Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeB18.1
Synchronization of Dynamical Networks by Network Control Tao Liu, David J. Hill and Jun Zhao
Abstract— In this paper, we study the locally controlled synchronization of dynamical networks by introducing a distributed controller which has a different network structure from the original network. We refer to this configuration as a feedback network. To reflect practical reality, a cost function is considered to constrain the controller. Then based on the master stability function method, the constrained controller design problem is transformed into a mixed-integer nonlinear optimization problem. In addition, when a single controller can not be found under the constraint, a switching controller is designed by a Lyapunov function method. The convex combination technique is used to design the synchronizing switching signal between the candidate controllers, and its coefficients are given by the solution of a convex optimization problem. We also provide a feasible way to construct the candidate controllers, and give a numerical example to demonstrate the effectiveness of the proposed results.
I. INTRODUCTION Dynamical networks composed of interconnected dynamical units are ubiquitous in nature and play very important roles in many different contexts including ecology, sociology and technology [1], [2], [3]. Examples of these networks are power grids, the Internet and the World Wide Web. Synchronization as an emergent phenomenon of dynamical networks has received a great deal of focus, and extensive numerical and analytical approaches to the problem are presented. In [4], the direct Lyapunov method was introduced to analyze the synchronization of coupled systems. Reference [5] developed the master stability function for coupled oscillators, and this method has been extended to networks with arbitrary topologies in [6], [7]. Synchronous phenomena of networks with time-varying coupling [8], time-delay coupling [9], switching topology [10], [11], and the relationship between synchronizability and network topological properties (see [12] and the references therein) have also been investigated. The topic is related to the study of consensus problems in swarms [13], [14] where independent agents are controlled over a network, i.e., the closed loop system becomes a kind of time-varying network. In fact, by putting these areas together we can see a more general formulation of feedback networks where a feedback network (or distributed controller) controls a network system. Meanwhile, synchronizing a network by controller design has also received attention in recent years. Adaptive control [15], pinning control [16] and impulsive control [17] have This work was supported by the Australian Research Council’s Discovery Projects FF 0455875. Tao Liu, David J. Hill and Jun Zhao are with Research School of Information Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia.
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978-1-4244-3872-3/09/$25.00 ©2009 IEEE
already been utilized to synchronize networks. These control strategies are of the decentralized control type, and have a common drawback, i.e., an extra signal is needed and required to be exactly the same for each controller. This will reduce the robustness of the network. On the contrary, a distributed control strategy which synchronizes the network by a feedback network to change the topology of the original network should have the advantage of producing synchronous solutions that are more robust. There is no critical reliance on any specific individual node, and the controller does not require the extra signal. However, very little attention has been paid on this subject. In this paper, we address how to give a systematic and effective method to design a distributed controller for dynamical networks. In view of these issues, we present a framework for the locally controlled synchronization of a given network by distributed or network controller design. The controller gains and structure are optimized subject to a constraint on the controller. With the help of the master stability function method, algebraic graph theory and optimization theory, we design the constrained controller by solving a mixed-integer nonlinear optimization problem. Moreover, if no single controller under the constraint can synchronize the network, then switching between two or more controllers which all satisfy the constraint is considered to solve the problem. A synthesis method which identifies the switching logic and produces the candidate controllers is given. The rest of this paper is organized as follows. We first give a framework of our problem in Section II, and investigate the synchronization of the network by network controller design in Section III. Section IV focuses on the problem of switching controller design. In section V, a numerical example is given to demonstrate the effectiveness of the theoretical results. Finally, conclusions are presented in Section VI. II. M ODEL D ESCRIPTION AND P RELIMINARIES Consider a dynamical network which consists of N linearly and diffusively coupled identical nodes. The state equations of the network are given below x˙ i (t) = f (xi (t)) + c
N X
aij Γxj (t), i = 1, 2, . . . , N, (1)
j=1
where xi = [xi1 , xi2 , . . . , xin ]⊤ ∈ Rn is the state variable of node i; f (·) : Rn → Rn is a continuously differentiable function; Γ ∈ Rn×n is the inner coupling matrix; c > 0 is the coupling strength; A ∈ RN ×N is the outer coupling matrix representing topology structure of the network, and if there is a connection between node i and node j (i 6= j), then
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WeB18.1 aij = aji = 1; otherwise aij = aji = 0, and the diagonal elements of matrix A are defined as aii = −
N X
aij = −
N X
aji ,
i = 1, 2, . . . , N.
(2)
e˙ i (t) = f (ei (t) + s(t)) − f (s(t)) + c
j=1 j6=i
j=1 j6=i
s(t) ˙ = f (s(t))
kxi (t0 ) − s(t0 )k ≤ δ ⇒ lim kxi (t; xi0 ; t0 ) − s(t)k = 0, t→∞
i = 1, 2, . . . , N
e˙ i (t) = Df (s(t))ei (t) + c
bij Γxj ,
(6)
j=1
and γ > 0 is the control gain, B ∈ RN ×N is controller outer coupling matrix representing the topological structure of the distributed controller. Suppose B has the same properties as A except for the assumption of irreducibility. Clearly, it is not realistic and meaningful in practice to design the controller without any constraint, so we use the following cost function to restrict parameters γ and B of the controller X ¯ γ bij ≤ d, (7) 1≤i 0 is the total cost constraint on the controller. Now, we will briefly review the master stability function method which plays a critical role in our design methods [5].
aij Γej (t)
(9)
ω˙ i (t) = Df (s(t))ωi (t) + cλi Γωi (t), i = 1, 2, . . . , N, (10) where Φ ∈ RN ×N is a unitary matrix such that Φ⊤ AΦ = Λ with Φ⊤ Φ = IN , Λ = diag{λ1 , λ2 , . . . , λN }. λi are the real eigenvalues of A. Because A is irreducible and has property (2), we can get 0 = λ1 > λ2 ≥ λ3 ≥ · · · ≥ λN . To study the stability of (10), consider the following system with a varying parameter α ω(t) ˙ = Df (s(t))ω(t) + αΓω(t).
(11)
By the definition of the largest Lyapunov exponent λmax , we know that any value of α which makes λmax < 0 will lead to the exponential stability of the equilibrium point ω = 0 of (11). So if we know the region S of α which guarantees the stability of system (11), then we can infer the stability of the systems (10) with α = cλi , i = 1, 2, . . . , N . In addition, as ω˙ 1 (t) = Df (s(t))ω1 (t) corresponds to the synchronization manifold x1 = x2 = · · · = xN = s(t), the synchronization of (1) is guaranteed if
aij Γxj (t)+ ui , i = 1, 2, . . . , N, (5)
where ui ∈ Rn is the controller for node i with the form:
N X
with Df (s(t)) = ∂f ∂x |x=s(t) being the Jacobian matrix of f (·). Then the synchronization of the network (1) is equivalent to the stability of the error systems (9). Define ω = (Φ⊤ ⊗ In )e ∈ RnN with e = ⊤ ⊤ ⊤ nN [e1 , e2 , · · · , e⊤ , “ ⊗ ” representing a Kronecker N] ∈ R product and In ∈ Rn×n being the identity matrix. We have the equivalent transformed systems
cλi ∈ S, i = 2, 3, . . . , N.
j=1
N X
(8)
j=1
(4)
where k · k is the Euclidean norm, then the network (1) is said to achieve local synchronization. Furthermore, if xi (t; xi0 ; t0 ) converge to s(t) for all xi (t0 ) ∈ Rn , then the network is said to achieve global synchronization. In this paper, we only consider local synchronization problems, thus after here “synchronization” means “local synchronization” without explicitly specified. We suppose that s(t) is a stable solution of (3), and it can be an equilibrium point, a periodic orbit, or a chaotic attractor. Our task in this paper is to investigate the controlled synchronization of the network (1), i.e., how to design distributed or switching distributed controllers to synchronize the network which can not synchronize by itself. Here the distributed controller is effectively adding some new links between unconnected nodes, i.e., control the network by a new feedback network. Rewrite (1) with the controller as follows
ui = γ
aij Γej (t).
Linearizing (8) at s(t) yields
(3)
with initial condition s(t0 ), and xi (t; xi0 ; t0 ), i = 1, 2, . . . , N , be a solution of the network (1), where xi0 = xi (t0 ) ∈ Rn are initial conditions. If there exists a δ > 0, such that
N X
N X j=1
Here we assume that A is irreducible, i.e., the network is connected. Definition 1: Let s(t) be a solution of an isolated node
x˙ i (t) = f (xi (t))+ c
Let ei (t) = xi (t)−s(t), and subtracting (3) from (1) gives the error dynamical systems
(12)
The λmax with the variable α is known as the master stability function of network (1) [5]. The region S of α is called the synchronized region, and it can be a bounded sector [α1 , α2 ], an unbounded region with (−∞, α1 ], an empty set ∅, or disconnected regions described by a combination of the first two cases [18]. III. N ETWORK D ISTRIBUTED C ONTROL In this section, we will discuss the locally controlled synchronization of the network (5), and our goal here is to design a distributed controller (6) which synchronizes the network when it is feasible under the constraint (7). Both bounded and unbounded synchronized regions are considered, and the distributed controller design problems are converted into solving mixed-integer nonlinear optimization problems. In algebraic graph theory [19], an undirected graph G = (V, E) with N nodes and M links consists of a node set V = V(xi )N ei )M ¯i i=1 and a link set E = E(¯ i=1 , where a link e
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WeB18.1 connects two distinct nodes if they are associated in some way. For the controller network, a link implies access to each other’s information. For a link e¯i connecting nodes k and l, we define the link vector hi ∈ RN as hik = 1, hil = −1 and all other entries 0. The incidence matrix H ∈ RN ×M of the graph G is the matrix with ith column hi . Then the Laplacian matrix L of G is the N × N matrix ⊤
L = HH
=
M X
hi h⊤ i .
(13)
i=1
Obviously, we have A = −L. Additionally, let G c (V, E c ) be the complement of G which has the same nodes set V c as G, but the complement link set E c = E c (¯ eci )M i=1 with N (N −1) c − M . And the Laplacian matrix of G c is M = 2 PM c Lc = H c H c ⊤ = i=1 hci hci ⊤ . Suppose that if a new link e¯ci is established by controller (6), then let e¯ci = 1, otherwise e¯ci = 0, so we can write B in the form of c
B=−
M X
e¯ci hci hci ⊤ .
(14)
i=1
From [20], we have that for any given connected undirected graph G of size N , the nonzero eigenvalues of its Laplacian matrix L grow monotonically with the number of added links, that is, for any added link e¯cl , λi (L + e¯cl ) ≥ λi (L), i = 2, 3, . . . N . It is therefore easy to see that only when cλ2 (A) > α1 for unbounded synchronized region and cλ2 (A) > α2 for the bounded one, it is possible to synchronize the network by design of controller (6). Thus, we only consider these two cases in the sequel. First, we consider the case of an unbounded synchronized region S = (−∞, α1 ]. The master stability function method shows that if we can Pdesign a distributed controller (6) with minimum d∗ = γ 1≤i 0 and a 0-1 vector e¯c = [¯ ec1 , e¯c2 , . . . , e¯cM c ] ∈ PM c c Mc ∗ ¯ R with minimum d = γ i=1 e¯i < d, such that λ2 (cA − γ
Mc X
e¯ci hci hci ⊤ ) ≤ α1 .
then the synchronization ofP the network (5) is achieved under ∗ the controller (6) u∗ = γ ∗ N j=1 bij Γxj . Proof: It is straightforward by applying master stability function theory. Remark 1: There is no efficient numerical algorithm to get the global optimized solution of such a mixed-integer nonlinear optimization problem in the literature. The limitation of available numerical algorithms will limit our method’s applicability to large networks right now. However, the local optimized solution of the problem (16) depends on the initial values, which means we can solve such a problem repetitively with different initial values to get a better solution. Further, the mixed-integer nonlinear optimization problem has attracted a lot of focus recently in the relevant field [21], and better solution methods can be anticipated in the future. For the bounded synchronized region S = [α1 , α2 ], the situation is more complicated than the unbounded region. We have to make sure that λ2 (cA + γB) ≤ α2 as well as λN (cA + γB) ≥ α1 . As we mentioned before, adding a new link into a connected network will never decrease the nonzero eigenvalues of it’s Laplacian matrix. This indicates that for some coupling matrix A with small c|λN (A)|, adding new links into the network by controller design can synchronize the network; but for the others, the effects of adding new links will drive λ2 (cA + γB) into S, and push the λN (cA + γB) out of S at the same time. For the latter situation, in order to synchronize the network, we have to add new links into the network, and additionally cut off some existing links from the original network. Next, we will study these two cases, respectively. Case a: c|λN (A)| is small enough, and controller (6) can adjust λ2 (cA+γB) and λN (cA+γB) into S simultaneously. With a similar procedure as the one for the unbounded case, we have the following theorem. Theorem 2: Consider the network (5) in Case a. If the ¯ solutions of the optimization problem (17) satisfy d∗ ≤ d, c
min d = γ
c
s.t.
M X
λ2 (cA − γ
e¯ci
i=1
s.t.
λ2 (cA − γ
e¯ci hci hci ⊤ )
≤ α1
γ>0
(17) e¯ci hci hci ⊤ ) ≥ α1
i=1
e¯ci ∈ {0, 1}, i = 1, 2, . . . , M c γ > 0, then the network is synchronized under the controller (6). Case b: the controller (6) can not guarantee λ2 (cA + γB) ≤ α2 and λN (cA + γB) ≥ α1 simultaneously. Then modify the controller as
(16)
ui = −c
N X j=1
i=1
e¯ci ∈ {0, 1}, i = 1, 2, . . . , M c
e¯ci hci hci ⊤ ) ≤ α2
λN (cA − γ
c
M X
M X
i=1 Mc X
i=1
c
e¯ci
i=1
(15)
This can be solved by a mixed-integer nonlinear optimization, which is addressed in the following theorem. Theorem 1: Consider the network (5). If there exist solutions γ ∗ and e¯c∗ of the mixed-integer nonlinear optimization ¯ problem (16) such that d∗ ≤ d, min d = γ
M X
acij Γxj + γ
N X
bij Γxj ,
(18)
j=1
where Ac = (acij ) ∈ RN ×N represents the links which should be cut off from network (1). Correspondingly, the
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WeB18.1 1 Lemma 1: Let P (t) > 0 ∈ P CnN ×nN . If the sets
cost function should be adapted as X X c acij + γ 1≤i 0 is the control gain, and Bk ∈ RN ×N is the controller outer coupling matrix. Here γk and Bk satisfy the constraint (22) with d¯ > 0, X ¯ γk bkij ≤ d. (22) 1≤i 0 ∈ P Cn×n satisfying P˙i + (Df (s(t)) + λi Γ)⊤ Pi (25) + Pi (Df (s(t)) + λi Γ) < 0, i = 1, 2, . . . , N, ¯ Then the synchronization where λi are the eigenvalues of A. of the network (5) is achieved under the switching law (23) and (24) with P = (Φ ⊗ In )P¯ (Φ⊤ ⊗ In ), P¯ = diag{P1 , P2 , . . . , PN } and Φ being a unitary matrix such ¯ = diag{λ1 , λ2 , . . . , λN }. that Φ⊤ AΦ Here P (t) : [0, ∞) → Rn×n is a time-varying matrix, and it is said to be positive definite (semi-definite), denoted by P > 0 (P ≥ 0), if there exists δ > 0 such that v ⊤ P (t)v ≥ δkvk2 (v ⊤ P (t)v ≥ 0) for any v ∈ Rn , 1 t ≥ 0. P Cn×n (P Cn×n ) is the linear space of the bounded continuous (continuously differentiable) real matrix-valued functions defined on [0, ∞). The proofs of Lemma 1 and Lemma 2 are straightforward applications of Proposition 4.1 and Theorem 4.3 in [11]; thus they are omitted. Lemma 1 gives a general principle on the synchronization of the network with switching controller, while Lemma 2 present a more detailed synchronization condition from the point of view of a convex combination method. Though the convex combination technique has been used to design stabilizing switching signals σ(t) in the field of switched systems for decades, there is no efficient way to compute coefficients of the convex combination θk for general switched systems, in particular when the number of the subsystems is greater than 2. Actually, this problem is known to be NP-hard [22]. In what follows, we will be able to use a convex combination method to design a synchronizing switching signal for the network (5). By combining the master stability function method and an inverse Lyapunov function for linear timevarying systems, we transform such a switching law design problem into a convex optimization problem which can be solved successfully by a mature and efficient numerical algorithm. Before giving the the main results of this section, we need the following assumption and lemma. Assumption 1: The equilibrium point xe = 0 of the timevarying system x(t) ˙ = Df (s(t))x(t)
(26)
is exponentially stable. Lemma 3 ([23]): Consider the linear time-varying system
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x(t) ˙ = H(t)x(t).
(27)
WeB18.1 Let xe = 0 be the exponentially stable equilibrium point of (27). Suppose H(t) ∈ P Cn×n . Let Q(t) > 0 ∈ P Cn×n . 1 Then there is a P (t) > 0 ∈ P Cn×n such that P˙ (t) + P (t)H(t) + H ⊤ (t)P (t) = −Q(t).
(28)
For unbounded synchronized region S = (−∞, α1 ], we can get the following theorem to find the synchronizing switching law. Theorem 4: Suppose Assumption 1 holds, and for given candidate controller set U, if the solutions γk∗ , θk∗ of the convex optimization (29) satisfy λ∗2 ≤ α1 , then the synchronization of the network (5) is achieved under the switching law (23) and (24) with the same P as defined in Lemma 2. m X θk γk Bk ) min λ2 (cA +
Similarly, for S = [α1 , α2 ], Theorem 5 gives the design procedure for the switching signal. Theorem 5: Suppose Assumption 1 holds, and for given candidate controller set U, if the solutions γk∗ , θk∗ of the convex optimization (33) satisfy λ∗2 ≤ α2 , then the synchronization of the network (5) is achieved under the switching law (23) and (24) with the same P as defined in Lemma 2. min λ2 (cA +
s.t.
s.t.
k=1
θk ∈ [0, 1], k = 1, 2, . . . , m Proof: P Similar to Proposition 3.2 in [24], we get that λ2 (cA + m k=1 θk γk Bk ) is a convex function which means that problem (29) is a convex optimization problem and there exists a global minimum solution λ∗2 . If λ∗2 ≤ α1 , based on master stability function method and Assumption 1, the following systems ¯ ω˙ i (t) = Df (s(t))ωi (t)+λi (A)Γω i (t), i = 1, 2, . . . , N. (30) are exponentially stable. Moreover, s(t) is a stable solution of (3), and f (·) is continuously differentiable, so Df (s(t)) is continuous and bounded. Then by the inverse Lyapunov function theorem (Lemma 3), there exist Pi (t) > 0 for each system in (30) with given Qi (t) > 0 such that P˙i +(Df (s(t)) + λi Γ)⊤ Pi +Pi (Df (s(t))+λi Γ) = −Qi , i = 1, 2, . . . , N.
(31)
Using Lemma 2, we can conclude that the network is synchronized under the switching law (23) and (24). Remark 2: So far, we only give a synchronizing switching law between the candidate controllers, but say nothing on those candidate controllers which satisfy the control constraint. It is natural to choose such controllers which minimize λ2 (cA + γk Bk ). One possible way to construct them is to solve the mixed-integer nonlinear optimization problem (32) repeatedly with different initial values, c
min λ2 (cA − γ
M X
e¯ci hci hci ⊤ )
i=1
c
s.t.
γ
M X
e¯ci
≤ d¯
i=1 c e¯i ∈ {0, 1},
(32) i = 1, 2, . . . , M c
γ > 0, and select some better ones in the feasible solutions. After that, solve (29) to check if there is a synchronizing switching law σ(t) for the selected controller group.
− λN (cA +
m X
θk γk Bk ) ≤ −α1
k=1 m X
(33)
θk = 1
k=1
θk ∈ [0, 1], k = 1, 2, . . . , m
(29)
θk = 1
θk γk Bk )
k=1
k=1
m X
m X
Proof: Based on [25], we have |λN (C)| + |λN (D)| ≥ |λN (C + D)| and |λN (aC)| = a|λN (C)| for any constant a > 0 and symmetric matrices C and D with appropriate dimensions. Here |λN (·)| is the spectral radius of a matrix. With Pm these properties, it is easy to verify that −λN (cA + k=1 θk γk Bk ) is a convex function, and (33) is a convex optimization problem. The rest of the proof is similar to that of Theorem 4. Remark 3: One can also solve similar optimization problems as (32) to get candidate controllers for bounded synchronized region, and it is omitted. V. E XAMPLE Next, we will give an example to show the effectiveness of the proposed results. The node dynamics considered is linear time-invariant, and s(t) = 0 ∈ Rn . Let c = 0.5, d¯ = 3, 0.25 0 1 1 0 0 , Γ = 0 0 0 , 0.4375 1 Df = 0 2 −3.6289 −1.4375 0 0 0 −1 0 0 1 0 0 0 0 0 0 0 −3 0 1 1 0 1 0 0 0 0 0 −1 1 0 0 0 0 0 0 1 1 1 −5 1 1 0 0 0 0 0 1 0 1 −5 1 0 1 0 1 . A= 0 0 0 1 1 −3 0 0 1 0 0 1 0 0 0 0 −1 0 0 0 0 0 0 0 1 0 0 −1 0 0 0 0 0 0 0 1 0 0 −1 0 0 0 0 0 1 0 0 0 0 −1 If a new link bij = bji = 1 is added, then describe it by bij = 1 for conciseness, and all the other elements of B are 0 if not mentioned. From [18], we see that the network (5) with the above Df and Γ has a S = (−∞, −1.1036], and can not synchronize with cλ2 (A) = −0.2929. Suppose γ = c for simplicity. By solving (16) 1000 times, we see that we can not get a single network controller under d∗ ≤ 3. So we turn to searching for a switching controller. We run the optimization problem (32) repeatedly
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coefficients which lead to the synchronizing switching signal between candidate controllers were obtained by solving a convex optimization problem. A possible way to design the candidate controllers was also discussed. It appears that the network control of networks structure, or feedback networks, is an interesting one to pursue further theoretically and for applications.
0
x11
−0.5 −1 −1.5 −2 0
50
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x
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0 −1 −2
R EFERENCES
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x
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Fig. 1.
The state response trajectories of x1 (t).
4
σ(t)
3
2
1 0
50
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t
Fig. 2.
The switching signal σ(t)
with different initial values, and select U = {u1 , u2 , u3 , u4 } with b11,9 = b13,7 = b17,8 = b17,9 = b18,10 = b19,10 = 1, b21,7 = b23,7 = b24,8 = b27,9 = b27,10 = b28,9 = 1, b31,3 = b31,9 = b33,7 = b33,8 = b37,9 = b37,10 = 1, b41,8 = b41,10 = b43,8 = b43,9 = b47,9 = b47,10 = 1. By calculating λ2 (cA + γBk ), it is clear that none of the single controllers uk ∈ U can synchronize the network. Solving (29) gives θ = [0.3487, 0.2176, 0.0627, 0.371] with ¯ = −1.155. According to Theorem 4, the network (5) λ2 (A) is synchronized under the switching signals (23) and (24). Fig. 1. shows the state response trajectories x1 (t) of the 1st node under the switching signal σ(t) which is shown in Fig. 2. The other nodes’ state response trajectories are similar with the ones of x1 (t). VI. C ONCLUSIONS This paper has investigated the local synchronization of dynamical networks by introducing a distributed controller of network form which has an overall control constraint cost function. The constrained controller design problem has been formulated as a mixed-integer nonlinear optimization problem. Furthermore, when no single controller can synchronize the network under the constraint, switching control was used to deal with such a problem. The convex combination
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