FLS-Based Adaptive Synchronization Control of ... - IEEE Xplore

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FLS-Based Adaptive Synchronization Control of Complex Dynamical Networks With Nonlinear Couplings and State-Dependent Uncertainties Xiao-Jian Li and Guang-Hong Yang, Senior Member, IEEE

Abstract—This paper is concerned with the problem of synchronization control of complex dynamical networks (CDN) subject to nonlinear couplings and uncertainties. An fuzzy logical system-based adaptive distributed controller is designed to achieve the synchronization. The asymptotic convergence of synchronization errors is analyzed by combining algebraic graph theory and Lyapunov theory. In contrast to the existing results, the proposed synchronization control method is applicable for the CDN with system uncertainties and unknown topology. Especially, the considered uncertainties are allowed to occur in the node local dynamics as well as in the interconnections of different nodes. In addition, it is shown that a unified controller design framework is derived for the CDN with or without coupling delays. Finally, simulations on a Chua’s circuit network are provided to validate the effectiveness of the theoretical results. Index Terms—Adaptive synchronization control, complex dynamical network (CDN), fuzzy logical system (FLS), graph theory, nonlinear coupling.

I. I NTRODUCTION N RECENT years, much attention has been paid to the complex dynamical network (CDN) due to their potential applications in various fields, such as physics, mathematics, engineering and automatic control [1]–[3]. In particular, the synchronization problem has been investigated by a number of researchers, since it is a typical collective behavior of complex networks, ranging from neural networks [4],

I

Manuscript received September 28, 2014; revised December 4, 2014 and January 20, 2015; accepted January 30, 2015. This work was supported in part by the National Science of China under Grant 61273148, Grant 61420106016, and Grant 61403070, in part by the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 201157, in part by the Fundamental Research Funds for the Central Universities under Grant N130405012 and Grant N140402002, in part by the China Post-Doctoral Science Foundation under Grant 2013M541241, in part by the Post-Doctoral Science Foundation of Northeastern University, and in part by the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries under Grant 2013ZCX01. This paper was recommended by Associate Editor M. J. Er. X.-J. Li is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China. G.-H. Yang is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China, and also with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2399334

impulsive dynamical networks [5], to chaotic circuits and systems [6], [7], and so on. In fact, the couplings which include the linear and nonlinear ones are important factors impacting the synchronization. By adjusting coupling gains or controller gains, some interesting synchronization analysis and synthesis results have been derived for CDN with linear couplings [8]–[13]. As to the nonlinear ones, an effective approach is to linearize the nonlinearities based on the norm-bounded conditions. For example, in [14] and [15], the problem of asymptotic synchronization of the CDN is studied through designing adaptive pinning control; in [16], a sufficient condition for the stability of uncertain CDN is derived in terms of linear matrix inequalities; in [17], the authors study robust impulsive synchronization of uncertain dynamical networks; and in [18], some simple synchronization criteria are provided via matrix measure. Besides, with the assumption that the nonlinear couplings are known beforehand, the online compensation strategy-based controller design schemes have also been given in [19] and [20]. Although the above methods have been proven to be capable of achieving synchronization for CDN, the topologies are generally required to be available for designing coupling gains or feedback gains. Note that the priori knowledge of a complex topology abstracted from real-world systems is usually unavailable and unmeasurable [21]. To overcome these difficulties, the adaptive techniques-based synchronization control approaches have been given in [21]–[23], where the topologies are allowed to be unknown. However, these results are derived in an uncertainty-free environment, and it is difficult to use the proposed adaptive techniques [21]–[23] to deal with system uncertainties, especially for the ones occurring in the interconnections of different nodes. Based on these observations, it is necessary to develop a new synchronization control method which is applicable for the CDN with system uncertainties and unknown topology, which motivates the study of this paper. On the other hand, due to the finite speed of information transmission and traffic congestions among the nodes, many large networks with communication inevitably encounter connection delays. Therefore, many approaches have been developed to address the synchronization problems of CDN with coupling delays. For instance, the authors in [24] and [25] investigate the exponential synchronization and robust stabilization problems of coupled networks with coupling delays; in [26], the synchronization issue of unknown coupled chaotic

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delayed fuzzy neural networks is studied; in [27], the problem of stochastic synchronization is investigated for a coupled stochastic complex networks; and in [28], the adaptive pinning control strategy in complex networks with nondelay and variable delay couplings is introduced. In spite of these efforts, the aforementioned problems are still not well resolved. Moreover, up to now, there is no unified framework to design synchronization controller for the CDN with or without coupling delays, which also motivates the present investigation. In this paper, the synchronization problem of CDN with nonlinear couplings and uncertainties is studied. Inspired by the works of [29]–[32], where the fuzzy logical system (FLS) are employed to model nonlinear functions, an FLS-based adaptive distributed controller design is presented to achieve the synchronization. By using the algebraic graph theory [33], a convex combination technique is developed to construct a global Lyapunov function, based on which the synchronization errors are proved to be asymptotically convergent. In contrast to the existing results, the proposed controller design method is applicable for the CDN with system uncertainties and unknown topology. Especially, the considered uncertainties are allowed to occur in the node local dynamics as well as in the interconnections of different nodes. In addition, a unified controller design framework is derived for the CDN with or without coupling delays. This paper is organized as follows. A CDN model and some necessary preliminaries as well as a communication protocol are presented in Section II. The adaptive distributed controller design for the uncertain CDN with or without coupling delays are summarized in Section III. In Section IV, an example on Chua’s circuit is given to illustrate the effectiveness of the proposed methods. Finally, the conclusion is given in Section V. II. P RELIMINARIES AND P ROBLEM S TATEMENT A. Preliminaries

T x1 x2 · · · xn , x = For a vector x = 2 2 x1 + x2 + · · · + xn2 , and x 1 =| x1 | + | x2 | + · · · + | xn | denotes the Euclidean norm and 1-norm, respectively. In addition, the following basic concepts and lemmas on graph theory are borrowed from [33]. A directed graph or digraph G = (V, E) contains a set V = 1, 2, . . . , N of vertices and a set E of arcs (edges) (i, j) leading from initial vertex i to terminal vertex j. A subgraph H of G is said to be spanning if H and G have the same vertex set. A digraph G is weighted if each arc ( j, i) is assigned a nonnegative weight lij . The weight w(H) of a subgraph H is the product of the weights on all its arcs. A directed path P in G is a subgraph with distinct vertices i1 , i2 , . . . , im such that its set of arcs is {(ik , ik+1 ): k = 1, 2, . . . , m − 1}. If im = i1 , we call P a directed cycle. A connected subgraph T is a tree if it contains no cycles, directed or undirected. A tree T is rooted at vertex i, called the root, if i is not a terminal vertex of any arcs, and each of the remaining vertices is a terminal vertex of exactly one arc. A subgraph Q is unicyclic if it is a disjoint union of rooted trees whose roots form a directed cycle.

Given a weighted digraph G with N vertices, define the weight matrix = (lij )N×N whose entry lij equals the weight of arc ( j, i). For our purpose, we denote a weighted digraph as (G, ), and it is used in this paper to model the topology of all links in the networks. A digraph G is strongly connected if, for any pair of distinct vertices, there exists a directed path from one to the other. A weighted digraph (G, ) is strongly connected if and only if the weight matrix is irreducible. The Laplacian matrix of (G, ) is defined as ⎤ ⎡ −l ··· −l1N k=1 l1k 12 ⎢ −l21 ··· −l2N ⎥ k=2 l2k ⎥ ⎢ L=⎢ ⎥. (1) .. .. .. . . ⎦ ⎣ . . . . −lN2 ··· −lN1 k=N lNk Let βi denote the cofactor of the ith diagonal element of L. The following result is standard in graph theory, and customarily called Kirchhoffs matrix tree theorem. We refer the reader to [34] for its proof. Lemma 1: Assume N ≥ 2. Then w(T ), i = 1, 2, . . . , N (2) βi = T ∈Ti

where Ti is the set of all spanning trees T of (G, ) that are rooted at vertex i, and w(T ) is the weight of T . In particular, if the digraph G is strongly connected, then βi > 0 for 1 ≤ i ≤ N. Based on the result of Lemma 1, a useful combinatorial identity has been derived in [33]. Lemma 2 [33]: Assume N ≥ 2. Let βi given in Lemma 1. Then the following identity holds: N

βi lij Fij xi , xj = w(Q)

i,j=1

Q∈Q

Frs (xr , xs ). (3)

(s,r)∈E(CQ )

Here Fij (xi , xj ), 1 ≤ i, j ≤ N, are arbitrary functions, Q is the set of all spanning unicyclic graphs of (G, ), w(Q) is the weight of Q, and CQ denotes the directed cycle of Q. B. Problem Statement In this paper, we consider a dynamical network on digraph G composed of identical nodes Gi , i = 1, 2, . . . , N with nonlinear coupling x˙ i (t) = f (xi (t), t) + c

N

aij g xj (t) + ui (t) + hi (x(t))

(4)

j=1 q where xi (t) ∈ RT is the state of note i, x(t) = x1 (t) x2 (t) · · · xN (t) is the state of the overall system; f ∈ Rq × R → Rq is continuously differentiable, g ∈ Rq → Rq is the nonlinear coupling function, and c and aij represent the unknown coupling strength and topological structure, respectively. Let lij = caij , thus L in (1) denotes the unknown coupling configuration matrix of the network. If lij > 0, then there is a connection between the ith node and the jth node; otherwise, lij = 0. ui (t) ∈ Rq is the control input; hi (x(t)) represents the system uncertainty, which may occur in the ith node local dynamics as well as in the interconnections of different nodes.

