5 Conclusions
In this paper, the problem of designing a feedback
control law to exponentially stabilization a class neural networks with various activation and mixed timevarying delay in state and control has been studied.
By using augmented Lyapunov-Krasovskii functionals,
a new delay-dependent condition for the global exponentially stabilization have been established in terms
of linear matrix inequalities which allows to compute
simultaneously the two bounds that characterize the
exponential stability of the solution. A numerical example is given to show the effectiveness of the obtained
results.

5 trang |

Chia sẻ: thucuc2301 | Ngày: 26/11/2020 | Lượt xem: 118 | Lượt tải: 1
Bạn đang xem nội dung tài liệu **Exponential stabilization of Neural Networks with mixed Time-Varying Delays in state and control - Mai V.Thuan**, để tải tài liệu về máy bạn click vào nút DOWNLOAD ở trên

Exponential stabilization of Neural Networks with
mixed Time-Varying Delays in state and control
MAI V. THUAN
1,∗
, N. T. T. HUYEN
1
and N. T. M. NGOC
2
1,∗
College of Sciences, Thainguyen university, Thainguyen, Vietnam
2
Thainguyen university of Technology, Thainguyen, Vietnam
∗
Corresponding author: maithuank1@gmail.com
Abstract. This paper presents some results on the
global exponential stabilization for cellular networks
with various activation functions and mixed time-
varying delays in state and control. Based on aug-
mented time-varying Lyapunov Krasovskii functionals,
new delay-dependent conditions for the global expo-
nential stabilization are obtained in terms of linear
matrix inequalities. Numerical examples are given to
illustrate the feasibleness of our results.
Key words. Cellular neutral networks, stabilization,
neural networks, mixed delay, Lyapunov function, Lin-
ear matrix inequalities.
1 Introduction
In the area of control, signal processing, pattern
recognition, image processing, and association, delayed
cellular neural networks have many usedful applica-
tions. Some of these applications require that the equi-
librium points of the designed network be stable. In
both biological and artificial neural systems, time de-
lays due to integration and communication are ubiqui-
tous, and often become a source of instability. The time
delays in electronic neural networks are usually time-
varying, and sometimes vary violently with respect to
time due to the finite switching speed of amplifiers
and faults in the electrical circuitry. Therefor, stabil-
ity analysis of delayed cellular neural network is a very
important issue, and many stability criteria have been
developed in the literature [2, 5, 8] and the references
cited therein.
Recently, the stabilization issue has been an im-
portant focus of research in the control society, and
several feedback stabilizing control design approaches
have been proposed in [3, 4]. Regarding stabilization of
neural netwoks, to the best of our knowledge, only a
few results are published. The papers [1, 6, 7] present
some stabilization criteria for delayed neural networks.
However, the results reported therein not only require
the only activation function, but the system matrices
are also strictly constrained.
In this paper, we consider a stabilization scheme
for a general class of delayed neural networks. The
novel features here are that the neural networks in con-
sideration are time-varying with mixed delay in state
and control and with various activation function. We
extend the results of [1, 2, 6, 7, 9] to exponential sta-
bilization of neural networks various activation funci-
tons and mixed time-varying delay in state and con-
trol. Using the Lyapunov stability theory and linear
matrix inequality (LMI) techniques, a control law with
an appropriate gain control matrix is derived to achieve
stabilization of the neural networks with mixed time-
varying delayed in state and control. The stabilization
criteria are obtained in terms of LMIs and hence the
gain control matrix is easily determined by numerical
Matlab's Control Toolbox.
2 Preliminaries
The following notation will be used in this paper:
R+ denotes the set of all real non-negative numbers;
Rn denotes the n−dimensional space with the scalar
product 〈., .〉 and the vector norm ‖ . ‖; Mn×r denotes
the space of all matrices of (n× r)−dimensions.
