i
T
-
h
i
-
+
h
i
+
2
e
i
e
i
T
e
i
e
i
T
]
[
e
(
k
)
h
(
k
)
]
=
[
e
(
k
)
h
(
k
)
]
T
[
V
H
1
-
V
H
2
-
V
H
2
V
]
[
e
(
k
)
h
(
k
)
]
≤
0
,
∑
i
=
1
n
‍
v
i
[
x
(
k
)
H
(
x
(
k
)
)
]
T
[
h
i
-
h
i
+
e
i
e
i
T
-
h
i
-
+
h
i
+
2
e
i
e
i
T
-
h
i
-
+
h
i
+
2
e
i
e
i
T
e
i
e
i
T
]
×
[
x
(
k
)
H
(
x
(
k
)
)
]
=
[
x
(
k
)
H
(
x
(
k
)
)
]
T
[
V
H
1
-
V
H
2
-
V
H
2
V
]
[
x
(
k
)
H
(
x
(
k
)
)
]
≤
0
,
where e
i denotes the unit column vector having one element on its rth row and zeros elsewhere.
According to (10), o
T
(
k
)
o
(
k
)
-
e
T
(
k
)
L
T
L
e
(
k
)
≤
0; the following equation can be concluded, where ϑ is a positive scalar:
(35)
ϑ
o
T
(
k
)
o
(
k
)
-
ϑ
e
T
(
k
)
L
T
L
e
(
k
)
≤
0
.
And from Assumption 3, we can obtain, for a positive scalar, σ
(36)
σ
δ
T
(
k
,
x
(
k
)
)
δ
(
k
,
x
(
k
)
)
-
σ
ξ
x
T
(
k
)
x
(
k
)
≤
0
.
Combining (27)–(36), we obtain
(37)
𝔼
{
Δ
V
(
e
^
(
k
)
,
k
)
}
≤
ζ
T
(
k
)
(
Γ
^
+
Φ
T
Λ
Φ
)
ζ
(
k
)
,
ζ
(
k
)
=
(
e
^
T
(
k
)
,
α
1
T
(
k
)
,
α
2
T
(
k
)
,
f
^
T
(
k
)
,
g
^
T
(
k
)
,
c
c
i
i
h
^
T
(
k
)
,
G
^
T
(
k
)
,
H
^
T
(
k
)
,
o
T
(
k
)
,
g
T
(
k
)
,
c
c
i
i
h
T
(
k
)
,
H
T
(
x
(
k
)
)
,
δ
T
(
k
,
x
(
k
)
)
)
T
,
where (38)
α
2
(
k
)
=
[
∑
j
=
-
d
1
(
k
)
-
1
‍
e
^
T
(
k
+
j
)
∑
j
=
-
d
2
(
k
)
-
1
e
^
T
(
k
+
j
)
⋯
∑
j
=
-
d
n
2
(
k
)
-
1
‍
e
^
T
(
k
+
j
)
]
T
,
Γ
^
=
[
Γ
1
Γ
1,2
0
Γ
1,4
Γ
1,5
Γ
1,6
Γ
1,7
Γ
1,8
Γ
^
1,9
Γ
1,10
Γ
1,11
Γ
1,12
0
*
Γ
2
0
0
Γ
2,5
0
0
0
0
0
0
0
0
*
*
Γ
3
0
0
0
0
0
0
0
0
0
0
*
*
*
Γ
4
Γ
4,5
Γ
4,6
Γ
4,7
Γ
4,8
Γ
^
4,9
0
0
0
0
*
*
*
*
Γ
5
Γ
5,6
Γ
5,7
Γ
5,8
Γ
^
5,9
0
0
0
0
*
*
*
*
*
Γ
6
Γ
6,7
Γ
6,8
Γ
^
6,9
0
0
0
0
*
*
*
*
*
*
Γ
7
Γ
7,8
Γ
^
7,9
0
0
0
0
*
*
*
*
*
*
*
Γ
8
Γ
^
8,9
0
0
0
0
*
*
*
*
*
*
*
*
Γ
^
9
0
0
0
0
*
*
*
*
*
*
*
*
*
