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Finite-time synchronization of delayed fractional-order heterogeneous complex networks
Finite-time synchronization of delayed fractional-order heterogeneous complex networks
Communicated by Dr Yan-Wu Wang
Journal Pre-proof
Finite-time synchronization of delayed fractional-order heterogeneous complex networks Ying Li, Yonggui Kao, Changhong Wang, Hongwei Xia PII: DOI: Reference:
S0925-2312(19)31645-5 https://doi.org/10.1016/j.neucom.2019.11.043 NEUCOM 21571
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
18 July 2019 24 October 2019 11 November 2019
Please cite this article as: Ying Li, Yonggui Kao, Changhong Wang, Hongwei Xia, Finite-time synchronization of delayed fractional-order heterogeneous complex networks, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.11.043
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1
Finite-time synchronization of delayed fractional-order heterogeneous complex networks Ying Li, Yonggui Kao, Changhong Wang and Hongwei Xia
Abstract—This paper is devoted to exploring the finitetime (FET) synchronization problem of time-varying delay fractional-order (FO) coupled heterogeneous complex networks (TFCHCNs) with external interference via a discontinuous feedback controller. Firstly, we propose a novel Lemma which is useful for discussing the FET stability and synchronization problem of FO systems. Secondly, based on the proposed Lemma, a discontinuous feedback controller is designed to guarantee the FET synchronization of TFCHCNs with external interference. Moreover, the upper bound of settling-time function is obtained. Finally, two simulation examples are provided to verify the practicability of our findings. Index Terms—Fractional-order systems, Heterogeneous complex networks, Finite-time synchronization, Discontinuous feedback controller.
I. I NTRODUCTION Complex networks have attracted more and more researchers from various fields, such as physics[1], mathematics, biomedical engineering, social management and so on [2]. Network synchronization is a significant dynamic character of complex networks owing to its extensive applications in secure communication, electronic circuits, biological systems and mechanism [3]. The optimal tracking problem for certain control system over a communication channel with packet dropouts has been researched in [4, 5]. In order to achieve network synchronization, lots of control methods have been proposed, such as feedback control [6], adaptive control [7], passive control [8], generalized projective control [9], sliding mode control [10], impulsive control [11], pinning control [12]. For heterogeneous complex networks, many classical theoretical results need to be extended. It is difficult for heterogeneous complex networks with different characteristics to find mixed Lyapunov functions with different properties, such as continuous and discrete, smooth and impulsive. However, most real networks are highly heterogeneous, and the roles of different nodes and different links in the evolution of networks are often very different. Therefore heterogeneous complex networks are more complicated and applicable than complex networks with the same nodes, there are few results on finite-time (FET) synchronization of heterogeneous complex networks. In [13], based on the eigenvalue-describing Yonggui Kao and Ying Li are with the Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, China. (e-mail: [email protected]). Changhong Wang and Hongwei Xia are with the Space Control and Inertial Technology Research Center, Harbin Institute of Technolog, Harbin, 150001, P.R. China. Supported by the National Natural Science Foundation of China No. 61873071 and Shandong Natural Science Foundation No. ZR2019MF006. Corresponding author: Yonggui Kao
coupling configuration matrix, Wang et al. discussed bounded synchronization of heterogeneous complex dynamic networks. Unlike integer-order (IO) systems, fractional-order (FO) systems have an advantage that they have memory, which is useful in depicting the memory and hereditary characters of numerous processes and systems [14, 15]. From the perspective of synchronization time, synchronization can be divided into asymptotic synchronization and FET synchronization. In practice, sometimes it is more important to study FET synchronization. For example, in confidential communication, if there is the longer time spent on synchronization, there is more possibility that the message will be taken over. FET synchronization owns meliorated properties of robustness and disturbance rejection than asymptotical synchronization. For FET synchronization of IO complex networks, in [16], Yang et al. researched FET synchronization of proposed IO complex networks with Lyapunov direct method. Recently, for FET synchronization of FO complex networks, in [17], A. Pratap et al. researched FET synchronization of timevarying delayed FO memristive neural networks using some common inequalities. In [18], Yang et al. discussed FET stability problem of FO neural networks with delay in virtue of Gronwall-Bellman inequality. Similarly, in [19], Li et al. probed FET synchronization of FO complex networks by some inequalities. In [20], rather than using Lyapunov direct method, Zheng et al. discussed the FET synchronization problems of memristor-based fractional-order fuzzy cellular neural networks in the light of Gronwall-Bellman inequality. By contrast, we obtain a new lemma similar with Lyapunov method that is more applicable and easier to generalize, which is also the innovation of this paper. Compared with the previous method, the method in this paper reduces the amount of computation and provides a new way to solve the FET synchronization problem. Base on the aforementioned concerns, we propose a novel and useful Lemma 4 which is applied to make the coupled heterogeneous complex networks (TFCHCNs) with external interference achieve FET synchronization. In Section 2, some definitions about FO equations, Lemmas and the Assumptions are presented. In Section 3, main results of FET synchronization are gained by using the novel lemma and a discontinuous feedback controller designed for the system. Two examples are presented to validate the effectiveness of our findings in Section 4. Finally, the conclusion is given in Section 5. Notations: In this paper, matrix A > B implies A − B is a positive definite matrix, where A and B are symmetric matrices. Let k · k be Euclid norm, Rη be the η-dimensional Euclidean space with norm k · k, R+ = [0, ∞). ς = dυe is an
2
integer which satisfies that ς −1 < υ ≤ ς. For a n-dimensional Pn column vector x = [x1 , x2 , . . . , xn ], |x| represents i=1 |xi |. II. P RELIMINARIES
C α t0 Dt ei (t)
Definition 1. [21] Let α ∈ R+ .The operator Jtα0 , defined on L1 [t0 , T ] by Z t 1 Jtα0 f (t) = (t − x)α−1 f (x)dx, (1) Γ(α) t0 for t0 ≤ t ≤ T , is α-order Riemann-Liouville fractional integral. Definition 2. [21] Let α ≥ 0 and m = dαe. Moreover assume that function f ∈ C n ([t0 , +∞), R) and n is a positive integer. α Operator C t0 Dt is defined by C α t0 Dt f (t)
=
1 Γ(m − α)
Z
t
t0
f (m) (s) ds, (t − s)α−m+1
(2)
α where Γ(·) is the Gamma function. The operator C t0 Dt is called Caputo fractional differential operator of order α. Especially, when 0 < α < 1, Z t (1) 1 f (s) C α (3) D f (t) = ds. t0 t Γ(1 − α) t0 (t − s)α
Consider FO time-varying delay complex networks coupled by m heterogeneous nodes, where every node is an ndimensional network. The slave system can be depicted as C α t0 Dt xi (t)
= fi (t, xi (t)) + bi (t) + c
m X
hij Υxj (t)
j=1
(4)
+ Axi (t − τ (t)) + ui (t), i = 1, 2, . . . , m. Where t0 ≥ 0 denotes the starting time, 0 < α < 1, xi (t) ∈ Rn denotes the state variable of the ith node, fi (t, x(t)) ∈ Rn denotes ith nonlinear function, bi (t) ∈ Rn is bounded and it represents external interference, τ (t) is time-varying delay. c is a positive constant and it represents the coupling strength of heterogeneous complex networks. Υ=diag(ε1 , ε2 , . . . , εn ) > 0 represents the inner-coupling matrix of heterogeneous networks. A = (aij )n×n is a constant P matric. Denote A¯ = n (|aij |)n×n , and row vector A¯sum = ( j=1 |aij |)1×n . H = (hij )m×m denotes the constant coupled configuration matrix and represents the topological structure of TFCHCNs, where hij is that : hij > 0(i 6= j)), if there exsits a direct link f rom node i to node j, h = 0(i = 6 j)), otherwise, ij Pm hii = − j=1,j6=i hij , i = 1, 2, . . . , m. With s(t) ∈ Rn , the master system is C α t0 Dt s(t)
The synchronization error is defined by ei (t) = xi (t)−s(t). Based on the equation (4) and equation (5), the synchronization error system is
= f0 (t, s(t)) + b0 (t) + As(t − τ (t)).