AT denotes the transpose of matrix A; A is symmet-
ric if A = AT ; I denotes the identity matrix; λ(A)
denotes the set of all eigenvalues of A; λ
max
(A) =
max{Reλ;λ ∈ λ(A)}.
xt := {x(t + s) : s ∈ [−h, 0]}, ‖ xt ‖= sups∈[−h,0] ‖
x(t+s) ‖; C([0, t], Rn) denotes the set of all Rn−valued
continuous functions on [0, t]; L2([0, t], Rm) denotes the
1
Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên
set of all the Rm−valued square integrable functions on
[0, t];
Matrix A is called semi-positive definite (A ≥ 0)
if 〈Ax, x〉 ≥ 0, for all x ∈ Rn;A is positive definite
(A > 0) if 〈Ax, x〉 > 0 for all x 6= 0;A > B means
A−B > 0. The notation diag{. . .} stands for a block-
diagonal matrix. The symmetric term in a matrix is
denoted by ∗.
Consider the following cellular neural networks with
mixed time-varying delays in state and control of the
form
x˙(t) = −Ax(t) +W0f(x(t)) +W1g(x(t− τ1(t)))
+W2
∫ t
t−τ2(t)
h(x(s)) ds+B0u(t) +B1u(t− τ3(t))
+B2
∫ t
t−τ4(t)
u(s) ds
x(t) = φ(t), t ∈ [−d, 0], d = max{τ1, τ2, τ3, τ4},
(2.1)
where x(t) = [x1(t), x2(t), . . . , xn(t)]T ∈ Rn is
the state vector of the neural networks; u(t) ∈
L2([0, s], Rm),∀s > 0, is the control input vector of
the neural networks; n is the number of neurals, and
f(x(t)) = [f1(x1(t)), f2(x2(t)), . . . , fn(xn(t))]T ,
g(x(t)) = [g1(x1(t)), g2(x2(t)), . . . , gn(xn(t))]T ,
h(x(t)) = [h1(x1(t)), h2(x2(t)), . . . , hn(xn(t))]T
are the neural activation functions.
The diagonal matrix A = diag(a1, a2, . . . , an) repre-
sents the self-feedback term and W0,W1,
W2, B0, B1, B2 are given real constant matrices with
appropriate dimensions. The time-varying delay func-
tions τ1(t), τ2(t), τ3(t), τ4(t) satisfy the condition
0 ≤ τ1(t) ≤ τ1, τ˙1(t) ≤ δ1 < 1, 0 ≤ τ2(t) ≤ τ2,
0 ≤ τ3(t) ≤ τ3, τ˙3(t) ≤ δ2 < 1, 0 ≤ τ4(t) ≤ τ4.
The initial functions φ(t) ∈ C([−d, 0], Rn), d =
max{τ1, τ2, τ3, τ4}, with the uniform norm ‖ φ ‖=
maxt∈[−d,0] ‖ φ(t) ‖ . We assume that the activation
functions f(x), g(x), h(x) are Lipschitz with the Lips-
chitz constains fi, gi, hi > 0 :
| fi(ξ1)− fi(ξ2) |≤ fi | ξ1 − ξ2 |,
| gi(ξ1)− gi(ξ2) |≤ gi | ξ1 − ξ2 |,
| hi(ξ1)− hi(ξ2) |≤ hi | ξ1 − ξ2 |,
∀i = 1, 2, . . . , n, ∀ξ1, ξ2 ∈ R.
(2.2)
Definition 2.1 [9] Given α > 0. The system (2.1) is
α−exponentially stable if there exist a positive number
N > 0 such that every solution x(t, φ) satisfies the
following condition:
‖ x(t, φ) ‖≤ Ne−αt ‖ φ ‖, ∀t ≥ 0.
Definition 2.2 [9] Given α > 0. The system (2.1)
is globally α−exponentially stabilizable if there is a
feedback control u(t) = Kx(t), such that the closed-
loop time-delay system
x˙(t) = −[A0 −B0K]x(t) +W0f(x(t)) +W1g(x(t− τ1(t)))
+B1Kx(t− τ3(t)) +W2
∫ t
t−τ2(t)
h(x(s)) ds
+B2
∫ t
t−τ4(t)
Kx(s) ds
x(t) = φ(t), t ∈ [−d, 0], d = max{τ1, τ2, τ3, τ4}
(2.3)
is α−exponentially stable.
We introduce the following technical well-known
propositions, which will be used in the proof of our
results.
Proposition 2.1 [9] Let P,Q be matrices of appro-
priate dimensions and Q is symmetric positive definite.
Then
2〈Py, x〉 − 〈Qy, y〉 ≤ 〈PQ−1PTx, x〉, ∀(x, y).
Proposition 2.2 [9] For any symmetric positive def-
inite matrix M > 0, scalar γ > 0 and vector function
ω : [0, γ] → Rn such that the integrations concerned
are well defined, the following inequality holds
(∫ γ
0
ω(s) ds
)T
M
(∫ γ
0
ω(s) ds
)
≤ γ
(∫ γ
0
ωT (s)Mω(s) ds
)
Proposition 2.3 (Schur complement lemma) [8]
Given constant symmetric matrices X,Y, Z with ap-
propriate dimensions satisfying X = XT , Y = Y T > 0.
Then X + ZTY −1Z < 0 if and only if(
X ZT
Z −Y
)
< 0 or
(−Y Z
ZT X
)
< 0.
2
Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên
3 Main result
Let us denote
Σ =−AP − PAT − (B0Y + Y TBT0 ) + 2αP
+W0D0WT0 + (1− δ1)−1e2ατ1W1D1WT1
+ (1− δ2)−1e2ατ3B1BT1 + τ2e2ατ2W2D2WT2
+ τ4e2ατ4B2BT2 ,
G = diag{gi, i = 1, 2, . . . , n},
H = diag{hi, i = 1, 2, . . . , n},
F = diag{fi, i = 1, 2, . . . , n},
g2 = max{g2i , i = 1, 2, . . . , n},
h2 = max{h2i , i = 1, 2, . . . , n},K = −Y P−1,
θ = 1 + τ4,
λ1 = λmin(P−1),
λ2 = λmax(P−1) + λmax(D−11 )g
2τ1 + λmax(D−12 )h
2τ22
+ (τ3 +
1
2
τ24 )λmax(K
TK).
Theorem 3.1. Given α > 0. System (2.1) is
α−exponentially stabilizable if there exist a symmet-
ric positive definite matrix P ∈ Rn×n, three diagonal
positive matrices Di, i = 0, 1, 2 and a matrix Y with ap-
propriate dimension such that the following LMI holds:
Ξ =
Σ PF PG τ2PH θY T
FP −D0 0 0 0
GP 0 −D1 0 0
τ2HP 0 0 −τ2D2 0
θY 0 0 0 −θIm
< 0.
(3.1)
The stabilizing feedback control is given by
u(t) = −Y P−1x(t),
and the solution x(t, φ) of the system satisfies
‖ x(t, φ) ‖≤
√
λ2
λ1
e−αt ‖ φ ‖, ∀t ≥ 0.
proof. Let us denote X = P−1,K = −Y P−1. With
the feedback control u(t) = −Y P−1x(t), we consider
the Lyapunov-Krasovskii functional for closed-loop sys-
tem (2.3)
V (t, xt) =
5∑
i=1
Vi(t, xt),
where
V1 = xT (t)Xx(t),
V2 =
∫ t
t−τ1(t)
e2α(s−t)gT (x(s))D−11 g(x(s)) ds,
V3 =
∫ 0
−τ2
∫ t
t+s
e2α(θ−t)hT (x(θ))D−12 h(x(θ)) dθ ds,
V4 =
∫ t
t−τ3(t)
e2α(s−t)xT (s)KTKx(s) ds,
V5 =
∫ 0
−τ4
∫ t
t+s
e2α(θ−t)xT (θ)KTKx(θ) dθ ds.
