DISTRIBUTED CONTROL FOR CONSENSUS OF NETWORKED

c 2009 Institute for Scientific ° Computing and Information

INTERNATIONAL JOURNAL OF INFORMATON AND SYSTEMS SCIENCES Volume 5, Number 2, Pages 161–178

DISTRIBUTED CONTROL FOR CONSENSUS OF NETWORKED MULTI-AGENT ROBOTIC SYSTEMS WITH TIME DELAYS LONG SHENG AND YA-JUN PAN Abstract. This paper deals with the consensus problem for the networked multi-agent robotic systems. Each member in the multi-vehicle group has its desired trajectory which is due to the group task. During the group movement, each robotic vehicle can move at the same velocity and keep a safety distance with others. Each robotic vehicle has its own coordinate system and it can exchange its relative position with its neighbors so that the group formation will be made according to the references after the system reaches consensus. A novel distributed feedback control scheme is proposed to stabilize the system and the sufficient conditions for the controller designs are given by using the Lyapunov method. The effects of the constant time delays have been considered in the controller design. Finally, the main results obtained in this paper are validated by simulation results. Key Words. consensus, multi-agent systems, time delays and robotics.

1. Introduction Recently, the study on multi-agent systems has been paid more attention due to its wide potential applications, such as platooning of vehicles in the urban transportation [1]-[2], the operation of the multiple robots [3], autonomous underwater vehicles [4]-[5] and the formation of aircrafts in military affairs [6]-[7]. Investigations for multi-agent systems begin with studying the behavior of a large number of interacting agents with a common group objective. These agents include fish, ants and bees. Take a flock of fish as an example, the desired track for each agent in the group has been already decided according to some external elements. Each agent should track the existing desired trajectory and acquire useful or necessary information, such as the velocities and relative positions, from its neighbors in order to: i) stay in a proper distance to nearby flock-mates; ii) avoid collisions; iii) match velocity with each other. Then all agents asymptotically move with the same velocity and form a cohesive flock without collisions (See Fig.1). This phenomena is called consensus. Notice that in the whole group movement, there is no one performs as the “leader” because of a widely accepted opinion by animal behavior scientists that “schools need no leaders” [8]. Consensus problems have a long history in the automata theory. In multi-agent systems, consensus means the states of all agents reach an agreement asymptotically regarding a certain quantity of interest that depends on the state of all agents. For example, in many applications such as multi-agent or multi-vehicle systems, groups Received by the editors September 1, 2008 and, in revised form, March 1, 2009. 2000 Mathematics Subject Classification. 70E55, 70E60, 15A39. This research was supported by NSERC, AUTO21-NCE and CFI Canada. 161

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L. SHENG AND Y.-J. PAN

Figure 1. The consensus of multiple agent systems of agents need to agree upon certain quantities of interest. Such quantities might or might not be related to the motion of the individual agents. A good literature review for consensus problem can be found in [9]. The most often used method to tackle the consensus problem with multi-agent systems is called graph theory (graph Laplacians) [10]-[11]. It plays a crucial role in the convergence analysis of consensus. In [12]-[13], the authors used graph theory to tackle the formation stabilization for groups of linear agents. This method for formation stabilization has not been used to systems with nonlinear systems that are not linearizable. Another approach is to use Lyapunov method to derive sufficient conditions for stabilization controllers. This Lyapunov function also can be called as the disagreement function, is a measure of group disagreement in a network. In [14], a common Lyapunov function guarantees asymptotic convergence to a group decision value in networks. Similar analysis idea was used in [15]. At first, the authors introduced two control algorithms which include the relative position term, the relative velocity term and the navigational feedback term. Then the disagreement functions were used to analyze the stability of the whole system. Compared with the graph method, Lyapunov method is more theoretic from the control aspect; it could visually give sufficient conditions for stabilizations and be closer to real applications. That is why our proposed paper limits its focus on this method to tackle the problem. In this paper, a peer-to-peer architecture without leaders is used to solve a kind of tracking problem of the linear multi-agent networked robotic vehicle systems. Fig.2 shows the architecture of the linear multi-agent networked robotic vehicle system. The agents in the systems are connected and share information with their neighbors via networks with constant networked induced time delays. Each robot has a local controller applied. It can sense its own position and orientation in a special coordinate (See Section.2 for details) by sensors and send those information to its neighbors via networks. Every agent in the system played the same role as the one in the flock of fish example. Each robot moves at the same velocity as its neighbors and also tracks its reference route which is in parallel with references of others in the flock. Then the whole flock will start moving from place A to place B. During the movement the flock should reach consensus to present the properties of “Flock Movement”. This model has wide application possibilities, such as the operation of the un-piloted combine harvester in the agriculture. The research work

DISTRIBUTED CONTROL OF NETWORKED MULTI-AGENT ROBOTIC SYSTEMS

163

Figure 2. The schematic diagram of multi-agent networked robotic systems

in this paper is to deal with this kind of operation. The harvester and the truck could be treated as two agents in the system. If using a lead-follower architecture to tackle this problem, the task could be done if the leader tracks its own desired route on the field and the follower tracks leader’s route with a certain following angle. However, researchers found that this architecture has poor disturbance rejection properties [16] which could cause failure for the task. Hence in this paper, a peer-topeer architecture is chosen to improve the stability of the whole group. According to the proposed design, then the harvester and the truck are supposed to have their own desired trajectories. The desired trajectories are made according to the working route on the field so that this architecture could avoid its dependency on one single agent and having poor performance in adversarial environments. Those two reference trajectories have fixed relative position with each other so the two agents could keep a fixed relative position during the movement. They need to share their state information with each other to drive the whole multiple system reaching consensus. In [15], the velocity and orientation of the flock movement are random. The aim is to make the flock reach consensus during the movement. But in our design the flock movement has known origin and destination. Each robot is supposed to exchange information with at least one of the other robots which is treated as a neighbor. In order to tackle the stabilization problem for our design, the system is modeled as error dynamics. The objective of this work is to design the distributed feedback control algorithm to drive the system into consensus so that: i) each agent can track its reference trajectory well; ii) all agents can move at the same velocity; iii) all agents can keep a proper and safe distance with each other to make a desired group formation. In order to tackle this problem, a distributed control algorithm which includes the relative position term and the navigational feedback term, has been introduced to stabilize the multiple agent system. A new term called collision avoidance has been added to the control algorithm to avoid collisions during the group movement. The Lyapunov analysis method has been used to derive the sufficient conditions for the stabilizing controller design. The effects caused by the communication delay have been considered in our design. Simulation results show that the multiple agents

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can reach consensus during the movement with the desired group formation in the two-dimensional spaces from an original place to the destination without collisions. The paper is organized as follows. In Section 2, the multiple coordinate system has been introduced; the model of the multi-agent system has been proposed. In Section 3, the distributed feedback control design is rigorously analyzed and as a result, the sufficient conditions for controller designs are provided. Section 4 shows the simulations results which demonstrate the validity of the proposed approach. Section 5 draws the conclusions and discussed future work. 2. Problem Formulation In this section, the multiple coordinate system is introduced. First we consider there are n mobile robots moving on a plane. Each robot is labeled as Ri . To position all the robots in a common coordinate system, a static coordinate system C0 is defined as the common position reference for all the robots. In order to make group formations, all the mobile robots need to know its relative position and relative orientation to others. So each robot needs its own coordinate system which is denoted with Ci where i = 1, · · · , m. With the help of modern communication techniques such as Global Position Systems (GPS) or relative positioning systems, C0 and Ci can be known to Ri . During the group movement, Ri senses its position and velocity in C0 and sends these information to its neighbor Rj , using the information and its own position and velocity in C0 , Rj (with j = 1, · · · , ni ) can get the relative position and velocity to Ri in Cj . All robots will sense the relative position to others in this way during the movement. The sketch map of multiple coordinates is as shown in Fig.3.

Figure 3. Multiple coordinate system The linear networked multi-agent robotic vehicle system (as shown in Fig.3) can be expressed in the following equation: X (1) x˙ i (t) = Aii xi (t) + Aij [xj (t − τ ) − ∆j ] + Bui (t), i ∈ m, j ∈ ni i6=j


165

where xi (t) ∈ 0, matrices Mi , Ni , i ∈ m with appropriate dimensions and a scalar L > 0, such that the following inequality holds    H= 

(6)

H11 H21 0 H41 0 0

∗ H22 0 H42 0 0

∗ ∗ H33 H43 0 H63

∗ ∗ ∗ H44 0 H64

∗ ∗ ∗ ∗ H55 0

∗ ∗ ∗ ∗ ∗ H66

  < 0, 

where H11

=

M1T Aii + ATii M1 + M1T BKii + KiiT B T M1 , H21 = P + M2T Aii + M2T BKii − M1 ,

H22

=

−M2T − M2 , H33 = diag[Q1 + τ R1 , . . . , Qm−1 + τ Rm−1 ]

=

T T + N11 , . . . , N1(m−1) + N1(m−1) ], +diag[N11    T M31 LTij    .. ..  · (Aii + BKii ) +   . .

H41

H42

=

H43

=

H44

=

T M3(m−1)  T M31  .. − .

H64

H66



  · M1 ,

T T B ATij + Kij 

LTij    .. +  · M2 , . T T T T M3(m−1) Aij + B Kij  T  0 N31 − N11 · · ·   .. .. ..  , . . . T 0 · · · N3(m−1) − N1(m−1)     T M31 Q1 · · · 0     .. .. .. −  ... +  · [Lij , . . . , Aij + BKij ] . . . T M3(m−1) 0 · · · Qm−1   T Lij   .. +  · [M31 , . . . , M3(m−1) ] . T T T Aij + Kij B   T 0 N31 + N31 · · ·   .. .. .. − , . . . 

H55





0

···

T N3(m−1) + N3(m−1)   T N21 − N11   ..  H63 =  .

R1 · · · 0 1 . .. . .. = −  .. . τ 0 0 · · · Rm−1   T N21 + N31 · · · 0   .. .. .. = − , . . . T 0 · · · N2(m−1) + N3(m−1)  T  N21 + N21 · · · 0   .. .. .. = (7) − , . . . T 0 · · · N2(m−1) + N2(m−1)

··· .. . ···

0 .. . T N2(m−1) − N1(m−1)

then system (1) is asymptotically stable, e.g. xi (t) tends to zero asymptotically which means Ri tracks its desired trajectory well.

  ,


Proof. Now consider the following Lyapunov functional candidate:

Vi (t)

(8)



Q1  .. T T ¯ j (s)  . = xi (t)P xi (t) + x t−τ 0  R1 · · · Z 0 Z t  .. ¯ Tj (s)  ... x + . Z

−τ

··· .. . ···

t

t+θ

0

···

0 .. . 

0 .. .

  ¯ j (s)ds x

Qm−1

 ¯ j (s)ds, x

Rm−1

Take the derivative of (8) and use Lemma 1 we have 

V˙ i (t)

Q1 · · ·  .. T T T .. ¯ j (t)  . = x˙ i (t)P xi (t) + xi (t)P x˙ i (t) + x . 0 ···   Q1 · · · 0   .. .. ¯ j (t − τ ) −¯ xTj (t − τ )  ... x . . 0



R1  .. T ¯ j (t)  . +τ x 0  Z t  ¯ Tj (s)  − x t−τ

··· ··· .. . ···

R1 .. . 0

 ¯ Tj (t)  +τ x

0

Rm−1

R1 .. . 0

··· ··· .. . ··· 

(9)

t

Qm−1



0 .. .

 ¯ j (s)ds x

Rm−1



0 .. .

  ¯ j (t) x

Qm−1

Qm−1  0  .. ¯ j (t) x .

Rm−1

R1 1  ¯ Tj (s)ds  ... x − τ t−τ 0 Z

 ¯ j (t) x

 ¯ j (t) x

Q1 · · ·  .. T T T .. ¯ j (t)  . ≤ x˙ i (t)P xi (t) + xi (t)P x˙ i (t) + x . 0 ···   Q1 · · · 0   .. .. ¯ j (t − τ ) −¯ xTj (t − τ )  ... x . . 



Qm−1 

0 .. . ··· .. . ···

0 .. .

··· .. . ···

0 .. . Rm−1

  

Z

t

¯ j (s)ds, x t−τ

Now with proper dimensions, the following equations holds: Φ1

Φ2

= 2Z1T M T {−x˙ i (t) + (Aii + Kii )xi (t) + ¯ j (t − τ )} = 0, [. . . , Lij , . . . , Aij + Kij , . . .]j6=i,j∈ni x Z t ¯˙ ij (s)ds − x ¯ ij (t − τ )] = 0, = 2Z2T [N1T , N2T , N3T ]T [¯ xij (t) − x t−τ

167

168


where

Z1

=

Z2

=

M

=

N1

=

¯ j (t − τ )]T , [xi (t), x˙ i (t), x Z t ¯˙ j (s)ds, x ¯ j (t − τ )]T , [¯ xj (t), x t−τ

{M1 , M2 , [M31 , . . . , M3(m−1) ]},   N11 , · · · 0  ..  .. ..  . , . . 

N2

N3

=

=

0

N21 ,  ..  . 0  N31 ,  ..  . 0

···

N1(m−1)

··· .. . ···

0 .. .

 ,

N2(m−1)

··· .. . ···



0 .. .

  .

N3(m−1)

Then we have

V˙ i (t) + Φ1 + Φ2 ≤ ZT HZ < 0,

(10)

¯ j (t), x ¯ j (t − τ ), where Z = [xi (t), x˙ i (t), x shown in (6).

Rt t−τ

¯ j (s)ds, x

Rt t−τ

¯˙ j (s)ds]T and H is as x ¤

The LMI condition in (6) is non-convex and hence the following theorem is proposed to be the equivalent sufficient condition. Theorem 2. For given scalars θi , i ∈ M , and a given time delay constant τ , if there ¯1, . . . , Q ¯ m−1 ], diag[R ¯1, . . . , R ¯ m−1 ], exist symmetric positive definite matrices diag[Q ¯ ¯ matrices Mi , Ni , i ∈ m with appropriate dimensions and a scalar L > 0, nonsingular matrix X with appropriate dimensions such that the following inequality holds,

 (11)

 ¯ = H 

¯ 11 H ¯ 21 H 0 ¯ 41 H 0 0

∗ ¯ 22 H 0 ¯ 42 H 0 0

∗ ∗ ¯ 33 H ¯ 43 H 0 ¯ 63 H

∗ ∗ ∗ ¯ 44 H 0 ¯ 64 H

∗ ∗ ∗ ∗ ¯ 55 H 0

∗ ∗ ∗ ∗ ∗ ¯ 66 H

   < 0, 


where ¯ 11 H ¯ 21 H ¯ 22 H

= = =

¯ 41 H

=

¯ 42 H

=

¯ 43 H

=

¯ 44 H

=

¯ 55 H

=

¯ 63 H

=

¯ 64 H

=

¯ 66 H

=

θ1T Aii X + θ1 X T ATii + θ1T BYii + θ1 YiiT B T P¯ + θ2 Aii X + θ2 BYii − θ1 X T ¯ 33 = diag[Q ¯1 + τ R ¯1, . . . , Q ¯ m−1 + τ R ¯ m−1 ] −θ2 X T − θ2 X, H ¯T + N ¯11 , . . . , N ¯T ¯ +diag[N 11 1(m−1) + N1(m−1) ]     θ1 X T LTij θ31     .. ..   · (Aii X + BYii ) +   . . T T T T θ3(m−1) θ1 X Aij + Yij B     T T θ2 X Lij θ31 X     .. .. − +  . . θ3(m−1) X θ2 X T ATij + YijT B T  ¯T  ¯11 · · · N31 − N 0   .. .. ..   . . . ¯T ¯1(m−1) 0 ··· N − N 3(m−1)     ¯ Q1 · · · 0 θ31     .. .. .. −  ... +  · [Lij X, . . . , Aij X + BYij ] . . . ¯ m−1 0 ··· Q θ3(m−1)   T T X Lij   .. +  · [θ31 , . . . , θ3(m−1) ] . T T T T X Aij + Yij B  ¯T  ¯31 · · · 0 N31 + N   .. .. .. −  . . . T ¯ ¯ 0 · · · N3(m−1) + N3(m−1)   ¯ R1 · · · 0 1  .. .. −  ...  . . τ ¯ 0 · · · Rm−1   ¯T ¯11 · · · N21 − N 0   .. .. ..   . . . ¯T ¯1(m−1) 0 ··· N − N 2(m−1)  ¯T  ¯ 0 N21 + N31 · · ·   .. .. .. −  . . . ¯T ¯3(m−1) 0 ··· N + N 2(m−1)  ¯T  ¯21 · · · N21 + N 0   .. .. .. − , . . . ¯T ¯2(m−1) 0 ··· N +N 2(m−1)

then the Kii and Kij matrices in Theorem 1 are obtained as (12)

Kii = Yii X −T ,

Kij = Yij X −T .

169

170


As a result, the system (1) is asymptotically stable, e.g. xi (t) tends to zero asymptotically which means Ri tracks its desired trajectory well. Proof: In order to transform the nonconvex LMI in (6) into a solvable LMI, we assume that we have some relations in Mi ’s, i ∈ M . One possibility is that Mi = θi M0 where M0 is nonsingular and θi is known and given. Define X = M0−1 , W = diag(X, X, X, X, X, X) and Yii = Kii X T , Yij = Kij X T . Then pre-multiplying the inequality in (6) by W and post-multiplying by W T , the inequality in (11) can be obtained. Note that the inequality in (11) is only a sufficient condition for the ¯ i = X T Qi X, R ¯ i = X T Ri X, solvability of (7) based on these derivations and Q T ¯ Nij = X Nij X. Remark 1. Note that i and j are finite numbers. If i and j are too big, there will be too many given scalars θi in (11) and it will be difficult to solve the LMI (11). The interaction topology is affected by the interactive weight matrix Aij ; if Aij = 0, then it means that there is no connection between agent i and j. If the LMI (11) can not be solved with relative weight matrices, then the system can not reach consensus. 4. Simulation Results

Figure 4. A three-Robot group configuration In the simulation, two cases of the consensus tasks of three and four mobile robots are studied respectively. As same as the simulation work done in [12], two types of configurations for the robot group are considered. The group in Fig.4 (a) and Fig.5 (b) and (c) has a configuration called “closed chain group”. The case in Fig.5 (c) is called full closed chain group compared with the one in Fig.5 (b). The group configuration in Fig.4 (b) and Fig.5 (a) is called “open chain group”. Both configurations are strongly connected graphs in graph theory. Our control scheme is independent of the configurations and it is applicable in groups which have other configurations. However, we just consider these closed and open chain configurations since the groups easily get tangled in other configurations. 4.1. Three-Robot Group. 4.1.1. Closed-Chain. For the case of a closed chain group as shown in Fig.4 (a), the robot group consists of three mobile robots free to move on a plane. The


171

Figure 5. A four-Robot group configuration dynamics for each robot is given in (1). The element values in the matrices are given as: · ¸ 0.2 0.1 Aii = 0.1 −0.3 · ¸ 0.1 0 Aij = 0.1 −0.1 · ¸ 0 B = , 1 where i, j ∈ [1, 2, 3], i 6= j. The distributed control scheme in (2) was applied in the simulation. The control gains are designed as: Kii = [−0.3244 − 1.2224], Kij = [−0.7463 − 0.3176], i, j ∈ [1, 2, 3]. to satisfy the conditions in Theorem 2. L = 0.1, τ = 1sec. The three robots start moving at different initial points which are located in a plane as [x0 , y0 ]R1 = [0.2, 0.2]; [x0 , y0 ]R2 = [0, 0.6]; [x0 , y0 ]R3 = [0.2, 0]. The desired trajectories for each robot are R1 : y = x; R2 : y = x + 0.2; R3 : y = x − 0.2. Fig.6 gives the evolution of the closed chain group movement in every 4 seconds. From the figure, in the first 8 seconds after moving, R1 and R3 stay too close with each other that there is a potential collision between them, then the collision avoidance device which was designed in the controller starts working to keep them in a safety distance. R2 is gradually attracted by its group mates and keeps a safe distance with them.

172


Figure 6. Evolution of the closed chain group movement

Figure 7. Evolution of the closed chain group formation Fig.7 gives the evolution of the closed chain group formation. At the beginning of the movement, each mobile robot in the group tries to track the references and keep a safety distance with others. Finally the group reaches consensus and keeps a stable group formation. 4.1.2. Open-Chain. As shown in Fig.4 (b), the open chain case is studied as well. For R1 the dynamics is the same as in the closed chain case. For R2 and R3 , · ¸ 0 0 A32 = A23 = 0 0 K22 = K33 = [−3.5431

−4.4995]

K23 = K32 = [0 0] K21 = K31 = [−0.0094

−0.1083] ,

and other parameters are the same as in the closed chain case.


173

Figure 8. Evolution of the open chain group movement in the first 10 seconds

Figure 9. Evolution of the open chain group movement

Fig.8 gives the evolution of an open chain group moving in the first 10 seconds. The three robots track their references well and make the group formation without collisions. Fig.9 shows the evolution of the group movement every 5 seconds. Fig.10 shows the evolution of the group formation. From the figures, the collision avoidance coefficients are designed well that the three mobile robots move and make group formation without any collisions. Finally, the open chain group has been stabilized by the distributed controller.

4.2. Four-Robot Group. For the full closed loop case of four robots, as shown in Fig.5 (c), The dynamics for each robot was given in (1) as well. L = 0.1, τ = 1sec.

174


Figure 10. Evolution of the open chain group formation

Figure 11. Evolution of the open chain group formation for four robots The element values in the matrices are given as: · ¸ 0 1 Aii = 0 0 · ¸ 0.1 0 Aij = 0.1 −0.1 · ¸ 0 B = , 1 The control gains are designed as: Kii = [−0.4235 − 1.2535], Kij = [−0.5483 − 0.2136]. where i, j ∈ [1, 2, 3, 4], i 6= j. For the close loop case in Fig.5 (b), most of the matrices are the same as in (c) except the related matrices of the two couples on the two diagonals. They are listed


175

Figure 12. Evolution of the closed chain group formation for four robots with diagonal communication

Figure 13. Evolution of the closed chain group formation for four robots in the first 15 seconds as follows:

· A13 = A31 = A24 = A42 =

0 0 0 0

¸ .

The control gains are designed as: Kii = [−0.3164 − 1.3412], Kij = [−0.4213 − 0.2135]. For the open loop case in Fig.5 (a), most of the matrices are the same as in (b) except the ones of R1 and R4 . They are listed as follows: · ¸ 0 0 A14 = A41 = . 0 0 The control gains are designed as: K11 = K44 = [−0.3254 − 1.4235], K22 = K33 = [−1.3176 − 0.5176].

176


Figure 14. Evolution of the closed chain group formation for four robots

K12,21 = K34,43 = [−0.2908 − 0.3504], K23 = K32 = [−0.2408 − 0.2004]. The four robots start moving at different initial points which are located in a plane as [x0 , y0 ]R1 = [0.2, 0.2]; [x0 , y0 ]R2 = [0.2, 0.4]; [x0 , y0 ]R3 = [0.4, 0.2];[x0 , y0 ]R4 = [0.6, 0]. The desired trajectories for each robot are R1 : y = x + 4; R2 : y = x + 5; R3 : y = x + 3; R4 : y = x + 2.5. Fig.11 and Fig.12 show the evolutions of the open chain group movement and closed chain group with diagonal communications respectively. From the figures, each robot in the group tries to track the reference trajectory and keep a safety distance with others. Fig.13 gives the details of the closed chain group evolution at the beginning of the movement. From the figure in the first 10 seconds of the movement, the trajectory for each robot in the group is disorganized. The group formation was being formed after 10 seconds. Since the collision avoidance coefficients are considered in the design, there is no collision between any of the group members during this period. Finally the closed chain group has been stabilized by the distributed controller and the final evolution was shown in Fig.14. 5. Conclusions and Future Work This paper dealt with the consensus problem for a class of networked multiagent robotic vehicle systems. The advantage of the peer-to-peer architecture which was used to model the system has been discussed. The multi-coordinate system has also been introduced to explain the system model and communications in the group. Each robotic vehicle has its own coordinate system, and can communicate its position and orientation information with its neighbors. The paper presents a novel distributed feedback control algorithm that guarantees the stability of the system by feedback, such as relative position to others. Sufficient conditions for controller design were given by analyzing the Lyapunov functional candidate. The system can be stabilized by the distributed feedback controller via the interaction among robotic vehicles. The future work will focus on testing this approach into real robotic vehicles located in our research lab.


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References [1] Dunbar, W. B. and R. M. Murray. Model predictive control of coordinated multi-vehicle formations. In: Proceedings of the Conference on Decision and Control, 2002. [2] Dunbar, W. B. and R. M. Murray. Distributed receding horizon control for multi-vehicle formation stabilization. Automatica 42,pp. 549–558, 2006. [3] Yamaguchi, H.. A cooperative hunting behavior by mobile robot troops. International Journal of Robotics Research. 18(9),pp. 931–940, 1999. [4] Curtin, T. B., J. G. Bellingham, J. Catipovic and D. Webb. Autonomous oceanographic sampling networks. pp. 86–94, 1993. [5] Smith, T. R., H. Hmann and N. E. Leonard. Orientation control of multiple underwater vehicles. In: In: Proceedings of 40th Conference on Decision and Control. pp. 4598–4603, 2001. [6] Buzogany, L. E., M. Pachter and J. J. Azzo. Automated control of aircraft in formation flight. In: Proceedings of AIAA Conference Guidance, Navigation and Control. pp. 1349–1370, 1993. [7] Wolfe, J. D., D. F. Chichka and J. L. Speyer. Decetntralized controllers for unmanned aerial vehicle formation flight. In: Proceedings of AIAA Conference Guidance, Navigation and Control. pp. 3833–3851, 1996. [8] Shaw, E.. Fish in schools. Natural History, 84(8),pp. 40–45, 1975. [9] Olfati-Saber, R., A. Fax and R. M. Murray. Consensus and cooperation in networked multiagent systems. In: Proceedings of the IEEE, 95 (1), pp. 215–233, 2007. [10] Mohar, B. The laplacian spectrum of graphs. In: Graph Theory, Combinatorics and Applications. pp. 871–898, 1991. [11] Merris, R. Laplacian matrices of a graph: A survey. Linear Algebra its Applications, 197,pp. 143–176, 1994. [12] Yamaguchi, H., T. Arai and B. Gerardo. A distributed control scheme for multiple robotic vehicles to make group formations. Robotics and Autonomous Systems 36,pp. 126–147, 2001. [13] Fax, A. and R. M. Murray. Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control 49(9),pp. 1465–1476, 2004. [14] Olfati-Saber, R. and R. M. Murray. Consensus problems in networks of agents with switching topology and time-dealys. IEEE Transactions on Automatic Control 49(9),pp. 1520–1533, 2004. [15] Olfati-Saber. Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Transactions on Automatic Control 51(3),pp. 401–420, 2006. [16] Yanakiev, D. and I. Kanellakopoulos. A simplified framework for string stability analysis in automated highway systems. In: Proceedings of 13th IFAC World Conference. Vol. Q. pp. 177–182, 1996.

Mr. Long Sheng, received the B.S. degree in control engineering from the school of electrical information and control engineering, Beijing University of Technology, Beijing, China, in 2003 and the M.S. degree in applied science from Saint Mary’s University, Nova Scotia, Canada, in 2006. He is currently a Ph.D candidate at department of mechanical engineering, Dalhousie University, Canada. His main research focus is on the networked control systems with issues arising from communication channels, and on the consensus problem for multi-agent systems.

Dr. Ya-Jun Pan, received the B.E. degree in Mechanical engineering from Yanshan University, P.R. China, in 1996, the M.E. degree in Mechanical engineering from Zhejiang University, P.R. China, in 1999 and the Ph.D degree in Electrical and Computer engineering from National University of Singapore, in 2003. After receiving the Ph.D. degree, she was a post-doctoral fellow of CNRS in the Laboratoire d’Automatique de Grenoble, France from 2003 to 2004. In 2004, she held post-doctoral position in the Department of Electrical and Computer Engineering at the University of Alberta, Canada. In January 2005, she joined the Faculty of the Mechanical Engineering Department at Dalhousie University, Canada

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and currently she is an Assistant Professor. Her research interests are in the fields of nonlinear systems, networked control systems, intelligent transportation control systems and tele-robotics. She is currently an Associate Editor of the Journal of Franklin Institute. She is a member of IEEE and a registered Professional Engineer in Nova Scotia, Canada. Department of Mechanical Engineering, Dalhousie University, Halifax, NS B3J 2X4, Canada E-mail: [email protected] and [email protected]