Γ
10
0
0
0
*
*
*
*
*
*
*
*
*
*
Γ
11
0
0
*
*
*
*
*
*
*
*
*
*
*
Γ
12
0
*
*
*
*
*
*
*
*
*
*
*
*
Γ
13
]
,
Γ
^
1,9
=
C
^
T
P
K
^
,
Γ
^
4,9
=
A
^
T
P
K
^
,
Γ
^
5,9
=
B
^
T
ρ
^
1
T
P
K
^
,
Γ
^
6,9
=
D
^
T
ξ
^
1
T
P
K
^
,
Γ
^
7,9
=
B
^
T
ρ
^
2
T
P
K
^
,
Γ
^
8,9
=
D
^
T
ξ
^
2
T
P
K
^
,
Γ
^
9
=
K
^
T
P
K
^
-
ϑ
I
,
Λ
=
h
τ
M
P
,
Φ
=
(
(
C
^
-
I
)
,
0,0
,
A
^
,
ρ
^
1
B
^
,
ξ
^
1
D
^
,
0,0
,
K
^
)
.
Using Schur complement, we can letΓ
^
+
Φ
T
Λ
Φ
<
0 be equal to
(39)
[
Γ
^
Φ
T
P
*
-
P
Λ
-
1
P
]
<
0
.
Pre- and postmultiplying (39), respectively, by diag
⁡
{
I
,
Λ
P
-
1
} and its transpose yield
(40)
[
Γ
^
Φ
T
Λ
*
-
Λ
]
<
0
.
Then, by denoting K
=
P
2
-
1
X, one gets that the LMI condition (40) can guarantee  (23).
It is obvious that(41)
𝔼
{
Δ
V
(
e
^
k
,
k
)
}
<
λ
max
⁡
(
Γ
)
𝔼
{
∥
ζ
(
k
)
∥
2
}
≤
-
λ
max
⁡
(
Γ
)
𝔼
{
∥
e
^
(
k
)
∥
2
}
.
The summation of both sides of (41) from 1 to N (let N be a positive integer) is equal to
(42)
𝔼
{
V
(
e
^
N
,
N
)
-
V
(
e
^
0
,
0
)
}
≤
λ
max
⁡
(
Γ
)
∑
k
=
1
N
‍
𝔼
{
∥
e
^
(
k
)
∥
2
}
.
So,
(43)
-
λ
max
⁡
(
Υ
)
∑
k
=
1
N
‍
𝔼
{
∥
e
^
(
k
)
∥
2
}
≤
𝔼
{
V
(
y
0
,
0
)
}
.
We can conclude that ∑
k
=
1
+
∞
𝔼
{
∥
e
^
(
k
)
∥
2
} is convergent and
(44)
lim
⁡
k
→
+
∞
𝔼
{
∥
e
(
k
)
∥
2
}
=
0
.
This completes the proof.Based on Theorem10, a further improved delay-dependent stability criterion of the system (15) is given in the following corollary by using Lemma 8.Corollary 11.
Under Assumptions1, 3, and 4, the system (15) is globally asymptotically stable in mean square, if there exist matrices P
=
P
T
=
diag
⁡
{
P
1
,
P
2
}
>
0, Q
i
=
Q
i
T
≥
0
(
i
=
1,2
,
3,4
), S
>
0, R
+
S
>
0, and W
=
diag
⁡
(
w
1
,
w
2
,
…
,
w
n
) such that the following LMI holds:
(45)
Ξ
=



[
Γ
+
∑
i
=
1
n
2
Sym
(
E
6
Q
i
)
*
*
*
*
d
1
Q
1
-
d
1
Q
1
*
*
*
d
2
Q
2
0
-
d
2
Q
2
*
*
⋮
⋮
⋮
⋱
*
d
n
1
Q
n
1
*
*
*
-
d
n
1
Q
n
1
]




<

0
,
where Γ is defined in Theorem 10 and E
6
=
(
0
2
n
,
0
(
n
1
+
1
)
n
,
0
n
2
n
,
0
2
n
,
0
n
1
n
,
I
n
2
,
0
n
1
n
,
0
n
2
n
,
0
n
,
0
n
,
0
n
,
0
n
,
0
n
)
T.Proof .
From Theorem10, we can know that
(46)
𝔼
{
Δ
V
5
(
e
k
,
k
)
}
=
𝔼
{
𝔼
{
V
5
(
e
k
+
1
,
k
+
1
)
∣
e
k
}
-
V
5
(
e
k
,
k
)
}
≤
∑
i
=
1
n
2
‍
{
-
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
T
(
k
+
j
)
T
i
h
¯
(
k
+
j
)
cccci
+
(
d
i
-
d
i
-
1
+
1
)
(
d
i
+
d
i
-
1
)
2
h
¯
T
(
k
)
T
i
h
¯
(
k
)
∑
j
=
-
d
i
(
k
)
-
1
‍
}
.
Then, Using Lemma 8 and d
i
(
k
)
∈
(
d
i
-
1
,
d
i
], we can deduce that, for any matrices Q
i with appropriate dimension,
(47)
-
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
T
(
k
+
j
)
T
i
h
¯
(
k
+
j
)
≤
d
i
(
k
)
ζ
T
(
k
)
Q
i
T
T
i
-
1
Q
i
ζ
(
k
)
+
2
ζ
T
(
k
)
Q
i
T
(
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
T
(
k
+
j
)
)
≤
d
i
-
1
ζ
T
(
k
)
Q
i
T
T
i
-
1
Q
i
ζ
(
k
)
+
2
ζ
T
(
k
)
Q
i
T
(
∑
j
=
-
d
i
(
k
)
-
1
‍
h
¯
T
(
k
+
j
)
)
.


## 4. Examples

In this section, a numerical example is given to illustrate the effectiveness and benefits of the developed methods.Example 1.
We consider the delayed stochastic DNNs (1) with the following parameters:
(48)
C
=
[
-
0.1
0
0
0.4
]
,
A
=
[
0.1
-
0.2
0
-
0.1
]
,
B
=
[
0
0.2
0.2
-
0.1
]
,
L
=
[
0.2
0
0
0.2
]
.
And the activation functions satisfy Assumption 1 with
(49)
F
1
=
[
-
0.64
0
0
0
]
,
F
2
=
[
0
0
0
0.2
]
,
G
1
=
[
-
0.6
0
0
0
]
,
G
2
=
[
-
0.2
0
0
-
0.1
]
.
For the parameters listed above, letting τ
m
=
1, τ
M
=
5, τ
1
=
3, and ρ
1
=
0.89, we can obtain the following feasible solution. Therefore, it is clear to see that our method is effective. Due to the limitation of the length of this paper, we only provide a part of the feasible solution here:
(50)
P
1
=
[
5.8618
2.1488
2.1488
3.9636
]
,
P
2
=
[
8.3545
3.4900
3.4900
3.9051
]
,
X
=
[
1.7218
0.4208
0.4208
0.9386
]
.
Therefore, according to Theorem10, the gain matrix of the desired estimator can be obtained as
(51)
K
=
P
2
-
1
X
=
[
0.2571
-
0.0798
-
0.1220
0.3117
]
.
When we define f
1
(
s
)
=
g
1
(
s
)
=
h
1
(
s
)
=
tanh
(
-
0.4
s
), f
2
(
s
)
=
g
2
(
s
)
=
h
2
(
s
)
=
tanh
(
0.4
s
)), O
(
k
,
x
(
k
)
)
=
0.2
sin
(
x
(
k
)
), δ
(
k
,
x
(
k
)
)
=
sin
(
x
(
k
)
), we get Figures 1 and 2. They represent the trajectories of x
(
k
) and the state estimator x
^
(
k
) with the initial condition x
(
k
)
=
[
-
1,0.5
], x
^
(
k
)
=
[
-
0.5
,
-
0.5
]. And from Theorem 10, it follows that the state estimator (46) is indeed a state estimator of the delayed neural network (1). Figure 3 further confirmed that the estimation error e
(
k
) tends to zero as k
→
∞.Figure 1
The trajectories ofx
(
k
) with J
=
0.Figure 2
The trajectories ofx
~
(
k
) with J
=
0.Figure 3
The evolution of estimation errorse
(
k
).


## 5. Conclusions

The robust stability for stochastic discrete-time NNs with mixed delays has been investigated in this research via the Lyapunov functional method. By employing delay partitioning and introducing a new Lyapunov functional, more general LMIs conditions on the stability of the stochastic discrete-time NNs are established. Finally, the feasibility and effectiveness of the developed methods and their less conservatism than most of the existing results have been shown by numerical simulation examples. The foregoing results have the potential to be useful for the study of stochastic discrete-time NNs. And the results can also been extended to complex networks with mixed time-varying delays.



---

*Source: 209486-2014-02-18.xml*