It easy to check that
λ1 ‖ x(t) ‖2≤ V (t, xt) ≤ λ2 ‖ xt ‖2, t ∈ R+. (3.2)
Taking derivative of V1 along solutions of the closed-
loop system (2.3), we get
V˙1 =xT (t)[−XA−ATX −X(B0Y + Y TBT0 )X]x(t)
+ 2xT (t)XW0f(x(t)) + 2xT (t)XW1g(x(t− τ1(t)))
+ 2xT (t)XB1u(t− τ3(t))
+ 2xT (t)XW2
∫ t
t−τ2(t)
h(x(s)) ds
+ 2xT (t)XB2
∫ t
t−τ4(t)
u(s) ds
Applying Proposition 2.1 and 2.2 gives
2xT (t)XW0f(x(t))
≤ xT (t)XW0D0WT0 Xx(t) + fT (x(t))D−10 f(x(t));
2xT (t)XW1g(x(t− τ1(t))
≤ (1− δ1)−1e2ατ1xT (t)XW1D1WT1 Xx(t)
+ (1− δ1)e−2ατ1gT (x(t− τ1(t))D−11 g(x(t− τ1(t)));
2xT (t)XB1u(t− τ3(t))
≤ (1− δ2)−1e2ατ3xT (t)XB1BT1 Xx(t)
+ (1− δ2)e−2ατ3 ‖ u(t− τ3(t)) ‖2;
2xT (t)XW2
∫ t
t−τ2(t)
h(x(s)) ds
≤ τ2e2ατ2xT (t)XW2D2WT2 Xx(t)
+
1
τ2
e−2ατ2
(∫ t
t−τ2(t)
h(x(s)) ds
)T
D−12
(∫ t
t−τ2(t)
h(x(s)) ds
)
≤ τ2e2ατ2xT (t)XW2D2WT2 Xx(t)
+ e−2ατ2
∫ t
t−τ2
hT (x(s))D−12 h(x(s)) ds;
3
Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên
2xT (t)XB2
∫ t
t−τ4(t)
u(s) ds
≤ τ4e2ατ4xT (t)XB2BT2 Xx(t)
+
1
τ4
e−2ατ4
(∫ t
t−τ4(t)
u(s) ds
)T(∫ t
t−τ4(t)
u(s) ds
)
≤ τ4e2ατ4xT (t)XB2BT2 Xx(t)
+ e−2ατ4
∫ t
t−τ4
‖ u(s) ‖2 ds.
Therefor
V˙1(t, xt)
≤ xT (t)[−XA−ATX −X(B0Y + Y TBT0 )X]x(t)
+ xTX(t)
[
W0D0W
T
0 + (1− δ1)−1e2ατ1W1D1WT1
+ (1− δ2)−1e2ατ3B1BT1 + τ2e2ατ2W2D2WT2
+ τ4e2ατ4B2BT2
]
Xx(t)
+ fT (x(t))D−10 f(x(t))
+ (1− δ1)e−2ατ1gT (x(t− τ1(t)))D−11 g(x(t− τ1(t)))
+ (1− δ2)e−2ατ3 ‖ u(t− τ3(t)) ‖2
+ e−2ατ2
∫ t
t−τ2
hT (x(s))D−12 h(x(s)) ds
+ e−2ατ4
∫ t
t−τ4
‖ u(s) ‖2 ds
(3.3)
Next, the derivatives of Vk, k = 2, . . . , 5 give
V˙2(t, xt)
≤ −2αV2 + (g(x(t)))TD−11 (g(x(t)))
− (1− δ1)e−2ατ1gT (x(t− τ1(t)))D−11 gT (x(t− τ1(t)));
V˙3(t, xt)
≤ −2αV3 + τ2hT (x(t))D−12 h(x(t))
− e−2ατ2
∫ t
t−τ2
hT (x(s))D−12 h(x(s)) ds;
V˙4(t, xt)
≤ −2αV4(t, xt) + xT (t)XY TY Xx(t)
− (1− δ2)e−2ατ3 ‖ u(t− τ3(t)) ‖2;
V˙5(t, xt)
≤ −2αV5(t,Xt) + τ4xT (t)XY TY Xx(t)
− e−2ατ4
∫ t
t−τ4
‖ u(s) ‖2 ds.
(3.4)
From (3.3)− (3.4), we obtain
V˙ + 2αV
≤ xT (t)
[
−XA−ATX −X(B0Y + Y TBT0 )X
+ 2αX +XW0D0WT0 X + (1− δ1)−1e2ατ1XW1D1WT1 X
+ (1− δ2)−1e2ατ3XB1BT1 X + τ2e2ατ2XW2D2WT2 X
+ τ4e2ατ4XB2BT2 X + (1 + τ4)XY
TY X
]
x(t)
+ fT (x(t))D−10 f(x(t)) + g
T (x(t))D−11 g(x(t))
+ τ2hT (x(t))D−12 h(x(t)).
(3.5)
Using the condition (2.2) and since the matrices Di >
0, i = 0, 1, 2 are diagonal, we have
fT (x(t))D−10 f(x(t)) ≤ xT (t)FD−10 Fx(t),
gT (x(t))D−11 g(x(t)) ≤ xTGD−11 Gx(t),
τ2h
T (x(t))D−12 h(x(t)) ≤ τ2xT (t)HD−12 Hx(t).
(3.6)
Since (3.5) and (3.6), we obtain
V˙ + 2αV ≤ xT (t)Ωx(t),
where
Ω =−XA−ATX −X(B0Y + Y TBT0 )X + 2αX
+XW0D0WT0 X + (1− δ1)−1e2ατ1XW1D1WT1 X
+ (1− δ2)−1e2ατ3XB1BT1 X + τ2e2ατ2XW2D2WT2 X
+ τ4e2ατ4XB2BT2 X + (1 + τ4)XY
TY X
+ FD−10 F +GD
−1
1 G+ τ2HD
−1
2 H.
We have, Ω < 0 if and only if PΩP < 0. And by Schur
complement lemma, PΩP < 0 if and only if Ξ < 0.
Thus
V˙ (t, xt) + 2αV (t, xt) ≤ 0
and hence
V (xt) ≤ V (φ)e−2αt ≤ λ2 ‖ φ ‖2 e−2αt, t ≥ 0.
Taking (3.2) into account we obtain
‖ x(t, φ) ‖≤
√
λ2
λ1
e−αt ‖ φ ‖, t ≥ 0,
which completes the proof.
4 Numerical example
Let us consider system (2.1) with τ1(t) =
sin
20.5t, τ3(t) = sin20.6t, and{
τ2(t) = 0.8 sin2 t if t ∈ I = ∪k≥0[2kpi, (2k + 1)pi]
τ2(t) = 0 if t ∈ R+ \ I,
4
Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên
{
τ4(t) = 0.5 cos2 t if t ∈ I = ∪k≥0[2kpi, (2k + 1)pi]
τ4(t) = 0 if t ∈ R+ \ I,
A =
(
32 0
0 21
)
,W0 =
(
1 0.15
1 2
)
,W1 =
(
0 1
1 0
)
,
W2 =
(
0.1 0.4
0.5 0.2
)
, B0 =
(
0
3
)
, B1 =
(
2
3
)
, B2 =
(
1
2
)
,
F =
(
0.1 0
0 0.4
)
, G =
(
0.5 0
0 0.8
)
, H =
(
0.1 0
0 0.6
)
.
We see that the time delay function τ2(t), τ4(t) are
bounded but non-differentiable and τ1 = 1, τ2 =
0.8, τ3 = 1, τ4 = 0.5, δ1 = 0.5, δ2 = 0.6. For α = 0.5, us-
ing MATLABs LMI Toolbox, the LMI (3.1) is feasible
with the following matrices:
P =
(
6.4893 2.2340
2.2340 15.5236
)
, D0 =
(
47.8388 0
0 47.6199
)
,
D1 =
(
47.6449 0
0 49.2631
)
, D2 =
(
52.2809 0
0 52.3712
)
,
Y =
[
0.1316 −0.0040] ,
and accordingly the feedback control is u(t) =[−0.0214 0.0033]x(t). The solution of the closed-loop
system satisfy
‖ x(t, φ) ‖≤ 2.9851 ‖ φ ‖ e−0.5t, ∀t ≥ 0.
5 Conclusions
In this paper, the problem of designing a feedback
control law to exponentially stabilization a class neu-
ral networks with various activation and mixed time-
varying delay in state and control has been studied.
By using augmented Lyapunov-Krasovskii functionals,
a new delay-dependent condition for the global expo-
nentially stabilization have been established in terms
of linear matrix inequalities which allows to compute
simultaneously the two bounds that characterize the
exponential stability of the solution. A numerical ex-
ample is given to show the effectiveness of the obtained
results.
T½nh ên ành ho¡ ÷ñc d¤ng mô cho h»
nìron th¦n kinh câ tr¹ tr¶n tr¤ng th¡i v i·u
khiºn
Tâm tt. Trong b i b¡o n y, chóng tæi ÷a ra mët
k¸t qu£ nghi¶n cùu mîi cho t½nh ên ành ho¡ ÷ñc d¤ng
mô cho lîp h» nìron th¦n kinh vîi c¡c h m k½nh ho¤t
kh¡c nhau v câ tr¹ tr¶n tr¤ng th¡i v i·u khiºn. B¬ng
vi»c sû döng h m Lyapunov-Krasovskii c£i ti¸n, chóng
tæi ¢ ÷a ra ÷ñc mët i·u ki»n õ mîi cho t½nh ên
ành ho¡ ÷ñc d¤ng mô cho lîp h» ÷ñc nâi ¸n ð tr¶n.
ﬃi·u ki»n õ cõa chóng tæi ÷ñc biºu di¹n d÷îi d¤ng
c¡c b§t ¯ng thùc ma trªn tuy¸n t½nh cho ph²p x¡c
ành ÷ñc c¡c ch¿ sè ên ành mô v cæng thùc i·u
khiºn ng÷ñc. Cuèi còng, v½ dö sè ÷ñc ÷a ra º minh
ho¤ cho t½nh hi»u qu£ cho k¸t qu£ cõa chóng tæi.
T i li»u
[1] J. Cao, S. Zhong, and Y. Wu, " Global stability
analysis for a class of neural networks with varying
delays and control input", Applied Math Comput.
, vol. 189, pp. 1480-1490, 2007.
[2] B. Chen and J. Wang, " Global exponential peri-
odicity and global exponential stability of a class
of recurrent neural networks with various activa-
tion functions and time-varying delays ", Neural
Netw., vol.20, no. 10, pp. 1067-1080, 2007.
[3] K. J. Hunt, D. Sbarbaro, R. Zbikowski and P. J.
Gawthrop, " Neural networks for control systems:
a survey ", Automatica, vol. 28, no. 6, pp. 1083-
1112, 1992.
[4] C. Hua and X. Guan, " Output feedback stabiliza-
tion for time-delay nonlinear interconnected sys-
tems using neural networks", IEEE Trans. Neural
Netw., vol. 19, no. 4, pp. 673-688, Apr. 2008.
[5] X. Liao and J. Yu, " Robust stability for interval
Hopfield neural networks ", IEEE Trans. Neural
Networks, vol. 9, pp. 1042-1046, 1998.
[6] M. Liu, "Delayed standard neural network mod-
els for control systems", IEEE Trans. Neural Net-
works, vol. 18, pp. 1376-1391, 2007.
[7] X. Lou and B. Cui, "On robust stabilization of
a class of neural networks with time-varying de-
lay,"In: Proc. of IEEE Int. Conf. Comput. Intel-
ligent and Security, pp. 437-440, Nov. 2006.
[8] J. H. Park, "An analysis of global robust sta-
bility of uncertain cellular neural networks with
discrete and distributed delays," Chaos, Solutions
and Fractal, vol. 32, pp. 800-807, 2007.
[9] V. N. Phat and H. Trinh, "Exponential stabi-
lization of neural netwoks with various activation
functions and mixed time-varying delays" IEEE
Transactions on neural networks, vol. 21, no. 7,
pp. 1180-1184, July 2010.
5
Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên

Các file đính kèm theo tài liệu này:

- brief_33438_37258_7920121061678_split_2_388_2052324.pdf