H∞ Consensus and Synchronization of Nonlinear Systems Based on ...

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 43, NO. 6, DECEMBER 2013

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H∞ Consensus and Synchronization of Nonlinear Systems Based on A Novel Fuzzy Model Yan Zhao, Bing Li, Jiahu Qin, Member, IEEE, Huijun Gao, Senior Member, IEEE, and Hamid Reza Karimi, Senior Member, IEEE Abstract—This paper investigates the H∞ consensus control problem of nonlinear multiagent systems under an arbitrary topological structure. A novel Takagi–Sukeno (T–S) fuzzy modeling method is proposed to describe the problem of nonlinear follower agents approaching a time-varying leader, i.e., the error dynamics between the follower agents and the leader, whose dynamics is evolving according to an isolated unforced nonlinear agent model, is described as a set of T–S fuzzy models. Based on the model, a leader-following consensus algorithm is designed so that, under an arbitrary network topology, all the follower agents reach consensus with the leader subject to external disturbances, preserving a guaranteed H∞ performance level. In addition, we obtain a sufficient condition for choosing the pinned nodes to make the entire multiagent network reach consensus. Moreover, the fuzzy modeling method is extended to solve the synchronization problem of nonlinear systems, and a fuzzy H∞ controller is designed so that two nonlinear systems reach synchronization with a prescribed H∞ performance level. The controller design procedure is greatly simplified by utilization of the proposed fuzzy modeling method. Finally, numerical simulations on chaotic systems and arbitrary nonlinear functions are provided to illustrate the effectiveness of the obtained theoretical results. Index Terms—H∞ consensus, nonlinear multiagent systems, synchronization, Takagi–Sukeno (T–S) fuzzy models.

I. I NTRODUCTION

C

ONTROL of nonlinear systems is regarded as a complex and difficult task in control theory. Many researchers have been dedicated to seeking effective methods and techniques to accomplish the task. This facilitates a rapidly growing interest in fuzzy model-based control of nonlinear systems. Takagi–Sugeno (T–S) fuzzy models can effectively approximate any smooth nonlinear function to any specified accuracy Manuscript received January 16, 2012; revised July 30, 2012; revised November 8, 2012; accepted January 9, 2013. Date of publication February 21, 2013; date of current version November 18, 2013. This work was supported in part by the National Natural Science Foundation of China under Grant 61105038 and Grant 51175105, by the 973 Project under Grant 2009CB320600, by the Key Laboratory Opening Funding of Technology of Micro-Spacecraft under Grant HIT.KLOF.2009099, by the China Postdoctoral Science Foundation under Grant 20110491068, and by the Basic Research Plan in Shenzhen City under Grant JC201105160523A. This paper was recommended by Associate Editor J. Q. Gan. Y. Zhao and B. Li are with the Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China (e-mail: [email protected]; [email protected]). J. Qin is with the Research School of Engineering, The Australian National University, Canberra ACT 2600, Australia (e-mail: [email protected]). H. Gao is with the Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, China (e-mail: [email protected]). H. R. Karimi is with Department of Engineering, Faculty of Engineering and Science, University of Agder N-4898 Grimstad, Norway (e-mail: hamid.r. [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.2013.2242197

within any compact set, which is realized by an interpolation fact that the local dynamics of each fuzzy rule is expressed by a linear or affine model, and the overall fuzzy model of the nonlinear system can be obtained by fuzzy “blending” of those local linear or affine models [36]. Owing to the local linearity, the stability analysis and controller synthesis can be simplified based on the well-known framework of Lyapunov functional approaches [34], by utilizing a parallel or nonparallel distributed compensation control scheme [13]. So far, a large number of results have been reported. To mention a few, the problem of stability analysis is investigated in [11], stabilization and H∞ control designs are reported in [15], state estimation is addressed in [6] and [35], and reliable control strategies are presented in [41]. Those results are concerned with various engineering constraints, including parameter uncertainties [15], time delay [3], [42], actuator saturation [2], [47], and singular perturbations [23]. Consensus of multiagent systems has attracted attention of experts from diverse disciplines, such as physics, biology, mathematics, and control science, due to its wide applications in formation control of air vehicles, cooperative control of robots, synchronization of complex networks, and design of distributed sensor networks. The consensus problem is to find control strategies that enable multiple agents to reach an agreement on certain criteria with a specified information exchange between an agent and its neighbors. Numerous results have been obtained during the last decade for both first-order and secondorder multiagent systems [10], [29], [31], [32], [38]. Various control methods have been utilized, such as H∞ control [18], [21], sampled-data control [45], and pinning control [4], and diverse engineer constraints have been considered, including time delay [45], data packet dropout [45], nonuniform input [22], and parameter uncertainties [21]. Among those methods, pinning state-feedback control is to enable the entire multiagent system to reach consensus by adding controllers to some nodes of the multiagent network. Therefore, pinning control has recently attracted increasing attention [27]. Those aforementioned studies are mostly concerned with multiple agents with linear dynamics. However, all physical systems are nonlinear in nature, and the consensus problem becomes more difficult when the agent dynamics is nonlinear. Many strategies have been developed to investigate the problem incorporated with the Lyapunov functional approach [8], [43], adaptive control, and other nonlinear control methods [28]. Noticed that the approaches for the consensus problem of multiagent systems can be used to solve the synchronization problem of complex networks. The pinning control strategies are designed to make the complex network reach synchronization with undirected

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graph topology in [26], [27], and [44], and the pinned nodes are freely chosen or chosen as those with a low or high degree [44]. However, most of the given research work copes with the multiagent network under an assumption that the network is connected. Few studies are directed to investigate the consensus problem of a nonlinear multiagent network under arbitrary topological structures. Recently, more research work in the multiagent system literature focuses on systems with external disturbances [46]. It was shown that the problem could be transformed into an H∞ control problem and solved by analyzing the H∞ control problem of a set of independent subsystems. Several results have been obtained for the consensus problem of multiagent systems by H∞ control. For instance, the consensus problem of a multiagent network with external disturbances on fixed and switching topologies is investigated in [20], and sufficient conditions are presented under which all agents reach consensus with a desired H∞ performance; the H∞ consensus problem of the second-order and high-dimensional multiagent systems with external disturbances is addressed in [21] and [24], respectively. In addition, the distributed H2 and H∞ control problems for multiagent systems are addressed in [17]. It is worth noting that the reported results are concerned with the multiagent systems with linear or linearized dynamics, and the H∞ consensus problem of nonlinear multiagent systems has not been investigated. As aforementioned, the T–S fuzzy model is always preferred by researchers when the nonlinearity is concerned. Therefore, an idea comes up that one can solve the H∞ consensus problem of nonlinear multiagent systems based on the T–S fuzzy model. In this case, not only the H∞ performance analysis can be carried out but also the complicated protocol design procedure can be greatly simplified without critical mathematical assumptions [28]. As noted earlier, the fuzzy logic system has been applied to multiple agents because it has learning ability and suits well the noisy environment [37]. For example, a learning fuzzy classifier system is designed and applied to learn both behaviors and their coordination for autonomous agents in [1], and fuzzy policies are developed for a leader–follower robotic system to make multiple robots keep a formation in [7] and [33]. Although those references provide an alternative approach for the consensus problem of multiagent systems, the H∞ consensus problem of nonlinear multiagent systems has not been investigated, partly due to the difficulties in systematic stability analysis and controller synthesis of the model-free fuzzy logic approach. If the coupling between the agents with nonlinear dynamics is not considered, the consensus problem is similar to the general synchronization problem of nonlinear systems. The T–S-fuzzy-model-based control approach has been borrowed to solve the synchronization problem [12], [16]. The linear matrix inequality method is used earlier on condition that the memberships of both response and drive chaotic systems are known [40], and the condition is alleviated later by using the H∞ tracking control approach [19]. Parameter uncertainties are dealt with by combining the adaptive control method [9], and the communication delay has been considered by giving a constraint to the membership function of the fuzzy controller

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[12]. It is noted that, in those studies, two T–S fuzzy models with different membership functions are used to describe the response system and the drive system, respectively. Because of the different membership functions, the controller design procedure is very complicated. Thus, an idea comes up whether we can describe the synchronization problem with a novel fuzzy model, eliminating the difference of the membership functions to simplify the controller design procedure. Motivated by the aforementioned discussions, in this paper, we aim to investigate the H∞ consensus problem of nonlinear multiagent systems. First, a novel T–S fuzzy modeling method is proposed to describe the problem of follower agents approaching the leader, i.e., the error dynamics between the agents and the leader, whose dynamics is evolving according to an isolated unforced nonlinear agent model, is described by a set of T–S fuzzy models. Based on the model, a leader-following consensus algorithm is designed so that, under external disturbances, all the follower agents reach consensus with the leader and preserve a guaranteed H∞ disturbance attenuation level. A sufficient condition for choosing the pinned nodes to make the entire multiagent network reach consensus is given. Moreover, the fuzzy modeling method is extended to solve the synchronization problem of nonlinear systems, and a fuzzy H∞ controller is designed so that the two nonlinear systems reach synchronization with a prescribed H∞ performance level. The controller design procedure is greatly simplified by utilization of the proposed fuzzy modeling method. Finally, numerical simulations are provided to show the effectiveness of the theoretical results. The rest of this paper is organized as follows. In Section II, the main idea of the proposed fuzzy modeling method is proposed. The H∞ consensus and synchronization problems are presented in Section III. Simulations are provided in Section IV, and some concluding remarks are given in Section V. The notation used throughout this paper is fairly standard. The superscript T stands for matrix transposition; Rn denotes the n-dimensional Euclidean space. The notation  ·  refers to the Euclidean vector norm, and diag{. . .} stands for a block-diagonal matrix. In symmetric block matrices or complex matrix expressions, we use an asterisk (∗) to represent a term that is induced by symmetry. In is referred to the identity matrix with dimension n × n, and if the subscript is not clarified, I denotes the identity matrix and 0 denotes the zero matrix with compatible dimensions; sym(P ) is defined as P + P T . Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. ⊗ denotes the Kronecker product. Let G = (V, ϑ, A) be a weighted diagraph of order N with a finite nonempty set of nodes V = {1, 2, . . . , N }, a set of edges ϑ ∈ V × A, and weighted adjacency matrix A = [aij ] ∈ RN ×N , where aij are the adjacency elements. The adjacency matrix of a weighted undirected graph is defined as aii = 0 and aij = aji > 0 if (j, i) ∈ ϑ, where i = j. The Laplacian matrix associated with G is defined as Li, j =

⎧ ⎨ ⎩

N 

−

k=1,k=j

aij ,

aik ,

j=i j = i.

ZHAO et al.: H∞ CONSENSUS AND SYNCHRONIZATION OF NONLINEAR SYSTEMS BASED ON NOVEL FUZZY MODEL

II. C ONSTRUCTING THE F UZZY M ODEL Here, we will introduce the fuzzy model of the error dynamics between the follower agents and the leader. Consider the multiagent system consisting of N identical follower agents and one leader. Each follower agent’s dynamics is described by the following continuous nonlinear function:  x˙ i (t) = f (xi (t), t)+ c aij Γ(xj (t)−xi (t)) , i = 1, . . . , N j∈Ni

(1) where xi (t) ∈ Rn is the state of node i, f (xi (t), t) is a nonlinear continuously differentiable vector function, Ni denotes the neighboring set of node i, c is the coupling strength, Γ ∈ Rn×n is a positive definite matrix describing the inner coupling structure, aij represents the coupling strength of the information flowing from node j to node i, and aij = aji > 0 when j ∈ Ni . Suppose the state of the leader evolves according to the following nonlinear dynamics: s˙ (t) = f (s(t), t)

(2)

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premise variables, whereas h(xi (t), ei (t)) cannot be approximated but can be regarded as a product of bounded ¯ i (t)) and ei (t), and then we reptime-varying matrix A(x resent the fuzzy model of g(xi (t), ei (t)) + h(xi (t), ei (t)) as the following rules: Rule p: If θi,1 (t) is Mp1 and θi,2 (t) is Mp2 and . . . and θi,κ (t) is Mpκ , then e˙ i (t) = Ap ei (t) + A¯ (xi (t)) ei (t),

p = 1, . . . , r

where θ i (t) = [θ Ti,1 (t), θ Ti,2 (t), . . . , θ Ti,κ (t)]T is the premise variable vector, Mpl for p = 1, 2, . . . , r and l = 1, 2, . . . , κ represent the fuzzy sets, r is the number of ifthen rules, Ap is the constant matrix. The compact form of the fuzzy model can be rewritten as e˙ i (t) =

r 

λp (θ i (t)) Ap ei (t) + A¯ (xi (t)) ei (t)

p=1

where λp (θ i (t)) = ω p (θ i (t)) /

r 

ωp (θ i (t))

p=1

where s(t) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Denote by ei (t) the error state between the follower agents and the leader, i.e., ei (t) = xi (t) − s(t), then the error dynamics can be represented as e˙ i (t) = f (xi (t), t) − f (s(t), t)  ai, j Γ (ej (t) − ei (t)) , +c

i = 1, . . . , N.

(3)

j∈Ni

r 

λp (xi (t)) Ap xi (t) −

p=1

+c



κ 

Mpl (θ i,l (t))

(5)

l=1

with Mpl (θi,l (t)) representing the grade of memberships of θi,l (t) in Mpl . Then, it can be seen that λp (θ i (t)) ≥ 0,

p = 1, 2, . . . , r,

r 

λp (θ i (t)) = 1

p=1

According to general fuzzy modeling methods, the error system in (4) is always established as e˙ i (t) =

ω p (θ i (t)) =

r 

λp (s(t)) Ap s(t)

p=1

ai, j Γ (ej (t) − ei (t)) ,

i = 1, . . . , N

(4)

j∈Ni

where λp (xi (t)) and λp (s(t)) represent the membership functions for establishing fuzzy models of f (xi (t), t) and f (s(t), t), respectively, and Ap is the constant fuzzy model matrix. Note that, here, we slightly abused notation Ap to denote the fuzzy system matrix and A to denote the adjacency matrix aforementioned. Because of the different membership functions λp (xi (t)) and λp (s(t)), the protocol or controller design process will be complicated with simultaneous consideration of H∞ performance evaluation. In the following, we will propose a novel fuzzy model to describe the error system f (xi (t), t) − f (s(t), t) in (4), which can make the protocol or controller design procedure more simplified. Two types of fuzzy models can be established with different number of fuzzy rules. 1) The first type has less rules by choosing less premise variables. In particular, we rewrite f (xi (t), t) − f (s(t), t) as g(xi (t), ei (t)) + h(xi (t), ei (t)), where g(xi (t), ei (t)) can be approximated by the T–S fuzzy model with the chosen

for all t. Since, in real world, all the physical variables are bounded, thus without loss of generality, we give the following assumption needed for the subsequent derivative process. ¯ i (t)) ≤ Q, where Q is a posiAssumption 1: A¯T (xi (t))A(x tive definite matrix, and can be represented as Q = RT R. 2) The first fuzzy model needs Assumption 1, which is critical from the mathematical viewpoint. Thus, we propose the following second type of the fuzzy model, which does not need the assumption. It has a bit larger number of fuzzy rules because the premise variables need to be freely chosen. Rewrite f (xi (t), t) − f (s(t), t) as f (xi (t), ei (t)), and based on the chosen premise variables, f (xi (t), ei (t)) can be approximated by the following T–S fuzzy model without the time-varying uncertainty in the first type of fuzzy model [36]: ˜ i,2 (t) is M ˜ i,1 (t) is M ˜ p1 and θ ˜ p2 and . . . and Rule p: If θ ˜ i,κ (t) is M ˜ pκ , then θ e˙ i (t) = Ap ei (t),

p = 1, . . . , r

˜ i (t) = [θ ˜ T (t), θ ˜ T (t), . . . , θ ˜ T (t)]T is the where θ i,1 i,2 i,κ ˜ pl for p = 1, 2, . . . , r and premise variable vector, M l = 1, 2, . . . , κ represents the fuzzy sets, and Ap is the constant matrix. The compact form of the fuzzy model can be analogously obtained. The second fuzzy model does

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not need the restrictive assumption that the trajectories are bounded. Both two modeling methods will be illustrated and verified in the simulations.

where

⎡ ⎢ Πp = ⎣

III. S OLVING P ROBLEMS BASED ON THE F UZZY M ODEL Here, based on the first type of the T–S fuzzy model, the H∞ consensus problem of the nonlinear multiagent systems and synchronization of general nonlinear systems will be investigated, respectively. The results for the second type of the T–S fuzzy model can be straightforwardly obtained. A. H∞ Consensus of Multiagent Systems This subsection is concerned with the H∞ consensus problem of the nonlinear multiagent system. Based on the obtained fuzzy model of the error system, we will mathematically formulate the problem of the leader-following consensus, and the protocol will be designed such that the nonlinear multiagent system can reach consensus with a prescribed H∞ performance level. To control the system with performance evaluation, some controlled outputs are given, which can be augmented as the following equation: e˙ i (t) =

r  p=1

+c

ai, j Γ (ej (t) − ei (t))

zi (t) =

0

0

⎤

⎥ ⎦. 0 λp (θ N (t))

t→∞

s(t) = 0 for i = 1, . . . , N , then the multiagent network in (1) is called reaching consensus. Definition 2: Given scalar γ > 0, if the system in (1) can reach consensus in the sense of Definition 1, and under zero initial conditions and nonzero external disturbances, the controlled output zi (t) satisfies zi (t)2 ≤ γwi (t)2 ; then, we say that the system in (1) reachs consensus with a prescribed H∞ performance level. Before proceeding further, we first introduce a lemma that will be used in the proof of the main results by setting F (t) = I and parameter  = 1 in Lemma 2.2 in [39] as follows. Lemma 1: Let E and L be real matrices of appropriate dimensions. Then, we have LE + E T LT ≤ LLT + E T E, Suppose that only a portion of the agents have access to s(t), i.e., the state of the leader. Let

j∈Ni r 

0 0

0 .. .

Some definitions describing the different aspects of the problems aforementioned are proposed in the following. Definition 1: If there exists a proper coupling strength c so that the states of the followers without external disturbances can reach consensus with the state of the leader, i.e., lim xi (t) −

λp (θ i (t)) (Ap ei (t) + Dp wi (t)) + A¯ (xi (t)) ei (t) 

λp (θ 1 (t))

λp (θ i (t)) (Czp ei (t)+Dzp wi (t)) ,

ui (t) = cdi Γ (s(t) − xi (t)) ,

i = 1, . . . , N

i = 1, . . . , N

(8)

p=1

(6) where Czp and Dzp are constant matrices; wi (t) is the external disturbance, and zi (t) is the controlled output of node (agent) i on which the control performance of the system will be evaluated. For brevity, the fuzzy state equation for the entire multiagent network can be rewritten as

where the control gain di > 0 if agent i has access to the leader’s state information, whereas di = 0 if otherwise. Then, the error dynamics in (6) in the leader-following framework becomes e˙ i (t) =

T w(t)=[w1T(t) . . . wN (t)]T

where and are the compact form of the state variables and external disturbance of the whole N agents, respectively; and L is the Laplacian matrix that characterizes the topological structure of the network. In this paper, we consider the case that the topology of the network is undirected. Further, we define A=

r 

(Πp ⊗ Ap ), A¯ (xi (t)) = IN ⊗ A¯ (xi (t))

p=1

D= Cz =

r  p=1 r  p=1

(Πp ⊗ Dp ) (Πp ⊗ Czp ), Dz =

r  p=1

(Πp ⊗ Dzp )

λp (θ i (t)) (Ap ei (t) + Dp wi (t)) + A¯ (xi (t)) ei (t)

p=1

+c

˙ e(t) = Ae(t) + A¯ (xi (t)) e(t) + Dw(t) + cL ⊗ Γe(t) z(t) = Cz e(t) + Dz w(t) (7) e(t)=[eT1 (t), . . . , eTN (t)]T

r 

N 

Li, j Γej (t) + ui (t)

j=1

zi (t) =

r 

λp (θ i (t)) (Czp ei (t)+Dzp wi (t)) ,

i = 1, . . . , N.

p=1

(9) In the following, a protocol will be designed such that the system in (1) reaches consensus with a prescribed disturbance attenuation level γ. ¯ as the graph consisting Label the leader as node 0, and G of node 0, G, and the edges between node 0 and the nodes in G, where there is an edge between node 0 and node i, i = 1, . . . , N , if and only if node i has access to the state information of node 0, i.e., the leader. Recall that L is the Laplacian matrix associated with G, and di > 0 whenever agent i has access to the leader, whereas di = 0 if otherwise. The following result will be useful in proving our first main result.

ZHAO et al.: H∞ CONSENSUS AND SYNCHRONIZATION OF NONLINEAR SYSTEMS BASED ON NOVEL FUZZY MODEL

Lemma 2: Let L be the Laplacian matrix associated with graph G (Proposition 1 in [30]) and L¯ = L − diag{d1 , d2 , . . . , dN } where di > 0 whenever agent i, i = 1, . . . , N , has access to the leader’s state information, whereas di = 0 if otherwise. Then, ¯ is connected. L¯ < 0 if and only if G Theorem 1: The follower agents, as described in (1), without the external disturbance can reach consensus with the leader ¯ is connected or, equivalently, at least one agent in each if G connected component of G has access to the state information of the leader, and further, c satisfies c > (λmax (A + AT ) + ¯ min (Γ)). In addition, an H∞ λmax (In + RT R))/(2λmax (L)λ performance level γ can be guaranteed if the linear matrix inequality (10) holds, which is shown at the bottom of the page. Proof: Choose the following Lyapunov functional candidate: V (t) =

N 

eTi (t)ei (t).

i=1

Observing Assumption 1 and Lemma 1, and then differentiating V (t) along the system (9) yield  r N   T ˙ V (t) =2 e (t) λp (θ i (t)) Ap ei (t) + A¯ (xi (t)) ei (t)

γw(t)2 . By simple computation, we have   J ≤ 2eT (t) A + cL¯ ⊗ Γ + IN ⊗ (In + RT R) e(t)   + 2eT (t)Dw(t) + eT (t) CzT Cz e(t)   + 2eT (t)CzT Dz w(t) + wT (t) DzT Dz − γ 2 w(t). Inequality (10) apparently guarantees the negativeness of J . The proof is completed.  In the derivative process, the property of convex additive systems holds that the whole system is stable if each local system is stable. The condition in Theorem 1 includes the membership functions, which cannot be directly employed for designing the protocol. To facilitate Theorem 1 for the convenient design, we let Γ be an identity matrix, and give the following theorem. Theorem 2: The follower agents, as described in (1), without the external disturbance can reach consensus with the ¯ is connected or, equivalently, at least one agent leader if G in each connected component of G has access to the state information of the leader, and further c satisfies c > (λmax (A + ¯ In addition, an H∞ perAT ) + λmax (In + RT R))/2λmax (L). formance level γ can be guaranteed if the following linear matrix inequalities hold for p, q = 1, . . . , r: ˜ max IN + 2cL¯ < 0 λ T Dzp Dzp − γ 2 I < 0

i

p=1

i=1

+c 

≤ e (t) A + A T

T



N 

Li, j Γej (t) − cdi Γei (t)

j=1

  e(t) + eT (t) IN ⊗ (In + RT R)

¯ is connected, it follows from Lemma 2 that L¯ < Since G 0. Note that Γ is a positive-definite matrix; thus, V˙ (t) < 0 if the coupling strength c satisfies c > (λmax (A + AT ) + ¯ min (Γ)), where λmax (·) and λmax (In + RT R))/(2λmax (L)λ λmin (·) denote the maximum and minimum eigenvalues of a matrix, respectively. This completes the proof for the first statement. We proceed to prove that the H∞ performance of the leaderfollowing multiagent systems. To this end, assume the zero initial condition and wi (t) = 0. An index is introduced as N  

 zTi (t)zi (t) − γ 2 wiT (t)wi (t) .

(11)

i=1

N only need to prove that J < 0, which indicates i=1 We ∞ T 2 T 0 (zi (t)zi (t) − γ wi (t)wi (t))dt < 0, implying z(t)2 ≤

⎡

(12)

where



× e(t) + 2ceT (t)(L¯ ⊗ Γ)e(t).

J = V˙ (t) +

2161

    ˜ max = λmax Ap +AT +λmax (In +RT R)+λmax C T Czp λ p zp    T   T ¯ C Dzq + Dp T Dzq + Dp λ + λmax Czp zp ¯ = |λmin ((DT Dzp − γ 2 I)−1 )|. and λ zp Proof: According to the Schur complement, the inequality in (10) is equivalent to A + AT + 2cL¯ ⊗ Γ + IN ⊗ (In + RT R) + CzT Cz  T  −1  D + C Tz Dz < 0 − D + C Tz Dz DzT Dz − γ 2 I DzT Dz − γ 2 I < 0. If Γ is an identity matrix, then we have A + AT + 2cL¯ ⊗ I + IN ⊗ (In + RT R) + CzT Cz  T  −1  D + C Tz Dz − D + C Tz Dz DzT Dz − γ 2 I ˜ max IN + 2cL) ¯ ⊗ In . ≤ (λ ˜ max IN + 2cL) ˜ max IN + 2cL¯ < 0, then (λ ¯ ⊗ In < 0, and If λ the condition in (10) is satisfied. The proof is completed. 

A + AT + 2c(L¯ ⊗ Γ) + IN ⊗ (In + RT R) D ⎣ ∗ −γ 2 In ∗ ∗

⎤ CzT DzT ⎦ < 0. −IN ×n

(10)

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When the second T–S fuzzy modeling method is adopted, we can give the following corollary by removing the uncertainties in Theorem 2. Corollary 1: The follower agents, as described in (1), without the external disturbance can reach consensus with ¯ is connected or, equivalently, at least one the leader if G agent in each connected component of G has access to the state information of the leader, and further c satisfies c > ¯ In addition, an H∞ performance λmax (A + AT )/2λmax (L). level γ can be guaranteed if the following linear matrix inequalities hold for p, q = 1, . . . , r: ˜ max IN + 2cL¯ < 0 λ T Dzp Dzp − γ 2 I < 0

(13)

where     ˜ max = λmax Ap + AT + λmax C T Czp λ p zp    T   T ¯ C Dzq + Dp T Dzq + Dp λ +λmax Czp zp ¯ = |λmin ((DT Dzp − γ 2 I)−1 )|. and λ zp B. Synchronization of Nonlinear Systems Here, we consider the application of the proposed fuzzy model of the error system in the synchronization problem of nonlinear systems. Suppose that the response system is ˙ x(t) = f (x(t), t) + g (x(t), t) u(t)

(14)

with the drive system in (2). By defining e(t) = x(t) − s(t), the fuzzy model of the error system can be obtained by utilizing the proposed fuzzy modeling method in Section II. Considering the external disturbance and controlled output, we augment the fuzzy model as follows: ˙ e(t) =

r 

λp (θ(t)) (Czp e(t) + Dzp w(t))

(15)

p=1

where Ap , Bp , Dp , Czp , and Dzp are the constant matrices, u(t) is the control input, and w(t) is the external disturbance. In this case, we consider the following fuzzy state-feedback controller: Controller Rule p: If θ1 (t) is Mp1 and θ2 (t) is Mp2 and . . . and θκ (t) is Mpκ , then up (t) = Kp e(t)

r  p=1

λp (θ(t)) Kp e(t).

λp (θ(t)) λq (θ(t))

p=1 q=1

× (Ap e(t) + Bp Kq e(t) + Dp w(t)) + A¯ (x(t)) e(t) (17) z(t) =

r 

λp (θ(t)) (Czp e(t) + Dzp w(t)) .

(16)

(18)

p=1

Before proceeding further, we first give the following definition to help the reader understand the synchronization problem. Definition 3: If there exists a controller in the form of (16) so that the system in (14) without the external disturbance asymptotically synchronizes to s(t), the solution of the unforced system s˙ (t) = f (s(t), t), i.e., limt→∞ x(t) − s(t) = 0. Thus, the nonlinear system in (1) is called reaching synchronization. Moreover, given a scalar γ > 0, under zero initial conditions and nonzero external disturbances, if the controlled output z(t) satisfies z(t)2 ≤ γw(t)2 , then we say that the system in (14) reaches synchronization with a prescribed H∞ performance level. The following theorem will present a criterion guaranteeing the synchronization of the nonlinear systems with a prescribed H∞ disturbance attenuation level γ. Theorem 3: Consider the fuzzy system in (15). For a given scalar γ > 0, there exists a state-feedback controller which guarantees the synchronization of the response system in (14) and the drive system in (2) with the H∞ disturbance attenuation level below γ, if there exist positive definite matrix X and matrix Fp so that the following set of linear matrix inequalities are feasible: p = 1, . . . r

(19)

p = 1, . . . , r − 1,

where ⎡

sym (Ap X + Bp Fq ) ⎢ ∗ ⎢ Φpq =⎢ ∗ ⎢ ⎣ ∗ ∗

Dp −γ 2 I ∗ ∗ ∗

q = p + 1, . . . r (20)

T XCzp T Dzp −I ∗ ∗

I 0 0 −I ∗

⎤ XRT 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −I

p, q = 1, . . . , r. Moreover, if the given conditions are satisfied, the controller gains are given by Kp = Fp X −1 ,

where Kp is the local controller gain. The compact form of the controller can be written as the following input–output form: u(t) =

r r  

Φpq + Φqp < 0,

+ A¯ (x(t)) e(t) z(t) =

˙ e(t) =

Φpp < 0,

λp (θ(t)) (Ap e(t) + Bp u(t) + Dp w(t))

p=1

r 

Then, from (15) and (16), the closed-loop system is given by

Proof: Choose candidate:

the

p = 1, . . . , r.

following

Lyapunov

V (t) = eT (t)P e(t)

(21) functional

ZHAO et al.: H∞ CONSENSUS AND SYNCHRONIZATION OF NONLINEAR SYSTEMS BASED ON NOVEL FUZZY MODEL

where P is a positive definite matrix. The time derivative of V (t) along the trajectories of (18) is V˙ (t) = 2eT (t)P

r r  

λp (θ(t)) λq (θ(t))

p=1 q=1



× (Ap e(t) + Bp Kq e(t) + Dp w(t)) + A¯ (x(t)) e(t) r  r 

=



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where ⎡ Φpq

⎢ ⎢ ⎢ =⎢ ⎢ ⎣

sym (Ap X + Bp Fq )

Dp

T XCzp

∗

−γ 2 I

T Dzp

∗

∗

−I

∗

∗

∗

λp (θ(t)) λq (θ(t))

I

⎤

⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎦ −I p, q = 1, . . . , r.

p=1 q=1

 × eT (t)sym (P (Ap + Bp Kq )) e(t) + 2eT (t)P A¯  × (x(t)) e(t) + 2eT (t)P Dp w(t) . (22) By observing Assumption 1 and according to Lemma 1, we have the following inequality: V˙ (t) ≤

r r  

λp (θ(t)) λq (θ(t)) eT (t)

p=1 q=1

 × sym (P (Ap + Bp Kq )) e(t) + eT (t)    × P P + RT R e(t) + 2eT (t)P Dp w(t) . (23) By considering (22), (23), and (18), index J = V˙ (t) + zT (t)z(t) − γ 2 wT (t)w(t) can be calculated as J=

r r  

λp (θ(t)) λq (θ(t)) η T (t)

p=1 q=1

 ×

Γ11 ∗

 T Czp Dzq + P Dp η(t) T Dzp Dzq − γ 2 I

Φpq + Φqp < 0,

p = 1, . . . r p = 1, . . . , r − 1,

Kp = Fp X −1 ,

p = 1, . . . , r.

(27)

Remark 1: It is noted that, here, we solve the synchronization problem using the common Lyapunov functional approach, which is simple with less computational cost. Readers can also refer to other Lyapunov functional methods such as the basis-dependent Lyapunov functional method and piecewise Lyapunov functional method, which can help reduce the conservatism of the results but may have larger computational cost [14], [48]. Remark 2: Here, the aim is to show the application of the proposed fuzzy modeling idea in the synchronization problem, which provides an alternative approach with less computational complexity for designing synchronization controllers when the response system and drive system have identical or similar structures. IV. I LLUSTRATIVE E XAMPLES

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where η(t) = [eT (t) wT (t)]T , and Γ11 = sym(P (Ap + T Czq . Bp Kq )) + P P + RT R + Czp −1 Defining P = X and pre- and postmultiplying both sides of the inequalities in (25) and (26) by diag{P T , I, I, I} and its transpose, we can conclude the negativeness of the inequality in (22). Assuming that w(t) ≡ 0, we can obtain V˙ (t) < 0; thus, limt→∞ x(t) − s(t) = 0. The asymptotic synchronization of the system in (18) is proven. Moreover, when w(t) = 0, index J < 0 implies that the H∞ performance of the controlled system in (18) is guaranteed. This completes the proof.  If the second type of the fuzzy model is adopted, we can give the following corollary by removing the uncertainties in Theorem 6. Corollary 2: Consider the second fuzzy system in (15) with¯ out the time-varying uncertainty A(x(t)). For a given scalar γ > 0, there exists a state-feedback controller that guarantees the synchronization of the response system in (14) and the drive system in (2) with the H∞ disturbance attenuation level below γ if there exist positive definite matrix X and matrix Fp , so that the following set of linear matrix inequalities are feasible: Φpp < 0,

Moreover, if the given conditions are satisfied, the controller gains are given by

(25) q = p + 1, . . . r (26)

Here, the illustrative examples will be given to show the effectiveness of the obtained theoretical results. The first two examples are concerned with the leader-following consensus problem, and the third is concerned with the synchronization problem. By applying the obtained protocol or controller to the initial nonlinear systems, the effectiveness of the proposed method will be verified. A. Example 1 Consider the multiagent network in Fig. 1 consisting of N identical systems. The dynamics of each node is described by the following chaotic equation:  ai, j Γ (xj (t)−xi (t)) , i = 1, . . . , N x˙ i (t) = f (x(t), t)+c j∈Ni

(28) where ⎡

−σxi1 (t) + σxi2 (t)

⎤

⎥ ⎢ f (x(t), t) = ⎣ η1 xi1 (t) − η2 xi2 (t) − xi1 (t)xi3 (t) ⎦ . xi1 (t)xi2 (t) − bxi3 (t) By choosing the values of (σ, η1 , η2 , b) as (10, 28, −1, 8/3) for chaos to emerge, the system in (28) becomes the Lorenz system.

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The error system between (28) and the isolated node s˙ (t) = f (s(t), t) can be derived as ⎤ ⎡ −σei1 (t) + σei2 (t) e˙ i (t) = ⎣ η1 ei1 (t) − η2 ei2 (t) − xi1 (t)ei3 (t) + ei1 (t)ei3 (t) ⎦ −bxi3 (t) + xi1 (t)ei2 (t) − ei1 (t)ei2 (t) ⎡ ⎤ N 0 0 0  + ⎣ −xi3 (t) 0 0 ⎦ ei (t) + c Li, j Γej (t) (29) j=1 xi2 (t) 0 0

Fig. 1.

Connected network topology.

Fig. 2.

Consensus of the multiagent network.

To illustrate the control performance clearly, here we choose N = 13, and N can be much larger even when the network is scale free. We use the first type of the T–S fuzzy model to describe the error system in (29), and the T–S fuzzy model can be established as follows: Rule jk : IF xi1 (t) is M1j and ei1 (t) is M ek THEN e˙ i (t) = Aˇjk ei (t) + A¯ (xi (t)) ei (t) +c

N 

Li, j Γej (t),

j, k = 1, 2

j=1

where xi1 (t) ∈ [M11 M12 ], ei1 (t) ∈ [M e1 M e2 ], M11 = −20, M12 = 30, M e1 = −50, and M e2 = 50. For simulation, by adding the external disturbance and controlled output, the augmented system becomes e˙ i (t) =

4 

λp (xi (t), ei (t)) can be calculated according to (6), and the membership functions are assumed to be in the following triangular form:

λp (xi (t), ei (t)) (Ap ei (t) + Dp wi (t))

p=1

+ A¯ (xi (t)) ei (t) + c

N 

Li, j Γej (t) M1 (xi1 (t)) =

j=1

zi (t) =

4 

λp (xi (t), ei (t))

M e1 (ei1 (t)) =

p=1

× (Czp ei (t) + Dzp wi (t)) ,

i = 1, . . . , N (30)

where A1 = Aˇ11 , A2 = Aˇ22 , A3 = Aˇ12 , A4 = Aˇ21 ⎤ ⎡ −σ σ 0 Aˇjk = ⎣ η1 η2 Mek − M1j ⎦ 0 M1j − Mek −b ⎤ ⎡ 0 0 0 A¯ (xi (t)) = ⎣ −xi3 (t) 0 0 ⎦ xi2 (t) 0 0 ⎡ ⎤ ⎡ ⎤ 0.001 0.001 D1,2 = ⎣ 0.002 ⎦ , D3,4 = ⎣ 0.003 ⎦ 0.008 0.008 Czp = [ 1

0 0 ] , Dzp = 0.1.

¯ i (t)) is bounded by RT R and ¯ i (t))T A(x A(x ⎡ ⎤ 0 −60 40 R = ⎣0 0 0 ⎦. 0 0 0

(31)

−xi1 (t) + M2 xi1 (t) − M1 , M2 (xi1 (t)) = M2 − M1 M2 − M1 −ei1 (t)+M e2 ei1 (t)−M e1 , M e2 (ei1 (t)) = . M e2 −M e1 M e2 −M e1

Here, we suppose the gain d1 = 1 and others dk = 0, for k = 2, . . . , 13, i.e., the first node is pinned. For a given H∞ performance level γ = 1.3, solving Theorem 2 based upon the fuzzy model in (30), we obtain the coupling strength in (8) as c = 20 918. Assuming the nonzero initial condition of ei (t) and wi (t) = 0, we apply the obtained protocol to the network in (28). The consensus of the network is shown in Fig. 2, which indicates that all the follower agents reach consensus with the signal s(t) in a short time. The consensus is verified again by the convergence of the errors between the states of the follower agents and s(t), shown in Fig. 3. Next, to illustrate the H∞ performance, we assume that the identical initial condition of ei (t) is zero and the external disturbance is  0.8, 0.02s ≤ t ≤ 0.03s wi (t) = w(t) = 0, elsewhere. By applying the protocol to the network with the disturbance, the error states of the nodes converging to zero are shown in Fig. 4, and the disturbance attenuation is shown in Fig. 5. By calculation, we obtain zi (t)22 w(t)22 = 0.0345/0.00054,

ZHAO et al.: H∞ CONSENSUS AND SYNCHRONIZATION OF NONLINEAR SYSTEMS BASED ON NOVEL FUZZY MODEL

Fig. 3.

Error states.

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Fig. 5. Signals zi (t) and w(t).

of the authors’ knowledge, so far, there are no explicit results concerning the problem of distributed H∞ control of nonlinear multiagent systems. Based on the proposed fuzzy model, the problem can be efficiently solved. Releasing the fuzzy model is possible but substantially more complicated formulas must be worked out, and it is difficult. Moreover, the conditions in [5] are obtained by assuming that the network is connected, The assumption is relaxed in this paper, and an arbitrary network topology is considered, which will be demonstrated in the following example. B. Example 2

Fig. 4.

Error states with disturbance.

In this example, we consider the unconnected multiagent network in Fig. 6 consisting of 16 identical nodes. The dynamics of each node is described by the following nonlinear system [36]:  x˙ i (t) = f (x(t), t) + c ai, j Γ (xj (t)−xi (t)) , i = 1, . . . , 16 j∈Ni

which yields γ ∗ = 0.0223 (below the prescribed γ = 1.3), indicating the effectiveness of the protocol design method well. The pinning control problem of a nonlinear network is investigated in [5]. Based on the assumptions that the vector field of the uncoupled nodes is Lipschitz and the function f (x, t) − σx (where σ is the coupling strength) is QUAD, a sufficient condition is presented to guarantee network consensus. By applying Theorem 6 of [5] to chaotic system (28) with the same control gain and pinned node, the network can achieve consensus attenuating the effect of disturbances with the strength σ = 320. The coupling strength is c = 20 918 obtained by Theorem 4, and c = 265 obtained by Corollary 5. In this case that the nonlinear function satisfies the Lipschitz condition, the method in [5] has less conservatism than Theorem 4 but is more conservative than Corollary 5. However, in this paper, the attention is to solve the consensus or synchronization problem for a class of general nonlinear systems that need not satisfy the Lipschitz assumption, for the proposed fuzzy model can deal with any nonlinear smooth functions. In addition, to the best

(32) where ⎤ xi2 (t) + sin (xi3 (t)) ⎢ xi1 (t) + 2xi2 (t) ⎥ f (x(t), t) = ⎣ ⎦. xi1 (t)2 xi2 (t) + xi1 (t) sin (xi3 (t)) + xi4 (t) ⎡

The error system between (32) and the isolated node s˙ (t) = f (s(t), t) can be derived as in (33), which is shown at the bottom of the next page. We use the second type of the T–S fuzzy model to describe the error system in (33), and the T–S fuzzy model can be established as Rule jklmq : IF θi1 (t) is M1j and θi2 (t) is M2k and θi3 (t) is M3l and θi4 (t) is M4m and θi5 (t) is M5q THEN e˙ i (t) = Aˇp ei (t) + c

N  j=1

Li, j Γej (t), j, k, l, m, q = 1, 2

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Fig. 6. Unconnected network topology.

where θi1 (t) = cos(2xi3 (t)−ei3 (t)/2), θi2 (t) = sin(ei3 (t)/2), θi3 (t) = xi1 (t), θi4 (t) = ei1 (t), and θi5 (t) = xi2 (t). By local nonlinearity idea, we can assume θi1 (t) ∈ [M11 , M12 ], θi2 (t) ∈ [M21 , M22 ], θi3 (t) ∈ [M31 , M32 ], θi4 (t) ∈ [M41 , M42 ], and θi5 (t) ∈ [M51 , M52 ], where M11 = cos(5), M12 = 1, M21 = sin(0.6)/0.6, M22 = 1, M31 = −0.8, M32 = 0.8, M41 = −1.6, M42 = 1.6, M51 = −0.1, and M52 = 0.1 [36]. The system matrices are given by

Fig. 7.

Error states.

Fig. 8.

Consensus in a 3-D plane.

A1 = Aˇ11111 , A2 = Aˇ11112 , A3 = Aˇ11121 , A4 = Aˇ11122 , . . . , A32 = Aˇ22222 where Aˇjklmq ⎡

0 1 ⎢ =⎣ 2M3l M5q −M4m M5q 0

1 2 (M3l −M4m )2 0

M1j M2k 0 0 M1j M2k

⎤ 0 0⎥ ⎦ 0 1

The membership functions are chosen to be in the triangular form and can be obtained analogously to those in Example 1. The aim is to make the multiagent system reach consensus compared with other methods. A similar problem has been investigated in [44], and a sufficient condition is given to guarantee the synchronization of the complex network with the undirected network topology based on an assumption that the network is connected. The assumption is also needed in many other studies [27]. Note that the methods in [27] and [44] cannot deal with the unconnected multiagent network shown in Fig. 6. For example, if the pinned nodes are chosen randomly according to [44] as 1, 5, 9, and 10 in Fig. 6, then the network cannot reach consensus at all. To make the multiagent network connected according to Corollary 5, we choose to pin nodes 1, 5, 9, and 16 in Fig. 6. By solving Corollary 5, we can obtain the coupling strength as c = 204. Fig. 7 depicts the error states ei (t), where the initial conditions of ei (t) are nonzero and

⎡

 ei2 (t) + 2 cos

wi (t) = 0. Fig. 8 demonstrates the trajectories of the followers approaching the leader in a 3-D plane. C. Example 3 In this example, we will show the effectiveness of the proposed method for the synchronization of general nonlinear systems. Suppose that the response system is x(t) ˙ = f (x(t), t) + Bu(t) where f (x(t), t) is described by the Chen equation in (28) with the values of (σ, η1 , η2 , b) changed to (35, −7, 28, 3). The error

2xi3 (t)−ei3 (t) 2



 sin

ei3 (t) 2



⎤

⎥ ⎢ N ⎥ ⎢  ei1 (t) + 2ei2 (t) ⎥ ⎢ Li,j Γej (t) e˙ i (t) = ⎢ ⎥+c ⎢ (xi1 (t) − ei1 (t))2 ei2 (t) + (2xi1 (t)xi2 (t) − ei1 (t)xi2 (t)) ei1 (t) ⎥ j=1 ⎦ ⎣     i3 (t) sin ei32(t) 2 cos 2xi3 (t)−e 2

(33)

ZHAO et al.: H∞ CONSENSUS AND SYNCHRONIZATION OF NONLINEAR SYSTEMS BASED ON NOVEL FUZZY MODEL

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system between the response system above and the drive system (2) is given in (34), which is shown at the bottom of the page. Similar to Example 2, we can obtain the fuzzy model of the given system: ˙ e(t) =

r 

λp (θ(t)) Ap e(t) + Bu(t)

p=1

where A1 = Aˇ1111 , A2 = Aˇ1112 , A3 = Aˇ1121 ⎡

A4 = Aˇ1122 , . . . , A16 = Aˇ2222

−σ Aˇjklm = ⎣ η1 − M3l M2k ⎡ ⎤ 1 B = ⎣0⎦. 0

M1j

σ η2 − Mem

⎤ 0 Mem − M1j ⎦ −b Fig. 9. Synchronization of the systems.

For simulation, the other matrices in (15) are arbitrarily chosen as Bp = [ 0.1 Czp = [ 1

0.1

0 ]T , Dp = [ 0.01

0 0],

Dzp = 0.1.

0.03

0.08 ]T

The membership functions can be analogously obtained with those in Example 1, where θ1 (t) = x1 (t), θ2 (t) = x2 (t), θ3 (t) = x3 (t), and θ4 (t) = e1 (t). Choosing the H∞ disturbance attenuation level γ = 1 and solving inequalities (25) and (26) in Corollary 2, we can obtain the controller gains by (27). First, by assuming the external disturbance is zero, the trajectories of the response system stimulated by the obtained controller and drive system are shown in Fig. 9, where the initial conditions are x0 = [−5, 4, −3] and s0 = [5, 3]. The synchronization error of the states is shown in Fig. 10, which confirms that the two nonlinear systems are synchronized. Next, for simulation, we assume that the initial condition of e(t) is zero and the external disturbance is  2.5, 0.5s ≤ t ≤ 1s (35) w(t) = 0, elsewhere. Fig. 11 shows the response of the signal e(t) converging to zero subject to the disturbance by the application of the obtained controller to the Chen system, and the disturbance attenuation is shown in Fig. 12 by signals z(t) and w(t). By calculation, we obtain z(t)22 = 0.663 and w(t)22 = 3118 in simulations; then, the simulated H∞ performance is γ ∗ = 0.0146, below the prescribed H∞ performance level of 1.

Fig. 10. Synchronization error of the states.

V. C ONCLUSION In this paper, the H∞ consensus problem of nonlinear multiagent systems has been investigated. First, a novel T–S fuzzy model is proposed to describe the error dynamics between the two nonlinear systems. Based on the model, the H∞ consensus algorithm has been designed so that all the follower agents reach consensus with the leader and preserve an H∞ performance level. Moreover, the fuzzy modeling method has been extended to formulating the synchronization problem of the nonlinear systems, and a fuzzy H∞ controller has been designed to guarantee the synchronization of the nonlinear systems. The effectiveness of the obtained theoretical results are finally verified by numerical simulations.

⎤ −σe1 (t) + σe2 (t) ˙ e(t) = ⎣ (η1 + x3 (t)) e1 (t) − η2 e2 (t) − x1 (t)e3 (t) + e1 (t)e3 (t) ⎦ + Bu(t) x2 (t)e1 (t) − bx3 (t) + x1 (t)e2 (t) − e1 (t)e2 (t) ⎡

(34)

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Fig. 11. Synchronization error of the states with disturbance.

Fig. 12. Signals z(t) and w(t).

Future work should include robust protocol design for the nonlinear multiagent system with parameter uncertainties and a solution for the synchronization problem when time delays exist in communication channels by improving the fuzzy modeling method proposed in this paper [12], [25]. R EFERENCES [1] A. Bonarini and V. Trianni, “Learning fuzzy classifier systems for multiagent coordination,” Inf. Sci., vol. 136, no. 1–4, pp. 215–239, Aug. 2001. [2] Y. Y. Cao and Z. L. Lin, “Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation,” IEEE Trans. Fuzzy Syst., vol. 11, no. 1, pp. 57–67, Feb. 2003. [3] B. Chen and X. P. Liu, “Delay-dependent robust H∞ control for T–S fuzzy systems with time delay,” IEEE Trans. Fuzzy Syst., vol. 13, no. 4, pp. 544–556, Aug. 2005. [4] F. Chen, Z. Chen, L. Xiang, Z. Liu, and Z. Yuan, “Reaching a consensus via pinning control,” Automatica, vol. 45, no. 5, pp. 1215–1220, May 2009. [5] P. Delellis, M. di Bernardo, and G. Russo, “On QUAD, Lipschitz, and contracting vector fields for consensus and synchronization of networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 3, pp. 576–583, Mar. 2011. [6] G. Feng, “Robust H∞ filtering of fuzzy dynamic systems,” IEEE Trans. Aerosp. Electron. Syst., vol. 41, no. 2, pp. 658–671, Apr. 2005.

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Yan Zhao received the B.S. degree in chemical engineering and equipment control and the M.S. degree in mechanical engineering from Inner Mongolia University of Technology, Hohhot, China, in 2002 and 2005, respectively, and the Ph.D. degree in control science and engineering from Harbin Institute of Technology, Harbin, China, in 2010. Her research interests include fuzzy control systems and robust control and networked control systems.

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Bing Li received the Ph.D. degree in mechanical design and theory from Harbin Institute of Technology, Harbin, China. He is currently a Professor in mechanical engineering and the Dean with the School of Mechanical Engineering and Automation, Shenzhen Graduate School, Harbin Institute of Technology. His research interests include mechanisms and robotics, parallel kinematic machines, vibration and control, etc.

Jiahu Qin (S’11–M’12) received the B.S. and M.S. degrees in mathematics from Harbin Institute of Technology, Harbin, China, in 2003 and 2005, respectively. He is currently working toward the Ph.D. degree at the Australian National University, Canberra, Australia. His research interests include consensus problems in multiagent coordination and synchronization of complex dynamical networks.

Huijun Gao (SM’09) received the Ph.D. degree in control science and engineering from Harbin Institute of Technology, Harbin, China, in 2005. From November 2003 to August 2004, he was a Research Associate with the Department of Mechanical Engineering, The University of Hong Kong, Hong Kong. From October 2005 to October 2007, he carried out his postdoctoral research with the Department of Electrical and Computer Engineering, University of Alberta, Canada. Since November 2004, he has been with Harbin Institute of Technology, where he is currently a Professor and the Director of the Research Institute of Intelligent Control and Systems. His research interests include networkbased control, robust control/filter theory, and time-delay systems and their engineering applications. Dr. Gao is an Associate Editor of Automatica; the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS ; the IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS —PART B: C YBERNETICS; the IEEE T RANSAC TIONS ON F UZZY S YSTEMS ; the IEEE T RANSACTIONS ON C IRCUITS AND S YSTEMS —PART I; etc. He has also been serving on the AdCom of the IEEE Industrial Electronics Society since 2013.

Hamid Reza Karimi (M’06–SM’09) received the B.Sc. degree in power systems engineering from Sharif University of Technology, Tehran, Iran, in 1998, and the M.Sc. and Ph.D. degrees in control systems engineering from the University of Tehran, Iran, in 2001 and 2005, respectively. He is currently a Professor of control systems with the Faculty of Engineering and Science, University of Agder, Kristiansand, Norway. His research interests include nonlinear systems, networked control systems, robust control/filter design, wavelets, and vibration control, with an emphasis on applications in engineering. Dr. Karimi serves as the Chair of the IEEE chapter on control systems at the IEEE Norway section. He is a member of the IEEE Technical Committee on Systems with Uncertainty and the Technical Committees on Robust Control and on Automotive Control of the International Federation of Automatic Control. He is also serving as an editorial board member for some international journals, such as Mechatronics, Information Sciences, Neurocomputing, Asian Journal of Control, Journal of The Franklin Institute, etc.