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Adaptive Fault-Tolerant Synchronization Control of a Class of Complex Dynamical Networks With General Input Distribution Matrices and Actuator Faults Xiao-Jian Li and Guang-Hong Yang, Senior Member, IEEE

Abstract— This paper is concerned with the problem of adaptive fault-tolerant synchronization control of a class of complex dynamical networks (CDNs) with actuator faults and unknown coupling weights. The considered input distribution matrix is assumed to be an arbitrary matrix, instead of a unit one. Within this framework, an adaptive fault-tolerant controller is designed to achieve synchronization for the CDN. Moreover, a convex combination technique and an important graph theory result are developed, such that the rigorous convergence analysis of synchronization errors can be conducted. In particular, it is shown that the proposed fault-tolerant synchronization control approach is valid for the CDN with both time-invariant and timevarying coupling weights. Finally, two simulation examples are provided to validate the effectiveness of the theoretical results. Index Terms— Adaptive fault-tolerant control, adaptive synchronization, complex dynamical networks (CDNs), graph theory, linear couplings.

I. I NTRODUCTION

I

N THE past decades, much attention has been paid on the complex dynamical network (CDN) due to many applications in physics, biology, and engineering [1]–[5]. The CDN usually consists of a number of nodes communicating over a web of interconnections, where the coupling strength and the topology structure are used to describe how nodes communicate with each other. A notable meeting point of the research is Manuscript received April 9, 2015; revised October 6, 2015 and December 2, 2015; accepted December 6, 2015. This work was supported in part by the National Science Foundation of China under Grant 61273148, Grant 61420106016, and Grant 61403070, in part by the Foundation for the Author of National Excellent Doctoral Dissertation, 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 China Post-Doctoral Science Foundation Special Funded Project under Grant 2015T80263, 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. (Corresponding author: Guang-Hong Yang.) X.-J. Li is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China (e-mail: [email protected]). 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/TNNLS.2015.2507183

the topic synchronization that widely exists in CDN, ranging from natural systems to man-made networks [6]–[9]. In the literature, an effective synchronization approach is to adjust the coupling strengths or design the feedback gains, such that all the systems in the network synchronize on the same evolution. For instance, an important pinning control approach with minimum number of controllers was given in [10] to force the coupled neural networks to the desired equilibrium point. The decentralized control idea was introduced in [11] for undirected CDN. In [12], a general pinning control scheme was given for CDN that includes strongly connected networks, networks with a directed spanning tree, and weakly connected networks. In [13], the cluster synchronization by a generalized pinning control strategy was studied for networked linear systems with more relaxed topological structure. On the other hand, a basic result in the synchronization of linear systems by output coupling was presented in [14]. An adaptive law of coupling strength for the global synchronization of CDN was provided in [15]. The analysis of the synchronization of networks of nonlinear oscillators through an innovative adaptive law of coupling strength was studied in [16]. The global synchronization of CDN with network failures was studied in [17] based on the switching system. The synchronization of a network under a possibly time-varying and directed interconnection structure was considered in [18]. A simple synchronization criterion that the coupling strength should be larger than some threshold was derived in [19]. In particular, the relaxed synchronization conditions were presented in [20] for general linear multiagent systems under a dynamic topology. Note that the topology structures in the above results are generally required to be available for designing the synchronization criteria. However, just as mentioned in [21], the prior knowledge of a complex topology abstracted from real-world systems is usually unavailable and unmeasurable. To tackle this difficulty, some controllers with an adaptive mechanism have been designed to compensate the coupling terms. The local and global adaptive synchronization controllers were designed in [22], where the topology structure and the coupling strength are allowed to be unknown. In addition, the corresponding adaptive controller design techniques were also given in [21] and [23] for CDN with delayed couplings or delayed nodes. These controllers are rather simple in form, which is very useful for practical engineering design.

