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Synchronization of fractional-order linear complex networks with directed coupling topology
Accepted Manuscript Synchronization of fractional-order linear complex networks with directed coupling topology Qingxiang Fang, Jigen Peng
PII: DOI: Reference:
S0378-4371(17)30785-9 http://dx.doi.org/10.1016/j.physa.2017.08.050 PHYSA 18496
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Physica A
Received date : 21 February 2017 Revised date : 17 June 2017 Please cite this article as: Q. Fang, J. Peng, Synchronization of fractional-order linear complex networks with directed coupling topology, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.08.050 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
>We investigate the synchronization of fractional-order complex networks with general linear dynamics under directed connected topology. > Real Jordan canonical form of matrix is employed to convert error system to an equivalent simultaneous stability problem of corresponding independent subsystems. >Sufficient conditions in terms of linear matrix inequalities for synchronization are established. >The effects of the derivative order on synchronization is clearly revealed by synchronization conditions. >The conditions adopted in this paper improve some existing results.
*Manuscript Click here to view linked References
Synchronization of fractional-order linear complex networks with directed coupling topology Qingxiang Fanga,b,∗, Jigen Penga a Department
of Applied Mathematics, Xi’an Jiaotong University, Xi’an 710049, China of Sciences, China Jiliang University, Hangzhou 310018, China
b College
Abstract The synchronization of fractional-order complex networks with general linear dynamics under directed connected topology is investigated. The synchronization problem is converted to an equivalent simultaneous stability problem of corresponding independent subsystems by use of a pseudo-state transformation technique and real Jordan canonical form of matrix. Sufficient conditions in terms of linear matrix inequalities for synchronization are established according to stability theory of fractional-order differential equations. In a certain range of fractional order, the effects of the fractional order on synchronization is clearly revealed. Conclusions obtained in this paper generalize the existing results. Three numerical examples are provided to illustrate the validity of proposed conclusions. Keywords: Fractional-order derivative, Complex networks, Synchronization, Real Jordan canonical form, Linear matrix inequality 2010 MSC: 15A21, 34D06
1. Introduction Fractional-order calculus has a history of more than 300 years. In the 19th century, some mathematicians, such as, Liouville, Riemann and Leibniz and so on, researched fractional-order calculus systematically [1, 2]. In recent decades, ∗ Corresponding
author Email address: [email protected] (Qingxiang Fang )
Preprint submitted to Physica A
June 17, 2017
5
more and more scholars have pointed out fractional-order calculus has wide application in physics and engineering [3, 4]. Differential equations with fractional order have attracted many researchers because of their useful applications in many fields such as physics [5], engineering [6], mathematical biology [7], finance [8] and social sciences [9]. Many fractional-order systems were found in
10
the real world, such as fractional-order Lorenz system [10], fractional-order Chua system [11], fractional-order Chen system [12] and fractional-order degenerate system [13]. It should be noted that the advantage of the fractional-order equation systems is that fractional-order derivative provides a good tool to describe memory and genetic properties of many materials and process [14, 15, 16]. As
15
a result, the fractional-order complex networks (FCNs) attracted much attention. At present, some excellent results about FCNs have been investigated in [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Synchronization phenomenon appeared in a wide range of real systems, such as physics, chemistry, biology, engineering, economy and social sciences [27, 28].
20
Network synchronization plays a very important role in the fields of nuclear magnetic resonance spectrometer, signal generator, granule crumbler, laser gear, superconducting materials, communication system and so on. In recent years, synchronization of FCNs became an active research field [29, 30, 31, 32]. Wu et al [33] show that the scenario of generalized synchronization can be real-
25
ized in FCNs with nonidentical nodes based on the stability theory of linear fraction-order systems. Yu et al [34] investigate global projective synchronization of fractional-order neural networks and obtain novel criteria via combining open loop control and adaptive control based on a new fractional-order differential inequality and a sufficient monotonicity condition for the continuous
30
and differential functions. Wang et al [35] concentrate on the synchronization problem of FCNs with general linear dynamics and the undirected coupling topology and establish sufficient conditions in terms of linear matrix inequalities (LMIs) for synchronization. Liang et al [36] investigate synchronization of general fractional-order uncertain complex networks with delay and derive some
35
sufficient asymptotical synchronization criteria under adaptive pinning control 2
by use of the inequality of the fractional derivative and the comparison principle of the linear fractional equation with delay. Huang et al [37] examine the issues of synchronization and anti-synchronization for fractional chaotic financial system with market confidence by taking advantage of active control approach and 40
derive sufficient conditions for synchronization and anti-synchronization. Despite these efforts, there are still many open problems in global synchronization of FCNs with general node dynamics (including linear and nonlinear dynamics) and general coupling topologies (including undirected and directed topologies). These concerns motivate the present study.
45
In this paper, we aim to investigate synchronization of a class of FCNs, in which the node dynamics is in the general linear type and the coupling topology between nodes is directed and weighted. Compared with the work in Wang et al [35], in which the coupling topology between nodes is undirected, our method has more extensive applicability. We transform the synchronization problem
50
into an equivalent simultaneous stability problem of corresponding independent subsystems taking advantage of a pseudo-state transformation technique and real Jordan canonical form of matrix. We establish two synchronization conditions in the form of LMIs according to different scope of derivative order. One of the conditions has nothing to do with the derivative order. The feedback gain
55
matrices are derived by solving LMIs. The remainder of the paper is organized as follows. In Section 2, the network model is described and some definitions and lemmas are given. The synchronization problem is converted to an equivalent simultaneous stability problem of corresponding subsystems and the synchronization conditions in the form of
60
LMIs are derived in Section 3. Section 4 shows the validity of the proposed synchronization conditions through three numerical examples. This paper is concluded in Section 5. Notation: In stands for the n × n identity matrix. The superscript T represents transpose. ⊗ denotes the Kronecker product. For a square matrix A,
65
Sym(A)= A + AT . For real symmetrical matrices A and B, A > B means that A − B is positive definite. The matrices, if not explicitly stated, are assumed to 3
have compatible dimensions. 2. Problem formulation and preliminaries 2.1. Fractional-order derivatives 70
The fractional-order integro-differential operator, the generalized concept of an integer order integro-differential operator, can be expressed as dα Re(α) > 0, α, dt α 1, Re(α) = 0, a Dt = R t (dτ )−α , Re(α) < 0, a
(1)
where α is fractional-order which may be a complex number, and a and t are limits of integration. There are different definitions for fractional-order derivatives. The frequent75
ly used definitions are the Gr¨ unwald-Letnikov, Riemann-Liouville and Caputo definitions [3, 38]. The Gr¨ unwald-Letnikov (GL) derivative with order α is defined as [(t−α)/h] X α −α GL α f (t − ih), (−1)i a Dt f (t) = lim h h→0 i i=0
(2)
where [·] denotes the integer part.
80
The Riemann-Liouville (RL) derivative with order α is defined as Z t dn 1 f (τ ) R α D f (t) = , n − 1 < α < n, a t dtn Γ(n − α) a (t − τ )α−n−1 R∞ where Γ(·) is the Gamma function given by Γ(z) = 0 tz−1 e−t dt.
The Caputo derivative with order α is defined as Z t 1 C α (t − τ )(n−α−1) f (n) (τ )dτ, n − 1 < α < n. D f (t) = a t Γ(n − α) a
(3)
(4)
where n ∈ Z+ .
Note here that as initial conditions are not taken into account, the results in this paper remain valid whatever the definition used, such as Gr¨ unwald85
Letnikov, Riemann-Liouville and Caputo definitions [38, 39]. In the following, we denote fractional-order derivative with order α by Dα . 4
2.2. Model description Consider a FCN consisting of N coupled nodes, in which each node is an n-dimensional fractional-order differential system. The entire FCN is described 90
by α
D xi (t) = Axi (t) + BK
N X j=1
cij (xj (t) − xi (t)), i = 1, 2, · · · , N,
(5)
where 0 < α < 2 is the fractional order, xi (t) = [xi1 (t), xi2 (t), · · · , xin (t)]T ∈
Rn is the pseudo-state of the ith node, i = 1, 2, · · · , N , A ∈ Rn×n and B ∈ Rn×m are constant matrices, K ∈ Rm×n is the feedback gain matrix to be
determined. cij ≥ 0 denotes the coupling coefficient from node j to node i, for 95
i, j = 1, · · · , N, i 6= j. cii satisfies cii = 0, i = 1, · · · , N . Remark 1. Vector xi (t) in (5) is called ”pseudo state” since knowledge of it at the initial time t0 is not enough to predict the future behavior of the system on account of the nonlocality of fractional-order derivative [40, 41, 42, 43, 44, 45]. For the FCN (5), if each node is regarded as a vertex and each commu-
100
nication link is regarded as an edge, then its coupling topology can be conveniently described by a simple graph G = (V, E, C), where V = {1, · · · , N } denotes the vertex set, E = {e(i, j)} denotes the directed edge set by that e(i, j) ∈ E if and only if there exists a directed edge from vertex j to vertex
i, and C = (cij ) ∈ RN ×N denotes the weighted adjacency matrix (for more 105
details on graph theory, the interested readers please refer to some textbooks [46]). The Laplacian matrix L = (lij ) ∈ RN ×N of the graph G is defined PN by L = D − C, where D=diag(d1 , d2 , · · · , dN ) and di = j=1,j6=i cij is the in-
degree of vertex i. A directed path from vertex j to i is a sequence of edges e(i, i1 ), e(i1 , i2 ), · · · , e(im , j) with distinct nodes ik , k = 1, 2, · · · , m. The graph 110
G is called strongly connected if there is a directed path from vertex i to j and a directed path from vertex j to i between every pair (i, j) of distinct nodes. Definition 1. ([35]) The FCN (5) is said to achieve synchronization if there holds lim kxi (t) − xj (t)k = 0,
t→∞
5
∀i, j = 1, 2, · · · , N.
(6)
Wang et al [35] considered synchronization problem of FCN (5) when cou115
pling matrix C is a symmetric matrix and obtained the solution of feedback gain K. The objective of this paper is to find appropriate feedback gain matrix K ∈
Rm×n such that FCN (5) achieves synchronization when C has no restricted condition of symmetry. 120
2.3. Stability theory of fractional-order linear systems To derive the main results in this paper, we give some useful lemmas about stability of fractional-order linear systems. Lemma 1. ([47]) Let A ∈ Rn×n and 0 < α < 2. The fractional-order system Dα x(t) = Ax(t) is asymptotically stable if and only if |arg(spec(A))| > απ/2,
125
(7)
where spec(A) is the spectrum of system matrix A. The following two lemmas establish LMI-based criteria for stability of fractionalorder linear systems. Lemma 2. ([48]) Let A ∈ Rn×n and 0 < α < 1. The fractional-order system Dα x(t) = Ax(t) is asymptotically stable if and only if there exist two real sym-
130
metric positive definite matrices Pk1 ∈ Rn×n (k = 1, 2) and two skew-symmetric matrices Pk2 ∈ Rn×n (k = 1, 2) such that 2 X 2 X i=1 j=1
and
where
Θ11 =
Θ21 =
P11
P12
−P12
P11
sin απ 2 cos απ 2 sin
απ 2
− cos
απ 2
Sym{Θij ⊗ (APij )} < 0,
> 0,
P21
P22
−P22
P21
− cos απ 2
cos
απ 2
sin
απ 2
sin απ 2
, Θ12 =
, Θ22 = 6
(8)
> 0,
(9)
cos απ 2
sin απ 2
− sin απ 2
cos απ 2
− cos − sin
απ 2 απ 2
sin
απ 2
,
− cos απ 2
.
(10)
Lemma 3. ([47]) Let A ∈ Rn×n and 1 ≤ α < 2. The fractional-order sys-
135
tem Dα x(t) = Ax(t) is asymptotically stable if and only if there exists a real symmetric positive definite matrix P ∈ Rn×n such that απ T (P A − AP ) cos (P AT + AP ) sin απ 2 2 < 0. απ απ T T (AP − P A ) cos 2 (P A + AP ) sin 2
(11)
3. Main results In this section, some conditions in the form of LMIs are presented to synchronize FCN (5) according to the topological structure of the network and the 140
properties of node dynamics. Let x = [xT1 , xT2 , · · · , xTN ]T , then FCN (5) can be rewritten as Dα x(t) = (IN ⊗ A − L ⊗ (BK))x(t),
(12)
where L is the Laplacian matrix of graph G. Denote ei (t) = xi+1 (t) − x1 (t), i = 1, 2, · · · , N − 1, be the synchronization
error and e(t) = [eT1 , eT2 , · · · , eTN −1 ]T = (E ⊗ In )x(t), then
Dα e(t) = (IN −1 ⊗ A − (ELF ) ⊗ (BK))e(t), 145
(13)
where E = [−1N −1 , IN −1 ], F = [0N −1 , IN −1 ]T , 1N −1 and 0N −1 denote N − 1 dimensional column vectors with all components 1 and 0, respectively. With above analysis, synchronization of the FCN (5) is equivalent to e(t) → 0(t → +∞). In the following, instead of investigating x(t), we investigate dynamical behaviors of e(t) directly.
150
In order to simplify the description, we assume that the graph G is strongly connected, then the algebraic multiplicity of eigenvalue 0 of L is equal to 1. Under this assumption, if λ is an eigenvalue of L and λ 6= 0, then Re(λ) > 0 by the Gerschgorin theorem [49]. Let W −1 LW = diag(0, J2 (λ2 ), · · · , Jp (λp ), C1 (α1 , β1 ), · · · , Cq (αq , βq )) 7
(14)
155
be real Jordan canonical form of L, where W is a real invertible matrix, Jk (λk ) is nk -order Jordan block with positive real diagonal element λk , k = 2, · · · , p, α1 + β1 i, · · · , αq + βq i are the non-real eigenvalues of L with α1 , · · · , αq , β1 , · · · , βq ∈ R, βj 6= 0, j = 1, 2, · · · , q,
c(αl , βl )
Cl (αl , βl ) =
160
I2 c(αl , βl )
..
.
..
.
I2 c(αl , βl )
c(αl , βl ) =
αl
βl
−βl
αl
∈ R2np+l ×2np+l ,
, l = 1, · · · , q,
(15)
(16)
and n2 + · · · + np + 2np+1 + · · · + 2np+q = N − 1.
˜ = E[W2 , · · · , WN ], then Denote W = (wij )N ×N = [W1 , W2 , · · · , WN ] and W
˜ is invertible. The matrix ELF has the W1 can be chosen as W1 = 1N and W following property. Lemma 4. Real Jordan canonical form of ELF can be expressed as ˜ −1 ELF W ˜ = diag(J2 (λ2 ), · · · , Jp (λp ), C1 (α1 , β1 ), · · · , Cq (αq , βq )). W 165
(17)
Proof. Denote ε = [1, 01×(N −1) ]T and J˜ = diag(J2 (λ2 ), · · · , Jp (λp ), C1 (α1 , β1 ), · · · , Cq (αq , βq )),
(18)
˜ from (14). then, EL[ε, F ]W = EW diag(0, J) Since L1N = 0, Lε = −L[0, 1TN −1]T = −LF 1N −1, ˜ = EW [0N −1 , J˜T ]T , that is, then ELF EW = EW diag(0, J˜) and ELF W
˜ =W ˜ J. ˜ (ELF )W Let
˜ −1 ⊗ In )e(t) = [˜ e˜(t) = (W eT1 (t), e˜T2 (t), · · · , e˜TN −1 (t)]T ∈ Rn(N −1) ,
8
(19)
170
then Dα e˜(t) = (IN −1 ⊗ A − J˜ ⊗ (BK))˜ e(t).
(20)
Denote s1 = 0, sk = sk−1 + nk , k = 2, · · · , p, sp+j = sp+j−1 + 2np+j , j = 1, 2, · · · , q, and In ⊗ A − Ji+1 (λi+1 ) ⊗ (BK), i = 1, 2, · · · , p − 1, i+1 Ai = I2ni+1 ⊗ A − Ci−p+1 (αi−p+1 , βi−p+1 ) ⊗ (BK), i = p, p + 1, · · · , p + q − 1,
(21)
then system (20) is composed of p + q − 1 independent subsystems Dα eˆi (t) = Ai eˆi (t),
(22)
where eˆi (t) = [˜ eTsi +1 (t), e˜Tsi +2 (t), · · · , e˜Tsi+1 (t)]T , i = 1, 2, · · · , p + q − 1. 175
Since Ai , i = 1, 2, · · · , p + q − 1 are all block upper triangular matrices, Ai
and A¯i have same eigenvalues, where A − λi+1 BK, i = 1, 2, · · · , p − 1, A¯i = A − α BK −β BK i−p+1 i−p+1 , i = p, p + 1, · · · , p + q − 1, βi−p+1 BK A − αi−p+1 BK (23)
if multiplicity of eigenvalues is ignored.
Lemma 1 shows that eˆi (t) → 0(t → +∞), i = 1, 2, · · · , p + q − 1 if and only
if |arg(spec(A¯i ))| > απ/2, i = 1, 2, · · · , p + q − 1. 180
In the following, we present synchronization conditions of the FCN (5) with 0 < α < 1 and 1 ≤ α < 2, respectively. Theorem 1. Assume that the coupling topology G is strongly connected. If
there exit matrices P ∈ Rn×n and X ∈ Rm×n , P > 0, such that
Φj = AP + P AT − λj BX − λj X T B T < 0, j = 2, 3, · · · , p, and
Ψl βl (BX − X T B T )
βl (X T B T − BX) Ψl 9
< 0, l = 1, 2, · · · , q,
(24)
(25)
185
where Ψl = AP + P AT − αl BX − αl X T B T ,
(26)
then the FCN (5) with 0 < α < 1 can achieve global synchronization. In this case, a feedback gain matrix K can be obtained as K = XP −1 . Proof. When (24) and (25) hold, let K = XP −1 , then A¯k P + P A¯Tk < 0, k = 1, 2, · · · , p − 1,
(27)
and A¯k diag(P, P ) + diag(P, P )A¯Tk < 0, k = p, p + 1, · · · , p + q − 1, 190
(28)
When 0 < α < 1, sin απ 2 > 0 is always true, so when k = 1, 2, · · · , p − 1, 0 (A¯k P + P A¯Tk ) sin απ 2 < 0, (29) απ T ¯ ¯ 0 (Ak P + P Ak ) sin 2
this means Sym{Θ11 ⊗ (A¯k P )} + Sym{Θ21 ⊗ (A¯k P )} < 0. P2 P2 Let P11 = P21 = P and P12 = P22 = 0n×n , then i=1 j=1 Sym{Θij ⊗ (A¯k Pij )} < 0 and P11 , P12 , P21 , P22 satisfy inequality (9).
According to Lemma 2, the system Dα y(t) = A¯k y(t) is asymptotically stable, 195
then |arg(spec(A¯k ))| > απ/2.
In the same way, we can prove that |arg(spec(A¯k ))| > απ/2 holds for k =
p, p + 1, · · · , p + q − 1 from (28). From Lemma 1, when t → +∞, eˆi (t) → 0, i = 1, 2, · · · , p + q − 1 and e(t) =
˜ ⊗ In )˜ (W e(t) → 0. This indicates that synchronization behavior of the FCN (5) 200
has been achieves.
Remark 2. Inequality (24) and 25 do not contain α. So Theorem 1 gives sufficient conditions which do not depend on order α. Theorem 2. Assume that the coupling topology G is strongly connected. If
there exit matrices P ∈ Rn×n and X ∈ Rm×n , P > 0, such that ˜ j cot απ Φ Φj 2 < 0, j = 2, 3, · · · , p, ˜ T cot απ Φ Φ j j 2 10
(30)
205
and
where
Υl
˜ l cot απ Υ 2
˜ T cot απ Υ l 2
Υl
< 0, l = 1, 2, · · · , q,
(31)
˜ j = P AT − AP + λj BX − λj X T B T , Φ Ψl βl (X T B T − BX) , Υl = βl (BX − X T B T ) Ψl ˜l Ψ βl (BX + X T B T ) ˜l = , Υ ˜l −βl (BX + X T B T ) Ψ
(32)
˜ l = P AT − AP + αl BX − αl X T B T , Ψ
and Φj and Ψl are respectively defined by (24) and (26), then the FCN (5) with 1 ≤ α < 2 can achieve global synchronization. In this case, a feedback gain matrix K can be obtained as K = XP −1 .
210
Proof. When (30) holds, let K = XP −1 , then A¯k P + P A¯Tk (P A¯Tk − A¯k P ) cot απ 2 < 0, k = 1, 2, · · · , p − 1. ¯k P + P A¯T A (P A¯Tk − A¯k P )T cot απ k 2 (33) By Schur complement [50], (33) is equivalent to ¯T − A¯k P ) cos απ (A¯k P + P A¯Tk ) sin απ (P A k 2 2 < 0, k = 1, 2, · · · , p − 1. ¯k P + P A¯T ) sin απ ( A (P A¯Tk − A¯k P )T cos απ k 2 2 (34) According to Lemma 3, when k = 1, 2, · · · , p−1, the system Dα y(t) = A¯k y(t)
is asymptotically stable, then |arg(spec(A¯k ))| > απ/2.
In the same way, we can prove that |arg(spec(A¯k ))| > απ/2 holds for k =
215
p, p + 1, · · · , p + q − 1 from (31).
Therefore, when t → +∞, e(t) → 0. This completes the proof.
Remark 3. On the basis of Schur complement, condition (30) is equivalent to ˜ T Φ−1 Φ ˜ j cot2 inequalities Φj < 0 and Φj − Φ j j
απ 2
< 0, j = 2, 3, · · · , p and condition
˜ T Υ−1 Υ ˜ l cot2 (31) is equivalent to inequalities Υl < 0 and Υl − Υ l l 11
απ 2
< 0, l =
220
1, 2, · · · , q. As a consequence, when 1 ≤ α < 2, with the increase of order α, there are less matrices A, B, C making the system (5) achieve synchronization. Remark 4. Compared with Theorem 1 and 2 in Wang et al [35], conditions (30) and (31) more clearly reveal the influence of order α on synchronization.
4. Applications 225
In this section, three numerical examples are presented to illustrate the efficiency of the conclusions developed in previous section. The first example considers the case when the individual fractional-order nodes are marginally stable, the second one considers the case when the individual fractional-order nodes are unstable and the third one considers the synchronization of networked
230
fractional-order inverted pendulums. Remark 5. The stability of equilibrium point of the system Dα x(t) = f (x(t)) can be divided into: stable, unstable, asymptotically stable and marginally stable. An equilibrium point is stable if all solutions starting at nearby points stay nearby; otherwise, it is unstable. It is asymptotically stable if all solutions s-
235
tarting at nearby points not only stay nearby, but also tend to the equilibrium point as time approaches infinity. It is marginally stable if it is stable but not asymptotically stable. In the following three examples, we employ same coupling topology for system (5) with 5 fractional-order nodes. The coupling topology among these 5
240
nodes is given in Figure 1 and the corresponding 5 −2 −2 −1 −1 5 −2 −1 L = 0 −1 6 −2 0 −3 −1 7 −1 −1 −2 −1
12
Laplacian matrix is 0 −1 −3 . −3 5
(35)
* Y H A HH HH A H A 2 5 * Y H A H H AK A HH A H A A H HH A U A AU j H 3 4 1
Figure 1: Topology structure of coupled 5 fractional-order nodes.
The real Jordan canonical form of L is 0 0 0 0 0 0 8 −1 0 0 JL = 0 1 8 0 0 . 0 0 0 6 1 0 0 0 0 6
(36)
The matrix L has a simple complex eigenvalue 8 + i and a real eigenvalue 6 with multiplicity 2. Since the synchronization criteria presented in [35] are only applicable to the 245
FCNs with symmetric Laplacian matrix, they cannot be used in the network represented by Figure 1. Example 1 (Synchronization of networked marginally stable fractionalorder nodes). Assume that the system matrices of each fractional-order node are
250
0 −3 −5
−1
4
(37) , B = −5 7 , α = 1.0. 0 −1 2 −3 2 0 √ √ Since the eigenvalues of A are 0, 2i and − 2i, the individual node system A= 0 0
is marginally stable for fractional order α = 1.0 ([51]). The solutions of LMIs (30) and (31) are P =
1.1287
0.1950
0.0012
0.1950
0.4497
−0.3789
0.0012
−0.3789
0.4970
,X = 13
"
−0.0198
−0.0306
−0.0874
−0.0058
−0.0040
−0.0483
#
, (38)
pseudo−state xi1(i=1,...,5)
10 0 −10
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
xi2(i=1,...,5)
5 0 −5
xi3(i=1,...,5)
5 0 −5
t
Figure 2: Evolution of pseudo-states xi (t)(i = 1, · · · , 5) in Example 1.
then the feedback gain is
K = XP −1 =
0.1110 −0.7400 −0.7402 0.0494 −0.3141 −0.3368
.
(39)
Therefore, the FCN (5) with system parameters (37) and coupling topology in 255
Figure 1 can achieve synchronization. The trajectories of pseudo-states xi (t)(i = 1, · · · , 5) and the error pseudo-states ei (t) = xi+1 (t)− x1 (t)(i = 1, · · · , 4) are displayed in Figures 2 and 3. The simulation results indicate that the FCN (5) of coupled marginally stable fractional-order nodes has achieved global synchronization.
260
Example 2 (Synchronization of networked unstable fractional-order nodes). Assume that the system matrices −1 −2 A = 0 −1 1 −1
of each fractional-order node are 0 1 2 , B = 1 , α = 1.2. 0 0
(40)
√ √ Since the eigenvalues of A are −2, 3i and − 3i, the individual node system
is unstable for fractional order α = 1.2. The solutions of LMIs (30) and (31)
14
error pseudo−state ei1(i=1,...,4)
5 0 −5
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
ei2(i=1,...,4)
5 0 −5
ei3(i=1,...,4)
5 0 −5
t
Figure 3: Evolution of error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) in Example 1.
are
P =
265
1.3304
0.4374
−0.5050
0.4374
0.1510
−0.1833
−0.5050
−0.1833
0.5077
then the feedback gain is K = XP −1 =
h
,X =
h
−0.1892
−0.0089
−2.9529 9.6204 0.9279
i
.
0.1988
i
, (41)
(42)
Therefore, the FCN (5) with system parameters (40) and coupling topology in Figure 1 can achieve synchronization. The trajectories of pseudo-states xi (t)(i = 1, · · · , 5) and the error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) are displayed in Figures 4 and 5. The simulation results indicate that the FCN (5) 270
of coupled unstable fractional-order nodes has achieved global synchronization.
Example 3 (Synchronization of networked fractional-order electrical circuit systems). Assume that the system matrices of each fractional-order electrical circuit system [20, 52] are A=
275
1
1
−4 0.52
,B =
0 1
, α = 0.8.
(43)
√ √ Since the eigenvalues of A are 3(1 + 7i)/4 and 3(1 − 7i)/4, the individual node system is unstable for fractional order α = 0.8. The solutions of LMIs (24) 15
xi1(i=1,...,5)
pseudo−state 20 a 0 −20 −40 0 xi2(i=1,...,5)
40
4
6
8
10
2
4
6
8
10
2
4
6
8
10
20 0 −20
xi3(i=1,...,5)
2 b
0 c
20 0 −20 0
t
Figure 4: Evolution of pseudo-states xi (t)(i = 1, · · · , 5) in Example 2. error pseudo−state ei1(i=1,...,4)
10
a
0 −10
0
ei2(i=1,...,4)
5
4
6
8
10
2
4
6
8
10
2
4
6
8
10
0 −5
0
10 ei3(i=1,...,4)
2 b
c
5 0 −5
0
t
Figure 5: Evolution of error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) in Example 2. pseudo−state 15
a
xi1(i=1,...,5)
10 5 0 −5 −10 −15
0
xi2(i=1,...,5)
20
5
10
15
10
15
b
10 0 −10 −20 0
5 t
Figure 6: Evolution of pseudo-states xi (t)(i = 1, · · · , 5) in Example 3.
16
error pseudo−state
ei1(i=1,...,4)
5
a
0
−5
0
5
10
15
10
15
10 ei2(i=1,...,4)
b 5
0
−5
0
5 t
Figure 7: Evolution of error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) in Example 3.
and (25) are
P =
253.15 −479.32 −479.32
1714.46
then the feedback gain is K = XP −1 =
,X = h
h
−2.3592 499.2066
1.1517 0.6132
i
i
,
.
(44)
(45)
Therefore, the FCN (5) with system parameters (43) and coupling topology in 280
Figure 1 can achieve synchronization. The trajectories of pseudo-states xi (t)(i = 1, · · · , 5) and the error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) are displayed in Figures 6 and 7. The simulation results indicate that the FCN (5) of coupled electrical circuit systems has achieved global synchronization.
5. Conclusion 285
In this paper, the synchronization problem of FCNs, in which the coupling topology between nodes is directed and weighted, has been investigated. Real Jordan canonical form of matrix was employed to implement error system decomposition. Synchronization conditions in the form of LMIs were established by use of stability theory of fractional-order differential equations and the feed-
290
back gain matrices were derived. 17
The significance of the paper, we think, can be summarized as follows: 1) to our knowledge, it seems to be the present paper that first employs real Jordan canonical form of matrix to investigate synchronization of complex networks; 2) the effects of the derivative order on synchronization is clearly revealed by 295
synchronization conditions; 3) the conditions adopted in this paper improve some existing results. However, there are many problems unsolved about the synchronization problem of FCNs. For example, in this paper only complex networks with linear and time-invariable coupling structure were treated, the complex networks with
300
nonlinear and time-variable coupling structure have not been involved; for the dynamical behaviors at each node, we only considered the fractional-order linear systems, the fractional-order nonlinear systems needs to be further studied.
Acknowledgments This work was supported by NCET (10531030), NSFC (11401549) and Nat305
ural Science Foundation of Zhejiang Province (Y1110036).
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23
PII: DOI: Reference:
S0378-4371(17)30785-9 http://dx.doi.org/10.1016/j.physa.2017.08.050 PHYSA 18496
To appear in:
Physica A
Received date : 21 February 2017 Revised date : 17 June 2017 Please cite this article as: Q. Fang, J. Peng, Synchronization of fractional-order linear complex networks with directed coupling topology, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.08.050 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
>We investigate the synchronization of fractional-order complex networks with general linear dynamics under directed connected topology. > Real Jordan canonical form of matrix is employed to convert error system to an equivalent simultaneous stability problem of corresponding independent subsystems. >Sufficient conditions in terms of linear matrix inequalities for synchronization are established. >The effects of the derivative order on synchronization is clearly revealed by synchronization conditions. >The conditions adopted in this paper improve some existing results.
*Manuscript Click here to view linked References
Synchronization of fractional-order linear complex networks with directed coupling topology Qingxiang Fanga,b,∗, Jigen Penga a Department
of Applied Mathematics, Xi’an Jiaotong University, Xi’an 710049, China of Sciences, China Jiliang University, Hangzhou 310018, China
b College
Abstract The synchronization of fractional-order complex networks with general linear dynamics under directed connected topology is investigated. The synchronization problem is converted to an equivalent simultaneous stability problem of corresponding independent subsystems by use of a pseudo-state transformation technique and real Jordan canonical form of matrix. Sufficient conditions in terms of linear matrix inequalities for synchronization are established according to stability theory of fractional-order differential equations. In a certain range of fractional order, the effects of the fractional order on synchronization is clearly revealed. Conclusions obtained in this paper generalize the existing results. Three numerical examples are provided to illustrate the validity of proposed conclusions. Keywords: Fractional-order derivative, Complex networks, Synchronization, Real Jordan canonical form, Linear matrix inequality 2010 MSC: 15A21, 34D06
1. Introduction Fractional-order calculus has a history of more than 300 years. In the 19th century, some mathematicians, such as, Liouville, Riemann and Leibniz and so on, researched fractional-order calculus systematically [1, 2]. In recent decades, ∗ Corresponding
author Email address: [email protected] (Qingxiang Fang )
Preprint submitted to Physica A
June 17, 2017
5
more and more scholars have pointed out fractional-order calculus has wide application in physics and engineering [3, 4]. Differential equations with fractional order have attracted many researchers because of their useful applications in many fields such as physics [5], engineering [6], mathematical biology [7], finance [8] and social sciences [9]. Many fractional-order systems were found in
10
the real world, such as fractional-order Lorenz system [10], fractional-order Chua system [11], fractional-order Chen system [12] and fractional-order degenerate system [13]. It should be noted that the advantage of the fractional-order equation systems is that fractional-order derivative provides a good tool to describe memory and genetic properties of many materials and process [14, 15, 16]. As
15
a result, the fractional-order complex networks (FCNs) attracted much attention. At present, some excellent results about FCNs have been investigated in [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Synchronization phenomenon appeared in a wide range of real systems, such as physics, chemistry, biology, engineering, economy and social sciences [27, 28].
20
Network synchronization plays a very important role in the fields of nuclear magnetic resonance spectrometer, signal generator, granule crumbler, laser gear, superconducting materials, communication system and so on. In recent years, synchronization of FCNs became an active research field [29, 30, 31, 32]. Wu et al [33] show that the scenario of generalized synchronization can be real-
25
ized in FCNs with nonidentical nodes based on the stability theory of linear fraction-order systems. Yu et al [34] investigate global projective synchronization of fractional-order neural networks and obtain novel criteria via combining open loop control and adaptive control based on a new fractional-order differential inequality and a sufficient monotonicity condition for the continuous
30
and differential functions. Wang et al [35] concentrate on the synchronization problem of FCNs with general linear dynamics and the undirected coupling topology and establish sufficient conditions in terms of linear matrix inequalities (LMIs) for synchronization. Liang et al [36] investigate synchronization of general fractional-order uncertain complex networks with delay and derive some
35
sufficient asymptotical synchronization criteria under adaptive pinning control 2
by use of the inequality of the fractional derivative and the comparison principle of the linear fractional equation with delay. Huang et al [37] examine the issues of synchronization and anti-synchronization for fractional chaotic financial system with market confidence by taking advantage of active control approach and 40
derive sufficient conditions for synchronization and anti-synchronization. Despite these efforts, there are still many open problems in global synchronization of FCNs with general node dynamics (including linear and nonlinear dynamics) and general coupling topologies (including undirected and directed topologies). These concerns motivate the present study.
45
In this paper, we aim to investigate synchronization of a class of FCNs, in which the node dynamics is in the general linear type and the coupling topology between nodes is directed and weighted. Compared with the work in Wang et al [35], in which the coupling topology between nodes is undirected, our method has more extensive applicability. We transform the synchronization problem
50
into an equivalent simultaneous stability problem of corresponding independent subsystems taking advantage of a pseudo-state transformation technique and real Jordan canonical form of matrix. We establish two synchronization conditions in the form of LMIs according to different scope of derivative order. One of the conditions has nothing to do with the derivative order. The feedback gain
55
matrices are derived by solving LMIs. The remainder of the paper is organized as follows. In Section 2, the network model is described and some definitions and lemmas are given. The synchronization problem is converted to an equivalent simultaneous stability problem of corresponding subsystems and the synchronization conditions in the form of
60
LMIs are derived in Section 3. Section 4 shows the validity of the proposed synchronization conditions through three numerical examples. This paper is concluded in Section 5. Notation: In stands for the n × n identity matrix. The superscript T represents transpose. ⊗ denotes the Kronecker product. For a square matrix A,
65
Sym(A)= A + AT . For real symmetrical matrices A and B, A > B means that A − B is positive definite. The matrices, if not explicitly stated, are assumed to 3
have compatible dimensions. 2. Problem formulation and preliminaries 2.1. Fractional-order derivatives 70
The fractional-order integro-differential operator, the generalized concept of an integer order integro-differential operator, can be expressed as dα Re(α) > 0, α, dt α 1, Re(α) = 0, a Dt = R t (dτ )−α , Re(α) < 0, a
(1)
where α is fractional-order which may be a complex number, and a and t are limits of integration. There are different definitions for fractional-order derivatives. The frequent75
ly used definitions are the Gr¨ unwald-Letnikov, Riemann-Liouville and Caputo definitions [3, 38]. The Gr¨ unwald-Letnikov (GL) derivative with order α is defined as [(t−α)/h] X α −α GL α f (t − ih), (−1)i a Dt f (t) = lim h h→0 i i=0
(2)
where [·] denotes the integer part.
80
The Riemann-Liouville (RL) derivative with order α is defined as Z t dn 1 f (τ ) R α D f (t) = , n − 1 < α < n, a t dtn Γ(n − α) a (t − τ )α−n−1 R∞ where Γ(·) is the Gamma function given by Γ(z) = 0 tz−1 e−t dt.
The Caputo derivative with order α is defined as Z t 1 C α (t − τ )(n−α−1) f (n) (τ )dτ, n − 1 < α < n. D f (t) = a t Γ(n − α) a
(3)
(4)
where n ∈ Z+ .
Note here that as initial conditions are not taken into account, the results in this paper remain valid whatever the definition used, such as Gr¨ unwald85
Letnikov, Riemann-Liouville and Caputo definitions [38, 39]. In the following, we denote fractional-order derivative with order α by Dα . 4
2.2. Model description Consider a FCN consisting of N coupled nodes, in which each node is an n-dimensional fractional-order differential system. The entire FCN is described 90
by α
D xi (t) = Axi (t) + BK
N X j=1
cij (xj (t) − xi (t)), i = 1, 2, · · · , N,
(5)
where 0 < α < 2 is the fractional order, xi (t) = [xi1 (t), xi2 (t), · · · , xin (t)]T ∈
Rn is the pseudo-state of the ith node, i = 1, 2, · · · , N , A ∈ Rn×n and B ∈ Rn×m are constant matrices, K ∈ Rm×n is the feedback gain matrix to be
determined. cij ≥ 0 denotes the coupling coefficient from node j to node i, for 95
i, j = 1, · · · , N, i 6= j. cii satisfies cii = 0, i = 1, · · · , N . Remark 1. Vector xi (t) in (5) is called ”pseudo state” since knowledge of it at the initial time t0 is not enough to predict the future behavior of the system on account of the nonlocality of fractional-order derivative [40, 41, 42, 43, 44, 45]. For the FCN (5), if each node is regarded as a vertex and each commu-
100
nication link is regarded as an edge, then its coupling topology can be conveniently described by a simple graph G = (V, E, C), where V = {1, · · · , N } denotes the vertex set, E = {e(i, j)} denotes the directed edge set by that e(i, j) ∈ E if and only if there exists a directed edge from vertex j to vertex
i, and C = (cij ) ∈ RN ×N denotes the weighted adjacency matrix (for more 105
details on graph theory, the interested readers please refer to some textbooks [46]). The Laplacian matrix L = (lij ) ∈ RN ×N of the graph G is defined PN by L = D − C, where D=diag(d1 , d2 , · · · , dN ) and di = j=1,j6=i cij is the in-
degree of vertex i. A directed path from vertex j to i is a sequence of edges e(i, i1 ), e(i1 , i2 ), · · · , e(im , j) with distinct nodes ik , k = 1, 2, · · · , m. The graph 110
G is called strongly connected if there is a directed path from vertex i to j and a directed path from vertex j to i between every pair (i, j) of distinct nodes. Definition 1. ([35]) The FCN (5) is said to achieve synchronization if there holds lim kxi (t) − xj (t)k = 0,
t→∞
5
∀i, j = 1, 2, · · · , N.
(6)
Wang et al [35] considered synchronization problem of FCN (5) when cou115
pling matrix C is a symmetric matrix and obtained the solution of feedback gain K. The objective of this paper is to find appropriate feedback gain matrix K ∈
Rm×n such that FCN (5) achieves synchronization when C has no restricted condition of symmetry. 120
2.3. Stability theory of fractional-order linear systems To derive the main results in this paper, we give some useful lemmas about stability of fractional-order linear systems. Lemma 1. ([47]) Let A ∈ Rn×n and 0 < α < 2. The fractional-order system Dα x(t) = Ax(t) is asymptotically stable if and only if |arg(spec(A))| > απ/2,
125
(7)
where spec(A) is the spectrum of system matrix A. The following two lemmas establish LMI-based criteria for stability of fractionalorder linear systems. Lemma 2. ([48]) Let A ∈ Rn×n and 0 < α < 1. The fractional-order system Dα x(t) = Ax(t) is asymptotically stable if and only if there exist two real sym-
130
metric positive definite matrices Pk1 ∈ Rn×n (k = 1, 2) and two skew-symmetric matrices Pk2 ∈ Rn×n (k = 1, 2) such that 2 X 2 X i=1 j=1
and
where
Θ11 =
Θ21 =
P11
P12
−P12
P11
sin απ 2 cos απ 2 sin
απ 2
− cos
απ 2
Sym{Θij ⊗ (APij )} < 0,
> 0,
P21
P22
−P22
P21
− cos απ 2
cos
απ 2
sin
απ 2
sin απ 2
, Θ12 =
, Θ22 = 6
(8)
> 0,
(9)
cos απ 2
sin απ 2
− sin απ 2
cos απ 2
− cos − sin
απ 2 απ 2
sin
απ 2
,
− cos απ 2
.
(10)
Lemma 3. ([47]) Let A ∈ Rn×n and 1 ≤ α < 2. The fractional-order sys-
135
tem Dα x(t) = Ax(t) is asymptotically stable if and only if there exists a real symmetric positive definite matrix P ∈ Rn×n such that απ T (P A − AP ) cos (P AT + AP ) sin απ 2 2 < 0. απ απ T T (AP − P A ) cos 2 (P A + AP ) sin 2
(11)
3. Main results In this section, some conditions in the form of LMIs are presented to synchronize FCN (5) according to the topological structure of the network and the 140
properties of node dynamics. Let x = [xT1 , xT2 , · · · , xTN ]T , then FCN (5) can be rewritten as Dα x(t) = (IN ⊗ A − L ⊗ (BK))x(t),
(12)
where L is the Laplacian matrix of graph G. Denote ei (t) = xi+1 (t) − x1 (t), i = 1, 2, · · · , N − 1, be the synchronization
error and e(t) = [eT1 , eT2 , · · · , eTN −1 ]T = (E ⊗ In )x(t), then
Dα e(t) = (IN −1 ⊗ A − (ELF ) ⊗ (BK))e(t), 145
(13)
where E = [−1N −1 , IN −1 ], F = [0N −1 , IN −1 ]T , 1N −1 and 0N −1 denote N − 1 dimensional column vectors with all components 1 and 0, respectively. With above analysis, synchronization of the FCN (5) is equivalent to e(t) → 0(t → +∞). In the following, instead of investigating x(t), we investigate dynamical behaviors of e(t) directly.
150
In order to simplify the description, we assume that the graph G is strongly connected, then the algebraic multiplicity of eigenvalue 0 of L is equal to 1. Under this assumption, if λ is an eigenvalue of L and λ 6= 0, then Re(λ) > 0 by the Gerschgorin theorem [49]. Let W −1 LW = diag(0, J2 (λ2 ), · · · , Jp (λp ), C1 (α1 , β1 ), · · · , Cq (αq , βq )) 7
(14)
155
be real Jordan canonical form of L, where W is a real invertible matrix, Jk (λk ) is nk -order Jordan block with positive real diagonal element λk , k = 2, · · · , p, α1 + β1 i, · · · , αq + βq i are the non-real eigenvalues of L with α1 , · · · , αq , β1 , · · · , βq ∈ R, βj 6= 0, j = 1, 2, · · · , q,
c(αl , βl )
Cl (αl , βl ) =
160
I2 c(αl , βl )
..
.
..
.
I2 c(αl , βl )
c(αl , βl ) =
αl
βl
−βl
αl
∈ R2np+l ×2np+l ,
, l = 1, · · · , q,
(15)
(16)
and n2 + · · · + np + 2np+1 + · · · + 2np+q = N − 1.
˜ = E[W2 , · · · , WN ], then Denote W = (wij )N ×N = [W1 , W2 , · · · , WN ] and W
˜ is invertible. The matrix ELF has the W1 can be chosen as W1 = 1N and W following property. Lemma 4. Real Jordan canonical form of ELF can be expressed as ˜ −1 ELF W ˜ = diag(J2 (λ2 ), · · · , Jp (λp ), C1 (α1 , β1 ), · · · , Cq (αq , βq )). W 165
(17)
Proof. Denote ε = [1, 01×(N −1) ]T and J˜ = diag(J2 (λ2 ), · · · , Jp (λp ), C1 (α1 , β1 ), · · · , Cq (αq , βq )),
(18)
˜ from (14). then, EL[ε, F ]W = EW diag(0, J) Since L1N = 0, Lε = −L[0, 1TN −1]T = −LF 1N −1, ˜ = EW [0N −1 , J˜T ]T , that is, then ELF EW = EW diag(0, J˜) and ELF W
˜ =W ˜ J. ˜ (ELF )W Let
˜ −1 ⊗ In )e(t) = [˜ e˜(t) = (W eT1 (t), e˜T2 (t), · · · , e˜TN −1 (t)]T ∈ Rn(N −1) ,
8
(19)
170
then Dα e˜(t) = (IN −1 ⊗ A − J˜ ⊗ (BK))˜ e(t).
(20)
Denote s1 = 0, sk = sk−1 + nk , k = 2, · · · , p, sp+j = sp+j−1 + 2np+j , j = 1, 2, · · · , q, and In ⊗ A − Ji+1 (λi+1 ) ⊗ (BK), i = 1, 2, · · · , p − 1, i+1 Ai = I2ni+1 ⊗ A − Ci−p+1 (αi−p+1 , βi−p+1 ) ⊗ (BK), i = p, p + 1, · · · , p + q − 1,
(21)
then system (20) is composed of p + q − 1 independent subsystems Dα eˆi (t) = Ai eˆi (t),
(22)
where eˆi (t) = [˜ eTsi +1 (t), e˜Tsi +2 (t), · · · , e˜Tsi+1 (t)]T , i = 1, 2, · · · , p + q − 1. 175
Since Ai , i = 1, 2, · · · , p + q − 1 are all block upper triangular matrices, Ai
and A¯i have same eigenvalues, where A − λi+1 BK, i = 1, 2, · · · , p − 1, A¯i = A − α BK −β BK i−p+1 i−p+1 , i = p, p + 1, · · · , p + q − 1, βi−p+1 BK A − αi−p+1 BK (23)
if multiplicity of eigenvalues is ignored.
Lemma 1 shows that eˆi (t) → 0(t → +∞), i = 1, 2, · · · , p + q − 1 if and only
if |arg(spec(A¯i ))| > απ/2, i = 1, 2, · · · , p + q − 1. 180
In the following, we present synchronization conditions of the FCN (5) with 0 < α < 1 and 1 ≤ α < 2, respectively. Theorem 1. Assume that the coupling topology G is strongly connected. If
there exit matrices P ∈ Rn×n and X ∈ Rm×n , P > 0, such that
Φj = AP + P AT − λj BX − λj X T B T < 0, j = 2, 3, · · · , p, and
Ψl βl (BX − X T B T )
βl (X T B T − BX) Ψl 9
< 0, l = 1, 2, · · · , q,
(24)
(25)
185
where Ψl = AP + P AT − αl BX − αl X T B T ,
(26)
then the FCN (5) with 0 < α < 1 can achieve global synchronization. In this case, a feedback gain matrix K can be obtained as K = XP −1 . Proof. When (24) and (25) hold, let K = XP −1 , then A¯k P + P A¯Tk < 0, k = 1, 2, · · · , p − 1,
(27)
and A¯k diag(P, P ) + diag(P, P )A¯Tk < 0, k = p, p + 1, · · · , p + q − 1, 190
(28)
When 0 < α < 1, sin απ 2 > 0 is always true, so when k = 1, 2, · · · , p − 1, 0 (A¯k P + P A¯Tk ) sin απ 2 < 0, (29) απ T ¯ ¯ 0 (Ak P + P Ak ) sin 2
this means Sym{Θ11 ⊗ (A¯k P )} + Sym{Θ21 ⊗ (A¯k P )} < 0. P2 P2 Let P11 = P21 = P and P12 = P22 = 0n×n , then i=1 j=1 Sym{Θij ⊗ (A¯k Pij )} < 0 and P11 , P12 , P21 , P22 satisfy inequality (9).
According to Lemma 2, the system Dα y(t) = A¯k y(t) is asymptotically stable, 195
then |arg(spec(A¯k ))| > απ/2.
In the same way, we can prove that |arg(spec(A¯k ))| > απ/2 holds for k =
p, p + 1, · · · , p + q − 1 from (28). From Lemma 1, when t → +∞, eˆi (t) → 0, i = 1, 2, · · · , p + q − 1 and e(t) =
˜ ⊗ In )˜ (W e(t) → 0. This indicates that synchronization behavior of the FCN (5) 200
has been achieves.
Remark 2. Inequality (24) and 25 do not contain α. So Theorem 1 gives sufficient conditions which do not depend on order α. Theorem 2. Assume that the coupling topology G is strongly connected. If
there exit matrices P ∈ Rn×n and X ∈ Rm×n , P > 0, such that ˜ j cot απ Φ Φj 2 < 0, j = 2, 3, · · · , p, ˜ T cot απ Φ Φ j j 2 10
(30)
205
and
where
Υl
˜ l cot απ Υ 2
˜ T cot απ Υ l 2
Υl
< 0, l = 1, 2, · · · , q,
(31)
˜ j = P AT − AP + λj BX − λj X T B T , Φ Ψl βl (X T B T − BX) , Υl = βl (BX − X T B T ) Ψl ˜l Ψ βl (BX + X T B T ) ˜l = , Υ ˜l −βl (BX + X T B T ) Ψ
(32)
˜ l = P AT − AP + αl BX − αl X T B T , Ψ
and Φj and Ψl are respectively defined by (24) and (26), then the FCN (5) with 1 ≤ α < 2 can achieve global synchronization. In this case, a feedback gain matrix K can be obtained as K = XP −1 .
210
Proof. When (30) holds, let K = XP −1 , then A¯k P + P A¯Tk (P A¯Tk − A¯k P ) cot απ 2 < 0, k = 1, 2, · · · , p − 1. ¯k P + P A¯T A (P A¯Tk − A¯k P )T cot απ k 2 (33) By Schur complement [50], (33) is equivalent to ¯T − A¯k P ) cos απ (A¯k P + P A¯Tk ) sin απ (P A k 2 2 < 0, k = 1, 2, · · · , p − 1. ¯k P + P A¯T ) sin απ ( A (P A¯Tk − A¯k P )T cos απ k 2 2 (34) According to Lemma 3, when k = 1, 2, · · · , p−1, the system Dα y(t) = A¯k y(t)
is asymptotically stable, then |arg(spec(A¯k ))| > απ/2.
In the same way, we can prove that |arg(spec(A¯k ))| > απ/2 holds for k =
215
p, p + 1, · · · , p + q − 1 from (31).
Therefore, when t → +∞, e(t) → 0. This completes the proof.
Remark 3. On the basis of Schur complement, condition (30) is equivalent to ˜ T Φ−1 Φ ˜ j cot2 inequalities Φj < 0 and Φj − Φ j j
απ 2
< 0, j = 2, 3, · · · , p and condition
˜ T Υ−1 Υ ˜ l cot2 (31) is equivalent to inequalities Υl < 0 and Υl − Υ l l 11
απ 2
< 0, l =
220
1, 2, · · · , q. As a consequence, when 1 ≤ α < 2, with the increase of order α, there are less matrices A, B, C making the system (5) achieve synchronization. Remark 4. Compared with Theorem 1 and 2 in Wang et al [35], conditions (30) and (31) more clearly reveal the influence of order α on synchronization.
4. Applications 225
In this section, three numerical examples are presented to illustrate the efficiency of the conclusions developed in previous section. The first example considers the case when the individual fractional-order nodes are marginally stable, the second one considers the case when the individual fractional-order nodes are unstable and the third one considers the synchronization of networked
230
fractional-order inverted pendulums. Remark 5. The stability of equilibrium point of the system Dα x(t) = f (x(t)) can be divided into: stable, unstable, asymptotically stable and marginally stable. An equilibrium point is stable if all solutions starting at nearby points stay nearby; otherwise, it is unstable. It is asymptotically stable if all solutions s-
235
tarting at nearby points not only stay nearby, but also tend to the equilibrium point as time approaches infinity. It is marginally stable if it is stable but not asymptotically stable. In the following three examples, we employ same coupling topology for system (5) with 5 fractional-order nodes. The coupling topology among these 5
240
nodes is given in Figure 1 and the corresponding 5 −2 −2 −1 −1 5 −2 −1 L = 0 −1 6 −2 0 −3 −1 7 −1 −1 −2 −1
12
Laplacian matrix is 0 −1 −3 . −3 5
(35)
* Y H A HH HH A H A 2 5 * Y H A H H AK A HH A H A A H HH A U A AU j H 3 4 1
Figure 1: Topology structure of coupled 5 fractional-order nodes.
The real Jordan canonical form of L is 0 0 0 0 0 0 8 −1 0 0 JL = 0 1 8 0 0 . 0 0 0 6 1 0 0 0 0 6
(36)
The matrix L has a simple complex eigenvalue 8 + i and a real eigenvalue 6 with multiplicity 2. Since the synchronization criteria presented in [35] are only applicable to the 245
FCNs with symmetric Laplacian matrix, they cannot be used in the network represented by Figure 1. Example 1 (Synchronization of networked marginally stable fractionalorder nodes). Assume that the system matrices of each fractional-order node are
250
0 −3 −5
−1
4
(37) , B = −5 7 , α = 1.0. 0 −1 2 −3 2 0 √ √ Since the eigenvalues of A are 0, 2i and − 2i, the individual node system A= 0 0
is marginally stable for fractional order α = 1.0 ([51]). The solutions of LMIs (30) and (31) are P =
1.1287
0.1950
0.0012
0.1950
0.4497
−0.3789
0.0012
−0.3789
0.4970
,X = 13
"
−0.0198
−0.0306
−0.0874
−0.0058
−0.0040
−0.0483
#
, (38)
pseudo−state xi1(i=1,...,5)
10 0 −10
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
xi2(i=1,...,5)
5 0 −5
xi3(i=1,...,5)
5 0 −5
t
Figure 2: Evolution of pseudo-states xi (t)(i = 1, · · · , 5) in Example 1.
then the feedback gain is
K = XP −1 =
0.1110 −0.7400 −0.7402 0.0494 −0.3141 −0.3368
.
(39)
Therefore, the FCN (5) with system parameters (37) and coupling topology in 255
Figure 1 can achieve synchronization. The trajectories of pseudo-states xi (t)(i = 1, · · · , 5) and the error pseudo-states ei (t) = xi+1 (t)− x1 (t)(i = 1, · · · , 4) are displayed in Figures 2 and 3. The simulation results indicate that the FCN (5) of coupled marginally stable fractional-order nodes has achieved global synchronization.
260
Example 2 (Synchronization of networked unstable fractional-order nodes). Assume that the system matrices −1 −2 A = 0 −1 1 −1
of each fractional-order node are 0 1 2 , B = 1 , α = 1.2. 0 0
(40)
√ √ Since the eigenvalues of A are −2, 3i and − 3i, the individual node system
is unstable for fractional order α = 1.2. The solutions of LMIs (30) and (31)
14
error pseudo−state ei1(i=1,...,4)
5 0 −5
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
ei2(i=1,...,4)
5 0 −5
ei3(i=1,...,4)
5 0 −5
t
Figure 3: Evolution of error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) in Example 1.
are
P =
265
1.3304
0.4374
−0.5050
0.4374
0.1510
−0.1833
−0.5050
−0.1833
0.5077
then the feedback gain is K = XP −1 =
h
,X =
h
−0.1892
−0.0089
−2.9529 9.6204 0.9279
i
.
0.1988
i
, (41)
(42)
Therefore, the FCN (5) with system parameters (40) and coupling topology in Figure 1 can achieve synchronization. The trajectories of pseudo-states xi (t)(i = 1, · · · , 5) and the error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) are displayed in Figures 4 and 5. The simulation results indicate that the FCN (5) 270
of coupled unstable fractional-order nodes has achieved global synchronization.
Example 3 (Synchronization of networked fractional-order electrical circuit systems). Assume that the system matrices of each fractional-order electrical circuit system [20, 52] are A=
275
1
1
−4 0.52
,B =
0 1
, α = 0.8.
(43)
√ √ Since the eigenvalues of A are 3(1 + 7i)/4 and 3(1 − 7i)/4, the individual node system is unstable for fractional order α = 0.8. The solutions of LMIs (24) 15
xi1(i=1,...,5)
pseudo−state 20 a 0 −20 −40 0 xi2(i=1,...,5)
40
4
6
8
10
2
4
6
8
10
2
4
6
8
10
20 0 −20
xi3(i=1,...,5)
2 b
0 c
20 0 −20 0
t
Figure 4: Evolution of pseudo-states xi (t)(i = 1, · · · , 5) in Example 2. error pseudo−state ei1(i=1,...,4)
10
a
0 −10
0
ei2(i=1,...,4)
5
4
6
8
10
2
4
6
8
10
2
4
6
8
10
0 −5
0
10 ei3(i=1,...,4)
2 b
c
5 0 −5
0
t
Figure 5: Evolution of error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) in Example 2. pseudo−state 15
a
xi1(i=1,...,5)
10 5 0 −5 −10 −15
0
xi2(i=1,...,5)
20
5
10
15
10
15
b
10 0 −10 −20 0
5 t
Figure 6: Evolution of pseudo-states xi (t)(i = 1, · · · , 5) in Example 3.
16
error pseudo−state
ei1(i=1,...,4)
5
a
0
−5
0
5
10
15
10
15
10 ei2(i=1,...,4)
b 5
0
−5
0
5 t
Figure 7: Evolution of error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) in Example 3.
and (25) are
P =
253.15 −479.32 −479.32
1714.46
then the feedback gain is K = XP −1 =
,X = h
h
−2.3592 499.2066
1.1517 0.6132
i
i
,
.
(44)
(45)
Therefore, the FCN (5) with system parameters (43) and coupling topology in 280
Figure 1 can achieve synchronization. The trajectories of pseudo-states xi (t)(i = 1, · · · , 5) and the error pseudo-states ei (t) = xi+1 (t) − x1 (t)(i = 1, · · · , 4) are displayed in Figures 6 and 7. The simulation results indicate that the FCN (5) of coupled electrical circuit systems has achieved global synchronization.
5. Conclusion 285
In this paper, the synchronization problem of FCNs, in which the coupling topology between nodes is directed and weighted, has been investigated. Real Jordan canonical form of matrix was employed to implement error system decomposition. Synchronization conditions in the form of LMIs were established by use of stability theory of fractional-order differential equations and the feed-
290
back gain matrices were derived. 17
The significance of the paper, we think, can be summarized as follows: 1) to our knowledge, it seems to be the present paper that first employs real Jordan canonical form of matrix to investigate synchronization of complex networks; 2) the effects of the derivative order on synchronization is clearly revealed by 295
synchronization conditions; 3) the conditions adopted in this paper improve some existing results. However, there are many problems unsolved about the synchronization problem of FCNs. For example, in this paper only complex networks with linear and time-invariable coupling structure were treated, the complex networks with
300
nonlinear and time-variable coupling structure have not been involved; for the dynamical behaviors at each node, we only considered the fractional-order linear systems, the fractional-order nonlinear systems needs to be further studied.
Acknowledgments This work was supported by NCET (10531030), NSFC (11401549) and Nat305
ural Science Foundation of Zhejiang Province (Y1110036).
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