Connectivity-Preserving Consensus Algorithms for Multi-agent Systems

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Connectivity-Preserving Consensus Algorithms for Multi-agent Systems ⋆ Jin Dai, Shanying Zhu, Cailian Chen, Xinping Guan Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, 200240, P. R. China (e-mail: daijin [email protected], [email protected], [email protected], [email protected]). Abstract: The consensus problem for multi-agent systems with double integrator dynamics is considered in this paper. We propose and analyze a connectivity-preserving consensus algorithm with position measurements only. Under the assumption that the initial interaction network is connected, the connectivity of the interaction network during the dynamical evolution can be preserved. Moreover, the consensus algorithm with a virtual leader is investigated. It is shown that all the agents can asymptotically attain a desired position and velocity even if only one agent in the team has access to the information of the virtual leader. Numerical simulations illustrate the effectiveness of the proposed methods. Keywords: Co-operative control, decentralized systems, autonomous mobile robots 1. INTRODUCTION The consensus problem for multi-agent systems concerns with how a team of agents can reach an agreement on a common value by negotiating with their neighbors (OlfatiSaber et al. [2007]). It arises in civilian and military applications including air traffic control, automated highway systems, sensor networks, mobile robots, and so on. Recently, it has received considerable research interests. See, for example, Olfati-Saber et al. [2007], Ren et al. [2007], and references therein. One of the main goals in the consensus problem is to design appropriate control laws such that the states of all of the agents in the system converge to a common value. Consensus algorithms have been studied extensively in the context of cooperative control of multi-agent systems. Current consensus algorithms are primarily studied for single integrator kinematics (Jadbabaie et al [2003], Fax and Murray [2004], Olfati-Saber and Murray [2004], Ren and Beard [2005]), and double integrator dynamics (Ren [2008], Hu and Lin [2010]). These algorithms and their variants are applied to rendezvous (Lin et al [2003], Dimarogonas and Kyriakopoulos [2007], Su et al [2010]), formation control (Lawton et al [2003], Fax and Murray [2004]), flocking (Tanner et al [2003], Olfati-Saber [2006]), and attitude alignment (Lawton and Beard [2002]). Existing results on distributed cooperative control problems critically rely on maintaining a connected network among the agents, either for all time as in Olfati-Saber and Murray [2004], Olfati-Saber [2006], Tanner et al [2007] or over bounded time intervals as in Jadbabaie et al [2003], Lin et al [2003], Ren and Beard [2005]. In Spanos and Mur⋆ This work was supported by the NSF of China under grant no. 60934003 and 60904123 and by Shanghai Jiao Tong University Innovation Fund for Postgraduates.

978-3-902661-93-7/11/$20.00 © 2011 IFAC

ray [2004], a measure of local connectivity is introduced which under certain conditions are sufficient for global connectivity. The rendezvous and the formation control problems over dynamic interaction graphs are considered by Ji and Egerstedt [2007]. The connectivity of the graphs is preserved by adding appropriate weights to the edges in the graphs. Artificial potential methods are proposed in Zavlanos and Pappas [2007], Zavlanos et al [2007] for maintaining connectivity of single integrator dynamics. Recently, Su and Wang [2008] proposed a new potential function for the coordinated control of double integrator dynamics with connectivity preserving. This coordinated control protocol relies on information about both the relative position and the relative velocity measurements. However, this assumption is unrealistic since the speed sensors are more expensive and thus can significantly increase the overall cost. Consensus algorithms and flocking algorithms using only position measurements are proposed in Ren [2008] and Su et al [2009], respectively. However, the consensus algorithms in Ren [2008] rely on the connectivity assumption of the network during the evolution. Motivated by the importance of network connectivity in multi-agent systems, we consider consensus algorithms with connectivity preserving for double integrator dynamics based only on relative position measurements. Combining the control input and connectivity control, we propose a connectivity-preserving consensus algorithm for double integrator dynamics with only position measurements and prove the global asymptotic stability by using LaSalle’s invariance principle. Moreover, we investigate the consensus algorithm with a virtual leader and show that all agents can asymptotically attain the desired position and velocity with the virtual leader. The rest of this paper is organized as follows. In Section 2, the problem is formulated, and preliminary results are

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

given. The connectivity preserving control design based only on position measurements is elaborated in Section 3. The consensus algorithm with a virtual leader is presented in Section 4. Simulation results are presented to illustrate the effectiveness of the proposed control strategies in Section 5. Finally, concluding remarks are drawn in Section 6.

T T T T T E(t)}, where qˆ(t) = [q11 , . . . , q1n , . . . , qn1 , . . . , qnn ] , and satisfies the following properties:

(i)

∂ψ(kqij k) ∂kqij k

(ii) (iii)

2. PROBLEM FORMULATION AND PRELIMINARIES Consider n agents with double integrator dynamics described by q˙i = pi , (1) p˙i = ui , i = 1, . . . , n, where qi , pi , ui ∈ Rm are, respectively, the position, velocity and control input of the ith agent. A weighted undirected dynamic graph G(t) = {N , E(t), A(t)} is used to model communication topology among the n agents, where G(t) consists of a set of vertices N = {1, . . . , n} indexed by the set of agents, a time varying set of links E(t) = {(i, j)|i, j ∈ N } satisfying the following conditions (i) initial links are generated by E(0) = {(i, j)|kqij (0)k < r, i, j ∈ N }, where qij (t) , qi (t)−qj (t) and r is the sensing radius; (ii) if (i, j) 6∈ E(t− ) and kqij (t)k < r − ε, where ε ∈ (0, r) is a given constant, then (i, j) is a new link to be added to E(t); (iii) if kqij (t)k ≥ r , then (i, j) 6∈ E(t), and a weighted adjacency matrix A(t) = (aij (t)) ∈ Rn×n defined by positive real numbers aij (t) = aji (t) if (i, j) ∈ E(t) and aij (t) = 0 otherwise. The Laplacian matrix L(t) = (lij (t)) ∈ Rn×n associated with the weighted adjacency matrix A(t) is defined as n P lii (t) = aij (t) and lij (t) = −aij (t) for i 6= j. j=1,j6=i

Some important properties of the Laplacian matrix L(t) are summarized as follows. Lemma 2.1[Merris (1994)] For an undirected graph G(t), the associated Laplacian matrix L(t) is symmetric positive semidefinite. Let λ1 (t) ≤ λ2 (t) ≤ · · · ≤ λn (t) be its ordered eigenvalues, then λ1 (t) = 0 for all t, with the corresponding eigenvector 1n = [1, . . . , 1]T ∈ Rn . Moreover, λ2 (t) > 0 if and only if G(t) is connected. A symmetric indicator function σij (t) ∈ {0, 1} is introduced by Ji and Egerstedt [2007] to determine whether or not there is a link between agent i and agent j at time t, which is defined as  0, if (σij (t− ) = 0 ∩ r − ε ≤ kqij (t)k < r)   ∪kqij (t)k ≥ r, σij (t) = − 1, if (σ  ij (t ) = 1 ∩ r − ε ≤ kqij (t)k < r)  ∪kqij (t)k ≤ r − ε.

The function σij (t) shows that there is a hysteresis in addition to new links in the graph G(t). The hysteresis is crucial in maintaining connectivity of the dynamic interaction network. As far as connectivity maintenance is concerned, artificial potential methods are widely used in the literature (see Zavlanos and Pappas [2007], Zavlanos et al [2007], Su and Wang [2008], Su et al [2009]). In this paper, we adopt the following nonnegative artificial potential ψ(kqij k) defined 2 on the domain Dq = {ˆ q (t) ∈ Rmn |kqij (t)k < r, (i, j) ∈

lim

kqij k→0

≥ 0 for kqij k ∈ [0, r);

∂ψ(kqij k) ∂kqij k

·

1 kqij k

is nonnegative and bounded;

lim ψ(kqij k) = +∞.

kqij k→r

The property (i) indicates that the potential ψ(kqij k) is an non-decreasing function for kqij k ∈ [0, r). The property (ii) indicates that the potential function can make all the agents move asymptotically to a common position. The property (iii) guarantees that no existing links in the network will be lost. An example of the artificial potential function ψ(kqij k) satisfying all the three properties is as follows: kqij kn1 , 0 ≤ kqij k < r, (2) ψ(kqij k) = (r − kqij k)n2 where n1 ≥ 2 and n2 ≥ 1 are integers. In this paper, the consensus means that for arbitrary qi (0) and pi (0), qi (t) → qj (t) and pi (t) → pj (t) as t → ∞. Our objective is to design the control inputs ui ∈ Rm so that consensus of the system is reached based on position measurements only, under the assumption that the initial network is connected. Under the circumstance, a virtual leader is introduced in the group of agents. The desired coordinated consensus control laws should be designed to enable all agents to track the virtual leader asymptotically. 3. CONNECTIVITY-PRESERVING CONSENSUS ALGORITHM WITH POSITION MEASUREMENTS To maintain the network connectivity in multi-agent systems, we apply the artificial potential function (2) to the coordinated control protocol. Moreover, to avoid using relative velocity measurements, we propose a new consensus algorithm based only on position measurements as follows: X ui = − (∇qi ψ(kqij k) + aij (t)(yi − yj )), j∈Ni (t)

x˙ i = T xi +

X

aij (t)qij ,

(3)

j∈Ni (t)

yi = P x˙ i , where Ni (t) = {j|σij (t) = 1, j 6= i, j = 1, . . . , n} is the neighborhood of agent i at time t, ∇ is the gradient operator, T ∈ Rm×m is a Hurwitz matrix, P ∈ Rm×m is a symmetric positive definite solution of the Lyapunov equation T T P + P T = −Q with the symmetric positive definite matrix Q ∈ Rm×m . By stacking the variables qi , pi , ui , xi , yi as follows q = [q1T , . . . , qnT ]T , p = [pT1 , . . . , pTn ]T , u = [uT1 , . . . , uTn ]T , x = [xT1 , . . . , xTn ]T , y = [y1T , . . . , ynT ]T , we can rewrite the control input (3) into X   − ∇q1 ψ(kq1j k)   j∈N1 (t)     . .. u=  − (L(t) ⊗ Im )y,   X − ∇qn ψ(kqnj k) 

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j∈Nn (t)

(4)

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

x˙ = (In ⊗ T )x + (L(t) ⊗ Im )q,

(5)

y = (In ⊗ P )x, ˙ (6) where ⊗ denotes the Kronecker product and In is the n×n identity matrix. In order to proceed the stability analysis of the nonlinear system (1) with control input (4)-(6), define the energy function of system (1) as V (t) =

n 1X X 1 (ψ(kqij k)+pTi pi )+ x˙ T (In ⊗P )x. ˙ (7) 2 i=1 2 j∈Ni (t)

Note that P is symmetric positive definite, then V (t) is a positive semidefinite function. The topology of the multi-agent system may change over time, then the system (1) under control input (3) results in a switching dynamical system. Let tk (k = 1, 2, . . .) denote the switching instants when the topology of the graph G(t) changes. It is ready to give the following result. Theorem 3.1 Consider the closed loop system (1) and (3), then E(tk ) ⊆ E(tl ) for any pair of switching instants tk < tl . Proof. For any c > 0, let Ωc = {(ˆ q (t), p(t), x(t)) ˙ ∈ Dq × Rmn × Rmn |V (t) ≤ c} denote the level set of V (t). Note that G(t) and E(t) are fixed in each time-interval [tk−1 , tk ). As a result, it is easy to verify that the derivative of V (t) in each interval [tk−1 , tk ) is

(i) G(t) will be connected for all t ≥ 0; (ii) all agents asymptotically achieve consensus, i.e., qi (t) → qj (t) and pi (t) → pj (t) for all i, j ∈ N as t → ∞. Proof. (i) According to Theorem 3.1, the graph G(t) is fixed in each time-interval [tk−1 , tk ). Particularly, it follows from (8) that the energy function V (t) is monotonically decreasing in the interval [t0 , t1 ), which implies that V (t) ≤ V0 , f or t ∈ [t0 , t1 ). (9) From Theorem 3.1, we know that no link will be lost at time t1 . Thus, new links must be added to the graph G(t) at switching time t1 . Since there is a hysteresis in the process of adding new links to the dynamic graph, then it is ensured that the associated potential is bounded. We have V (t1 ) ≤ V0 + m1 ψ(r − ε) if m1 new links are added to the interaction network at time t1 , where 0 < m1 ≤ M . Similar to the above analysis, the function V (t) is monotonically decreasing in the interval [tk−1 , tk ) and V (t) ≤ V (tk−1 ), f or t ∈ [tk−1 , tk ). (10) Moreover, no link will be lost at time tk , and V (tk ) ≤ V0 + (m1 + · · · + mk )ψ(r − ε) (11) if mk new links are added to graph G(t) at time tk . Since the graph G(0) is connected and no link in E(0) will be lost, G(t) will be connected for all t ≥ 0.

(ii) Since mk new links can be added to the network G(t) at the switching time tk , then it follows from Corollary X 1 T 3.2 that k ≤ M , m1 + · · · + mk ≤ M and V (t) ≤ ˙ ˙ V (t) = ψ(kqij k) + pi u i 2 i=1 V 0 + M ψ(r − ε) , Vmax for all t ≥ 0. The interaction i=1 j∈Ni (t) network G(t) turns to be fixed after time tM . Thus, 1 T 1 T + x ¨ (In ⊗ P )x˙ + x˙ (In ⊗ P )¨ x the discussion about the steady-state of the system (1) 2 2 can be restricted on the time-interval (tM , ∞). Clearly, 1 T T T = −p (L ⊗ Im )y + p (L ⊗ Im )(In ⊗ P )x˙ − x˙ (In ⊗ Q)x˙ Ω q (t), p(t), x(t)) ˙ ∈ Dq × Rmn × Rmn |V (t) ≤ Vmax = {(ˆ 2 1 T Vmax } is positively invariant. From claim (i), G(t) remains = − x˙ (In ⊗ Q)x. ˙ connected for all t ≥ 0. For any two agents i and agent 2 (8) j, kqij k ≤ (n − 1)r holds. Hence, kˆ qk ≤ n(n − 1)r. Since n P Since Q is symmetric positive definite, then V˙ (t) ≤ 0 for V (t) ≤ Vmax for all t ≥ 0, then pTi pi ≤ 2Vmax and i=1 all t ∈ [tk−1 , tk )(k = 1, 2, . . .). Hence, the level set Ωc is T ˙ T x˙ ≤ max . It follows that λmin (P )x positively invariant. It implies that for any (i, j) ∈ E(t), x˙ (In ⊗ P )x˙ ≤ 2Vp ˙ ≤ 2Vmax /λmin (P ), where λmin (P ) is ψ(kqij k) remains bounded. On the other hand, if for some 2Vmax , i.e., kxk (i, j) ∈ E(t), kqij k → r, then ψ(kqij k) → +∞. Thus, by the smallest eigenvalue of the symmetric positive definite continuity of V (t) in Dq , it follows that kqij k < r for all matrix P . Therefore, the set ΩVmax is compact. (i, j) ∈ E(t) and t ∈ [tk−1 , tk ). In other words, all links Applying LaSalle’s invariance principle Khalil [2002] to in the graph G(t) are preserved during switching instants the system (1) described by the vector field [q˙T , p˙ T ]T , tk−1 and tk . It implies that E(tk−1 ) ⊆ E(tk ). Applying it follows that every solution starting in ΩVmax asymprecursively this argument completes the proof. 2 totically converges to the largest invariant set in S = Corollary 3.2 Under control input (3), the total number {(ˆ q (t), p(t), x(t)) ˙ ∈ Dq × Rmn × Rmn |V˙ (t) = 0}. From (8), M of switching instants of the system (1) is finite, and V˙ (t) = 0 if and only if x(t) ˙ = 0. It implies that x ¨(t) = 0. M ≤ 21 (n − 1)(n − 2). ˙ From (5) and L(t) = 0, in [tk−1 , tk ), we have Proof. The proof is similar to that of Corollary 3.2 in x ¨ = (In ⊗ T )x˙ + (L(t) ⊗ Im )p, (12) Zavlanos et al [2007], and is omitted here. 2 thus (L(t) ⊗ Im )p = 0 holds. Since the dynamic network As stated in Corollary 3.2, the graph G(t) will be fixed is connected all the time, it follows from Lemma 2.1 that after finite switches. Based on the previous discussion we p converges asymptotically to 1n ⊗ α, for some constant can now give the following result. vector α ∈ Rm . It implies that kpi −pj k → 0 for all i, j ∈ N as t → ∞. Theorem 3.3 Consider the closed loop system (1) with (3). Assume that the initial graph G(0) is connected and It follows from (L(t) ⊗ Im )p = 0 that (L(t) ⊗ Im )u = 0, the initial energy V0 = V (0) is finite, then the following which implies that u ∈ span(1n ⊗ β), for some constant statements hold: vector β ∈ Rm . n X

n X

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

On the other hand, from (6) and x˙ = 0, we have y = 0. Hence, it follows from (4) that  X  ∇q1 ψ(kq1j k)  j∈N1 (t)      . .. (13) u = − .  X    ∇qn ψ(kqnj k) j∈Nn (t)

Since ψ(kqij k) = ψ(kqji k), then (14) ∇qi ψ(kqij k) = −∇qj ψ(kqij k) = −∇qj ψ(kqji k). Combining (13) with (14) gives X X ∇q1 ψ(kq1j k) + · · · + ∇qn ψ(kqnj k) = 0, j∈N1 (t)

j∈Nn (t)

then (1n ⊗ β)T u = 0. It implies that u ∈ span(1n ⊗ β)⊥ , hence, u ∈ span(1n ⊗ β) ∩ span(1n ⊗ β)⊥ . Thus u = 0 holds. According to (13), we have X X ∂ψ(kqij k) (qi − qj ) ∇qi ψ(kqij k) = · = 0. ∂kqij k kqij k j∈Ni (t)

j∈Ni (t)

(15) It can be further rewritten in the following matrix form ˆ ⊗ Im )q = 0, (L(t) (16) ˆ = (ˆlij (t)) ∈ Rn×n , where the matrix L(t)  n X ∂ψ(kqij k)  1   · , j = i,  kqij k ˆlij (t) = j=1,j6=i ∂kqij k  1  ∂ψ(kqij k)  · , j 6= i. − ∂kqij k kqij k

ui = −

X

(∇qi ψ(kqij k) + aij (t)(yi − yj )) − hi yi ,

j∈Ni (t)

x˙ i = T xi +

X

aij (t)qij + hi (qi − qη ),

j∈Ni (t)

yi = P x˙ i , (17) where qη = pη t is the position of the virtual leader. If agent i is the informed one, i.e., the virtual leader, then hi = 1; otherwise, hi = 0. Without loss of generality, we assume that hi = 1 for i = 1 and hi = 0 for all the others. Accordingly, the energy function of system (1) without leaders is modified as n 1X X (ψ(kqij k) + (pi − pη )T (pi − pη )) 2 i=1 j∈Ni (t) 1 + x˙ T (In ⊗ P )x. ˙ 2 (18) Clearly, W (t) is a positive semidefinite function. The convergence result which resembles with the leaderless multi-agent system, is summarized as follows.

W (t) =

Theorem 4.1 Consider the closed loop system (1) with (17). Assume that the initial network G(0) is connected and the initial energy W0 = W (0) is finite, then the following statements hold: (i) G(t) will be connected for all t ≥ 0; (ii) all agents asymptotically reach consensus with the virtual leader. Proof. (i) Let q˜i = qi − qη , p˜i = pi − pη , then the dynamic equations of the system (1) can be rewritten as

From the properties (i) and (ii) of the potential function ˆ ψ(kqij k), L(t) can be viewed as a Laplacian matrix. Since G(t) remains connected for all t ≥ 0, it follows from Lemma 2.1 that q converges asymptotically to 1n ⊗ γ, for some constant vector γ ∈ Rm , which implies that kqi − qj k → 0 for all i, j ∈ N as t → ∞. This completes the proof. 2 4. CONSENSUS ALGORITHM WITH A VIRTUAL LEADER

q˜˙i = p˜i , p˜˙ i = ui , i = 1, . . . , n,

(19)

With the definition of ψ(kqij k), we have ψ(kqij k) = ψ(k˜ qij k), where q˜ij = q˜i − q˜j . The control input (17) can be replaced by X ui = − (∇q˜i ψ(k˜ qij k) + aij (t)(yi − yj )) − hi yi , j∈Ni (t)

x˙ i = T xi + In practical systems, only few individuals in a bioticcommunity may have the guiding information, such as the location of the food source and the route of migration. Agents in a multi-agent network are called virtual leaders if they can access to such kind of guiding information, such as the desired velocity or position. In this section, the consensus algorithm with virtual leaders is investigated. For simplicity but without loss of generality, the situation of only one virtual leader is considered. Certainly, the result can be generalized to the case with multiple virtual leaders. Assume that the virtual leader moves along a fixed direction with a constant velocity pη , and all the agents of the system (1) are expected to converge asymptotically to the same position of the virtual leader with the desired velocity pη . The control input for agent i used in the previous section is modified to take the leadership of some agents into account.

X

aij (t)˜ qij + hi q˜i ,

j∈Ni (t)

yi = P x˙ i , (20) and the energy function (18) now becomes W (t) =

n 1X X 1 (ψ(k˜ qij k) + p˜Ti p˜i ) + x˙ T (In ⊗ P )x. ˙ 2 i=1 2 j∈Ni (t)

(21) Similar to (8), we can obtain the derivative of W (t) in [tk−1 , tk ) as follows ˙ (t) = − 1 x˙ T (In ⊗ Q)x˙ ≤ 0. W 2

(22)

It implies that the function W (t) is monotonically decreasing in the interval [tk−1 , tk ). Similar to the proof of part (i) of Theorem 3.3, it follows that G(t) will be connected for all t ≥ 0.

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

initial position and velocity

T T T T T q˜(t) = [˜ q11 , . . . , q˜1n , . . . , q˜n1 , . . . , q˜nn ] , p˜ = [˜ pT1 , . . . , p˜Tn ]T .

steady−state position and velocity

4.5

31.7

4

31.6

3.5

31.5

3

31.4 Y−position

Let ΩW = {(˜ q (t), p˜(t), x(t)) ˙ ∈ Dq˜×Rmn ×Rmn |W (t) ≤ Wmax }, where 2 Dq˜ = {˜ q (t) ∈ Rmn |k˜ qij (t)k < r, (i, j) ∈ E(t)},

velocity. Fig.1 (c) and (d) depict the convergence of the positions for x-axis and y-axis, respectively. Fig.1 (e) and (f) show the convergence of the velocities. All these simulation results are in accordance with the theoretical analysis proceeded in Section 3.

Y−position

(ii) Similar to the proof of part (ii) of Theorem 3.3, it follows that W (t) ≤ W0 + M ψ(r − ε) , Wmax for all t ≥ 0.

2.5 2 1.5

We can prove that the set ΩW is positively invariant, and compact. It follows from LaSalle’s invariance principle that every solution starting in ΩW asymptotically converges to the largest invariant set in S = {(˜ q (t), p˜(t), x(t)) ˙ ∈ Dq˜ × mn mn ˙ ˙ R ×R |W (t) = 0}. From (22), W (t) = 0 if and only if x(t) ˙ = 0, which ¨(t) = 0. It follows from P implies that x x˙ i = T xi + aij (t)˜ qij + hi q˜i and a˙ ij (t) = 0 that

31

0.5

30.9

0

0.5

1

1.5 2 X−position

2.5

3

30.8 22.4

3.5

22.5

22.6

(a)

22.7 22.8 X−position

22.9

23

23.1

(b)

Position Convergence (X−axis)

Position convergence (Y−axis) 35 30

20 25

Y position

(23)

20

15

10

15 10 5

j∈Ni (t)

5 0

It implies that ((L(t) + Hn ) ⊗ Im )˜ p = 0, where Hn = diag(h1 , . . . , hn ). Since the dynamic network G(t) is connected all the time and one agent is the virtual leader, it follows from Lemma 3 in Hong et al [2006] that the symmetric matrix (L(t) + Hn ) ⊗ Im is positive definite. Then p˜ converges asymptotically to the zero vector, which implies that k˜ pi k → 0 for all i as t → ∞, hence, pi → pη for all i as t → ∞.

0

In this section, we present numerical simulations to illustrate the effectiveness of the proposed methods in the sections 3 and 4. All the simulations are performed with 10 agents moving in 2-dimensional space. The initial positions and velocities without a leader are chosen randomly within [0, 6] × [0, 6] and [0, 1] × [0, 1], respectively. The initial positions are selected to guarantee that the initial interaction network is connected. Let the sensing radius be r = 4, the given positive parameter ε = 0.5, and the weight of the link aij (t) = 1 for all (i, j) ∈ E(t). Choose T = −0.5I2 and Q = I2 , then the Lyapunov equation T T P + P T = −Q has a unique symmetric positive definite solution P = I2 . The artificial potential field ψ(kqij k) is shown in (2) with n1 = 2, and n2 = 1. Fig.1 (a) and (b) show the initial and the steady states of the agents respectively, where the black lines with the arrows represent the direction and the magnitude of velocities. Applying the connectivity-preserving consensus algorithm (3), we can see that 10 agents asymptotically converge to the same position and move with the same

10

15

20 time

25

30

35

−5

40

0

5

10

(c)

15

20 time

25

30

35

40

(d) Velocity Convergence (Y−axis) 25

15

20 10

15

X direction velocity

10 5

0

−5

5 0 −5 −10 −15

−10

−20 −15

0

5

10

15

20 time

25

30

35

40

−25

0

5

10

(e)

15

20 time

25

30

35

40

(f)

Fig.1 Consensus of 10 agents under the control input (3)

j∈Ni (t)

5. SIMULATION STUDIES

5

Velocity Convergence (X−axis)

In steady state, it is clear that yi = 0 from x˙ i (t) = 0 for all i. From p˙i (t) = p˜˙ i (t) = p˙ η = 0, we have ui = 0 for all i, i.e., X ∂ψ(kqij k) (qi − qj ) · = 0. ui = − ∂kqij k kqij k Similar to the proof of part (ii) of Theorem 3.3, it follows that kqi −qj k → 0 for all i, j ∈ N as t → ∞. This completes the proof.

0

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aij (t)(˜ pi − p˜j ) + hi p˜i .

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The desired initial position and velocity of the virtual leader are set to be qη (0) = [0, 0]T and pη (0) = [2, 2]T . The initial positions and velocities with a virtual leader are chosen randomly within [0, 4] × [0, 4] and [0, 1] × [0, 1], respectively. Applying the connectivity-preserving consensus algorithm (17) with a virtual leader, the satisfactory convergence behavior is shown in Figure 2. Fig.2 (a) and (b) show the initial and the steady states of the agents respectively, where the solid lines with the arrows represent the direction and the magnitude of velocities. From Fig.2 (c)(d) and Fig.2 (e)(f), we can see that all the agents eventually follow the virtual leader and attain the same position and velocity with the virtual leader. This demonstrates the performance of the consensus algorithm given in (17). The other simulations are performed with the different link weights. The convergence of the proposed methods is similar. The performance of the consensus algorithms is not affected almost by the choice of the link weights. 6. CONCLUSION In this paper, a connectivity-preserving consensus algorithm based only on position measurements has been presented. Under the assumption that the initial network is connected, we have proved that under the proposed control

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

initial position and velocity (the bold arrow represents the leader)

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Fig.2 Consensus of 10 agents with a virtual leader under the control input (17)

law the network remains connected all the time, and all agents asymptotically converge to the same position and with the same velocity. Moreover, the connectivitypreserving consensus algorithm with a virtual leader has been proposed. The simulation studies have been given to demonstrate the desired behavior of all the agents with the desired velocity even if only one agent in the team has the information about the virtual leader. REFERENCES R. Olfati-Saber, J.A. Fax, and R.M. Murray. Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95:215-233, 2007. W. Ren, R.W. Beard, and E.M. Atkins. Information consensus in multivehicle cooperative control: Collective group behavior through local interaction. IEEE Control Syst. Mag. , 27:71-82, 2007. A. Jadbabaie, J. Lin, and A.S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control, 48:9881001, 2003. J.A. Fax and R.M. Murray. Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control, 49:1465-1476, 2004. R. Olfati-Saber and R.M. Murray. Consensus problems in networks of agents with switching topology and timedelays. IEEE Trans. Autom. Control, 49:1520-1533, 2004. W. Ren and R.W. Beard. Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control, 50:655-661, 2005.

W. Ren and E.M. Atkins. Distributed multi-vehicle coordinated control via local information exchange. Int. J. Robust Nonlinear Contr., 17:1002-1033, 2007. W. Ren. On consensus algorithms for double-integrator dynamics. IEEE Trans. Autom. Control, 53:1503-1509, 2008. J. Hu and Y.S. Lin. Consensus control for multi-agent systems with double-integrator dynamics and time delays. IET Control Theory Appl., 4:109-118, 2010. J. Lin, A.S. Morse, and B.D.O. Anderson. The multiagent rendezvous problem. Proc. IEEE Conf. Decision and Control, pages 1508-1513, 2003. D.V. Dimarogonas and K.J. Kyriakopoulos. On the rendezvous problem for multiple nonholonomic agents. IEEE Trans. Autom. Control, 52:916-922, 2007. H. Su, X. Wang and G. Chen. Rendezvous of multiple mobile agents with preserved network connectivity. Systems & Control Letters, 59:313-322, 2010. J.R. Lawton, R.W. Beard, and B.J. Young. A decentralized approach to formation maneuvers. IEEE Trans. Robot. Automat. , 19:933-941, 2003. H.G. Tanner, A. Jadbabaie, and G.J. Pappas. Stable flocking of mobile agents, part i: Fixed topology. Proc. IEEE Conf. Decision and Control, papes 2010-2015, 2003. H.G. Tanner, A. Jadbabaie, and G.J. Pappas. Stable flocking of mobile agents, part ii: Dynamic topology. Proc. IEEE Conf. Decision and Control, pages 20162021, 2003. R. Olfati-Saber. Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Trans. Autom. Control, 51:401-420, 2006. J.R. Lawton and R.W. Beard. Synchronized multiple spacecraft rotations. Automatica , 38:1359-1364, 2002. H.G. Tanner, A. Jadbabaie, and G.J. Pappas. Flocking in fixed and switching networks. IEEE Trans. Autom. Control, 52:863-868, 2007. D.P. Spanos and R.M. Murray. Robust connectivity of networked vehicles. Proc. IEEE Conf. Decision and Control, pages 2893-2898, 2004. M. Ji and M. Egerstedt. Distributed coordination control of multiagent systems while preserving connectedness. IEEE Trans. Robot., 23:693-703, 2007. M.M. Zavlanos and G.J. Pappas. Potential fields for maintaining connectivity of mobile networks. IEEE Trans. Robot., 23:812-816, 2007. M.M. Zavlanos, A. Jadbabaie, and G.J. Pappas. Flocking while preserving network connectivity. Proc. IEEE Conf. Decision and Control, pages 2919-2924, 2007. H. Su and X. Wang. Coordinated control of multiple mobile agents with connectivity preserving. Proc. the 17th IFAC World Congress, pages 3725-3730, 2008. H. Su, X. Wang and G. Chen. A connectivty-preserving flocking algorithm for multi-agent systems based only on position measurements. Int. J. Control, 82:1334-1343, 2009. R. Merris. Laplacian matrices of graphs: A survey. Linear Algebra Appl., 197-198:143-176, 1994. H. K. Khalil. Nonlinear Systems(3rd ed.). Prentice Hall, Upper Saddle River, NJ, 2002. Y. Hong, J. Hu, and L. Gao. Tracking control for multiagent consensus with an active leader and variable topology. Automatica, 42:1177-1182, 2006.

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