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On the other hand, some adaptive synchronization control approaches have been developed for CDN with system uncertainties and network faults. For example, the robust adaptive synchronization methods for an uncertain CDN with network faults were given in [24] and [25]. The robust synchronization problem of fuzzy CDNs was investigated in [26], and the fuzzy logical system-based adaptive synchronization control method was introduced in [27] for CDN with an unknown topology structure and system uncertainties. A common feature of [21]–[27] is that the input distribution matrix is required to be an identical one. This implies that the unknown couplings, uncertainties, network faults, and control inputs share the same channels, and then, one can design adaptive controllers to compensate these unknown factors. However, the above requirement on the input distribution matrix is strict and it cannot be satisfied for some practical control systems. Recently, a synchronization controller was designed in [28] for stochastic CDN based on linear matrix inequality (LMI) and adaptive techniques. The major contribution of [28] is that the coefficient matrix of the adaptive controller is an arbitrary one. Nevertheless, if the coupling weights are unknown and time-variant, the proposed controller design approach cannot be used because they are involved in the LMI conditions. In particular, how to compensate the actuator faults, which usually occur in the local dynamics of the isolated node [29], is not considered in [28]. In fact, some effective decentralized fault-tolerant control schemes have been given in the literature [30], [31] for interconnected systems, where the coefficient of the single input is reversible. In this case, the interconnections can be compensated by designing an appropriate controller. In addition, the fault-tolerant control problems have also been studied in [32] and [33] for multiagent systems; however, the coupling weights are required to be known and time-invariant, and the control input is single input. To the best of our knowledge, the fault-tolerant synchronization control problem of CDN has not been solved under the preconditions that the input distribution matrix is an arbitrary one and the coupling weights are unknown and time-varying. The main difficulties are summarized in the following. First, since the control inputs and the couplings do not share the same channels, how to address the coupling terms is a challenging task. Second, the actuator fault compensation and the convergence analysis of synchronization errors in the presence of unknown coupling weights also present theoretical challenges. Based on these observations, it is necessary to develop some new techniques to resolve these difficulties, which motivates the current investigation. This paper studies the fault-tolerant synchronization control problem of CDN with unknown time-invariant or time-varying coupling weights. The input distribution matrix is assumed to be an arbitrary matrix, instead of a unit one. Within this framework, the major contribution is that an adaptive fault-tolerant controller is designed to achieve synchronization for the CDN subject to actuator faults. The proposed fault-tolerant synchronization control approach relies on the following key techniques. First, by using the algebraic graph theory [34], it is proved that, whether the unknown coupling

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weights are time-invariant or not, the linear couplings can generate a positive effect on the synchronization, and they do not need to be compensated. Second, based on the above result, a new adaptive fault-tolerant control scheme and a convex combination technique are developed, such that the synchronization errors are proved to be asymptotically convergent. This paper is organized as follows. A CDN model and some necessary preliminaries are presented in Section II. The adaptive fault-tolerant controller design approach for the CDN with unknown coupling weights and node faults is summarized in Section III. In Section IV, two examples are given to illustrate the effectiveness of the proposed method. Finally, the conclusions are given in Section V. II. P RELIMINARIES AND P ROBLEM S TATEMENT A. Preliminaries For a vector x = [x 1 x 2 · · · x n ]T , x = + x 22 + · · · + x n2 )1/2 denotes the Euclidean norm, and for a scalar y, |y| represents the absolute value. A block diagonal matrix with matrices X 1 , X 2 , . . . , X n on its main diagonal is denoted as diag(X 1 , X 2 , . . . , X n ). In addition, the following basic concepts and lemmas on graph theory are borrowed from [34]. 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 li j . 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 i 1 , i 2 , . . . , i m , such that its set of arcs is {(i k , i k+1 ) : k = 1, 2, . . . , m − 1}. If i m = i 1 , 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 = (li j ) N×N whose entry li j 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 the 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 ⎤ ⎡ l1k −l12 ··· −l1N ⎥ ⎢ k=1 ⎢ −l l2k · · · −l2N ⎥ 21 ⎥ ⎢ ⎥ ⎢ k =2 L=⎢ (1) ⎥. .. .. .. ⎥ ⎢ .. . ⎥ ⎢ . . . ⎦ ⎣ −l N1 −l N2 · · · l Nk (x 12

k = N

Let βi denote the cofactor of the i th diagonal element of L. The following result is standard in graph theory, and

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customarily called Kirchhoff’s matrix tree theorem. We refer the reader to [35] for its proof. Lemma 1 [35]: 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 [34]. Lemma 2 [34]: Assume N ≥ 2. Let βi be given in Lemma 1. Then, the following identity holds: N

βi li j Fi j (x i , x j ) =

Q∈Q

i, j =1

w(Q)

Frs (xr , x s )

(3)

(s,r)∈E(CQ )

where Fi j (x i , x j ), 1 ≤ i, j ≤ N, are the 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.

3

Denote

T F (t) = αih u i (t) + δih (t) u hi F (t) = u hi,1F (t), u hi,2F (t), . . . , u hi,m where

h h h , . . . , αi,k , . . . , αi,m αih = diag αi,1

h T h h δih (t) = δi,1 (t), . . . , δi,k (t), . . . , δi,m (t) i = 1, . . . , N, k = 1, 2, . . . , m, h = 1, 2, . . . , H. (7)

For simplicity of presentation, for all H possible faulty modes, the uniform actuator fault model is formulated by u iF (t) = αi u i (t) + δi (t)

(8)

where αi ∈ {αi1 , . . . , αiH } and δi ∈ {δi1 , . . . , δiH }. Then, the dynamics of node (4) with actuator faults (8) is described by x˙i (t) = Ax i (t) + f (x i (t)) + B(αi u i (t) + δi (t)) N + li j (x j (t) − x i (t)).

(9)

j =1

B. Problem Statement In this paper, we consider a dynamical network on digraph G composed of identical nodes G i , i = 1, 2, . . . , N with linear coupling x˙i (t) = Ax i (t) + f (x i (t)) + Bu i (t) + c

N

ai j (x j (t) − x i (t))

j =1

(4) where x i (t) ∈ Rq is the state of node i , f ∈ Rq → Rq is a nonlinear function, and u i ∈ Rm is the control input. A and B are arbitrary matrices with appropriate dimensions. c and ai j represent the unknown coupling strength and topological structure, respectively. Let li j = cai j , and thus li j represent the unknown coupling weights and L in (1) denotes the coupling configuration matrix of the network. If li j > 0, then there is a connection between the i th node and the j th node; otherwise, li j = 0. The following actuator fault model is considered: h h u hi,kF (t) = αi,k u i,k (t) + δi,k (t)

(5)

where i = 1, 2, . . . , N, k = 1, 2, . . . , m, h = 1, 2, . . . , H , and h are the unknown actuator efficiency factors satisfying αi,k h h 0 < α hi,k ≤ αi,k ≤ α i,k ≤ 1.

(6)

h (t) is the unknown time-varying bounded signal, the δi,k index h represents the hth faulty mode, and H is the number of all faulty modes. Note that (5) implies the following three cases. h h = 1 and δi,k (t) = 0. This means the 1) α hi,k = α i,k fault-free case. h < 1 and δ h (t) = 0. This indicates the 2) 0 < α hi,k ≤ α i,k i,k partial loss of effectiveness. h = 1 and δ h (t) = 0. This indicates the bias 3) α hi,k = α i,k i,k fault.

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

(10)

where s(t) can be an equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit. Denote ei (t) = x i (t)−s(t), and then the synchronized error dynamics can be described as e˙i (t) = Aei (t) + f (x i (t)) − f (s(t)) + B(αi u i (t) + δi (t)) N + li j (x j (t) − x i (t)). (11) j =1

On the other hand, the following Assumptions 1-3 are required in this paper. Assumption 1: The digraph G is strongly connected. Assumption 2: Nonlinearity function f (x i (t)) satisfies the following equality: f (x i (t)) − f (s(t)) = Mei (x i (t) − s(t))

(12)

where Mei = (m ei (t))q×q is a bounded matrix, in which the elements m ei (t) satisfy m ei ≤ m ei (t) ≤ m¯ ei , and m ei , m¯ ei are known constants. Assumption 3: The actuator bias faults are bounded, that is, there exist an unknown constant δ¯i , such that δi (t) ≤ δ¯i .

(13)

Remark 1: Assumption 1 is typical in the CDN literature [10], [12], [19], [27], [34]. According to Lemma 1, Assumption 1 is introduced for guaranteeing βi > 0, which plays an important role in the convergence analysis. On the other hand, Assumptions 2 and 3 are given for the technical reasons on the controller design. In fact, some concrete examples satisfying Assumption 2 have been given in [36], where the nonlinear function f (x i (t)) is only required



to be continuous. Moreover, according to the differential mean value theorem [37], Assumption 2 also holds when f (x i (t)) is differentiable. In addition, Assumption 3 has been widely used in the existing fault-tolerant control results [29], [33], [38]–[41]. Now, the considered problem is summarized as follows. Problem: Given the CDN consisting of nodes (9) with unknown coupling weights, the main objective is to design an adaptive fault-tolerant controller, such that the synchronization errors ei (t) in (11) converge to zero asymptotically. III. A DAPTIVE FAULT-T OLERANT C ONTROLLER D ESIGN In this section, an adaptive fault-tolerant synchronization controller design approach is presented. A. CDN With Time-Invariant Coupling Weights We first consider the CDN with time-invariant coupling weights li j . Using Lemma 2 and matrix factorization technique, an important property on linear couplings will be given in this section. The derived property reveals that the linear couplings can generate positive effect on the synchronization, and they do not need to be compensated. Thus, one can own more design freedom to compensate the actuator faults even when the input matrix is an arbitrary one. To this end, the following adaptive control laws are designed for node (9) with actuator faults: u i (t) = u i1 (t) + u i2 (t)

(14)

Subsequently, by (ei , k˜i ), we denote a solution of the closed-loop system and the error system. Then, we can obtain Theorem 1. Theorem 1: Suppose that Assumptions 1–3 hold. Consider the synchronized error system (11) with adaptive fault-tolerant controller (14). If there exists a positive definite matrix P such that the following matrix inequality: P(A + Bαi K i + Mei ) + (A + Bαi K i + Mei )T P < 0

is solvable, then the synchronization errors ei (t) (i = 1, 2, . . . , N) given in (11) converge to zero asymptotically under the adaptive control laws (15). Proof: First, we define a Lyapunov function candidate for node i k2 (20) Vi (t) = eiT Pei + α i ηi i . γi Then, according to (11), the time derivative of Vi (t) for t ≥ t0 satisfies V˙i (t) = 2eiT P((A + Bαi K i )ei + ( f (x i , t) − f (s, t))) k˙ i ki + 2eiT PB(αi u i2 (t) + δi (t)) + 2α i ηi γi N +2 li j eiT P(x j − x i ) j =1

≤ 2eiT P(A + Bαi K i + Mei )ei + 2α i ηi + 2ηi

where u i1 (t) = K i ei (t)

T u i2 (t) = −ηi sgn eiT PB 1 . . . , sgn eiT PB m kî (t) m

T (15) ei PB k sgn eiT PB k . k˙î (t) = γi k=1

Here, K i are the fixed controller gain matrices to be determined, P is a positive definite matrix that will be chosen later, (eiT PB)k denotes the kth element of vector eiT PB, γi and ηi are given positive scalars, and the sign function is defined as follows:

T 1 if eiT PB k ≥ 0 T sgn ei PB k = (16) −1 if ei PB k < 0. Denote αi = diag(αi1 , . . . , αik , . . . , αim ). Together with (6), we know that there must exist an unknown constant α i > 0, such that α i ≤ αik for all k ∈ {1, 2, . . . , m}. From Assumption 3, there also exists an unknown constant ki satisfying δ¯i ≤ ki ηi α i .

(17)

kî (t) in (15) is an estimation of ki , which satisfies kî (t0 ) > 0. Let k˜i (t) = kî (t) − ki , we obtain that k˙˜i (t) = γi

m k=1

eiT PB k sgn eiT PB k .

(18)

(19)

m

k˙ i ki γi

eiT PB k αik − sgn eiT PB k kî (t)

k=1 N li j eiT P(e j − ei ). + 2eiT PBδi (t) + 2 j =1

From the initial condition kî (t0 ) > 0 and (15), we know that kî (t) > 0 for all t > t0 . Therefore, together with the inequality α i ≤ αik , it follows that: ki k˙ i V˙i (t) ≤ λmax (Q)ei 2 + 2α i ηi γi m T T ei PB k sgn ei PB k − 2α i ηi kî (t) k=1 N + 2eiT PBδi (t) + 2 li j eiT P(e j − ei ) j =1

k˙ i ki ≤ λmax (Q)ei 2 + 2α i ηi γi m T e PB − 2α ηi kî (t) i

i

k

k=1

+2

m N T e PB δ¯i + 2 li j eiT P(e j − ei ) (21) i k k=1

j =1

where λmax (Q) denotes the maximum eigenvalue of Q with Q = P(A + Bαi K i + Mei ) + (A + Bαi K i + Mei )T P. Note that


T the inequality eiT PB ≤ m k=1 |(ei PB)k | has been used to obtain (21). T From (17), we have −2α i ηi kî (t) m k=1 |(ei PB)k | + m m 2 k=1 |(eiT PB)k |δ¯i ≤ −2α i ηi k˜i (t) k=1 |(eiT PB)k |. Together with (18), we have 2α i ηi ( ki /γi ) − 2α i ηi kî (t) k˙ i m m T T ¯ k=1 |(ei PB)k | + 2 k=1 |(ei PB)k |δi ≤ 0. Thus, the following inequality can be derived: V˙i (t) ≤ λmax (Q)ei 2 + 2

N

li j eiT P(e j − ei ).

βi Vi (t)

(22)

i=1

where βi is given in Lemma 1. According to Assumption 1, it is known that βi > 0. Then, we obtain V˙ (t) ≤ λmax (Q)

N

βi ei 2 + 2

N N

βi li j eiT P(e j − ei ).

i=1 j =1

i=1

N N

Next, the objective is to show 1 2

i=1

(23)

T j =1 βi l i j ei

P(e j − ei ) ≤ 0. To this end, let z i = P ei , and we have 2

N

βi li j eiT P(e j − ei ) = 2

i, j =1

≤

N

βi li j z Tj z j − z iT z i .

Using Lemma 2, it is derived that

i, j =1

z sT z s −z rT z r .

(s,r)∈E(CQ )

Q∈Q

N

βi i x˜i (t) ≤ V (t) ≤ V (t0 ) +

t

V˙ (τ )dτ

t0

i=1

t N

βi ei (τ )2 dτ

≤ V (t0 )

(25) Without loss of generality, for any directed cycle CQ , the set E(CQ ) can be denoted by E(CQ ) = {(i k , i k+1 )|k = 1, 2, . . . , m − 1, m ≤ N, i m = i 1 }. (26) Based on (26), we obtain T

z s z s − z rT z r (s,r)∈E(CQ ) = z iT1 z i1 − z iT2 z i2 + z iT2 z i2 − z iT3 z i3 + · · · z iTm−1 z im−1 − z iTm z im + z iTm z im − z iT1 z i1 = 0.

(29)

which means that the solutions of the synchronization error system described by (11) and adaptive estimation error system described by (18) are uniformly bounded. In addition, (29) implies that t N (−λmax (Q))βi ei (τ )2 dτ ≤ V (t0 ). (30) lim t →∞ t 0 i=1

Since x˜i (t) is uniformly bounded, from Assumptions 1–3, it follows that ei (t) and e˙i (t) are uniformly bounded, which implies that ei (t) is uniformly continuous. Therefore, N 2 i=1 (−λmax (Q))βi ei (t) is also uniformly continuous. Using Barbalat’s Lemma [42] to (30), the following inequality N (−λmax (Q))βi ei (t)2 = 0, can be derived limt →∞ i=1 that is: lim ei (t) = 0

(24)

βi li j z Tj z j −z iT z i = w(Q)

0≤

t →∞

i, j =1

N

then

βi li j z iT (z j − z i )

i, j =1 N

0 ≤ i x˜i (t) ≤ Vi (t)

t0 i=1

Subsequently, a convex combination technique is introduced to construct a global Lyapunov function V (t) =

of Lyapunov functions given in (20), there always exists a positive constant i such that

≤ V (t0 ) + λmax (Q)

j =1

N

5

(27)

which completes the proof. Remark 2: In (24), = holds if and only if z j = z i , which is equivalent to e j = ei . Then, by combining (24)–(27), it is N N T known that the double sum i=1 j =1 βi l i j ei P(e j − ei ) is strictly negative if e j = ei . Together with (23), this means that the linear couplings can generate a positive effect on synchronization, and they do not need to be compensated in the controller design. In other words, the synchronization controller can own more design freedom to achieve other performances, such as the reliable performance. Remark 3: By multiplying on the left-hand side and right-hand side of (19) with P −1 and P −T , we obtain (AY + Bαi L i + Mei Y ) + (AY + Bαi L i + Mei Y )T < 0

(31)

P −1

and L i = K i Y . Note that the inequality where Y = condition (31) is linearly dependent on αi and Mei , and thus from [43, Lemma 3.1], (31) holds for all αik ∈ [α ik , α¯ ik ] and m ei ∈ [m ei , m¯ ei ], if (31) holds for all αik ∈ {α ik , α¯ ik } and m ei ∈ {m ei , m¯ ei }, where [·] and {·} represent interval and set, respectively. In other words, the positive definite matrix P and controller gains K i can be computed by solving the inequality condition (31) by the LMI control toolbox [43].

From (23)–(27), we have V˙ (t) ≤ λmax (Q)

N

B. CDN With Time-Varying Coupling Weights βi ei 2 .

(28)

i=1

The rest of the proof is similar to that in [27]. Let x˜i (t) = [eiT (t) kiT (t)]T , and thus according to the definition

In this section, the time-varying coupling weights li j (t) are considered, and Assumption 4 is required. Assumption 4: The coupling weights li j (t) satisfy l˙i j (t) ≤ 0.



Assumption 4 is introduced for technical reason, and at least it is satisfied for the constant coupling weights. Moreover, the Laplacian matrix with time-varying weights is in the following form: ⎡ ⎤ l1k (t) −l12 (t) ··· −l1N (t) ⎢ k=1 ⎥ ⎢ −l (t) l2k (t) · · · −l2N (t) ⎥ 21 ⎢ ⎥ ⎢ ⎥ k =2 L(t) = ⎢ ⎥. (32) .. .. .. .. ⎢ ⎥ . ⎢ ⎥ . . . ⎣ ⎦ −l N2 (t) · · · l Nk (t) −l N1 (t) k = N

Similar to the time-invariant case, the cofactor of the i th diagonal element of L(t) is denoted by βi (t). From Lemma 1, we have βi (t) = w(T ), i = 1, 2 . . . , N. (33) T ∈Ti

Similar to Theorem 1, using the adaptive control laws (15) and inequality (37), it is known that the time derivative of Vi (t) satisfies V˙i (t) ≤ λmax (Q)ei 2 + 2

(34)

V (t) =

=

Q∈Q

wt (Q)

βi (t)Vi (t)

(40)

where βi (t) is given in (33). the chain rule, it is easy to obtain V˙ (t) Using N N ˙ ˙ (t)V β i (t) + i=1 i i=1 βi (t) Vi (t). Since β˙i (t) ≤ 0, we have V˙ (t) ≤

N

βi (t)V˙i (t) ≤ λmax (Q)

i=1

+2

N

=

βi (t)ei 2

i=1 N

βi (t)li j (t)eiT P(e j − ei ).

(41)

i, j =1

As before, let z i = P 1/2 ei , we have 2

N

βi (t)li j (t)eiT P(e j − ei )

i, j =1

≤

N

βi (t)li j (t) z Tj z j − z iT z i .

(42)

i, j =1

Frs (xr , x s ).

(35)

(s,r)∈E(CQ )

Compared with Lemma 2, wt (Q) is introduced in (35) to represent the time-varying weight of the directed cycle of Q. Subsequently, based on the above analysis, Theorem 2 can be derived for the considered CDN with time-varying coupling weights e˙i (t) = Aei (t) + f (x i (t)) − f (s(t)) + B(αi u i (t) + δi (t)) N + li j (t)(x j (t) − x i (t)). (36) j =1

Theorem 2: Suppose that Assumptions 1–4 hold. Consider the synchronized error system (36) with an adaptive fault-tolerant controller (14). If there exists a positive definite matrix P such that the following matrix inequality: P(A + Bαi K i + Mei ) + (A + Bαi K i + Mei )T P < 0

(37)

is solvable, then the synchronization errors ei (t) (i = 1, 2, . . . , N) given in (36) converge to zero asymptotically under the adaptive control laws (15). Proof: The key idea for proving Theorem 2 is the same as that of Theorem 1 except for addressing the time-varying coupling weights. We define the following Lyapunov function candidate for node i : Vi (t) = eiT Pei + α i

N i=1

βi (t)li j (t)Fi j (x i , x j )

i, j =1

(39)

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

Moreover, using the proof of Lemma 2 [34, pp. 3–4], it is known that the time-varying version of identity (3) still holds N

li j (t)eiT P(e j − ei ).

j =1

Note that, different from (2), w(T ) in (33) consists of the product of time-varying weights li j (t); together with Assumption 4, this implies β˙i (t) ≤ 0.

N

ki2 . γi

(38)

Using (35), it is derived that N

βi (t)li j (t) z Tj z j − z iT z i

i, j =1

=

Q∈Q

wt (Q)

z sT z s − z rT z r = 0.

(43)

(s,r)∈E(CQ )

From (41)–(43), we obtain V˙ (t) ≤ λmax (Q)

N

βi ei 2 .

(44)

i=1

The rest of the proof can then be completed in a manner similar to that in Theorem 1. Remark 4: This paper is mainly concerned with the faulttolerant control problem of CDN with actuator faults. Note that the strong connection of graph G given in Assumption 1 may be unsatisfied if the link faults occur. On the other hand, the fault compensation laws u 2i (t) in (15) are difficult to be used for the CDN with sensor faults. Therefore, new techniques should be developed to design a fault-tolerant controller in these two cases, and this problem will be studied in the further research. Remark 5: The results of (42) and (43) tell us that the linear couplings with time-varying weights li j (t) can still generate a positive effect on the synchronization, provided that Assumption 4 holds. Therefore, the proposed adaptive faulttolerant synchronization controller (14) is still applicable for error system (36).


Fig. 1.

Chua’s circuit.

IV. S IMULATION E XAMPLES A. Example 1 In this section, we consider a CDN consisting of five Chua’s circuits coupled by the following topology: ⎡ ⎤ 1 −1 0 0 0 ⎢ −1 2 0 0 −1 ⎥ ⎢ ⎥ ⎢ −1 1 0 0 ⎥ L=⎢ 0 ⎥. ⎣ 0 0 −1 1 0 ⎦ 0 0 0 −1 1 The single Chua’s circuit shown in Fig. 1 is described by the following differential equations [44]: 1 (−v 1 + v 2 ) − f (v 1 ) R 1 C2 v˙2 (t) = (v 1 − v 2 ) + i 3 R L i˙3 (t) = −(v 2 + R0 i 3 ) C1 v˙1 (t) =

where the descriptions of v 1 , v 2 , C1 , C2 , i 3 L, R0 , and R can also be found in [44]. The term f (v 1 ) represents the current through the nonlinear resistor N R , which is a piecewise-linear function expressed as f (v 1 ) = G b1 v 1 + 0.5(G a1 − G b1 ) (|v 1 + 1| − |v 1 − 1|). According to the result in [45], a feedback control is added in series with the inductor a voltage source u(t), thus the networked system is defined as follows: ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎡ ⎤ −p p 0 x i1 −ω f (x i1 ) x˙i1 ⎣ x˙i2 ⎦ = ⎣ q ⎦ −q r ⎦ ⎣ x i2 ⎦ + ⎣ 0 0 −v −z x˙i3 x i3 0 ⎡ ⎤ 5 0 + ⎣ 0 ⎦ u(t) + li j (x j (t) − x i (t)) 1 j =1, j =i where x i1 = v i1 , x i2 = v i2 , x i3 = i i3 , p = (1/RC1 ), q = (1/RC2 ), r = (1/C2 ), v = (1/L), z = (R0 /L), and ω = (1/C1 ), i = 1, 2, . . . , 5. Note that the coupling weights are unknown; therefore, the fault-tolerant control schemes in [32] and [33] fail to work. Moreover, as the input distribution matrix is not a unit one, the synchronization control methods in [21]–[27] cannot be used in this example.

7

Fig. 2. Synchronization errors with time-invariant coupling weights in Case I.

In the following, it is shown that the proposed faulttolerant synchronization controller is valid. First, similar to [24] and [27], the system parameters are selected as p = 9.1, q = 1, r = 1, ω = 9.1, G b1 = −0.7559, G a1 = −1.39386, v = 16.5811, and z = 0.138083. In [36], it has been proved that f (x i1 ) satisfies Assumption 2. In particular, f (x i1 ) − f (si1 ) = m ei1 (x i1 − si1 ) with G a1 ≤ m ei1 ≤ G b1 . In addition, it is assumed that h h αi,k in (6) satisfies 0.1 ≤ αi,k ≤ 1 for i = 1, 2, . . . , 5 and k = 1, 2, and 3. Therefore, according to Remark 3, the positive definite matrix P and K i can be derived and given as K i = [−0.0429 0.3268 − 0.4023], i = 1, 2, . . . , 5 and ⎡ ⎤ 0.0118 0.0032 0.0008 P = ⎣ 0.0032 0.1707 0.0006 ⎦. 0.0008 0.0006 0.0106 For nodes 1–5, γi and ηi in (15) are set to be 250 and 1, respectively. The initial conditions of system states are x 1 (0) = [1 −1 0]T , x 2 (0) = [3 −2 −5]T , x 3 (0) = [7 −2 −3]T , x 4 (0) = [1 −2 0]T , x 5 (0) = [3 2 −5]T , s(0) = [5 1 −3]T , and kî (0) = 5 for i = 1, 2, . . . , 5. In addition, the following two types of actuator faults are considered. Case I: Bias fault δ1 (t) = 10 occurs in node 1 at t = 20 s. Case II: The actuator of node 5 loses 80% of its effectiveness at t = 20 s. In Case I, the synchronized errors ei (t) (i = 1, 2, . . . , 5) are shown in Fig. 2. It can be seen that, without using the prior information of coupling weights, the evolution of each node in the CDN is synchronized to s(t) before and after the occurrence of a fault by using the proposed faulttolerant controller, which verifies the theoretical result in Theorem 1 well. Moreover, the adaptive estimations kî (t) are displayed in Fig. 3, from which we know that these estimates are all bounded. For Case II, the synchronized errors ei (t) (i = 1, 2, . . . , 5) are shown in Fig. 4, where the state x i (t) also synchronizes to the desired orbit s(t). Since the control input matrix is not a unit one, only the method in [28] is applicable in this example, and the corresponding synchronization errors are plotted in Fig. 5 with the topology given in L. Note that the synchronization cannot be


Fig. 3.


Estimates of kî (t) in Case I. Fig. 6. Synchronization errors with time-varying coupling weights in Case I.

In this case, by using the method in Theorem 2, the synchronization errors are plotted in Fig. 6 with the actuator fault given in Case I. It can be observed that the synchronization is achieved even in the presence of time-varying coupling weights. In other words, although the evolutions of the synchronization errors in Figs. 2 and 6 are slightly different, the proposed fault-tolerant synchronization control approach is always valid for the CDN with both time-invariant coupling weights and time-varying ones. Fig. 4. Synchronization errors with time-invariant coupling weights in Case II.

Fig. 5.

Synchronization errors by using the existing method [27] in Case I.

guaranteed due to the lack of fault compensation mechanism. In particular, if the coupling weights li j are unknown, the method in [28] also fail to work. Therefore, the comparison results between Figs. 2 and 5 illustrate the advantages of the proposed synchronization fault-tolerant control method. On the other hand, we further consider the timevarying coupling weights, and the following topology is given: ⎤ ⎡ 1 −1 0 0 0 ⎢−1 2 0 0 −1 ⎥ ⎥ ⎢ ⎥. 0 −1 1 0 0 L(t) = ⎢ ⎥ ⎢ ⎦ ⎣0 0 −1 − e−0.1t 1 + e−0.1t 0 0 0 0 −1 − e−0.1t 1 + e−0.1t

B. Example 2 In this example, to further illustrate the potential applications of the proposed fault-tolerant synchronization control approach, the CDN consisting of five Rossler systems [36], [46]–[48] is investigated, where the topology L is assumed to be the same as that of Example 1. According to [36], [47], and [48], the networked Rossler systems can be described as follows: ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎡ ⎤ 0 −1 −1 x i1 x˙i1 0 ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎢ ⎦ 0 ⎦ ⎣ x i2 ⎦ + 0 ⎣ x˙ i2 ⎦ = ⎣ 1 a x i1 x i3 + b x˙i3 x i3 0 0 −c ⎡ ⎤ 5 u 1 (t) li j (x j (t) − x i (t)) + ⎣ u 2 (t) ⎦ + u 3 (t) j =1, j =i where x i = [x i1 x i2 x i3 ]T , i = 1, 2, . . . , 5. Let a = 0.2, b = 0.2, c = 5.7, f i (x i ) = [0 0 x i1 x i3 + b]T , ei = [ei1 ei2 ei3 ]T , and s = [s1 s2 s3 ]T ; then by [36], we have ⎤⎡ ⎤ ⎡ ei1 0 0 0 ⎥⎢ ⎥ ⎢ f (x i ) − f (s) = ⎣ 0 0 0 ⎦ ⎣ ei2 ⎦ ei3 s3 0 x i1 with 0 < s3 < 24 and −10 < x i1 < 13. h in (6) satisfies 0.2 ≤ α 1 ≤ 0.8 Here, it is assumed that αi,k i,k 2 ≤ 0.9 for i = 1, 2, and 3 and k = 1, 2, and 3, and 0.1 ≤ αi,k for i = 4, and 5 and k = 1, 2, and 3. Therefore, according to Remark 3, the controller gain matrix K i and the positive


Fig. 7. case.

Synchronization errors using discontinuous control inputs in faulty

Fig. 8. Synchronization errors using discontinuous control inputs in fault-free case.

definite matrix P can be derived as follows: ⎡ ⎤ −74.7786 −2.5988 −28.4314 −12.7629 0.4652 ⎦, i = 1, 2, 3 K i = ⎣ 0.5657 −29.6884 −0.8436 −27.3880 ⎡ ⎤ −69.4802 −1.9655 −26.4604 K i = ⎣ −1.9399 −13.3194 −0.5815 ⎦, i = 4, 5 −26.1287 −0.4558 −25.4953 ⎡ ⎤ 0.1134 0.0040 0.0426 P = ⎣ 0.0040 0.0216 0.0014 ⎦. 0.0426 0.0014 0.0431 Moreover, γi and ηi in (15) are set to be 250 and 50, respectively. The initial conditions of system states are x 1 (0) = [1 −1 0]T , x 2 (0) = [3 −2 −5]T , x 3 (0) = [7 −2 −3]T , x 4 (0) = [1 −2 0]T , x 5 (0) = [3 2 −5]T , s(0) = [2 2 −1]T , and kî (0) = 5 for i = 1, 2, . . . , 5. Suppose that the actuator bias fault δ1 (t) = 10 occurs in node 1 at t = 15 s, and the synchronized errors ei (t) (i = 1, 2, . . . , 5) are shown in Fig. 7. In addition, the synchronized errors are also plotted in Fig. 8 for the faultfree case. From Figs. 7 and 8, it can be seen that, the states of each node can track the desired target orbit s(t) for both faulty and fault-free cases. However, the tracking errors suffer from chattering, since the discontinuous sign functions are introduced in the control law u i2 (t) (15), and (eiT PB)k frequently crosses the zero point.

9

Fig. 9. Synchronization errors using continuous control inputs in faulty case.

Fig. 10. Synchronization errors using continuous control inputs in fault-free case.

Fig. 11.

Desired target orbit s(t) in Example 1.

According to [49] and [50], the chattering phenomenon can be reduced using an arctangent function that approximates the sign function. In fact, sgn(x) ≈ (2/π)arctan(βx), where β is a given positive constant, and the larger β can reduce the approximated errors. Set β = 5, the synchronized errors for faulty and fault-free cases are shown in Figs. 9 and 10. It can be observed that the chattering suppression is achieved and the synchronization performances have also been improved. Here, it should be pointed out that the chattering problem does not occur in Example 1, because the steady-state values of s(t) (see Fig. 11) are constants and x i (t) can track s(t). In this case, (eiT PB)k also approach constants. On the contrary, the steady-state values of s(t) in Example 2 (see Fig. 12) are timevarying and frequently cross zero, which lead to the chattering phenomenon.


Fig. 12.


Desired target orbit s(t) in Example 2.

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Xiao-Jian Li received the B.S. and M.S. degrees 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. He is currently an Associate Professor with the College of Information Science and Engineering, Northeastern University. His current research interests include fault diagnosis, fault-tolerant control, fuzzy control, and complex networks.

Guang-Hong Yang (SM’04) received the B.S. and M.S. degrees from the Northeast University of Technology, Jinzhou, China, in 1983 and 1986, respectively, and the Ph.D. degree in control engineering from Northeastern University, Shenyang, China, in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University, Singapore, in 1996, as a Post-Doctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with 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. Dr. Yang is an Associate Editor of the International Journal of Control, Automation, and Systems, the International Journal of Systems Science, the IET Control Theory & Applications, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS .