Multitarget Tracking Control for Coupled Heterogeneous ... - IEEE Xplore

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Multitarget Tracking Control for Coupled Heterogeneous Inertial Agents Systems Based on Flocking Behavior Shiming Chen, Huiqin Pei , Qiang Lai, and Huaicheng Yan

Abstract—In this paper, the flocking behavior and targets consensus tracking problems of heterogeneous multiple inertial agents with limited communication ranges are investigated. Consider the inertial effect of real agents, a distributed control protocol is designed such that the agents can achieve stable group behaviors. Using the decomposition approach, the stability of a multiple inertial agents system is proved. Combining with the flocking behavior of multiple inertial agents, agents themselves are in charge of searching and choosing the target in an autonomous and individual way according to the relationship between velocities of agents and targets. An augmented distributed multiflocking method is proposed to guarantee that the multitarget consensus tracking can be reached for heterogeneous multiagent systems. It shows that the proposed control approach can not only ensure local flocking with pursuing agents, but also make heterogeneous agents consistently track the multitarget. Index Terms—Consensus tracking, distributed control, heterogeneous multiagent systems, inertial effect.

I. I NTRODUCTION N RECENT years, the multiagent distributed coordination control not only has received much attention, but also has made significant research progress on account of its extensive applications, including system science [1]–[13], intelligent control [14]–[17], computer science [18], [19], robotics [20], [21], and biology communities. According to the individual characteristics or function, multiagent can be divided into homogeneous group and heterogeneous group. The idea of coordinated control is usually derived from the

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Manuscript received September 20, 2017; revised November 28, 2017; accepted December 27, 2017. This work was supported in part by the Natural Science Foundation of China under Grant 61364017 and Grant 11662002, in part by the Innovation Team Project of Jiangxi Provincial Innovation Drive 5511 Advantaged Science and Technology under Grant 20165BCB19011, in part by the Natural Science Foundation of Jiangxi Provincial Science and Technology Department under Grant 20171BAB202029, and in part by the Key Research and Development Project of Jiangxi Provincial Technology Department under Grant 20161BBE53008. This paper was recommended by Associate Editor X. Zhao. (Corresponding author: Huiqin Pei.) S. Chen, H. Pei, and Q. Lai are with the School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang 330013, China (e-mail: [email protected]; [email protected]; [email protected]). H. Yan is with the School of information Science and Engineering, East China University of Science and Technology, Shanghai 200237, 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/TSMC.2017.2789335

biological behavior, such as migratory birds, foraging ants, fish and bees, etc. [22]–[24]. Flocking has the characteristics of universality, which is that a group of simple intelligent agents form into a coordinated motion through only limited environment information and simple rules. Reynolds [24] proposed a flocking model which follows three simple rules: 1) separation; 2) cohesion; and 3) alignment. Olfati-Saber [25] came up with a theoretical framework on the design and analysis of multiple flocking algorithms of multiagent systems. Cucker and Smale [26] put forward a continuous and discrete time model describing the evolution of flocks. On that basis, Shang [27] investigated the emergent behavior of four types of generic dynamical systems under random environmental perturbations. Antonelli et al. [28] presented null-space-based behavioral control architecture to let a group of robots flock. Shang and Bouffanais [29] studied influence of the number of topologically interacting neighbors on swarm dynamics. Then, based on these results, a host of researchers examined the flocking of multiagents from different points of view (see [30]–[34]). Nevertheless, as far as we know many results have mainly discussed the flocking of homogeneous groups without inertias. However, most of actuators are able to only affect the acceleration via the agents’ inertias. So it is indispensable to think about the inertial effect in the design of multiagent distributed coordination control. In [35] and [36], the inertial effect was considered primarily in simple flocking of multiple inertial agents, namely only including cohesion and alignment. In many practical applications, individual heterogeneity exists widely, and the rule of separation is indispensable. Therefore, the complex flocking of heterogeneous multiple inertial agents has not been fully investigated and there is still more research space. In our previous work, multitarget consensus circle pursuit has been studied for multiagent systems via a distributed multiflocking method [37]. Zhang and Lewis [38] investigated adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics. For a class of strict feedback nonlinear systems with parametric uncertainties, adaptive consensus tracking control was designed using the backstepping technique [39], [40]. Then, Wang [41] proposed a new distributed adaptive tracking control for unknown model parameters and unknown dynamics with the informed agent. However, there has been little research on the multitarget tracking control of heterogeneous systems.

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

The main contributions of this paper can be summarized as follows. 1) Motivated by the existing research work, the complex flocking of multiple inertial agents is analyzed via the decomposition approach. 2) A distributed control protocol is designed to achieve stable group behaviors of multiple inertial agents. 3) Combining with the flocking behavior of multiple inertial agents, an augmented distributed multiflocking method is proposed to ensure that the multitarget consensus tracking can be reached for heterogeneous multiagent systems. The rest of this paper is organized as follows. Section II presents quite a few preliminaries in graph theory and problem formulation. Section III describes our main results, including the proposed distributed control protocol and the augmented distributed multiflocking method. Illustrative examples are provided to evaluate the proposed method in Section IV. Finally, Section V gives some concluding remarks. Notations: Denote R as the set of real numbers, Rn as the set of real vectors and Rn×n as the n × n real matrix space. · represents the Euclidean norm of a matrix. And diag[ · ] represents a diagonal matrix.

II. P RELIMINARIES AND P ROBLEM S TATEMENT A. Preliminaries of Graph Theory The information interactions among agents can be represented by a weighted undirected graph G = (V, E, A), where V = {1, 2, . . . , n} is the set of nodes, i.e., agents; E ⊆ V × V is the set of edges, namely, ordered pairs of the nodes; A = [aij ] ∈ Rn×n is a weighted adjacency matrix, which is defined as follows: aij = 1, if (i, j) ∈ E, aij = 0 otherwise, and (i, j) denotes an edge of G, aij denotes the weight of (i, j). n×n is defined as D = diag(d), where The degree matrix D ∈ R as di = j∈V aij . The Laplacian matrix L ∈ Rn×n is defined L = L(G) = [lij ], namely, L = D − A, where lii = np=i aip and lij = −aij , ∀i = j. B. Problem Statement Consider a multiple inertial agents system composed of n agents and b targets. The position, velocity and mass of agent i ∈ V = {1, 2, . . . , n} are denoted by ri , vi , and mi , respectively, and the position and velocity of target k ∈ W = {1, 2, . . . , b} are denoted by rtk , and vtk , respectively, and assume that n > b. The control protocol/acceleration of agent i and target k are denoted by ui , and utk . The agent i has an observation area with a radius rc , and the neighbor set of agent i is described as Ni (t) = j ∈ V : rj − ri < rc . The dynamic model of agent i is governed by

r˙i = vi mi v˙ i = ui

i∈V

(1)

where mi > 0 denotes the mass of agent i, and ui is described as the following form ui = kp aij rj − ri − ds eij + kv aij vj − vi j∈N

j∈Ni

i rj − ri eij = rj − ri

where kp , kv > 0 are the stiffness and damping gains, respectively. It is usually merely possible to directly control the acceleration. Hence, the agents’ inertias are incorporated into the flocking model. The closed-loop dynamics of agent i is described as follows: − kp aij ri − rj − ds sgn ri − rj − kv aij r˙i − r˙j mi r¨i = j∈Ni

(2) where mi > 0, kp , kv > 0, and ds is the desired distance between agents. Moreover, the dynamic model of target k is governed by r˙tk = vtk k∈W (3) mtk v˙ tk = utk where mtk > 0 denotes the mass of target k. The goal of this paper is to deal with the flocking and targets consensus tracking problems of heterogeneous multiple inertial agents under local information interaction environment. Due to the inertial effect of agents causing unstable group behaviors, the agents’ inertias are considered in the flocking model and the distributed coordination control design with multiple agents. On the basis of the flocking of multiple inertial agents, an augmented distributed multiflocking method is presented to ensure that heterogeneous agents can consistently track multitargets for heterogeneous multiagent systems. In the conclusion, the effectiveness of the control method is illustrated via numerical examples. III. M AIN R ESULTS A. Stable Flocking of Multiple Inertial Agents In this section, we investigate stable flocking of multiple inertial agents through the passive decomposition [41]. According to (2), the closed-loop group dynamics is described as M¨r + kv L˙r + kp Lr = kp DDs

(4)

where M = diag[m1 , m2 , . . . , mn ] ∈ Rn×n , r = [r1 , r2 , . . . , rn ]T ∈ Rn×1 , and Ds = ds 1n×1 ∈ Rn×1 . The closed-loop group dynamics (4) can be decomposed into a shape system and a locked system. The coordinate transformation is defined as z = Tr, where T ∈ Rn×n is the full rank transformation matrix as follows: ⎡ m1 ⎤ nm2 nm3 · · · nmn m n m m m i i i i i=1 i=1 i=1 ⎢ i=1 −1 0 ··· 0 ⎥ ⎢ 1 ⎥ ⎢ 0 1 −1 ··· 0 ⎥ T=⎢ ⎥ ⎢ . .. ⎥ .. .. .. ⎣ .. . . . . ⎦ 0

···

···

1

−1

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and z = [z1 , z2 , . . . , zn ]T ∈ Rn×1 is the transformed vector. Suppose that zτ = [z2 , z3 , . . . , zn ]T ∈ R(n−1)×1 , and then z = [z1 , zTτ ]T , where z1 describes the motion of the over group as z 1 = n

1

i=1 mi

(m1 r1 + m2 r2 + · · · + mn rn )

Using z = Tr, the dynamics (4) can be rewritten as T −T MT −1 z¨ + kv T −T LT −1 z˙ + kp T −T LT −1 z = T −T kp DDs (6) where the inverse of T is described as ⎤ ⎡ 1 ϕ2 ϕ3 ··· ϕn ⎢ 1 ϕ2 − 1 ϕ3 ··· ϕn ⎥ ⎥ ⎢ ⎢ ··· ϕn ⎥ T −1 = ⎢ 1 ϕ2 − 1 ϕ3 − 1 ⎥ ⎥ ⎢ .. .. .. . . .. ⎦ ⎣. .. . . ··· ϕn − 1 1 ϕ2 − 1 ϕ3 − 1 thereinto ϕi = nk=i mk / nj=1 mj . ˜ where ml = (6), we have T −T MT −1 = diag[ml , M], From n (n−1)×(n−1) ˜ is a symmetric positive i=1 mi , and M ∈ R definite matrix, as well as ˜T 0 D −T −1 T LT = 0(n−1)×1 L˜ ˜ ∈ R(n−1)×1 including where D n ˜j = − D (L1k + L2k + · · · + Lnk ) j ∈ {1, 2, . . . , n − 1} and L˜ ∈ R(n−1)×(n−1) including n n

Lst ,

i, j ∈ {1, 2, . . . , n − 1}.

s=i+1 t=j+1

Therefore, the closed-loop group dynamics (4) can be decomposed as follows: ˜ T z˙τ + kp D ˜ T zτ = kp Dl ml z¨1 + kv D ˜ zτ + kp Lz ˜ τ = kp D ¯D ¯s ˜ zτ + kv L˙ M¨

˜ M P= ˜ ςM

˜ ςM kp + ς kv L˜ s

∈ R2(n−1)×2(n−1)

(9)

ς > 0 is positive definite. Then, we have T dV z˙τ z˙τ =− ¯ s Q zτ − D ¯s zτ − D dt where

˜ kv L˜ s − ς M Q= − 12 kp − ς kv L˜ w

1 2

kp − ς kv L˜ w ς kp L˜ s

(10)

where Q ∈ R2(n−1)×2(n−1) , ς = kp /kv is chosen, then Q ˜ > 0 is positive definite when and only when kv2 L˜ s − kp M in (10). Similarly, P is positive definite when and only when ˜ in the (9). According to the stability crite2kv2 L˜ s > kp M rion, when both P and Q are positive definite, (˙zτ (t), (zτ (t) − ¯ s )) → 0 exponentially when t → ∞. From (5), we have D (˙ri (t) − r˙j (t), ri (t) − rj (t) − ds ) → 0, i, j ∈ {1, 2, . . . , n}. (i = 1, 2, . . . , n) is obtained in terms Moreover, z˙1 (t) → r˙i (t), of z˙1 (t) = ni=1 mi r˙i (t)/ ni=1 mi . B. Targets Consensus Tracking of Heterogeneous Multiagent Systems In this section, combining with the above flocking analysis of multiple inertial agents, we will present an augmented distributed multiflocking method. It guarantees that the multitarget consensus tracking can be achieved for heterogeneous multiagent systems. In Section III-A, the control protocol (or the acceleration) of each agent is given in the following form: (11) aij rj − ri − ds eij + kv aij vj − vi ui = kp

k=j+1

˜j + L˜ ij = ϕi+1 D

strongly connected and balanced. Here, a Lyapunov function is chosen as T 1 z˙τ z˙τ V= ¯ s P zτ − D ¯s 2 zτ − D where

and zτ describes the internal group shape, which is governed by T zτ = r1 − r2 , r2 − r3 , . . . , rn−1 − rn . (5)

3

(7) (8)

¯ D ¯ ∈ R(n−1)×(n−1) , Dl = d1 = where D = diag[Dl , D], (n−1)×1 . Here, the dynam¯ j∈V a1j , and Ds = ds 1(n−1)×1 ∈ R ics of z1 represents the centroid motion, and the dynamics of zτ describes the internal group shape. Theorem 1: Assume that the graph G is balanced and strongly connected, set the stiffness gain kv > 0, the damping ˜ > 0, where L˜ s = (1/2)(L˜ + L˜ T ) gain kp > 0 s.t. kv2 L˜ s − kp M ¯ s )) → 0, D ¯s = is positive definite. Thus, (˙zτ (t), (zτ (t) − D ds 1(n−1)×1 ∈ R(n−1)×1 , exponentially, and r˙i (t) → z˙1 (t), i = 1, 2, . . . , n when t → ∞. Proof: Consider the closed-loop group dynamics (4) as well as their locked and shape dynamics (7) and (8). Let L˜ s.t. L˜ = L˜ s + L˜ w , where L˜ w = (1/2)(L˜ − L˜ T ). From in [42, Proposition 2], L˜ s is positive definite as long as G is

j∈Ni

where

j∈Ni

rj − ri , k , k > 0. eij = rj − ri p v

In the augmented distributed multiflocking method, the design of corresponding distributed control protocols is based on (11). The search and tracking of the target is mainly determined by two factors including the number of pursuing agents for each target, which is denoted by nt and the velocity relation of pursuing agents and targets. The augmented distributed multiple flocking method is set up through the following procedures. 1) When agent i is in target searching mode, agent i seeks a target with required maximum number of pursuing agents. Meanwhile, the velocity relation of agent i and the target needs to satisfy the following condition vi vt max

=

vi [angle(vi )−angle(vt max )] e vt max



where vi = vi eangle(vi ) denotes the velocity of agent i, and vt max = vt max eangle(vt max ) denotes the velocity of the target with required maximum number of pursuing agents, as well as [angle(vi ) − angle(vt max )] ∈ [0, (π/2)], vi /vt max ∈ (0, 1]. Then the control protocol of agent i is (11). 2) When agent i is in target tracking mode, the distributed control protocol of agent i is described as follows: aij rj − ri − ds eij + kv aij vj − vi ui = kp j∈Ni

j∈Ni

+ c13 (rtmax − ri ) + c23 (vt max − vi )

(12)

where rt max denotes the position of the target with required maximum number of pursuing agents, and c13 > 0, c23 > 0. If the agent i searches for multiple targets, then the distributed control protocol of agent i is changed as follows: aij rj − ri − ds eij + kv aij vj − vi ui = kp j∈Ni

+

m

j∈Ni γ

ui,tk

(13)

k=1

where uγi,tk =

c12 (rtk − ri ) + c22 (vtk − vi ) , i ∈ V, k ∈ W ntk

c12 > 0, c22 > 0, and ntk represents the number of pursuing agents for any target k. 3) When agent i enters into the circle region of the target as the center and dc as the radius, and the number of pursuing agents meets requirement for the target. Then the distributed control protocol of agent i is described as aij rj − ri − ds eij + kv aij vj − vi ui = kp j∈Ni

j∈Ni

− c14 (rt − ri − dc )eit − c24 (vt − vi )

(14)

where c14 > 0, c24 > 0, and eit = [(rt − ri )/(rt − ri )]. Ultimately, the connected graph G of pursuing agents can be guaranteed for the same target. Lemma 1: Consider the agents given by (1) with the distributed control protocol, as the following properties have: collision can be avoided between agents, that is ri − rj > 0, i, j = 1, 2, . . . , n when t ≥ 0; all agents make up local flocks with tracking targets; the velocities approach consensus for agents with the common target, i.e., lim (vi − vj ) = 0, t→∞ i, j = 1, 2, . . . , ntk . Proof: From Theorem 1, under the condition of rc ≥ 2dc (namely subgraph Gk is connected by pursuing agents with the target), we further know (˙ri (t) − r˙j (t)) → 0, (ri (t) − rj (t) − ds ) → 0, i, j ∈ {1, 2, . . . , ntk } for agents in the circle region of the target k as the center and dc as the radius, and the target to be coincident with the mass centroid of these pursuing agents.

Fig. 1.

Target consensus tracking with three agents.

Theorem 2: Assume that the desired distance between pursuing agents is expressed as ds , namely rj − ri = ds . Then dc cos(π/ntk ) ≥ (1/2)ds is the sufficient condition of consistency tracking with the target, where ntk ≥ 4, and dc denotes the radius of a round with the target as the center. Proof: By Lemma 1, it is known that ntk agents track congruously target k, that is v1 ≡ v2 ≡ . . . ≡ vntk ≡ vtk . If ntk agents form a round with the radius dc , then β = (2π/ntk ). Using the mathematical induction method, there are the following steps. 1) When ntk = 4, β = (2π/4), dc cos(π/4) = (1/2)ds , and then for the square, ttarget = tagent , as shown in Fig. 1. 2) When ntk = k, β = (2π/k) (k ∈ N + ), assume that dc cos(π/k) > (1/2)ds , and then for regular k-angle form, there is ttarget > tagent . 3) When ntk = k + 1, β = [2π/(k + 1)], it is known that cos [π/(k + 1)] > cos(π/k), and then for (k + 1)-angle form, there are dc cos [π/(k + 1)] > dc cos(π/k) > (1/2)ds , and ttarget > tagent . In Fig. 1, if the distance between target k and the center of agent i, j is more than or equal to the distance from one of them to the center, i.e., dc cos(π/ntk ) ≥ (1/2)ds , (ntk ≥ 4), then ttarget > tagent . Hence, the consensus tracking of target k is achieved. From Theorem 2, if ntk < 4, then the tracking of the target is failed. Under this situation, pursuing agents can move around the target, that is clockwise/anticlockwise rotation of the target, as shown in Fig. 1. When the number of pursuing agents is ntk = 3, then β = (π/3), dc cos(π/3) < (1/2)ds . Through the computational analysis, it is known√that the target k is successfully tracked if and only if vi ≥ 3vtk = 1.732vtk , i ∈ {1, 2, . . . , ntk }. However, using moving around the target, there are (1/2)l = (π/6)dc = 0.523dc , where l denotes arc length, and if vi = vtk , then ttarget > tagent , i.e., for the target to mid point of arc, time is greater than the individual time required. Through the above analysis, pursuing agents can more easily track the target k in the red circular region in Fig. 1. Remark 1: In addition, the required number of pursuing agent have a close relationship with the velocity of each target.


Fig. 4.

Initial distribution of inertial agents and targets.

Fig. 5.

Consensus surround targets with 15 inertial agents.

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Fig. 2. Stable convergence of four inertial agents with the condition (ds = 0).

Fig. 3.

Stable flocking of four inertial agents with the condition (ds = 0).

For multiagent tracking, an augmented distributed multiflocking method can not only ensure local flocking with pursuing agents, but also make the heterogeneous agent system to consistently track multitargets. Under severe environment situations, inertial agents are able to adaptively adjust tracking heterogeneous targets. IV. S IMULATION E XAMPLES In this section, illustrative examples are presented to verify the effectiveness of the proposed methods. Consider the ˜ > 0. Using cyclic graph of four inertial agents, kv2 L˜ s − kp M Theorem 1, the group behavior is stable, namely, the desired internal group shape is realized, and the velocities of agents

tend to the same value, as shown in Figs. 2 and 3. We set parameters kv = 1, kp = 0.9, and M = [1, 0.4, 1, 0.4]. Combining with the above flocking analysis of multiple inertial agents, we consider the multiple inertial agent system composed of three targets and fifteen agents. The initial velocity coordinates are chosen randomly from the box [−1, 1]2 . The other parameters are provided as follows: ds = 5, rc = 5, dc = 4, M = [1.2, 1, 1, 1, 1, 1, 1.2, 1.2, 1, 1, 1, 1, 1, 1, 1.2], rt1 = [−50, 20], rt2 = [0, −20], rt3 = [40, 40], and utk = 0. The initial positions of 15 agents submit to the normal distribution as shown in Fig. 4. The red and blue solid rounds with arrows denote heterogeneous pursuing agents, and the green solid stars denote dynamic targets. Furthermore, the initial interactive topology is disconnected. Using the augmented distributed multiflocking method, Theorems 1 and 2 and Lemma 1, the inertial agents are ultimately divided into three groups to accomplish the consistency tracking of dynamic targets as shown in Fig. 5. From Theorem 2, it is known that the number of pursuing agents with any target, namely ntk = 5 when the speeds of targets are given as vt1 = vt2 = vt3 = 1.3. The corresponding simulation result is shown in Fig. 5. It is easy to see from Fig. 5 that each target is surrounded by five heterogeneous pursuing agents finally. Similarly, when the speeds of targets are given as vt1 = 1.0, vt2 = 1.38, and vt3 = 0.6, it is not difficult to analyze that the required minimum number of pursuing agents is four,



R EFERENCES

Fig. 6.

Consensus surround heterogeneous targets with 15 inertial agents.

Fig. 7.

Consensus surround heterogeneous targets with 15 inertial agents.

five, and three for heterogeneous targets. As shown in Fig. 6, it shows that fifteen agents are assigned to ultimately form the different formation surrounding multitarget, and there are three inertial agents in the state of target searching. In addition, when the speeds of targets are given as vt1 = 1.3, vt2 = 1.55, and vt3 = 1.0, the corresponding simulation result is shown in Fig. 7. V. C ONCLUSION In this paper, the flocking behavior and targets consensus tracking problems of heterogeneous multiple inertial agents have been investigated. Considering the agents’ inertial effect, a distributed control protocol has been designed to achieve stable group behaviors for inertial agents. The stability of the multiple inertial agents system has been proved via the decomposition approach. Based on the flocking behavior of multiple inertial agents, an augmented distributed multiflocking method has been proposed to ensure that the multitarget consensus tracking can be achieved for the heterogeneous multiagent system. It is shown that the proposed distributed control method can not only guarantee local flocking with pursuing agents, but also realize the heterogeneous agent system to consistently track multitargets by numerical examples. In the future, the unknown information with targets and existing external disturbances of heterogeneous agent systems can be further researched.

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Shiming Chen received the Ph.D. degree in control theory and control engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2006. He is currently a Professor with the School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang, China. His current research interests include swarm dynamics and cooperative control, complex networks, and particle swarm optimization algorithms.

Huiqin Pei received the Ph.D. degree in control science and engineering from East China Jiaotong University, Nanchang, China, in 2017. She is currently a Lecturer with the School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang. Her current research interests include swarm dynamics and cooperative control and multiagent systems.

Qiang Lai received the Ph.D. degree in control theory and control engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2014. He is currently an Associate Professor with the School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang, China. His current research interests include nonlinear dynamics, chaos control, and complex networks.

Huaicheng Yan received the Ph.D. degree in control theory and control engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2007. He is currently a Professor with the School of Information Science and Engineering, East China University of Science and Technology, Shanghai, China. His current research interests include networked control systems, multiagent systems, and fuzzy systems.