(5)
Our work is to synchronize system (4) to system (5) in a finite time via an appropriate controller.
= fi (t, xi (t)) − f0 (t, s(t)) + bi (t) − b0 (t) m X (6) +c hij Υej (t) + Aei (t − τ (t)) + ui (t), j=1
for i = 1, 2, . . . , m. Definition 3. [22] Define that the TFCHCN 4 is synchronized to TFCHCN 5 in a FET via a designed controller, if there exists a constant t1 > 0 satisfying lim kxi (t) − s(t)k = 0 and kxi (t) − s(t)k ≡ 0,
t→t1
for all t > t1 , i = 1, 2, . . . , m, where t1 is named the settling time. Lemma 1. [23] If f (t) ∈ C 1 ([a, +∞), R), then for any 0 < α < 1, there exists the inequality C α a Dt |f (t)|
α ≤ sign(f (t))C a Dt f (t),
almost everywhere. Lemma 2. [24] Given matrices A, B, positive definite matrix P with appropriate dimensions and a constant ρ > 0, such that AT B + B T A ≤ ρAT P A + ρ−1 B T P −1 B. Lemma 3. [21] Assume that α ≥ 0, ς = dαe, and f ∈ Aς [a, b]. Then α Jaα C a Dx f (x) = f (x) −
ς−1 X Dk f (a)
k=0
k!
(x − a)k .
Lemma 4. Assume that there exists a continuous function V : D → R leading to that the following conditions are tenable: (1) V (t) is positive definite. (2) There exist real numbers µ > 0 such that C α t0 D t V
(t) ≤ −µ, t ∈ [t0 , ∞).
(7)
Then V (t) fulfils that V (t) ≤ V (t0 ) −
µ(t − t0 )α , t0 ≤ t ≤ T (t0 ), Γ(1 + α)
(8)
and V (t) ≡ 0, for t ≥ T (t0 ). Moreover, T (t0 ) is the settlingtime function, T (t0 ) = t0 + (
V (t0 )Γ(1 + α) 1 )α , µ
(9)
and T (t0 ) is continuous. Proof. There exists a nonnegative function M (t) such that C α t0 D t V
(t) + M (t) = −µ.
Taking the αth order integral on both sides from t0 to t α α α Jtα0 C t0 Dt V (t) + Jt0 (M (t)) = Jt0 (−µ).
3
Applying lemma 3, we get Z
t
1 V (t) = V (t0 ) − (t − s)α−1 M (s)ds Γ(α) t0 Z t 1 − µ(t − s)α−1 ds. Γ(α) t0
As we know M (t) ≥ 0, therefore, Z t 1 (t − s)α−1 M (s)ds ≥ 0. Γ(α) t0
And compute the following integral, Z t 1 µ(t − t0 )α µ(t − s)α−1 ds = . Γ(α) t0 Γ(1 + α)
Theorem 1. Assume that the above Assumption 1, Assumption 2 and Assumption 3 are tenable. If the following inequalities hold, m X Li + cε hji − σi ≤ 0, (11) j=1
and 1 ω(t) + 2bmax + A¯sum A¯T − θ(t) ≤ 0. 2
Then synchronization of TFCHCNs (4) and TFCHCNs (5) via the discontinuous feedback controller ui (t) (10) will be realized in FET T (t0 ), and the settling time is
Then,
T (t0 ) = t0 + ( µ(t − t0 )α V (t) ≤ V (t0 ) − . Γ(1 + α)
V (t) =
V (t0 )Γ(1 + α) 1 )α . t ≤ T (t0 ) = t0 + ( µ
C α t0 Dt V
α (t) = C t0 Dt
= ≤
|fi (t, y) − fi (t, x)| < Li |y − x|,
for every x, y ∈ Rn .
=
Assumption 3. For external interference bi (t) there exists a positive constant bmax leading to that
C α t0 Dt V
(t) ≤
ei (t) = [|ei1 (t)|, |ei2 (t)|, . . . , |ein (t)|]T .
m X
m X
|ei (t)| i=1 m X n X C α t0 Dt |eij (t)| i=1 j=1 m X n X α sign(eij (t))C t0 Dt eij (t) i=1 j=1 m X α signT (ei (t))C t0 Dt ei (t). i=1
signT (ei (t))(fi (t, xi (t)) − f0 (t, s(t)))
i=1 m X
+
signT (ei (t))(bi (t) − b0 (t))
i=1 m X
III. M AIN RESULTS
sign(ei (t)) = [sign(ei1 (t)), sign(ei2 (t)), . . . , sign(ein (t))]T ,
(14)
|ei (t)|,
Then,
||bi (t)|| ≤ bmax , f or i = 0, 1, 2, . . . , m. In this section, we explore FET synchronization of FO heterogeneous complex network (4) and network (5). For each network node, we design the discontinuous feedback controller as follows: ui (t) = −σi ei (t) − θ(t)sign(ei (t)) − µi sign(ei (t)) (10) 1 − sign(ei (t)) eTi (t − τ (t))ei (t − τ (t)) , 2 where σi are tunable positive constants, K is a gain matrix, θ(t) is undetermined function and θ(t) > 0, µ is an given positive constant, 0 < β < 1. Moreover,
(13)
and take the α-order derivative of V (t) in virtue of Lemma 1,
Assumption 1. For all nodes i = 1, 2, . . . , m, the vector function fi (t, x), there exists a positive matrix L = diag(l1 , l2 , . . . , ln ) leading to that
||fi (t, x) − f0 (t, x)|| ≤ ω(t), f or i = 1, 2, . . . , m.
m X i=1
If f (t, x) is locally bounded and is Lipschitz in x, then the existence and uniqueness of solution to the FO network can be ensured [25].
Assumption 2. There exists a bounded function ω(t) > 0 leading to that
V (t0 )Γ(1 + α) 1 )α . µm
Proof. Construct a Lyapunov functional
It is known that V (t) ≥ 0 , so we obtain that
Moreover, it is easily to get that limt→T0 V (t) = 0. And when t ≥ T (t0 ), V (t) ≡ 0.
(12)
+c +
i=1 m X i=1
+
m X
signT (ei (t))
m X
hij Υej (t)
j=1
signT (ei (t))Aei (t − τ (t)) signT (ei (t))ui (t).
i=1
Denote W1 (t) =
m X
signT (ei (t))(fi (t, xi (t)) − f0 (t, s(t)))
i=1 m X
+
i=1
signT (ei (t))(bi (t) − b0 (t)).
4
Using Assumptions 1, 2 and 3, we gain W1 (t) ≤
m X i=1
|fi (t, xi (t)) − fi (t, s(t)) + fi (t, s(t)) m X
− f0 (t, s(t))| + .≤
m X i=1
i=1
On the basis of the above work, we can drive that m m m X X X C α Li |ei (t)| + ω(t) + 2 bmax t0 Dt V (t) ≤ i=1
|bi (t) − b0 (t)|
|fi (t, xi (t)) − fi (t, s(t))|
−
i=1
≤
i=1
Li |ei (t)| +
m X
ω(t) + 2
i=1
m X
−
bmax .
i=1
=
e = [ε1 , ε2 , . . . , εn ] and a constant Denoting a row vector Υ ε = max{εi , i = 1, 2, . . . , n}, we obtain that W2 (t) = c
m X
signT (ei (t))
=c ≤c
i=1 j=1 m X m X
According to
≤ ≤
i=1 m X
1 2
+
1 2
=
m X
i=1 m X i=1
1 ω(t) + 2bmax + A¯sum A¯T − θ(t) 2
m X
µi .
i=1
A¯sum A¯Tsum |eTi (t − τ (t))||ei (t − τ (t))|.
signT (ei (t)){−σi ei (t) − θ(t)sign(ei (t))
1 − sign(ei (t))(eTi (t − τ (t))ei (t − τ (t)))} 2 m m m X X X =− σi |ei (t)| − θ(t) − µi i=1
Finally, by Lemma 4, V (t) ≤ V (t0 ) −
signT (ei (t))ui (t)
i=1 m
µi
i=1
j=1
m X
C α t0 Dt V
Substituting the controller into the derivative equation, we have W4 (t) =
m X
hji ) − σi ≤ 0,
1 ω(t) + 2bmax + A¯sum A¯T − θ(t) ≤ 0. 2 Denoting µ = min{µi , i = 1, 2, . . . , m}, we have
A¯sum |ei (t − τ (t))|
i=1
i=1
θ(t) −
m X Li + εi ( hij ) − σi |ei (t)|
and
hji )|ei (t)|.
sign (ei (t))Aei (t − τ (t))
i=1 m X
i=1
m X
|eTi (t − τ (t))ei (t − τ (t))|
j=1
T
i=1 m X
1 2
Li + cε
With the help of Lemma 2 W3 (t) =
i=1 m X
m X
−
i=1 j=1
m X
σi |ei (t)| −
i=1
hij ε|ej (t)|
(
m X
+
hij Υej (t)
e j (t) hij Υe
i=1 j=1 m X m X
= cε
m X
1X T |e (t − τ (t))||ei (t − τ (t))| 2 i=1 i
i=1 m X
j=1
i=1
m X m X
i=1
m
+
m X + |fi (t, s(t)) − f0 (t, s(t))| + (|bi (t)| + |b0 (t)|) m X
i=1
m m X m X 1X ¯ Asum A¯Tsum + cε ( hji )|ei (t)| + 2 i=1 i=1 j=1
i=1
1X T − |e (t − τ (t))ei (t − τ (t))|. 2 i=1 i
(t) ≤ −µm.
µ(t − t0 )α , t0 ≤ t ≤ T (t0 ), Γ(1 + α)
and V (t) ≡ 0, for t ≥ T (t0 ). The upper bound of settling time function is V (t0 )Γ(1 + α) 1 T (t0 ) = t0 + ( )α . µm Based on the above discussions and the continuity of V (t) about x(t), we can conclude that limt→T (t(0)) ei (t) = 0 and ei (t) ≡ 0 when t ≥ T (t(0)). This means the error system achieves synchronization in FET. Remark 1. In Theorem 1, a discontinuous controller (10) is designed for heterogeneous complex networks (4) to get the FET synchronization. The controller consists of four parts: −σi ei (t) is used to overcome quasi-linear growth part of linear term; − 12 sign(ei (t)) eTi (t − τ (t))ei (t − τ (t)) is for eliminating the effects of time-varying delay; −θ(t)sign(ei (t)) is used to eliminate the influence of heterogeneity difference on the synchronization of the network, which is caused by the nodes non-linear function and uncertain interference; −µi sign(ei (t)) is served to guarantee synchronization in FET, which enhances the practicability of this Theorem.
5
Remark 2. In most papers, the coupled configuration matrix H is requested to be symmetric or irreducible. The coupling matrix in this paper is no longer required to be symmetric or irreducible, which increases the applicability of the conclusion. If the nonlinear functions in systems satisfies that f0 = f1 = f2 = . . . , = fm = f . Denote L = Li , for i = 1, 2, . . . , m. Then the heterogeneous complex network is C α t0 Dt xi (t)
= f (t, xi (t)) + bi (t) + c
m X
hij Υxj (t)
j=1
(15)
+ Axi (t − τ (t)) + ui (t),
for i = 1, 2, . . . , m. The master system is C α t0 Dt s(t)
= f (t, s(t)) + b0 (t) + As(t − τ (t)).
(16)
The error equation of controlled heterogeneous complex networks can be written in the following form, for i = 1, 2, . . . , m, C α t0 Dt ei (t)
= f (t, xi (t)) − f (t, s(t)) + bi (t) − b0 (t) m X hij Υej (t) + Aei (t − τ (t)) + ui (t), +c
The master system is C α t0 Dt s(t)
Construct the discontinuous feedback controller as the above ui (t) = −σi ei (t) − θ(t)sign(ei (t)) − µi sign(ei (t)) (18) 1 − sign(ei (t)) eTi (t − τ (t))ei (t − τ (t)) , 2 where the coefficients are as described above. Then we present a corollary on the FET synchronization of above error network (17) according to Theorem 1. Corollary 1. Assume that the above Assumption 1 and Assumption 3 are tenable. If the following inequalities hold, L + cε
m X j=1
hji − σi ≤ 0,
(19)
C α t0 Dt ei (t)
= f (t, xi (t)) − f (t, s(t)) + c + Aei (t − τ (t)) + ui (t),
1 2bmax + A¯sum A¯T − θ(t) ≤ 0. (20) 2 Then synchronization of network (15) and network (16) under the discontinuous feedback controller ui (t) (18) will reached in FET T (t0 ), the settling time T (t0 ) = t0 + (
V (t0 )Γ(1 + α) 1 )α . µm
(21)
If the nonlinear functions in systems satisfies that f0 = f1 = f2 = . . . , = fm = f , and there is no external interference in networks, that is bi (t) = 0, for i = 0, 1, . . . , m. Then the heterogeneous complex network is C α t0 Dt xi (t)
= f (t, xi (t)) + c
m X j=1
hij Υxj (t) + Axi (t − τ (t))
+ ui (t), f or i = 1, 2, . . . , m.
(22)
m X
hij Υej (t)
j=1
(24)
Construct the discontinuous feedback controller ui (t) = −σi ei (t) − θ(t)sign(ei (t)) − µi sign(ei (t)) (25) 1 − sign(ei (t)) eTi (t − τ (t))ei (t − τ (t)) , 2 where the coefficients are the same as the above mentioned. Then we present a corollary on the FET synchronization of the above error system (24) according to Theorem 1. Corollary 2. Assume that the above Assumption 1 is tenable. If the following inequalities hold, L + cε
m X j=1
and
(26)
hji − σi ≤ 0,
1¯ Asum A¯T − θ(t) ≤ 0. (27) 2 Then synchronization of network (22) and network (23) under the discontinuous feedback controller ui (t) (25) could be achieved in FET T (t0 ), and the settling time is T (t0 ) = t0 + (
V (t0 )Γ(1 + α) 1 )α . µm
(28)
If the nonlinear functions in systems satisfies that f0 = f1 = f2 = . . . , = fm = f , and there is no external interference in networks. Moreover, there is no time-varying delay. Then the heterogeneous complex network is C α t0 Dt xi (t)
and
(23)
The error equation of controlled heterogeneous complex networks can be written in the following form, for i = 1, 2, . . . , m,
j=1
(17)
= f (t, s(t)) + As(t − τ (t)).
= f (t, xi (t)) + c
m X
hij Υxj (t) + ui (t),
(29)
j=1
for i = 1, 2, . . . , m. The master system is C α t0 Dt s(t)
(30)
= f (t, s(t)).
The error equation of controlled heterogeneous complex networks can be written in the following form, for i = 1, 2, . . . , m, C α t0 Dt ei (t)
= f (t, xi (t)) − f (t, s(t)) + c
m X
hij Υej (t) + ui (t).
j=1
(31)
Construct the discontinuous feedback controller ui (t) = −σi ei (t) − µi sign(ei (t)),
(32)
where the coefficients are mentioned above. Then we present a corollary on the FET synchronization of above error system (31) with the help of Theorem 1.
6
Corollary 3. Suppose that the above Assumption 1 is tenable. If the following inequality holds, (33)
e13 e14 e15
−0.5
Then, synchronization of network (29) and network (30) under the discontinuous feedback controller ui (t) (32) will be achieved in FET T (t0 ), and the settling time is T (t0 ) = t0 + (
e12
0
V (t0 )Γ(1 + α) 1 )α . µm
1
j=1
hji − σi ≤ 0.
e11
e
L + cε
m X
α=0.85 0.5
−1
−1.5
(34) −2
IV. N UMERICAL EXAMPLES −2.5 −0.5
Example 1. Consider a TFCHCN consisting of five nodes with external interference. The subsystem equation of the ith node is m X C α hij Υxj (t) t0 Dt xi (t) = fi (t, xi (t)) + bi (t) + c
The nonlinear function is fi (t, xi (t)) = [0.6xi2 (t), − 0.6xi2 (t) − 0.02isin(xi1 (t))]T , and interference bi (t) = [0.1sin(t), 0.1sin(t)]T . Denote time delay τ = 0.2, coupling strength c = 2, matrix A = [− 41 , 0; 0, 34 ]. The master system is C α t0 Dt s(t) = f (t, s(t)) + b0 (t) + As(t − τ (t)). The initial value is s(0) = [1, 1]T . The nonlinear function is f0 (t, s(t)) = [0.6s2 (t), −0.6s2 (t) − 0.05sin(s1 (t))]T , b0 (t) = [0, 0]T . Then ω(t) = 0.1, L = [1, 0; 0, 1]. The controller is designed as equation (10), σ = [1, 1, 1, 1, 2]0 , θ(t) = 0.62, µ = 1. Figure 1 shows that the error system is synchronized in FET T = 0.78. In the light of Theorem 1, the FET upper bound could be computed, 13.0425 × Γ(1.85) 1 ) 0.85 = 2.8927. 1×5 The system achieves synchronization within the theoretical value. This stimulation example indicates that the effectiveness T (t0 ) = (
2
2.5
3
e
21
0.4
e
0.2
e23
0
e25
22
e24
−0.2 2
The inner-coupling matrix and coupled configuration matrix of time-varying delay complex networks coupled by 5 heterogeneous nodes is respectively, 1 0 Υ= , 0 1 −1 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 H= . 0 0 1 −1 0 0 0 0 0 0
1.5
α=0.85
e
0.9 −0.3 0 . 1.5 −0.2
1
0.6
j=1
0.5
t
+ Axi (t − τ (t)) + ui (t), i = 1, 2, . . . , m. (35)
The initial value of the state is 0.6 1.2 X(0) = −0.9 −1.2 1.05
0
−0.4 −0.6 −0.8 −1 −1.2 −1.4 −0.5
0
0.5
1
1.5
2
2.5
3
t
Fig. 1. state response of model (35) with control
of Theorem 1 and the conservativeness is relatively low. This property is urgently needed in reality for the predicted upper bound of FET is close to what is presented in numerical simulation. Example 2. This numerical example shows the practicability of corollary 3. Discussing a complex network comprising of five nodes with linear coupling. The equation of the node of ith subsystem is C α t0 Dt xi (t)
= f (t, xi (t)) + c
m X
hij Υxj (t), i = 1, 2, . . . m.
j=1
(36)
The master system is C α t0 Dt s(t)
= f (t, s(t)).
Based on example 1, we let nonlinear function is f (t, xi (t)) = [0.6xi2 (t), − 0.8xi2 (t) − 0.1sin(xi1 (t))]T . Then L = [1, 0; 0, 1]. The remaining parameters remain unchanged. The controller is designed as (18), σ = [2, 2, 2, 2, 2]0 , µ = 1. Fig. 2 reveals that the error system is synchronized in FET T = 0.83. In the light of Theorem 1, the FET upper bound can be computed, T (t0 ) = (
13.0425 × Γ(1.85) 1 ) 0.85 = 2.8927. 1×5
7
α=0.85 0.5 e11 e12
0
e13 e14 e15
e1
−0.5
−1
−1.5
−2
−2.5
0
0.5
1
1.5 t
2
2.5
3
α=0.85 0.6 e
21
0.4
e
0.2
e23
0
e25
22
e24
e2
−0.2 −0.4 −0.6 −0.8 −1 −1.2 −1.4
0
0.5
1
1.5 t
2
2.5
3
Fig. 2. state response of model (36) with control
The system achieves synchronization within the theoretical value. This stimulation example indicates that the effectiveness of corollary 1 and the conservativeness is relatively weakened. V. C ONCLUSION In this paper, we have researched the FET synchronization problem of TFCHCNs with external interference. Our FO complex network model considers time-varying delay and heterogeneity. We propose a novel Lemma which is useful for probing the FET stability and synchronization problem of FO complex networks. We also construct a novel discontinuous feedback controller to synchronize the complex network onto the proposed system in FET. At last, the setting time is obtained. Two numerical examples are presented to show the effectiveness of our findings. Our next work will focus on the FET synchronization problem of TFCHCNs with impulses [26–29]. R EFERENCES [1] A. Lancichinetti, S. Fortunato, and J. Kertsz, “Detecting the overlapping and hierarchical community structure in complex networks,” New Journal of Physics, vol. 11, no. 3, pp. 19-44, 2009.
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in 1983, 1986 and 1991, respectively. He is presently a full professor and the deputy dean of Academy of Science and Technology, Harbin Institute of Technology. His research interests include intelligent control and intelligent system, inertial technology, robotics, and precision servo system.
Ying Li received her B.S. degree from Henan Normal University, Xinxiang, China, in 2017. And she received the master’s degree from Harbin Institute of Technology (Weihai), Weihai, China, in 2019. Her present research interests include control theory of fractional-order complex networks.
Yonggui Kao received the B.E. degree from Beijing Jiaotong University in 1996. He received M. E. and Ph. D. degrees from Ocean University of China in 2005 and 2008, respectively. He now is a Professor at Department of Mathematics, Harbin Institute of Technology(Weihai). His research interest covers stochastic systems, impulsive systems, neural networks, stability theory and sliding mode control.
Changhong Wang received B.E., M. E., and Ph. D. degrees from Harbin Institute of Technology, Harbin, China
Hongwei Xia received B.E., M.E., and Ph.D. degrees from Harbin Institute of Technology, Harbin, China in 2002, 2004 and 2008, respectively. He is presently a full professor, Harbin Institute of Technology. His research interests include robust control, intelligent control and intelligent system.
Conflict of Interest Form We wish to draw the attention of the Editor to the following facts which may be considered as potential conflicts of interest and to significant financial contributions to this work. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He/she is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author and which has been configured to accept email from [email protected]. Signed by all authors as follows:
Ying Li, Email: [email protected] Yonggui Kao, Email:[email protected] Changhong Wang, Email: [email protected] Hongwei Xia, Email: [email protected]
Communicated by Dr Yan-Wu Wang
Journal Pre-proof
Finite-time synchronization of delayed fractional-order heterogeneous complex networks Ying Li, Yonggui Kao, Changhong Wang, Hongwei Xia PII: DOI: Reference:
S0925-2312(19)31645-5 https://doi.org/10.1016/j.neucom.2019.11.043 NEUCOM 21571
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
18 July 2019 24 October 2019 11 November 2019
Please cite this article as: Ying Li, Yonggui Kao, Changhong Wang, Hongwei Xia, Finite-time synchronization of delayed fractional-order heterogeneous complex networks, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.11.043
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1
Finite-time synchronization of delayed fractional-order heterogeneous complex networks Ying Li, Yonggui Kao, Changhong Wang and Hongwei Xia
Abstract—This paper is devoted to exploring the finitetime (FET) synchronization problem of time-varying delay fractional-order (FO) coupled heterogeneous complex networks (TFCHCNs) with external interference via a discontinuous feedback controller. Firstly, we propose a novel Lemma which is useful for discussing the FET stability and synchronization problem of FO systems. Secondly, based on the proposed Lemma, a discontinuous feedback controller is designed to guarantee the FET synchronization of TFCHCNs with external interference. Moreover, the upper bound of settling-time function is obtained. Finally, two simulation examples are provided to verify the practicability of our findings. Index Terms—Fractional-order systems, Heterogeneous complex networks, Finite-time synchronization, Discontinuous feedback controller.
I. I NTRODUCTION Complex networks have attracted more and more researchers from various fields, such as physics[1], mathematics, biomedical engineering, social management and so on [2]. Network synchronization is a significant dynamic character of complex networks owing to its extensive applications in secure communication, electronic circuits, biological systems and mechanism [3]. The optimal tracking problem for certain control system over a communication channel with packet dropouts has been researched in [4, 5]. In order to achieve network synchronization, lots of control methods have been proposed, such as feedback control [6], adaptive control [7], passive control [8], generalized projective control [9], sliding mode control [10], impulsive control [11], pinning control [12]. For heterogeneous complex networks, many classical theoretical results need to be extended. It is difficult for heterogeneous complex networks with different characteristics to find mixed Lyapunov functions with different properties, such as continuous and discrete, smooth and impulsive. However, most real networks are highly heterogeneous, and the roles of different nodes and different links in the evolution of networks are often very different. Therefore heterogeneous complex networks are more complicated and applicable than complex networks with the same nodes, there are few results on finite-time (FET) synchronization of heterogeneous complex networks. In [13], based on the eigenvalue-describing Yonggui Kao and Ying Li are with the Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, China. (e-mail: [email protected]). Changhong Wang and Hongwei Xia are with the Space Control and Inertial Technology Research Center, Harbin Institute of Technolog, Harbin, 150001, P.R. China. Supported by the National Natural Science Foundation of China No. 61873071 and Shandong Natural Science Foundation No. ZR2019MF006. Corresponding author: Yonggui Kao
coupling configuration matrix, Wang et al. discussed bounded synchronization of heterogeneous complex dynamic networks. Unlike integer-order (IO) systems, fractional-order (FO) systems have an advantage that they have memory, which is useful in depicting the memory and hereditary characters of numerous processes and systems [14, 15]. From the perspective of synchronization time, synchronization can be divided into asymptotic synchronization and FET synchronization. In practice, sometimes it is more important to study FET synchronization. For example, in confidential communication, if there is the longer time spent on synchronization, there is more possibility that the message will be taken over. FET synchronization owns meliorated properties of robustness and disturbance rejection than asymptotical synchronization. For FET synchronization of IO complex networks, in [16], Yang et al. researched FET synchronization of proposed IO complex networks with Lyapunov direct method. Recently, for FET synchronization of FO complex networks, in [17], A. Pratap et al. researched FET synchronization of timevarying delayed FO memristive neural networks using some common inequalities. In [18], Yang et al. discussed FET stability problem of FO neural networks with delay in virtue of Gronwall-Bellman inequality. Similarly, in [19], Li et al. probed FET synchronization of FO complex networks by some inequalities. In [20], rather than using Lyapunov direct method, Zheng et al. discussed the FET synchronization problems of memristor-based fractional-order fuzzy cellular neural networks in the light of Gronwall-Bellman inequality. By contrast, we obtain a new lemma similar with Lyapunov method that is more applicable and easier to generalize, which is also the innovation of this paper. Compared with the previous method, the method in this paper reduces the amount of computation and provides a new way to solve the FET synchronization problem. Base on the aforementioned concerns, we propose a novel and useful Lemma 4 which is applied to make the coupled heterogeneous complex networks (TFCHCNs) with external interference achieve FET synchronization. In Section 2, some definitions about FO equations, Lemmas and the Assumptions are presented. In Section 3, main results of FET synchronization are gained by using the novel lemma and a discontinuous feedback controller designed for the system. Two examples are presented to validate the effectiveness of our findings in Section 4. Finally, the conclusion is given in Section 5. Notations: In this paper, matrix A > B implies A − B is a positive definite matrix, where A and B are symmetric matrices. Let k · k be Euclid norm, Rη be the η-dimensional Euclidean space with norm k · k, R+ = [0, ∞). ς = dυe is an
2
integer which satisfies that ς −1 < υ ≤ ς. For a n-dimensional Pn column vector x = [x1 , x2 , . . . , xn ], |x| represents i=1 |xi |. II. P RELIMINARIES
C α t0 Dt ei (t)
Definition 1. [21] Let α ∈ R+ .The operator Jtα0 , defined on L1 [t0 , T ] by Z t 1 Jtα0 f (t) = (t − x)α−1 f (x)dx, (1) Γ(α) t0 for t0 ≤ t ≤ T , is α-order Riemann-Liouville fractional integral. Definition 2. [21] Let α ≥ 0 and m = dαe. Moreover assume that function f ∈ C n ([t0 , +∞), R) and n is a positive integer. α Operator C t0 Dt is defined by C α t0 Dt f (t)
=
1 Γ(m − α)
Z
t
t0
f (m) (s) ds, (t − s)α−m+1
(2)
α where Γ(·) is the Gamma function. The operator C t0 Dt is called Caputo fractional differential operator of order α. Especially, when 0 < α < 1, Z t (1) 1 f (s) C α (3) D f (t) = ds. t0 t Γ(1 − α) t0 (t − s)α
Consider FO time-varying delay complex networks coupled by m heterogeneous nodes, where every node is an ndimensional network. The slave system can be depicted as C α t0 Dt xi (t)
= fi (t, xi (t)) + bi (t) + c
m X
hij Υxj (t)
j=1
(4)
+ Axi (t − τ (t)) + ui (t), i = 1, 2, . . . , m. Where t0 ≥ 0 denotes the starting time, 0 < α < 1, xi (t) ∈ Rn denotes the state variable of the ith node, fi (t, x(t)) ∈ Rn denotes ith nonlinear function, bi (t) ∈ Rn is bounded and it represents external interference, τ (t) is time-varying delay. c is a positive constant and it represents the coupling strength of heterogeneous complex networks. Υ=diag(ε1 , ε2 , . . . , εn ) > 0 represents the inner-coupling matrix of heterogeneous networks. A = (aij )n×n is a constant P matric. Denote A¯ = n (|aij |)n×n , and row vector A¯sum = ( j=1 |aij |)1×n . H = (hij )m×m denotes the constant coupled configuration matrix and represents the topological structure of TFCHCNs, where hij is that : hij > 0(i 6= j)), if there exsits a direct link f rom node i to node j, h = 0(i = 6 j)), otherwise, ij Pm hii = − j=1,j6=i hij , i = 1, 2, . . . , m. With s(t) ∈ Rn , the master system is C α t0 Dt s(t)
The synchronization error is defined by ei (t) = xi (t)−s(t). Based on the equation (4) and equation (5), the synchronization error system is
= f0 (t, s(t)) + b0 (t) + As(t − τ (t)).
(5)
Our work is to synchronize system (4) to system (5) in a finite time via an appropriate controller.
= fi (t, xi (t)) − f0 (t, s(t)) + bi (t) − b0 (t) m X (6) +c hij Υej (t) + Aei (t − τ (t)) + ui (t), j=1
for i = 1, 2, . . . , m. Definition 3. [22] Define that the TFCHCN 4 is synchronized to TFCHCN 5 in a FET via a designed controller, if there exists a constant t1 > 0 satisfying lim kxi (t) − s(t)k = 0 and kxi (t) − s(t)k ≡ 0,
t→t1
for all t > t1 , i = 1, 2, . . . , m, where t1 is named the settling time. Lemma 1. [23] If f (t) ∈ C 1 ([a, +∞), R), then for any 0 < α < 1, there exists the inequality C α a Dt |f (t)|
α ≤ sign(f (t))C a Dt f (t),
almost everywhere. Lemma 2. [24] Given matrices A, B, positive definite matrix P with appropriate dimensions and a constant ρ > 0, such that AT B + B T A ≤ ρAT P A + ρ−1 B T P −1 B. Lemma 3. [21] Assume that α ≥ 0, ς = dαe, and f ∈ Aς [a, b]. Then α Jaα C a Dx f (x) = f (x) −
ς−1 X Dk f (a)
k=0
k!
(x − a)k .
Lemma 4. Assume that there exists a continuous function V : D → R leading to that the following conditions are tenable: (1) V (t) is positive definite. (2) There exist real numbers µ > 0 such that C α t0 D t V
(t) ≤ −µ, t ∈ [t0 , ∞).
(7)
Then V (t) fulfils that V (t) ≤ V (t0 ) −
µ(t − t0 )α , t0 ≤ t ≤ T (t0 ), Γ(1 + α)
(8)
and V (t) ≡ 0, for t ≥ T (t0 ). Moreover, T (t0 ) is the settlingtime function, T (t0 ) = t0 + (
V (t0 )Γ(1 + α) 1 )α , µ
(9)
and T (t0 ) is continuous. Proof. There exists a nonnegative function M (t) such that C α t0 D t V
(t) + M (t) = −µ.
Taking the αth order integral on both sides from t0 to t α α α Jtα0 C t0 Dt V (t) + Jt0 (M (t)) = Jt0 (−µ).
3
Applying lemma 3, we get Z
t
1 V (t) = V (t0 ) − (t − s)α−1 M (s)ds Γ(α) t0 Z t 1 − µ(t − s)α−1 ds. Γ(α) t0
As we know M (t) ≥ 0, therefore, Z t 1 (t − s)α−1 M (s)ds ≥ 0. Γ(α) t0
And compute the following integral, Z t 1 µ(t − t0 )α µ(t − s)α−1 ds = . Γ(α) t0 Γ(1 + α)
Theorem 1. Assume that the above Assumption 1, Assumption 2 and Assumption 3 are tenable. If the following inequalities hold, m X Li + cε hji − σi ≤ 0, (11) j=1
and 1 ω(t) + 2bmax + A¯sum A¯T − θ(t) ≤ 0. 2
Then synchronization of TFCHCNs (4) and TFCHCNs (5) via the discontinuous feedback controller ui (t) (10) will be realized in FET T (t0 ), and the settling time is
Then,
T (t0 ) = t0 + ( µ(t − t0 )α V (t) ≤ V (t0 ) − . Γ(1 + α)
V (t) =
V (t0 )Γ(1 + α) 1 )α . t ≤ T (t0 ) = t0 + ( µ
C α t0 Dt V
α (t) = C t0 Dt
= ≤
|fi (t, y) − fi (t, x)| < Li |y − x|,
for every x, y ∈ Rn .
=
Assumption 3. For external interference bi (t) there exists a positive constant bmax leading to that
C α t0 Dt V
(t) ≤
ei (t) = [|ei1 (t)|, |ei2 (t)|, . . . , |ein (t)|]T .
m X
m X
|ei (t)| i=1 m X n X C α t0 Dt |eij (t)| i=1 j=1 m X n X α sign(eij (t))C t0 Dt eij (t) i=1 j=1 m X α signT (ei (t))C t0 Dt ei (t). i=1
signT (ei (t))(fi (t, xi (t)) − f0 (t, s(t)))
i=1 m X
+
signT (ei (t))(bi (t) − b0 (t))
i=1 m X
III. M AIN RESULTS
sign(ei (t)) = [sign(ei1 (t)), sign(ei2 (t)), . . . , sign(ein (t))]T ,
(14)
|ei (t)|,
Then,
||bi (t)|| ≤ bmax , f or i = 0, 1, 2, . . . , m. In this section, we explore FET synchronization of FO heterogeneous complex network (4) and network (5). For each network node, we design the discontinuous feedback controller as follows: ui (t) = −σi ei (t) − θ(t)sign(ei (t)) − µi sign(ei (t)) (10) 1 − sign(ei (t)) eTi (t − τ (t))ei (t − τ (t)) , 2 where σi are tunable positive constants, K is a gain matrix, θ(t) is undetermined function and θ(t) > 0, µ is an given positive constant, 0 < β < 1. Moreover,
(13)
and take the α-order derivative of V (t) in virtue of Lemma 1,
Assumption 1. For all nodes i = 1, 2, . . . , m, the vector function fi (t, x), there exists a positive matrix L = diag(l1 , l2 , . . . , ln ) leading to that
||fi (t, x) − f0 (t, x)|| ≤ ω(t), f or i = 1, 2, . . . , m.
m X i=1
If f (t, x) is locally bounded and is Lipschitz in x, then the existence and uniqueness of solution to the FO network can be ensured [25].
Assumption 2. There exists a bounded function ω(t) > 0 leading to that
V (t0 )Γ(1 + α) 1 )α . µm
Proof. Construct a Lyapunov functional
It is known that V (t) ≥ 0 , so we obtain that
Moreover, it is easily to get that limt→T0 V (t) = 0. And when t ≥ T (t0 ), V (t) ≡ 0.
(12)
+c +
i=1 m X i=1
+
m X
signT (ei (t))
m X
hij Υej (t)
j=1
signT (ei (t))Aei (t − τ (t)) signT (ei (t))ui (t).
i=1
Denote W1 (t) =
m X
signT (ei (t))(fi (t, xi (t)) − f0 (t, s(t)))
i=1 m X
+
i=1
signT (ei (t))(bi (t) − b0 (t)).
4
Using Assumptions 1, 2 and 3, we gain W1 (t) ≤
m X i=1
|fi (t, xi (t)) − fi (t, s(t)) + fi (t, s(t)) m X
− f0 (t, s(t))| + .≤
m X i=1
i=1
On the basis of the above work, we can drive that m m m X X X C α Li |ei (t)| + ω(t) + 2 bmax t0 Dt V (t) ≤ i=1
|bi (t) − b0 (t)|
|fi (t, xi (t)) − fi (t, s(t))|
−
i=1
≤
i=1
Li |ei (t)| +
m X
ω(t) + 2
i=1
m X
−
bmax .
i=1
=
e = [ε1 , ε2 , . . . , εn ] and a constant Denoting a row vector Υ ε = max{εi , i = 1, 2, . . . , n}, we obtain that W2 (t) = c
m X
signT (ei (t))
=c ≤c
i=1 j=1 m X m X
According to
≤ ≤
i=1 m X
1 2
+
1 2
=
m X
i=1 m X i=1
1 ω(t) + 2bmax + A¯sum A¯T − θ(t) 2
m X
µi .
i=1
A¯sum A¯Tsum |eTi (t − τ (t))||ei (t − τ (t))|.
signT (ei (t)){−σi ei (t) − θ(t)sign(ei (t))
1 − sign(ei (t))(eTi (t − τ (t))ei (t − τ (t)))} 2 m m m X X X =− σi |ei (t)| − θ(t) − µi i=1
Finally, by Lemma 4, V (t) ≤ V (t0 ) −
signT (ei (t))ui (t)
i=1 m
µi
i=1
j=1
m X
C α t0 Dt V
Substituting the controller into the derivative equation, we have W4 (t) =
m X
hji ) − σi ≤ 0,
1 ω(t) + 2bmax + A¯sum A¯T − θ(t) ≤ 0. 2 Denoting µ = min{µi , i = 1, 2, . . . , m}, we have
A¯sum |ei (t − τ (t))|
i=1
i=1
θ(t) −
m X Li + εi ( hij ) − σi |ei (t)|
and
hji )|ei (t)|.
sign (ei (t))Aei (t − τ (t))
i=1 m X
i=1
m X
|eTi (t − τ (t))ei (t − τ (t))|
j=1
T
i=1 m X
1 2
Li + cε
With the help of Lemma 2 W3 (t) =
i=1 m X
m X
−
i=1 j=1
m X
σi |ei (t)| −
i=1
hij ε|ej (t)|
(
m X
+
hij Υej (t)
e j (t) hij Υe
i=1 j=1 m X m X
= cε
m X
1X T |e (t − τ (t))||ei (t − τ (t))| 2 i=1 i
i=1 m X
j=1
i=1
m X m X
i=1
m
+
m X + |fi (t, s(t)) − f0 (t, s(t))| + (|bi (t)| + |b0 (t)|) m X
i=1
m m X m X 1X ¯ Asum A¯Tsum + cε ( hji )|ei (t)| + 2 i=1 i=1 j=1
i=1
1X T − |e (t − τ (t))ei (t − τ (t))|. 2 i=1 i
(t) ≤ −µm.
µ(t − t0 )α , t0 ≤ t ≤ T (t0 ), Γ(1 + α)
and V (t) ≡ 0, for t ≥ T (t0 ). The upper bound of settling time function is V (t0 )Γ(1 + α) 1 T (t0 ) = t0 + ( )α . µm Based on the above discussions and the continuity of V (t) about x(t), we can conclude that limt→T (t(0)) ei (t) = 0 and ei (t) ≡ 0 when t ≥ T (t(0)). This means the error system achieves synchronization in FET. Remark 1. In Theorem 1, a discontinuous controller (10) is designed for heterogeneous complex networks (4) to get the FET synchronization. The controller consists of four parts: −σi ei (t) is used to overcome quasi-linear growth part of linear term; − 12 sign(ei (t)) eTi (t − τ (t))ei (t − τ (t)) is for eliminating the effects of time-varying delay; −θ(t)sign(ei (t)) is used to eliminate the influence of heterogeneity difference on the synchronization of the network, which is caused by the nodes non-linear function and uncertain interference; −µi sign(ei (t)) is served to guarantee synchronization in FET, which enhances the practicability of this Theorem.
5
Remark 2. In most papers, the coupled configuration matrix H is requested to be symmetric or irreducible. The coupling matrix in this paper is no longer required to be symmetric or irreducible, which increases the applicability of the conclusion. If the nonlinear functions in systems satisfies that f0 = f1 = f2 = . . . , = fm = f . Denote L = Li , for i = 1, 2, . . . , m. Then the heterogeneous complex network is C α t0 Dt xi (t)
= f (t, xi (t)) + bi (t) + c
m X
hij Υxj (t)
j=1
(15)
+ Axi (t − τ (t)) + ui (t),
for i = 1, 2, . . . , m. The master system is C α t0 Dt s(t)
= f (t, s(t)) + b0 (t) + As(t − τ (t)).
(16)
The error equation of controlled heterogeneous complex networks can be written in the following form, for i = 1, 2, . . . , m, C α t0 Dt ei (t)
= f (t, xi (t)) − f (t, s(t)) + bi (t) − b0 (t) m X hij Υej (t) + Aei (t − τ (t)) + ui (t), +c
The master system is C α t0 Dt s(t)
Construct the discontinuous feedback controller as the above ui (t) = −σi ei (t) − θ(t)sign(ei (t)) − µi sign(ei (t)) (18) 1 − sign(ei (t)) eTi (t − τ (t))ei (t − τ (t)) , 2 where the coefficients are as described above. Then we present a corollary on the FET synchronization of above error network (17) according to Theorem 1. Corollary 1. Assume that the above Assumption 1 and Assumption 3 are tenable. If the following inequalities hold, L + cε
m X j=1
hji − σi ≤ 0,
(19)
C α t0 Dt ei (t)
= f (t, xi (t)) − f (t, s(t)) + c + Aei (t − τ (t)) + ui (t),
1 2bmax + A¯sum A¯T − θ(t) ≤ 0. (20) 2 Then synchronization of network (15) and network (16) under the discontinuous feedback controller ui (t) (18) will reached in FET T (t0 ), the settling time T (t0 ) = t0 + (
V (t0 )Γ(1 + α) 1 )α . µm
(21)
If the nonlinear functions in systems satisfies that f0 = f1 = f2 = . . . , = fm = f , and there is no external interference in networks, that is bi (t) = 0, for i = 0, 1, . . . , m. Then the heterogeneous complex network is C α t0 Dt xi (t)
= f (t, xi (t)) + c
m X j=1
hij Υxj (t) + Axi (t − τ (t))
+ ui (t), f or i = 1, 2, . . . , m.
(22)
m X
hij Υej (t)
j=1
(24)
Construct the discontinuous feedback controller ui (t) = −σi ei (t) − θ(t)sign(ei (t)) − µi sign(ei (t)) (25) 1 − sign(ei (t)) eTi (t − τ (t))ei (t − τ (t)) , 2 where the coefficients are the same as the above mentioned. Then we present a corollary on the FET synchronization of the above error system (24) according to Theorem 1. Corollary 2. Assume that the above Assumption 1 is tenable. If the following inequalities hold, L + cε
m X j=1
and
(26)
hji − σi ≤ 0,
1¯ Asum A¯T − θ(t) ≤ 0. (27) 2 Then synchronization of network (22) and network (23) under the discontinuous feedback controller ui (t) (25) could be achieved in FET T (t0 ), and the settling time is T (t0 ) = t0 + (
V (t0 )Γ(1 + α) 1 )α . µm
(28)
If the nonlinear functions in systems satisfies that f0 = f1 = f2 = . . . , = fm = f , and there is no external interference in networks. Moreover, there is no time-varying delay. Then the heterogeneous complex network is C α t0 Dt xi (t)
and
(23)
The error equation of controlled heterogeneous complex networks can be written in the following form, for i = 1, 2, . . . , m,
j=1
(17)
= f (t, s(t)) + As(t − τ (t)).
= f (t, xi (t)) + c
m X
hij Υxj (t) + ui (t),
(29)
j=1
for i = 1, 2, . . . , m. The master system is C α t0 Dt s(t)
(30)
= f (t, s(t)).
The error equation of controlled heterogeneous complex networks can be written in the following form, for i = 1, 2, . . . , m, C α t0 Dt ei (t)
= f (t, xi (t)) − f (t, s(t)) + c
m X
hij Υej (t) + ui (t).
j=1
(31)
Construct the discontinuous feedback controller ui (t) = −σi ei (t) − µi sign(ei (t)),
(32)
where the coefficients are mentioned above. Then we present a corollary on the FET synchronization of above error system (31) with the help of Theorem 1.
6
Corollary 3. Suppose that the above Assumption 1 is tenable. If the following inequality holds, (33)
e13 e14 e15
−0.5
Then, synchronization of network (29) and network (30) under the discontinuous feedback controller ui (t) (32) will be achieved in FET T (t0 ), and the settling time is T (t0 ) = t0 + (
e12
0
V (t0 )Γ(1 + α) 1 )α . µm
1
j=1
hji − σi ≤ 0.
e11
e
L + cε
m X
α=0.85 0.5
−1
−1.5
(34) −2
IV. N UMERICAL EXAMPLES −2.5 −0.5
Example 1. Consider a TFCHCN consisting of five nodes with external interference. The subsystem equation of the ith node is m X C α hij Υxj (t) t0 Dt xi (t) = fi (t, xi (t)) + bi (t) + c
The nonlinear function is fi (t, xi (t)) = [0.6xi2 (t), − 0.6xi2 (t) − 0.02isin(xi1 (t))]T , and interference bi (t) = [0.1sin(t), 0.1sin(t)]T . Denote time delay τ = 0.2, coupling strength c = 2, matrix A = [− 41 , 0; 0, 34 ]. The master system is C α t0 Dt s(t) = f (t, s(t)) + b0 (t) + As(t − τ (t)). The initial value is s(0) = [1, 1]T . The nonlinear function is f0 (t, s(t)) = [0.6s2 (t), −0.6s2 (t) − 0.05sin(s1 (t))]T , b0 (t) = [0, 0]T . Then ω(t) = 0.1, L = [1, 0; 0, 1]. The controller is designed as equation (10), σ = [1, 1, 1, 1, 2]0 , θ(t) = 0.62, µ = 1. Figure 1 shows that the error system is synchronized in FET T = 0.78. In the light of Theorem 1, the FET upper bound could be computed, 13.0425 × Γ(1.85) 1 ) 0.85 = 2.8927. 1×5 The system achieves synchronization within the theoretical value. This stimulation example indicates that the effectiveness T (t0 ) = (
2
2.5
3
e
21
0.4
e
0.2
e23
0
e25
22
e24
−0.2 2
The inner-coupling matrix and coupled configuration matrix of time-varying delay complex networks coupled by 5 heterogeneous nodes is respectively, 1 0 Υ= , 0 1 −1 1 0 0 0 1 −2 1 0 0 0 1 −2 1 0 H= . 0 0 1 −1 0 0 0 0 0 0
1.5
α=0.85
e
0.9 −0.3 0 . 1.5 −0.2
1
0.6
j=1
0.5
t
+ Axi (t − τ (t)) + ui (t), i = 1, 2, . . . , m. (35)
The initial value of the state is 0.6 1.2 X(0) = −0.9 −1.2 1.05
0
−0.4 −0.6 −0.8 −1 −1.2 −1.4 −0.5
0
0.5
1
1.5
2
2.5
3
t
Fig. 1. state response of model (35) with control
of Theorem 1 and the conservativeness is relatively low. This property is urgently needed in reality for the predicted upper bound of FET is close to what is presented in numerical simulation. Example 2. This numerical example shows the practicability of corollary 3. Discussing a complex network comprising of five nodes with linear coupling. The equation of the node of ith subsystem is C α t0 Dt xi (t)
= f (t, xi (t)) + c
m X
hij Υxj (t), i = 1, 2, . . . m.
j=1
(36)
The master system is C α t0 Dt s(t)
= f (t, s(t)).
Based on example 1, we let nonlinear function is f (t, xi (t)) = [0.6xi2 (t), − 0.8xi2 (t) − 0.1sin(xi1 (t))]T . Then L = [1, 0; 0, 1]. The remaining parameters remain unchanged. The controller is designed as (18), σ = [2, 2, 2, 2, 2]0 , µ = 1. Fig. 2 reveals that the error system is synchronized in FET T = 0.83. In the light of Theorem 1, the FET upper bound can be computed, T (t0 ) = (
13.0425 × Γ(1.85) 1 ) 0.85 = 2.8927. 1×5
7
α=0.85 0.5 e11 e12
0
e13 e14 e15
e1
−0.5
−1
−1.5
−2
−2.5
0
0.5
1
1.5 t
2
2.5
3
α=0.85 0.6 e
21
0.4
e
0.2
e23
0
e25
22
e24
e2
−0.2 −0.4 −0.6 −0.8 −1 −1.2 −1.4
0
0.5
1
1.5 t
2
2.5
3
Fig. 2. state response of model (36) with control
The system achieves synchronization within the theoretical value. This stimulation example indicates that the effectiveness of corollary 1 and the conservativeness is relatively weakened. V. C ONCLUSION In this paper, we have researched the FET synchronization problem of TFCHCNs with external interference. Our FO complex network model considers time-varying delay and heterogeneity. We propose a novel Lemma which is useful for probing the FET stability and synchronization problem of FO complex networks. We also construct a novel discontinuous feedback controller to synchronize the complex network onto the proposed system in FET. At last, the setting time is obtained. Two numerical examples are presented to show the effectiveness of our findings. Our next work will focus on the FET synchronization problem of TFCHCNs with impulses [26–29]. R EFERENCES [1] A. Lancichinetti, S. Fortunato, and J. Kertsz, “Detecting the overlapping and hierarchical community structure in complex networks,” New Journal of Physics, vol. 11, no. 3, pp. 19-44, 2009.
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in 1983, 1986 and 1991, respectively. He is presently a full professor and the deputy dean of Academy of Science and Technology, Harbin Institute of Technology. His research interests include intelligent control and intelligent system, inertial technology, robotics, and precision servo system.
Ying Li received her B.S. degree from Henan Normal University, Xinxiang, China, in 2017. And she received the master’s degree from Harbin Institute of Technology (Weihai), Weihai, China, in 2019. Her present research interests include control theory of fractional-order complex networks.
Yonggui Kao received the B.E. degree from Beijing Jiaotong University in 1996. He received M. E. and Ph. D. degrees from Ocean University of China in 2005 and 2008, respectively. He now is a Professor at Department of Mathematics, Harbin Institute of Technology(Weihai). His research interest covers stochastic systems, impulsive systems, neural networks, stability theory and sliding mode control.
Changhong Wang received B.E., M. E., and Ph. D. degrees from Harbin Institute of Technology, Harbin, China
Hongwei Xia received B.E., M.E., and Ph.D. degrees from Harbin Institute of Technology, Harbin, China in 2002, 2004 and 2008, respectively. He is presently a full professor, Harbin Institute of Technology. His research interests include robust control, intelligent control and intelligent system.
Conflict of Interest Form We wish to draw the attention of the Editor to the following facts which may be considered as potential conflicts of interest and to significant financial contributions to this work. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He/she is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author and which has been configured to accept email from [email protected]. Signed by all authors as follows:
Ying Li, Email: [email protected] Yonggui Kao, Email:[email protected] Changhong Wang, Email: [email protected] Hongwei Xia, Email: [email protected]