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Furthermore, let s(t) be a solution of the isolate node of the network, which is assumed to be unique and satisfies s˙(t) = f (s(t), t)

(5)

where s(t) can be an equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit. Denote ei (t) = xi (t) − s(t), then according to N j=1 lij g(s) = 0, the synchronized error dynamics can be described as e˙ i (t) = f (xi (t), t) − f (s(t), t) N

+ lij g xj (t) − g(s(t)) + ui (t) + hi (x(t)).

3

Now, the problem considered in this paper is formulated as follows. Problem: Given the CDN (4) with system uncertainty hi (x(t)) and unknown topology L, the main objective is to design an FLS-based adaptive distributed controller such that the synchronization errors ei (t) in (6) converge to zero asymptotically. III. FLS-BASED A DAPTIVE S YNCHRONIZATION C ONTROL

(6)

In this section, an FLS-based adaptive synchronization control scheme is presented.

j=1

Note that the uncertainty hi (x(t)) is more general than hi (xi (t)) and hi (t), which denote the uncertainties only occurring in the local dynamics of the ith node [14], [21]. To accommodate the uncertainty hi (x(t)), the state information should be allowed to exchange among nodes. However, in many practical applications of large-scale systems which include the CDNs, there is a need to minimize communication between controllers while maintaining a high level of performance [35]. Therefore, we refer to the communication protocol given in [35]. More specifically, the ith node can receive the state information xj (t) whenever the synchronized error ej (t) exceeds a given constant dj ( xj (t) − s(t) > dj ), otherwise, the ith node is only able to obtain the state variables xi (t) and s(t). Without loss of generality, define xj (t) if xj (t) − s(t) > dj x¯ j (t) = (7) s(t) if xj (t) − s(t) ≤ dj . Then, according to the above communication protocol T x¯ (t) = x¯ 1 (t) x¯ 2 (t) · · · x¯ n (t) (8) is available for all nodes. On the other hand, the controller design requires the following assumptions. Assumption 1: Nonlinearity f (xi (t), t) satisfies a Lipshitz condition, that is, there exists an unknown constant ηi for any two different constants xi , s ∈ Rn such that (xi − s)T P( f (xi , t) − f (s, t)) ≤ ηi xi − s 2

(9)

where P = diag{p1 , p2 , . . . , pn } is a positive definite matrix. Assumption 2: Suppose that there exists an unknown constant γj > 0 such that

(10) g xj (t) − g(s(t)) ≤ γj xj − s . Assumption 3: The digraph G is strongly connected. Assumption 4: The state xi (t) of the ith node is confined to a compact set Xi . Remark 1: Assumption 1 is the so-called one-side Lipschitz condition, which is mild in the CDN literature [8]–[10], [14]–[16]. For example, all linear and piecewise-linear time-invariant continuous functions satisfy this condition. Especially, the condition is satisfied if ∂fi /∂xj (i, j = 1, 2, . . . , n) are uniformly bounded. Similar conclusions also hold for Assumption 2. As to Assumptions 3 and 4, they are required for the technical reasons.

A. Fuzzy Logic Systems The FLS is usually used as a tool for modeling nonlinear functions due to their good capabilities in function approximation [36]. It is a collection of fuzzy IF-THEN rules of the form R(n) : IF z1 (t) is F1n and z2 (t) is F2n , . . . and zl (t) is Fln THEN y(t) is Bn , n = 1, . . . , M where zι (t) and y(t) are the input and output of the FLS, respectively. Fιn and Bn are fuzzy sets and M is the number of the fuzzy rules. Through singleton function, centeraverage defuzzification, and product inference, the FLS can be expressed as follows: M l n (zι ) y μ n F n=1 ι=1 ι (11) y(z) = M l n=1 ι=1 μFιn (zι ) T where z = z1 z2 · · · zl , μFιn (zι ) is the membership function of linguistic variable zι , yn is the point at which μBn reaches its maximum, and we assume that μBn (yn ) = 1. Then, the FLS (11) can be rewritten as follows: y(z) = ξ T (z)θ T y1 y2 · · · yM

and ξ(z) where θ T ξ1 (z) ξ2 (z) · · · ξM (z) with l μFιn (zι ) ι=1 , n = 1, 2, . . . , M. ξn (z) = M l n=1 ι=1 μFιn (zι )

(12)

Lemma 3 [36]: Let f (z) be a continuous function, which is defined on a compact set z . Then, for any constant δ0 , there exists an FLS (12), such as supz∈ z ξ T (z)θ ∗ − f (z) ≤ δ0

(13)

where θ ∗ is the ideal constant parameter. Then considering the system (6) in the presence of uncertainty hi (x(t)), one can assume that hi (x(t)) = ξiT (x(t))θi∗ + δi (x(t))

(14)

where the fuzzy-basis function ξi (x(t)) and constant parameter θi∗ are defined according to Lemma 3, and δi (x(t)) is the approximation error, which is bounded by the unknown constant δ0 . Also, it has been pointed out in [37] and [38] that



θi∗ and ξi (x(t)) usually satisfy the following conditions on the compact set Xi : θi∗ ≤ θ¯i , ξi (x(t)) − ξi (¯x(t)) ≤ mi x(t) − x¯ (t)

(15)

where θ¯i and mi are known positive constants, and x¯ is defined in (8). B. Synchronization Control for CDN Without Coupling Delay Before introducing the main results, the following notations are introduced: ⎞ ⎛ N lij γj2 + P−1 2 ⎠, αi = 2δ0 . ki := 2 ⎝ηi P−1 2 + j=1

(16) It is worth pointing out that the constants ηi , lij , γj , and δ0 are unknown, thus ki and αi are all unknown. Consider the dynamics described by (6), the following adaptive synchronization control law is designed for node i: 1 ui (t) = − ei (t) + ui1 (t) + ui2 (t) 2

(17)

θ˙˜i (t) = eTi P 1 i−1 ξi (¯x(t)).

Next, by (ei (t), ki (t), αi (t), θ˜i (t)), we denote a solution of the closed-loop CDN system. Then the following theorem can be derived. Theorem 1: Suppose that Assumptions 1–4 hold. Then, the synchronization errors ei (t) (i = 1, 2, . . . , N) described by (6) converge to zero asymptotically under the adaptive control laws (17)–(24). Proof: First, we define a Lyapunov function candidate for node i k2 α2 (28) Vi (t) = eTi Pei + γi1−1 i + γi2−1 i + θ˜iT i θ˜i . 2 2 Then, according to (6), the time derivative of Vi (t) for t > 0 satisfies V˙ i (t) = 2eTi P( f (xi , t) − f (s, t)) + 2

ρi (t) = sgn eTi P 1

− eTi Pei + 2eTi P −

T sgn eTi P n .

(18)

(19)

(21)

N

⎞

⎞

dj⎠ pi (t)⎠

Here, σ¯ iϑ are positive bounded constants. In (19), θî is the estimate of θi∗ , updated by the following adaptive law: (24)

where i is a given positive definite matrix. In (20), the notation sgn denotes the sign function, and (eTi P)i represents the ith element of vector eTi P. αi (t) = αˆ i − αi , θ˜i (t) = θî (t) − θi∗ , Denote ki (t) = kˆ i − ki , then we have the following error systems: (25) (26)

N

lij γj eTi P

j=1

× + ×

(22)

t→∞ t 0

ki (t)σi1 (t) − γi1 ki (t)σi1 (t) k˙ i (t) = γi1 eTi (t)P 2 −γi1 T α˙ i (t) = γi2 ei (t)P −γi2 αi (t)σi2 (t) − γi2 αi (t)σi2 (t)

≤ −λmin (P) ei 2 + 2ηi ei 2 + 2

(20)

γi1 and γi2 are any positive constants, and kˆ i (t0 ) and αˆ i (t0 ) are finite. σi1 (t) ∈ R+ and σi2 (t) ∈ R+ are any positive uniform continuous and bounded functions which satisfy t lim σiϑ (τ )dτ ≤ σ¯ iϑ < +∞, ϑ = 1, 2. (23)

θ˙î = eTi P 1 i−1 ξi (¯x(t))

+ 2eTi P ⎝hi (x(t)) + ⎝−θîT ξi (¯x(t)) − θ¯i mi

j=1

In (18), kˆ i (t) and αˆ i (t) are the estimates of ki and αi , updated by the following adaptive laws: ˙ kˆ i (t) = γi1 eTi (t)P 2 − γi1 kˆ i (t)σi1 (t) αˆ˙ i (t) = γi2 eTi (t)P − γi2 αˆ i (t)σi2 (t)

1 Pei eTi P 2 kˆ i2 2 eTi P 2 kˆ i + σi1

k˙ i α˙ i αi ki + γi2−1 + 2θ˜iT i θ˙˜i + γi1−1 ⎛ ⎛

j=1

···

lij eTi P g xj − g(s)

Pei αˆ i2 1 − 2 eTi P αˆ i + σi2

1 Pei (t) eTi (t)P 2 kˆ i2 (t) 2 eTi (t)P 2 kˆ i (t) + σi1 (t)

Pei (t)αˆ i2 (t) 1 − 2 eTi (t)P αˆ i (t) + σi2 (t) ⎛ ⎞ N ui2 (t) = ⎝−θîT ξi (¯x(t)) − θ¯i mi dj ⎠ ρi (t)

N j=1

where ui1 (t) = −

(27)

+

eTi P 4 kˆ i2 eTi P 2 αˆ i2 − ej − eTi P αˆ i + σi2 eTi P 2 kˆ i + σi1 k˙ i ki + γi2−1 2θ˜iT i θ˜˙i + γi1−1 αi + 2 eTi P α˙ i

∗T θi ξi (x(t)) + δi (x) ⎛ ⎞ N 2 eTi P 1 ⎝−θîT ξi (¯x(t)) − θ¯i mi dj ⎠ j=1

≤ −λmin (P) ei + 2ηi ei N + lij γj2 eTi PPei + eTi ei + eTj ej − eTi ei + 2θ˜iT i θ˙˜i 2

2

j=1

eTi P 4 kˆ i2 eTi P 2 αˆ i2 − eTi P αˆ i + σi2 eTi P 2 kˆ i + σi1 k˙ i α˙ i ki + γi2−1 + 2δ0 eTi P + γi1−1 αi ⎛ −

+ 2 eTi P 1⎝θi∗T ξi (x(t)) − θîT ξi (¯x(t)) − θ¯i mi ⎞ + θi∗T ξi (¯x(t)) − θi∗T ξi (¯x(t))⎠

N

dj

j=1

(29)

where λmin (P) denotes the minimum eigenvalue of P. In fact, the inequality eTi P ≤ eTi P 1 has been used to get (29).


By (16), (25)–(27), we have

5

Now, we construct a global Lyapunov function V(t)

V˙ i (t) ≤ −λmin (P) ei 2 + ki eTi P 2 + αi eTi P N eTi P 4 kˆ i2 + lij eTj ej − eTi ei − eTi P 2 kˆ i + σi1 j=1 eTi P 2 αˆ i2 − + eTi P 2 − ki σi1 − ki σi1 ki T ei P αˆ i + σi2

αi + eTi P − αi σi2 − αi σi2 ⎞ ⎛ N + 2 eTi P 1 ⎝θi∗T ξi (x(t)) − θ¯i mi dj + θi∗T ξi (¯x(t))⎠

V(t) =

N

≤ −λmin (P) ei 2 +

lij eTj ej − eTi ei

˙ V(t) ≤ −λmin (P)

N

βi ei 2 +

i=1

j=1

(30) Next, in terms of the following inequality 0 < ab/a + b < a, ∀a, b > 0, from (30), we can obtain: V˙ i (t) ≤ −λmin (P) ei + + σi1 ⎛

ki2

i=1 j=1

+

4

+ 1 + σi2

× ⎝θi∗T ξi (x(t)) − θ¯i mi

αi2 4

N

≤ −λmin (P) ei 2 +

dj + θi∗T ξi (¯x(t))⎠

+ μi (σi1 + σi2 ) + 2 eTi P1 ⎛ ⎞ N × ⎝θi∗T ξi (x(t)) − θ¯i mi dj + θi∗T ξi (¯x(t))⎠ (31) j=1

j=1

then, we have N

=

w(Q)

Q∈Q

T es es − eTr er . (34)

(s,r)∈E(CQ )

Without loss of generality, for any directed cycle CQ , the set E(CQ ) can be described as E(CQ ) = {(ik , ik+1 ) | k = 1, 2, . . . , m − 1, m ≤ N, im = i1 }. (35)

+ · · · eTim−1 eim−1 − eTim eim + eTim eim − eTi1 ei1 = 0.


j=1

+ μi (σi1 + σi2 ).

(36)

From (33)–(36), it follows that: ˙ V(t) ≤ −λmin (P)

N

βi ei 2 +

i=1

N

βi μi (σi1 + σi2 ).

(37)

i=1

T Let x˜ i (t) = eTi (t) kiT (t) αiT (t) θiT (t) , thus according to the definition of Lyapunov functions given in (28), there always exists a positive constant i such that 0 ≤ i x˜ i (t) ≤ Vi (t)

0≤

θi∗T ξi (x(t)) − θi∗T ξi (¯x(t)) ≤ θi∗T ξi (x(t)) − ξi (¯x(t)) ≤ θ¯i mi x(t) − x¯ (t)

V˙ i (t) ≤ −λmin (P) ei 2 +

i=1 j=1

then

where μi = max(ki2 /4 + 1, αi2 /4 + 1). Note that

dj

(33)

βi lij eTj ej − eTi ei

⎞


N

N N

+ 1 + 2 eTi P 1

j=1

≤ θ¯i mi


(s,r)∈E(CQ )

j=1 N

N

Based on (35), we have eTs es − eTr er = eTi1 ei1 − eTi2 ei2 + eTi2 ei2 − eTi3 ei3


j=1

βi lij eTj ej − eTi ei

By using Lemma 2, we have

σi2 eTi P αˆ i σi1 eTi P 2 kˆ i + + eTi P αˆ i + σi2 eTi P 2 kˆ i + σi1

+ − ki σi1 − ki σi1 αi σi2 − αi σi2 ) αi ki + (− ⎞ ⎛ N + 2 eTi P 1 ⎝θi∗T ξi (x(t)) − θ¯i mi dj + θi∗T ξi (¯x(t))⎠.

N

N N

i=1

j=1

2

(32)

where βi is given in Lemma 1. According to the Assumption 3, it is known that βi > 0. Then, we have

j=1 N

βi Vi (t)

i=1

N

βi i x˜ i (t) ≤ V(t) ≤ V(t0 ) +

˙ )dτ V(τ

t0

i=1

≤ V(t0 ) − λmin (P) +

t

t N

t N

βi ei (τ ) 2 dτ

t0 i=1

βi μi (σi1 (τ ) + σi2 (τ ))dτ

t0 i=1 N

≤ V(t0 ) +

βi μi (σ¯ i1 + σ¯ i2 )

(38)

i=1

which means that the solutions of the synchronization error system described by (6) and adaptive estimation error systems



described by (25)–(27) are uniformly bounded. Also, (38) implies that t N λmin (P)βi ei (τ ) 2 dτ lim t→∞ t 0 i=1

≤ V(t0 ) +

N

where τ is an unknown constant delay, and other parameters and variables are the same as those of system (4). By (41), we get the synchronized error dynamics

e˙ i (t) = f (xi (t), t) − f (s(t), t) N

lij g xj (t − τ ) − g(s(t − τ )) + ui (t) + hi (x(t)). + (39)

βi μi (σ¯ i1 + σ¯ i2 ).

j=1

i=1

Since x˜ i (t) is uniformly bounded, from Assumptions 1 and 2 it follows that ei (t) and e˙ i (t) are uniformly bounded, which implies ei (t) is uniformly continuous. Therefore, N λ (P)βi ei (t) 2 is also uniformly continumin i=1 ous. Applying the Barbalat’s Lemma [39] to (39) yields λ (P)βi ei (t) 2 = 0, that is limt→∞ N min i=1 lim ei (t) = 0

t→∞

which completes the proof. Remark 2: It should be pointed out that the ui2 (t) in (17) is introduced for the purpose of compensating the uncertainty hi (x(t)). If hi (x(t)) = 0, the control law can be designed as 1 ui (t) = − ei (t) − ui1 (t) 2

(42) Based on (42), the following theorem can be derived. Before giving the main result, we introduce the notations ei (t) ei , ei (t − τ ) eiτ for simplification. Theorem 2: Suppose that Assumptions 1–4 hold. Then, the synchronization errors ei (t)(i = 1, 2, . . . , N) given in (42) converge to zero asymptotically under the adaptive control laws (17)–(24). Proof: Consider the Lyapunov functional candidate for node i k2 α2 Vi (t) = eTi Pei + γi1−1 i + γi2−1 i + θ˜iT i θ˜i 2 2 t N + lij eTi ei ds. (43)

with

t−τ

j=1

ui1 (t) = −

1 Pei (t) eTi (t)P 2 kˆ i2 (t) . 2 eTi (t)P 2 kˆ i (t) + σi1 (t)

Then, according to (42), the time derivative of Vi (t) for t > 0 satisfies

Furthermore, let σi1 (t) = 0 and P = I, then it follows that ui (t) = −1/2ei (t) −1/2ei (t)kˆ i (t). Define di (t) = 1/2 +1/2kˆ i (t) and φi = 1/2γi1 , we have

V˙ i (t) = 2eTi P ( f (xi , t) − f (s, t)) + 2

ui (t) = −di (t)ei (t) d˙ i (t) = φi ei 2.

− eTi Pei + 2eTi P −

In fact, the system (4) with hi (x(t)) = 0 has been studied in [23], where the control law (40) has also been given based on adaptive techniques. In this sense, the Theorem 1 extends the result of [23] to a more general case. Subsequently, according to (17)–(24), the controller design procedures are summarized in the following. Step 1: Determine a positive definite matrix Pi such that Assumption 1 is satisfied. Give σi1 (t) and σi2 (t) according to (23), then the adaptive estimations κˆ i (t) and αˆ i (t) can be derived with the appropriate choice of constants γi1 and γi2 . Step 2: Choose a fuzzy basis function ξi (¯x(t)) and give a positive definite matrix i , then the adaptive estimation θî (t) can be derived. Step 3: On the basis of Steps 1 and 2, one can obtain ui1 (t) and ui2 (t) in (17). C. Synchronization Control for CDN With Coupling Delay Here, it is shown that the proposed adaptive control laws (17)–(24) are still valid for the following CDN with coupling delay: x˙ i (t) = f (xi (t), t) N

+c aij g xj (t − τ ) + ui (t) + hi (x(t)) (41) j=1

lij eTi P g xjτ − g(sτ )

j=1

(40)

N

1 Pei eTi P 2 kˆ i2 2 eTi P 2 kˆ i + σi1

Pei αˆ i2 1 − T 2 ei P αˆ i + σi2 α˙ i αi ki + γi2−1 + 2θ˜iT i θ˙˜i + γi1−1 k˙ i ⎛ ⎛ + 2eTi P⎝hi (x(t)) + ⎝−θîT ξi (¯x(t)) − θ¯i mi

N

⎞

dj ⎠ pi (t)⎠

j=1

+

N

lij eTi ei − eTiτ eiτ

j=1

≤ − λmin (P) ei 2 +2ηi ei 2 +2

N

lij γj eTi P

j=1

× +

eTi P 4 kˆ i2 eTi P 2 αˆ i2 − ejτ − eTi P αˆ i + σi2 eTi P 2 kˆ i + σi1 k˙ i α˙ i 2θ˜iT i θ˙˜i + γi1−1 αi + 2eTi P(hi (x(t)) ki + γi2−1 ⎛

+ ⎝−θîT ξi (¯x(t)) − θ¯i mi

N

⎞

dj ⎠ ki (t))

j=1

+

N

lij eTi ei − eTiτ eiτ

j=1

≤ −λmin (P) ei 2 + 2ηi ei 2

⎞


+

N

7

lij γj2 eTi PPei + eTi ei + eTjτ ejτ − eTiτ eiτ

j=1

eTi P 4 kˆ i2 eTi P 2 αˆ i2 − eTi P αˆ i + σi2 eTi P 2 kˆ i + σi1 α˙ i ki + γi2−1 + 2θ˜iT i θ˜˙i + γi1−1 αi + 2δ0 eTi P k˙ i ⎛ −

+ 2 eTi P 1 ⎝θi∗T ξi (x(t)) − θîT ξi (¯x(t)) − θ¯i mi

Fig. 1. N

⎞

Topological structure.

dj

j=1

+ θi∗T ξi (¯x(t)) − θi∗T ξi (¯x(t))⎠ where λmin (P) denotes the minimum eigenvalue of P. By Theorem 1, we have V˙ i (t) ≤ −λmin (P) ei 2 +

N

lij eTjτ ejτ − eTiτ eiτ

j=1

+ μi (σi1 + σi2 ). Now, we construct a global Lyapunov function V(t) V(t) =

N

βi Vi (t)

Fig. 2.

(44)

i=1

where βi is given in Lemma 1. Then, we have ˙ V(t) ≤ −λmin (P)

N

+

N βi lij eTjτ ejτ − eTiτ eiτ + βi μi (σi1 + σi2 ).

i=1 j=1

i=1

(45) By using Lemma 2 and Theorem 1, we have N N

βi lij eTjτ ejτ − eTiτ eiτ

i=1 j=1

=

w(Q)

Q∈Q

T esτ esτ − eTrτ erτ = 0

(s,r)∈E(CQ )

which follows that: ˙ V(t) ≤ −λmin (P)

N i=1

Consider a network of ten Chua’s circuits coupled via the digraph shown in Fig. 1. The single Chua’s circuit shown in Fig. 2 is described by the following differential equations [40]: 1 (−v1 + v2 ) − f (v1 ) R 1 C2 v˙ 2 (t) = (v1 − v2 ) + i3 R L˙i3 (t) = −(v2 + R0 i3 )

C1 v˙ 1 (t) =

βi ei 2

i=1 N N

Chuas circuit.

βi ei + 2

N


(46)

i=1

The rest of the proof follows the same steps as those in Theorem 1. Remark 3: Combining Theorems 1 and 2, it can be concluded that the proposed synchronization control method is valid not only for the system (4) without coupling delay, but also for the system (41) with coupling delay. IV. E XAMPLE In this section, an example is given to demonstrate the effectiveness of the proposed adaptive synchronization method.

where v1 and v2 are the voltages across the capacitors C1 and C2 , respectively, i3 denotes the current through the inductances L, R0 and R are linear resistors, the term f (v1 ) represents the current through the nonlinear resistor NR which is a piecewiselinear function expressed as f (v1 ) = Gb1 v1 + 0.5(Ga1 − Gb1 ) ( v1 + 1 − v1 − 1 ). The networked system is then defined as follows: ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ x˙ i1 −p p 0 xi1 −ωf (xi1 ) ⎣ x˙ i2 ⎦ = ⎣ q ⎦ −q r ⎦ ⎣ xi2 ⎦ + ⎣ 0 0 −v −z x˙ i3 xi3 0 +

10

lij g(xj (t)) + ui (t)

(47)

j=1

where xi1 = vi1 , xi2 = vi2 , xi3 = ii3 , p = 1/RC1 , q = 1/RC2 , r = 1/C2 , v = 1/L, z = R0 /L, ω = 1/C1 , i = 1, 2, . . . , 5. Similar to [14], the system parameters are selected as p = 9.1, q = 1, r = 1, ω = 9.1, Gb1 = −0.7559, Ga1 = −1.39386, v = 16.5811, z = 0.138083. To illustrate the effectiveness of the proposed method, the simulation results are considered and the following parameters related with controller design are given: P = I, γi1 = 1, γi2 = 50, σi1 (t) = e−10t , σi2 (t) = e−100t , κˆ i (0) = αˆ i (0) = 0,



Fig. 3.

Synchronization error of system (47) without coupling delay.

Fig. 4.

Estimate of ki .

Fig. 5.

Estimate of αi .

Fig. 6.

Control inputs of node 1.

Fig. 7.

Control inputs of node 2.

i = I, θ¯i = 5, mi = 1, di = 0.5, i = 1, 2, . . . , 10: T T x1 (0) = 4 −1 −4 , x2 (0) = 1 −1 −0.5 T T x3 (0) = 2 −0.5 −0.6 , x4 (0) = −3 1 1 T T x5 (0) = 0 −1 −2 , x6 (0) = 0 −0.1 −0.2 T x7 (0) = −2.6 −1 1.7 T x8 (0) = 0.8 1.1 −2.6 T T x9 (0) = 0.6 2.1 2.2 , x10 (0) = 0 −1 −2 T s(0) = 5 5 −3 . In addition, the system uncertainties and coupling functions are assumed to be h1 (x(t)) = 2log2 (| x2 (t) |), h2 (x(t)) = 2log2 (| x1 (t) |) h3 (x(t)) = 2log2 (| x2 (t) |), log2 | xi (t) | T = log2 | xi1 (t) | log2 | xi2 (t) | log2 | xi3 (t) | i = 1, 2, 3.

hi (x(t)) = 0, i = 4, 5, 6, 7

hi (x(t)) = 5xi (t), i = 8, 9, 10. g xj (t) = 15sin xj (t) . The FLS in (19) is defined as follows. For i = 1, 2, 3, we define three fuzzy sets over each axis, which label as Fιn , ι = 1, 2, 3, n = 1, 2, 3. The fuzzy mem2 bership functions are μFι1 (zι ) = e−(zι +2) , μFι2 (zι ) = e−zι , μFι3 (zι ) = e−(zι −2) , zι = x¯ jι . For i = 8, 9, 10, μFι1 (zι ) = 2

2

e−(zι +2) , μFι2 (zι ) = e−zι , μFι3 (zι ) = e−(zι −2) , zι = x¯ iι . The error trajectories under the distributed adaptive controller described by (17)–(24) are depicted in Fig. 3, from which it can be observed that the asymptotical synchronization can be achieved for the CDN with nonlinear couplings and uncertainties, which verifies the theoretical result in Theorem 1 well. In addition, from Figs. 4 and 5, it can be seen that the estimations κˆ i (t) and αˆ i (t) converge to finite steady-state values. Moreover, the control inputs for the nodes 1 and 2 are displayed in Figs. 6 and 7, and others are omitted here for simplification. These two figures illustrate that the synchronization is achieved at a price of control chattering, and this is due to the fact that the sign functions are introduced in u2i (t). On the other hand, assume that the coupling delay is τ = 10, the synchronization errors are plotted in Fig. 8, from which we get the following conclusion: the proposed adaptive synchronization control method is valid not only for the CDN without 2

2

2

coupling delay, but also for the ones with coupling delay. Yet, the existing results in the literature cannot provide such unified controller design approach, which further shows the significance of this paper. Also, the estimate values κˆ i (t) and αˆ i (t)


9

have been given to illustrate the effectiveness of the proposed methods. R EFERENCES

Fig. 8.

Synchronization error of system (47) with coupling delay τ = 10.

Fig. 9.

Estimate of ki .

Fig. 10.

Estimate of αi .

are displayed in Figs. 9 and 10. Similar to Figs. 4 and 5, it is known that these estimates are all convergent and bounded. V. C ONCLUSION In this paper, the FLS-based adaptive synchronization control of CDNs with nonlinear couplings and uncertainties has been considered. A new adaptive distributed controller has been designed to achieve the synchronization. Via the algebraic graph theory and Lyapunov theory, the rigorous convergence analysis of synchronization errors has also been conducted. In terms of the ability of addressing system uncertainties, which are allowed to occur in the node local dynamics as well as in the interconnections of different nodes, this paper can be seen as a significant extension of the existing results. Especially, a unified synchronization controller design framework has been derived for the CDNs with or without coupling delays. The simulation results on a Chua’s circuit network

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Xiao-Jian Li received the B.S. and M.S. degrees, both in mathematics from Northeast Normal University, Changchun, China, in 2003 and 2006, respectively, and the Ph.D. degree in control theory and engineering from Northeastern University, Shenyang, China, in 2011. His current research interests include fault diagnosis, fault-tolerant control, fuzzy systems, and complex networks.

Guang-Hong Yang (SM’04) received the B.S. and M.S. degrees from Northeast University of Technology, Shenyang, China, in 1983 and 1986, respectively, and the Ph.D. degree in control engineering from Northeastern University, Shenyang (formerly Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor at Northeastern University from 1986 to 1995. In 1996, he joined Nanyang Technological University, Singapore, as a Post-Doctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist at the National University of Singapore, Singapore. He is currently a Professor with the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, nonfragile control systems design, and robust control. Prof. Yang is an Associate Editor of the International Journal of Control, Automation, and Systems, the International Journal of Systems Science, the IET Control Theory and Applications, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS.