Synchronization of Coupled Harmonic Oscillators ... - Semantic Scholar
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThBIn2.7
Synchronization of Coupled Harmonic Oscillators in a Dynamic Proximity Network Housheng Su
Xiaofan Wang
Abstract— In this paper, we revisit the synchronization problems for coupled harmonic oscillators in a dynamic proximity network. Unlike many existing algorithms for distributed control of complex dynamical networks that require explicit assumptions on the network connectivity, we show that the coupled harmonic oscillators can always be synchronized, without imposing any network connectivity assumption. Moreover, we also investigate the synchronization with a leader and show that all harmonic oscillators can asymptotically attain the position and velocity of the leader, again without any assumption on connectivity of the followers. Numerical simulation illustrates the theoretical results. Keywords: Synchronization, distributed control, coupled harmonic oscillators, multi-agent systems.
I. I NTRODUCTION In recent years, there has been significant interest in the study of synchronization from different fields (see, for example, [1], [2] and the references therein). Two main lines of research on the problems of synchronization have emerged from this study. On the one hand, the two pioneering papers on synchronization in coupled systems [3] and synchronization in chaotic systems [4] have stimulated a great deal of interest in the study of complete synchronization of coupled nonlinear dynamical systems. On the other hand, there has been much interest in the study of synchronization in dynamical networks with complex topologies in the past few years due to the discovery of the small-world and scale-free properties of many natural and artificial complex networks ([5], [6]). One of the most important contributions to the problem of synchronization in coupled systems is the Kuramoto model [7], which was established based on the phenomenon of collective synchronization. In the Kuramoto model, the information of oscillators are assumed to be global, i.e., the underlying topology is fully connected. The Kuramoto model was later modified for the scenarios of nearest neighbor interaction [8], switching topologies and presence of nonhomogeneous delays [9]. In contrast to the above models where the coupled systems are described by single integrator This work was supported in part by National Natural Science Foundation of China under grants 60731160629 and 60674045. The work was performed while the third author was visiting Shanghai Jiao Tong University as a Cheung Kong Scholarship Chair Professor. Housheng Su and Xiaofan Wang are with Department of Automation, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai 200240, China. Email: [email protected], [email protected] Zongli Lin is with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P.O. Box 400473, Charlottesville, VA 22904-4743, U.S.A.. Email: [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
Zongli Lin
dynamics, Ren in [10] recently investigated coupled secondorder linear harmonic oscillator models. The oscillators investigated in [10] are modeled as point masses on a real line. Under some rather mild network connectivity assumptions, the positions and velocities of coupled harmonic oscillators can be synchronized in both fixed and switching networks, with or without a leader. Related to the synchronization of coupled harmonic oscillators are second-order consensus problems ([11], [12], [13]) and flocking problems ([14], [15], [16], [17]) in multiagent systems. In order to achieve second-order consensus in a multi-agent system, the underlying topology must contain a directed spanning tree in fixed networks, or must have a directed spanning tree at each time instant in switching networks ([11], [12], [13]). In the case of tracking a virtual leader, one of the followers should have the information of the virtual leader in a fixed network [13]. Stimulated by Reynolds’ model [18], flocking algorithms have been proposed by a combining a local artificial potential field with a velocity consensus component ([14], [15], [16], [17]). The convergence condition for the flocking algorithms in ([14], [15], [16]) is that the underlying topology is connected at each time instant. In order to track a virtual leader, the followers should get in touch, directly or indirectly, with the virtual leader from time to time [17]. Synchronization of coupled harmonic oscillators, secondorder consensus and flocking are all characterized by second order dynamics, distributed control, local interactions and self-organization. A key difference among these problems lies in the intrinsic dynamics of the uncoupled systems. The intrinsic dynamics of the uncoupled systems in the second-order consensus and flocking problems are that of double integrators, while in the synchronization of coupled harmonic oscillators, the dynamics are that of second order oscillators. A direct consequence of this difference is that the consensus and flocking equilibria for the velocities are zero or nonzero constants, while the synchronization equilibrium for the velocities is time varying. In the synchronization [10], second-order consensus ([11], [13]) and flocking ([14], [15], [16], [17]) algorithms, certain network connectivity assumptions play a crucial role in the stability analysis. This is because exchanging sufficient information among agents is necessary for cooperation. However, in practice, such kinds of network connectivity assumptions are usually very difficult to verify and may not hold even if the initial network is well connected. On the other hand, we observe that the intrinsic dynamics of the harmonic oscillators will cause the agents to meet with each other from
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ThBIn2.7 time to time, even if the initial velocities and positions are different. Motivated by this observation and inspired by the recent work [10], we revisit in this paper the coupled second-order linear harmonic oscillator models in a dynamic proximity network. The topology of proximity network depends on the relative distances of the harmonic oscillators. The harmonic oscillators are coupled by their velocity information. We will examine the synchronization of coupled harmonic oscillators in the dynamic proximity network without any connectivity assumption. We will also examine the synchronization of coupled harmonic oscillators with a leader and in the absence of any connectivity assumption on the followers. The remainder of the paper is organized as follows. Section II states the problems to be solved in this paper. Section III establishes synchronization results, both without and with a leader. Section IV presents the simulation results. Finally, Section V draws a brief conclusion to the paper. II. P ROBLEM S TATEMENT We consider N agents moving in a one-dimensional Euclidean space. The behavior of each agent is described by a harmonic oscillator of the form q˙i = pi , p˙ i = −ω 2 qi + ui , i = 1, 2, · · · , N,
(1)
where qi ∈ R is the position of agent i, p i ∈ R is its velocity vector, ui ∈ R is its control input and ω is the frequency of the oscillator. For notational convenience, we also define ⎡ ⎤ ⎤ ⎡ p1 q1 ⎢ p2 ⎥ ⎢ q2 ⎥ ⎢ ⎥ ⎥ ⎢ q = ⎢ . ⎥, p = ⎢ . ⎥. . . ⎣ . ⎦ ⎣ . ⎦ qN pN The problem of synchronization is to design a control input ui to cause lim qi (t) − qj (t) = 0, t→∞
and lim pi (t) − pj (t) = 0,
t→∞
for all i and j. In the situation where a leader, labeled as agent N + 1, is present, the goal is then to design a control input ui to cause lim qi (t) − qγ (t) = 0,
t→∞
and lim pi (t) − pγ (t) = 0,
t→∞
for all i, where q γ and pγ are the position and velocity of the leader, respectively. The dynamic of the leader satisfies q˙γ = pγ , p˙ γ = −ω 2 qγ .
(2)
In [10], N coupled harmonic oscillators are connected by dampers, i.e., ui = −
N i=1
aij (t) (pi − pj ) ,
i = 1, 2, · · · , N,
(3)
where aij (t) characterizes the interaction between agents i and j at time t. Under certain network connectivity assumptions and the influence of the control input (3), synchronization of the positions and velocities in both fixed and switching networks was established in [10]. In this paper, we investigate the system in a dynamic proximity network. Each agent has a limited communication capability which allows it to communicate only with agents within its neighborhood. The neighboring agents of agent i at time t is denoted as: Ni (t) = {j : qi − qj < r, j = 1, 2, · · · , N, j = i} ,
(4)
where · is the Euclidean norm. In the above definition, we have assumed that all agents have an identical influencing/sensing radius r. During the course of motion, the relative distances between agents may vary with time, so the neighbors of each agent may change. We define the neighboring graph G(t) = {V, E(t)} to be an undirected graph consisting of a set of vertices V = {1, 2, · · · , N }, whose elements represent agents in the group, and a set of edges E(t) = {(i, j) ∈ V × V : i ∼ j}, containing unordered pairs of vertices that represent neighboring relations at time t. Vertices i and j are said to be adjacent at time t if (i, j) ∈ E(t). III. S YNCHRONIZATION OF C OUPLED H ARMONIC O SCILLATORS A. Synchronization without a Leader Let the control input for agent i be given by aij (q) (pi − pj ) , i = 1, 2, · · · , N, ui = −
(5)
j∈Ni (t)
where A(q) = (aij (q))N ×N is the adjacent matrix which is defined in [14] as 0, if j = i, (6) aij (q) = ρh (qj − qi σ /rσ ) , if j = i, with the bump function ρ h (z), h ∈ (0, 1), being ⎧ ⎪ ⎨ 1, if z ∈ [0, h), z−h 0.5 1 + cos π 1−h , if z ∈ [h, 1], ρh (z) = ⎪ ⎩ 0, otherwise,
(7)
where the σ-norm · σ , defined for a general n-dimensional vector as a map R n → R≥0 , 1 zσ = 1 + εz2 − 1 , ε for some constant ε > 0. It is clear that a ij (q) = aji (q) > 0 if (j, i) ∈ E(t) and aij (q) = aji (q) = 0 if j = i but (j, i) ∈ E(t). Note that the adjacent matrix A(q) is continuous even if the network is switching. We define the Laplacian of graph G(t) with adjacent matrix A(q) as L(q) = ∆(A(q)) − A(q), where the degree matrix ∆(A(q)) is a diagonal matrix with N the ith diagonal element being j=1 aij (q). Denote the eigenvalues of L(q) as λ1 (L(q)) ≤ λ2 (L(q)) ≤ · · · ≤ λN (L(q)). Then, λ1 (L(q)) = 0 and (1, 1, · · · , 1)T ∈ RN is the corresponding eigenvector. Moreover, if G(t) is a
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ThBIn2.7 connected graph, then λ 2 (L(q)) > 0 [19]. Denote the position and velocity of the center of mass (COM) of all agents in the group as N N qi pi q¯ = i=1 , p¯ = i=1 . N N Theorem 1: Consider a system of N mobile agents with dynamics (1), each being steered by the control input (5). Then, 1 qi (t) → q¯(0) cos(ωt) + p¯(0) sin(ωt) ω and pi (t) → −ω q¯(0) sin(ωt)¯ q (0) + p¯(0) cos(ωt), as t → ∞, where q¯(0) and p¯(0) are respectively the initial position and velocity of the COM of the group. Proof. Denote the position difference and the velocity difference between agent i and the COM as q˜i = qi − q¯ and p˜i = pi − p¯, respectively. Then, q˜˙ i = p˜i , q ) (˜ pi − p˜j ) . p˜˙ i = −ω 2 q˜i − j∈Ni (t) aij (˜
(8)
For notational convenience, we also define ⎤ ⎤ ⎡ ⎡ q˜1 p˜1 ⎢ q˜2 ⎥ ⎢ p˜2 ⎥ ⎥ ⎥ ⎢ ⎢ q˜ = ⎢ . ⎥ , p˜ = ⎢ . ⎥ . ⎣ .. ⎦ ⎣ .. ⎦ q˜N p˜N
1 1 2 T ω q˜ q˜ + p˜T p˜, 2 2
(9)
which is positive definite function and radially unbounded with respect to p˜ and q˜. The derivative of Q along the trajectories of the agents is given by, Q˙
= ω 2 p˜T q˜ + p˜T p˜˙ q)˜ p) = ω 2 p˜T q˜ + p˜T (−ω 2 q˜ − L(˜ = −˜ pT L(˜ q )˜ p ≤ 0.
˘ q ) = P (t)L(˜ L(˜ q )P T (t) ⎡ L1 (˜ q) · · · 0 ··· 0 ⎢ .. . . .. . .. .. .. ⎢ . . ⎢ (˜ q ) · · · 0 0 · · · L = ⎢ k ⎢ ⎢ .. .. .. .. .. ⎣ . . . . . 0 ··· 0 · · · Ln(t) (˜ q)
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
where Lk (˜ q ) ∈ RNk (t)×Nk (t) is the Laplacian matrix associated with the kth connected subgraph of the graph G(t).The indices of the state vector can be rearranged such that ⎡ ⎤ ⎤ ⎡ p˘k1 p˘1 ⎢ p˘2 ⎥ ⎢ p˘k2 ⎥ ⎢ ⎥ ⎥ ⎢ k p, p˘ = ⎢ p˘ = ⎢ .. ⎥ = P (t)˜ ⎥, .. ⎣ . ⎦ ⎦ ⎣ . k n(t) p˘Nk (t) p˘ where p˘k is the velocity difference vector of the N k (t) agents within the kth connected subgraph. Therefore ˘ q )˘ p −˜ pT L(˜ q )˜ p = −˘ pT L(˜
T T T T 1 =− p˘ · · · p˘k · · · p˘n(t) ⎤⎡ ⎡ p˘1 L1 (˜ q) · · · 0 · · · 0 ⎢ ⎥ ⎢ .. . . . . .. .. .. .. ⎥⎢ .. ⎢ . . ⎥⎢ k ⎢ ⎢ q) · · · 0 ⎥ · · · Lk (˜ ×⎢ ⎥⎢ p˘ ⎢ 0 ⎢ . ⎥ ⎢ .. .. .. . . .. .. ⎦⎣ .. ⎣ . . . q ) p˘n(t) 0 · · · 0 · · · Ln(t) (˜
⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦ (11)
Consider the Lyapunov function candidate Q(˜ q , p˜) =
that L(˜ q) can be transformed into a block diagonal matrix as follows,
Clearly, −˜ pT L(˜ q )˜ p = 0 if and only if k T q ))˘ pk = 0, k = 1, 2, · · · , n(t). p˘ (Lk (˜ By the sum of squares property [14], 2 k T 1 p˘ (Lk (˜ q ))˘ pk = aij (˜ q ) p˘ki − p˘kj . 2 (i,j)∈E(t)
Therefore, equation (12) is equivalent to p˘k1 = p˘k2 = · · · = p˘kNk (t) ,
(10)
It then follows from the LaSalle Invariance Principle [20] that all trajectories of the agents converge to the largest invariant set inside the region T S = [˜ q T , p˜T ] ∈ R2N : Q˙ = 0 . We now consider the trajectories that lie entirely in S. Suppose that G(t) consists of n(t), 1 ≤ n(t) ≤ N , connected subgraphs and there are N k (t) agents in the kth, 1 ≤ k ≤ n(t), connected subgraph at time t. For any time t, there exists an orthogonal permutation matrix P (t) ∈ R N ×N such
(12)
k = 1, 2, · · · , n(t),
(13)
which implies that the velocities of the agents in the same subgroup are synchronized. We next show that n(t) = 1. This would imply that all agents approach the same velocity asymptotically. From (1), (5) and (13), we have q˙i = pi , p˙i = −ω 2 qi ,
i = 1, 2, · · · , N,
(14)
from which we obtain qi (t) = c1i cos(ωt) + c2i sin(ωt)
(15)
pi (t) = −c1i ω sin(ωt) + c2i ω cos(ωt),
(16)
and
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ThBIn2.7 where c1i and c2i depend on the initial positions and velocities of all the agents in the group. From (15), every pair of agents in the group must collide during each uniformly bounded time-interval π (17) [ti , ti+1 ), ti+1 − ti = . ω Assume that there are at least two connected subgroups in the system. Then, from earlier analysis, we know that agents from two different connected subgroups must collide at some time t , i.e., at time t , an agent i is connected with another agent j. In this case, we have p˜i (t ) = p˜j (t ). It then follows from (12) that ˙ ) < 0, Q(t ˙ which contradicts with the fact that Q(t) = 0. Thus, we conclude that the interaction network will not switch for all t. Consequently, n(t) = 1 for all t, i.e., p˜1 = p˜2 = · · · = p˜N = 0,
(18)
p1 = p2 = · · · = pN = p¯,
(19)
or which implies that p˙ 1 = p˙2 = · · · = p˙ N .
B. Synchronization with a Leader In this section, we investigate the synchronization algorithm for the situation when a leader is present. In this case, the control input for agent i is given by aij (q) (pi − pj ) ui = − j∈Ni (t)
−aiγ (q) (pi − pγ ) ,
as t → ∞, where qγ (0) and pγ (0) are respectively the initial position and velocity of the leader.
(20)
(22) (23)
In conclusion, it follows from the LaSalle Invariance Principle that qi (t) → q¯(0) cos(ωt) +
q˜˙ i = p˜i , p˜˙ i = −ω 2 q˜i − j∈Ni (t) aij (˜ q ) (˜ pi − p˜j ) − aiγ (˜ q )˜ pi . (25) For notational convenience, we also define ⎡ ⎤ ⎤ ⎡ p˜1 q˜1 ⎢ p˜2 ⎥ ⎢ q˜2 ⎥ ⎢ ⎥ ⎥ ⎢ q˜ = ⎢ . ⎥ , p˜ = ⎢ . ⎥ . ⎣ .. ⎦ ⎣ .. ⎦ q˜N
Q˙ where
pi (t) → −ω q¯(0) sin(ωt) + p¯(0) cos(ωt),
=
ω 2 p˜T q˜ + p˜T (−ω 2 q˜ − L(˜ q )˜ p − H(˜ q )˜ p)
=
−˜ pT (L(˜ q ) + H(˜ q ))˜ p ≤ 0, ⎡
⎢ ⎢ H(˜ q) = ⎢ ⎣
Remark 1: Unlike many existing algorithms, Theorem 1 shows that the coupled harmonic oscillators can be synchronized even without any network connectivity assumption. This is because there exist an infinite sequence of contiguous, nonempty and uniformly bounded time-intervals [tj , tj+1 ), j = 0, 1, 2, · · ·, such that across each time interval
p˜N
Consider the Lyapunov function candidate 1 1 Q(˜ q , p˜) = ω 2 q˜T q˜ + p˜T p˜, (26) 2 2 which is positive definite function and radially unbounded with respect to p˜ and q˜. The derivative of Q along the trajectories of the agents is given by,
1 p¯(0) sin(ωt) ω
and
as t → ∞.
(24)
Theorem 2: Consider a system of N mobile agents with dynamics (1) and a leader with dynamic (2). Let each follower be steered by the control input (24). Then, 1 qi (t) → qγ (0) cos(ωt) + pγ (0) sin(ωt) ω and pi (t) → −ωqγ (0) sin(ωt) + pγ (0) cos(ωt),
It follows from (1), (5) and the symmetry of A(t) that N i=1 p˙ i = −ω 2 q¯, p¯˙ = N which in turn implies that q¯˙ = p¯, (21) p¯˙ = −ω 2 q¯. The solution of (21) can be obtained as 1 q¯(t) = q¯(0) cos(ωt) + p¯(0) sin(ωt), ω and p¯(t) = −ω q¯(0) sin(ωt) + p¯(0) cos(ωt).
i = 1, 2, · · · , N.
Proof. Denote the position difference and the velocity difference between agent i and the leader as q˜i = qi − qγ and p˜i = pi − pγ , respectively. Then
Then, from (1) and (5) q1 = q2 = · · · = qN .
each pair of agents in the group must naturally collide, i.e., they exchange velocity information with each other when the relative distances between them are within the influencing/sensing radius r across each time interval.
q) 0 ··· 0 a1γ (˜ 0 a2γ (˜ q) · · · 0 .. .. .. .. . . . . q) 0 0 · · · aN γ (˜
(27) ⎤ ⎥ ⎥ ⎥. ⎦
It then follows from the LaSalle Invariance Principle [20] that all trajectories of the agents converge to the largest invariant set inside the region T S = [˜ q T , p˜T ] ∈ R2N : Q˙ = 0 .
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ThBIn2.7 We now consider the trajectories that lie entirely in S. Suppose that G(t) consists of n(t), 1 ≤ n(t) ≤ N , connected subgraphs and there are N k (t) agents in the kth, 1 ≤ k ≤ n(t), connected subgraph at time t. Similar to the analysis in the proof of Theorem 1, we have
T T T T q ) + H(˜ q ))˜ p = − p˘1 · · · p˘k · · · p˘n(t) −˜ pT (L(˜ ⎡ ⎤ ⎤⎡ p˘1 L1 (˜ q) · · · 0 · · · 0 ⎢ .. ⎥⎢ .. ⎥ .. .. .. .. ⎢ . ⎥⎢ . ⎥ . . . . ⎢ ⎥ ⎥⎢ ⎥⎢ p˘k ⎥ (˜ q ) · · · 0 0 · · · L ×⎢ k ⎢ ⎥ ⎥⎢ ⎢ .. ⎥⎢ .. ⎥ .. .. .. .. ⎣ ⎣ . ⎦ . . ⎦ . . . n(t) q) 0 · · · 0 · · · Ln(t) (˜ p˘
T T T T − p˘1 · · · p˘k · · · p˘n(t) ⎤ ⎤⎡ ⎡ p˘1 H1 (˜ q) · · · 0 ··· 0 ⎥⎢ .. ⎥ ⎢ .. .. .. .. .. ⎥⎢ . ⎥ ⎢ . . . . . ⎥ ⎥⎢ ⎢ ⎥⎢ p˘k ⎥ , ⎢ q) · · · 0 · · · Hk (˜ (28) ×⎢ 0 ⎥ ⎥⎢ ⎢ ⎥ ⎥ ⎢ .. . . .. .. . .. .. ⎦⎣ .. ⎦ ⎣ . . . 0 ··· 0 · · · Hn(t) (˜ q) p˘n(t) where p˘k is the velocity difference within the kth connected subgraph. Clearly, −˜ p T (L(˜ q ) + H(˜ q ))˜ p = 0 if and only if k T p˘ (Lk (˜ q ) + Hk (˜ q ))˘ pk = 0, k = 1, 2, · · · , n(t), (29) which implies that if one agent in the subgroup k is the neighbor of the leader, then the velocity difference between the agents in this subgroup and the virtual leader is zero, otherwise, the velocity difference between the agents in this subgroup is synchronized but may not be zero. As in the proof of Theorem 1, we can show that n(t) = 1, and the leader is a neighbor of the agents of the connected group. From (29), (30) p˜1 = p˜2 = · · · = p˜N = 0, from which it follows that p˜˙ 1 = p˜˙ 2 = · · · = p˜˙ N = 0. Then, from (25) it follows that q˜1 = q˜2 = · · · = q˜N = 0.
(31)
Finally, from (2), (30) and (31), we have qi (t) → qγ (0) cos(ωt) +
1 pγ (0) sin(ωt), ω
[tj , tj+1 ), j = 0, 1, 2, · · ·, such that during each time interval all agents in the group and the leader must naturally collide, i.e., each agent in the group exchange velocity information with the leader when the relative distance between them are below the influencing/sensing radius r across each time interval. IV. S IMULATION S TUDY All the simulations are performed with 10 agents whose initial positions and velocities are randomly chosen within [−4, 4] and [−2, 2], respectively. In the simulation, we choose the influencing/sensing radius r = 0.5, the parameter ω = 3.162, and ε = 0.1 for the σ-norm. Fig. 1 shows the synchronization of coupled harmonic oscillators under control input (5). In particular, Plots (a) and (b) are respectively the evolutions of the position and velocity of the ten agents. It is obvious from these plots that the control input (5) is capable of achieving stable synchronization motion in the absence of any network connectivity assumption. Fig. 2 shows the synchronization with a leader of coupled harmonic oscillators under control input (24). In this figure, initial position and velocity of the leader are set to be qγ (0) = 1 and pγ (0) = 1. Shown in Plots (a) and (b) are respectively the evolution of the position difference and velocity differences between the ten agents and the leader. These simulation results show that all agents asymptotically attain the position and velocity of the leader in the absence of any connectivity assumption on the followers. V. C ONCLUSIONS In this paper, we have investigated synchronization of coupled harmonic oscillators, both without and with a leader. By exploiting the inherent dynamical properties of the uncoupled systems, i.e., the coupled harmonic oscillators will naturally exchange information with each other from time to time through the coupling, we were able to design control laws that achieve synchronization without requiring any connectivity assumption on the agents. This is in contrast with existing algorithms for the same problem where certain network connectivity assumptions are needed. In the situation when a leader is present, all agents in the group will track the leader in the absence of any connectivity assumption on the followers. R EFERENCES
and pi (t) → −ωqγ (0) sin(ωt) + pγ (0) cos(ωt), as t → ∞.
Remark 2: Unlike many existing algorithms with a leader, Theorem 2 shows that all harmonic oscillators can asymptotically attain the position and velocity of the leader even without any connectivity assumption on the followers. This is because there exists an infinite sequence of contiguous, nonempty and uniformly bounded time-intervals
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[1] X. Wang, “Complex networks: topology, dynamics, and synchronization,” International Journal of Bifurcation and Chaos, Vol. 12, No. 5, pp. 885-916, 2002. [2] C. W. Wu, Synchronization in Coupled Chaotic Circuits and Systems, World Scientific, Singapore, 2002. [3] H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,” Progress of Theoretical Physics, Vol. 69, No. 1, pp. 32-47, 1983. [4] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, Vol. 64, No. 8, pp. 821-824, 1990. [5] X. Wang and G. Chen, “Synchronization in scale-free dynamical networks: robustness and fragility,” IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, Vol. 49, No. 1, pp. 54-62, 2002.
ThBIn2.7 Position Convergence
Velocity Convergence
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8
6 2 4
1
2
p
q
0 0
−2
−1
−4
−6 −2 −8
−3
0
5
10
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15
0
5
t
10
15
10
15
t
(a)
(b)
Fig. 1.
Synchronization of 10 agents under control input (5).
Position Difference
Velocity Difference
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8
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1 4 0.5 2 p
q
0
−0.5
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−1 −2 −1.5 −4
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t
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5 t
(a)
Fig. 2.
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(b)
Synchronization with a leader of 10 agents under control input (24) with a leader.
[6] X. Wang and G. Chen, “Complex networks: small-world, scale-free and beyond,” IEEE Circuits and Systems Magzine, Vol. 3, No. 1, pp. 6-20, 2003. [7] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, 1984. [8] A. Jadbabaie, N. Motee and M. Barahona, “On the stability of the Kuramoto model of coupled nonlinear oscillators,” In Proceedings of the American Control Conference, pp. 4296-4301, 2004. [9] A. Papachristodoulou and A. Jadbabaie, “Synchronization in oscillator networks: switching topologies and presence of nonhomogeneous delays,” Proc. of the 44th IEEE Conference on Decision and Control, pp. 5692-5697, 2005. [10] W. Ren, “Synchronization of coupled harmonic oscillators with local interaction,” Automatica, Vol. 44, No. 12, pp. 3195-3200, 2008. [11] W. Ren and E. Atkins, “Distributed multi-vehicle coordinated control via local information exchange,” International Journal of Robust and Nonlinear Control, Vol. 17, No. 10, pp. 1002-1033, 2007. [12] G. Xie and L. Wang, “Consensus control for a class of networks of dynamic agents,” Internatonal Journal of Robust and Nonlinear Control, Vol. 17, No. 10-11, pp. 941-959, 2007. [13] W. Ren, “On consensus algorithms for double-integrator dynamics,” IEEE Transactions on Automatic Control, Vol. 53, No. 6, pp. 1503-
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1509, 2008. [14] R. Olfati-Saber, “Flocking for multi-agent dynamic systems: algorithms and theory,” IEEE Transactions on Automatic Control, Vol. 51, No. 3, pp. 401-420, 2006. [15] H. G. Tanner, A. Jadbabaie and G. J. Pappas, “Stable flocking of mobile agents, part I: fixed topology,” Proc. the 42nd IEEE Conference on Decision and Control, pp. 2010-2015, December 2003. [16] H. G. Tanner, A. Jadbabaie and G. J. Pappas, “Stable flocking of mobile agents, part II: dynamic topology,” Proc. of the 42nd IEEE Conference on Decision and Control, pp. 2016-2021, December 2003. [17] H. Su, X. Wang and Z. Lin, “Flocking of multi-agents with a virtual leader,” IEEE Transactions on Automatic Control, Vol. 54, No. 2, pp. 293-307, 2009. [18] C. W. Reynolds, “Flocks, herds, and schools: a distributed behavioral model,” Computer Graphics, ACM SIGGRAPH 87 Conference Proceedings, Vol. 21, No. 4, pp. 25-34, 1987. [19] C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, Vol. 207, Springer, New York, 2001. [20] H. K. Khalil, Nonlinear Systems, Third Edition, Prentice Hall, Upper Saddle River, New Jersey, 2002.
ThBIn2.7
Synchronization of Coupled Harmonic Oscillators in a Dynamic Proximity Network Housheng Su
Xiaofan Wang
Abstract— In this paper, we revisit the synchronization problems for coupled harmonic oscillators in a dynamic proximity network. Unlike many existing algorithms for distributed control of complex dynamical networks that require explicit assumptions on the network connectivity, we show that the coupled harmonic oscillators can always be synchronized, without imposing any network connectivity assumption. Moreover, we also investigate the synchronization with a leader and show that all harmonic oscillators can asymptotically attain the position and velocity of the leader, again without any assumption on connectivity of the followers. Numerical simulation illustrates the theoretical results. Keywords: Synchronization, distributed control, coupled harmonic oscillators, multi-agent systems.
I. I NTRODUCTION In recent years, there has been significant interest in the study of synchronization from different fields (see, for example, [1], [2] and the references therein). Two main lines of research on the problems of synchronization have emerged from this study. On the one hand, the two pioneering papers on synchronization in coupled systems [3] and synchronization in chaotic systems [4] have stimulated a great deal of interest in the study of complete synchronization of coupled nonlinear dynamical systems. On the other hand, there has been much interest in the study of synchronization in dynamical networks with complex topologies in the past few years due to the discovery of the small-world and scale-free properties of many natural and artificial complex networks ([5], [6]). One of the most important contributions to the problem of synchronization in coupled systems is the Kuramoto model [7], which was established based on the phenomenon of collective synchronization. In the Kuramoto model, the information of oscillators are assumed to be global, i.e., the underlying topology is fully connected. The Kuramoto model was later modified for the scenarios of nearest neighbor interaction [8], switching topologies and presence of nonhomogeneous delays [9]. In contrast to the above models where the coupled systems are described by single integrator This work was supported in part by National Natural Science Foundation of China under grants 60731160629 and 60674045. The work was performed while the third author was visiting Shanghai Jiao Tong University as a Cheung Kong Scholarship Chair Professor. Housheng Su and Xiaofan Wang are with Department of Automation, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai 200240, China. Email: [email protected], [email protected] Zongli Lin is with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P.O. Box 400473, Charlottesville, VA 22904-4743, U.S.A.. Email: [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
Zongli Lin
dynamics, Ren in [10] recently investigated coupled secondorder linear harmonic oscillator models. The oscillators investigated in [10] are modeled as point masses on a real line. Under some rather mild network connectivity assumptions, the positions and velocities of coupled harmonic oscillators can be synchronized in both fixed and switching networks, with or without a leader. Related to the synchronization of coupled harmonic oscillators are second-order consensus problems ([11], [12], [13]) and flocking problems ([14], [15], [16], [17]) in multiagent systems. In order to achieve second-order consensus in a multi-agent system, the underlying topology must contain a directed spanning tree in fixed networks, or must have a directed spanning tree at each time instant in switching networks ([11], [12], [13]). In the case of tracking a virtual leader, one of the followers should have the information of the virtual leader in a fixed network [13]. Stimulated by Reynolds’ model [18], flocking algorithms have been proposed by a combining a local artificial potential field with a velocity consensus component ([14], [15], [16], [17]). The convergence condition for the flocking algorithms in ([14], [15], [16]) is that the underlying topology is connected at each time instant. In order to track a virtual leader, the followers should get in touch, directly or indirectly, with the virtual leader from time to time [17]. Synchronization of coupled harmonic oscillators, secondorder consensus and flocking are all characterized by second order dynamics, distributed control, local interactions and self-organization. A key difference among these problems lies in the intrinsic dynamics of the uncoupled systems. The intrinsic dynamics of the uncoupled systems in the second-order consensus and flocking problems are that of double integrators, while in the synchronization of coupled harmonic oscillators, the dynamics are that of second order oscillators. A direct consequence of this difference is that the consensus and flocking equilibria for the velocities are zero or nonzero constants, while the synchronization equilibrium for the velocities is time varying. In the synchronization [10], second-order consensus ([11], [13]) and flocking ([14], [15], [16], [17]) algorithms, certain network connectivity assumptions play a crucial role in the stability analysis. This is because exchanging sufficient information among agents is necessary for cooperation. However, in practice, such kinds of network connectivity assumptions are usually very difficult to verify and may not hold even if the initial network is well connected. On the other hand, we observe that the intrinsic dynamics of the harmonic oscillators will cause the agents to meet with each other from
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ThBIn2.7 time to time, even if the initial velocities and positions are different. Motivated by this observation and inspired by the recent work [10], we revisit in this paper the coupled second-order linear harmonic oscillator models in a dynamic proximity network. The topology of proximity network depends on the relative distances of the harmonic oscillators. The harmonic oscillators are coupled by their velocity information. We will examine the synchronization of coupled harmonic oscillators in the dynamic proximity network without any connectivity assumption. We will also examine the synchronization of coupled harmonic oscillators with a leader and in the absence of any connectivity assumption on the followers. The remainder of the paper is organized as follows. Section II states the problems to be solved in this paper. Section III establishes synchronization results, both without and with a leader. Section IV presents the simulation results. Finally, Section V draws a brief conclusion to the paper. II. P ROBLEM S TATEMENT We consider N agents moving in a one-dimensional Euclidean space. The behavior of each agent is described by a harmonic oscillator of the form q˙i = pi , p˙ i = −ω 2 qi + ui , i = 1, 2, · · · , N,
(1)
where qi ∈ R is the position of agent i, p i ∈ R is its velocity vector, ui ∈ R is its control input and ω is the frequency of the oscillator. For notational convenience, we also define ⎡ ⎤ ⎤ ⎡ p1 q1 ⎢ p2 ⎥ ⎢ q2 ⎥ ⎢ ⎥ ⎥ ⎢ q = ⎢ . ⎥, p = ⎢ . ⎥. . . ⎣ . ⎦ ⎣ . ⎦ qN pN The problem of synchronization is to design a control input ui to cause lim qi (t) − qj (t) = 0, t→∞
and lim pi (t) − pj (t) = 0,
t→∞
for all i and j. In the situation where a leader, labeled as agent N + 1, is present, the goal is then to design a control input ui to cause lim qi (t) − qγ (t) = 0,
t→∞
and lim pi (t) − pγ (t) = 0,
t→∞
for all i, where q γ and pγ are the position and velocity of the leader, respectively. The dynamic of the leader satisfies q˙γ = pγ , p˙ γ = −ω 2 qγ .
(2)
In [10], N coupled harmonic oscillators are connected by dampers, i.e., ui = −
N i=1
aij (t) (pi − pj ) ,
i = 1, 2, · · · , N,
(3)
where aij (t) characterizes the interaction between agents i and j at time t. Under certain network connectivity assumptions and the influence of the control input (3), synchronization of the positions and velocities in both fixed and switching networks was established in [10]. In this paper, we investigate the system in a dynamic proximity network. Each agent has a limited communication capability which allows it to communicate only with agents within its neighborhood. The neighboring agents of agent i at time t is denoted as: Ni (t) = {j : qi − qj < r, j = 1, 2, · · · , N, j = i} ,
(4)
where · is the Euclidean norm. In the above definition, we have assumed that all agents have an identical influencing/sensing radius r. During the course of motion, the relative distances between agents may vary with time, so the neighbors of each agent may change. We define the neighboring graph G(t) = {V, E(t)} to be an undirected graph consisting of a set of vertices V = {1, 2, · · · , N }, whose elements represent agents in the group, and a set of edges E(t) = {(i, j) ∈ V × V : i ∼ j}, containing unordered pairs of vertices that represent neighboring relations at time t. Vertices i and j are said to be adjacent at time t if (i, j) ∈ E(t). III. S YNCHRONIZATION OF C OUPLED H ARMONIC O SCILLATORS A. Synchronization without a Leader Let the control input for agent i be given by aij (q) (pi − pj ) , i = 1, 2, · · · , N, ui = −
(5)
j∈Ni (t)
where A(q) = (aij (q))N ×N is the adjacent matrix which is defined in [14] as 0, if j = i, (6) aij (q) = ρh (qj − qi σ /rσ ) , if j = i, with the bump function ρ h (z), h ∈ (0, 1), being ⎧ ⎪ ⎨ 1, if z ∈ [0, h), z−h 0.5 1 + cos π 1−h , if z ∈ [h, 1], ρh (z) = ⎪ ⎩ 0, otherwise,
(7)
where the σ-norm · σ , defined for a general n-dimensional vector as a map R n → R≥0 , 1 zσ = 1 + εz2 − 1 , ε for some constant ε > 0. It is clear that a ij (q) = aji (q) > 0 if (j, i) ∈ E(t) and aij (q) = aji (q) = 0 if j = i but (j, i) ∈ E(t). Note that the adjacent matrix A(q) is continuous even if the network is switching. We define the Laplacian of graph G(t) with adjacent matrix A(q) as L(q) = ∆(A(q)) − A(q), where the degree matrix ∆(A(q)) is a diagonal matrix with N the ith diagonal element being j=1 aij (q). Denote the eigenvalues of L(q) as λ1 (L(q)) ≤ λ2 (L(q)) ≤ · · · ≤ λN (L(q)). Then, λ1 (L(q)) = 0 and (1, 1, · · · , 1)T ∈ RN is the corresponding eigenvector. Moreover, if G(t) is a
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ThBIn2.7 connected graph, then λ 2 (L(q)) > 0 [19]. Denote the position and velocity of the center of mass (COM) of all agents in the group as N N qi pi q¯ = i=1 , p¯ = i=1 . N N Theorem 1: Consider a system of N mobile agents with dynamics (1), each being steered by the control input (5). Then, 1 qi (t) → q¯(0) cos(ωt) + p¯(0) sin(ωt) ω and pi (t) → −ω q¯(0) sin(ωt)¯ q (0) + p¯(0) cos(ωt), as t → ∞, where q¯(0) and p¯(0) are respectively the initial position and velocity of the COM of the group. Proof. Denote the position difference and the velocity difference between agent i and the COM as q˜i = qi − q¯ and p˜i = pi − p¯, respectively. Then, q˜˙ i = p˜i , q ) (˜ pi − p˜j ) . p˜˙ i = −ω 2 q˜i − j∈Ni (t) aij (˜
(8)
For notational convenience, we also define ⎤ ⎤ ⎡ ⎡ q˜1 p˜1 ⎢ q˜2 ⎥ ⎢ p˜2 ⎥ ⎥ ⎥ ⎢ ⎢ q˜ = ⎢ . ⎥ , p˜ = ⎢ . ⎥ . ⎣ .. ⎦ ⎣ .. ⎦ q˜N p˜N
1 1 2 T ω q˜ q˜ + p˜T p˜, 2 2
(9)
which is positive definite function and radially unbounded with respect to p˜ and q˜. The derivative of Q along the trajectories of the agents is given by, Q˙
= ω 2 p˜T q˜ + p˜T p˜˙ q)˜ p) = ω 2 p˜T q˜ + p˜T (−ω 2 q˜ − L(˜ = −˜ pT L(˜ q )˜ p ≤ 0.
˘ q ) = P (t)L(˜ L(˜ q )P T (t) ⎡ L1 (˜ q) · · · 0 ··· 0 ⎢ .. . . .. . .. .. .. ⎢ . . ⎢ (˜ q ) · · · 0 0 · · · L = ⎢ k ⎢ ⎢ .. .. .. .. .. ⎣ . . . . . 0 ··· 0 · · · Ln(t) (˜ q)
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
where Lk (˜ q ) ∈ RNk (t)×Nk (t) is the Laplacian matrix associated with the kth connected subgraph of the graph G(t).The indices of the state vector can be rearranged such that ⎡ ⎤ ⎤ ⎡ p˘k1 p˘1 ⎢ p˘2 ⎥ ⎢ p˘k2 ⎥ ⎢ ⎥ ⎥ ⎢ k p, p˘ = ⎢ p˘ = ⎢ .. ⎥ = P (t)˜ ⎥, .. ⎣ . ⎦ ⎦ ⎣ . k n(t) p˘Nk (t) p˘ where p˘k is the velocity difference vector of the N k (t) agents within the kth connected subgraph. Therefore ˘ q )˘ p −˜ pT L(˜ q )˜ p = −˘ pT L(˜
T T T T 1 =− p˘ · · · p˘k · · · p˘n(t) ⎤⎡ ⎡ p˘1 L1 (˜ q) · · · 0 · · · 0 ⎢ ⎥ ⎢ .. . . . . .. .. .. .. ⎥⎢ .. ⎢ . . ⎥⎢ k ⎢ ⎢ q) · · · 0 ⎥ · · · Lk (˜ ×⎢ ⎥⎢ p˘ ⎢ 0 ⎢ . ⎥ ⎢ .. .. .. . . .. .. ⎦⎣ .. ⎣ . . . q ) p˘n(t) 0 · · · 0 · · · Ln(t) (˜
⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦ (11)
Consider the Lyapunov function candidate Q(˜ q , p˜) =
that L(˜ q) can be transformed into a block diagonal matrix as follows,
Clearly, −˜ pT L(˜ q )˜ p = 0 if and only if k T q ))˘ pk = 0, k = 1, 2, · · · , n(t). p˘ (Lk (˜ By the sum of squares property [14], 2 k T 1 p˘ (Lk (˜ q ))˘ pk = aij (˜ q ) p˘ki − p˘kj . 2 (i,j)∈E(t)
Therefore, equation (12) is equivalent to p˘k1 = p˘k2 = · · · = p˘kNk (t) ,
(10)
It then follows from the LaSalle Invariance Principle [20] that all trajectories of the agents converge to the largest invariant set inside the region T S = [˜ q T , p˜T ] ∈ R2N : Q˙ = 0 . We now consider the trajectories that lie entirely in S. Suppose that G(t) consists of n(t), 1 ≤ n(t) ≤ N , connected subgraphs and there are N k (t) agents in the kth, 1 ≤ k ≤ n(t), connected subgraph at time t. For any time t, there exists an orthogonal permutation matrix P (t) ∈ R N ×N such
(12)
k = 1, 2, · · · , n(t),
(13)
which implies that the velocities of the agents in the same subgroup are synchronized. We next show that n(t) = 1. This would imply that all agents approach the same velocity asymptotically. From (1), (5) and (13), we have q˙i = pi , p˙i = −ω 2 qi ,
i = 1, 2, · · · , N,
(14)
from which we obtain qi (t) = c1i cos(ωt) + c2i sin(ωt)
(15)
pi (t) = −c1i ω sin(ωt) + c2i ω cos(ωt),
(16)
and
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ThBIn2.7 where c1i and c2i depend on the initial positions and velocities of all the agents in the group. From (15), every pair of agents in the group must collide during each uniformly bounded time-interval π (17) [ti , ti+1 ), ti+1 − ti = . ω Assume that there are at least two connected subgroups in the system. Then, from earlier analysis, we know that agents from two different connected subgroups must collide at some time t , i.e., at time t , an agent i is connected with another agent j. In this case, we have p˜i (t ) = p˜j (t ). It then follows from (12) that ˙ ) < 0, Q(t ˙ which contradicts with the fact that Q(t) = 0. Thus, we conclude that the interaction network will not switch for all t. Consequently, n(t) = 1 for all t, i.e., p˜1 = p˜2 = · · · = p˜N = 0,
(18)
p1 = p2 = · · · = pN = p¯,
(19)
or which implies that p˙ 1 = p˙2 = · · · = p˙ N .
B. Synchronization with a Leader In this section, we investigate the synchronization algorithm for the situation when a leader is present. In this case, the control input for agent i is given by aij (q) (pi − pj ) ui = − j∈Ni (t)
−aiγ (q) (pi − pγ ) ,
as t → ∞, where qγ (0) and pγ (0) are respectively the initial position and velocity of the leader.
(20)
(22) (23)
In conclusion, it follows from the LaSalle Invariance Principle that qi (t) → q¯(0) cos(ωt) +
q˜˙ i = p˜i , p˜˙ i = −ω 2 q˜i − j∈Ni (t) aij (˜ q ) (˜ pi − p˜j ) − aiγ (˜ q )˜ pi . (25) For notational convenience, we also define ⎡ ⎤ ⎤ ⎡ p˜1 q˜1 ⎢ p˜2 ⎥ ⎢ q˜2 ⎥ ⎢ ⎥ ⎥ ⎢ q˜ = ⎢ . ⎥ , p˜ = ⎢ . ⎥ . ⎣ .. ⎦ ⎣ .. ⎦ q˜N
Q˙ where
pi (t) → −ω q¯(0) sin(ωt) + p¯(0) cos(ωt),
=
ω 2 p˜T q˜ + p˜T (−ω 2 q˜ − L(˜ q )˜ p − H(˜ q )˜ p)
=
−˜ pT (L(˜ q ) + H(˜ q ))˜ p ≤ 0, ⎡
⎢ ⎢ H(˜ q) = ⎢ ⎣
Remark 1: Unlike many existing algorithms, Theorem 1 shows that the coupled harmonic oscillators can be synchronized even without any network connectivity assumption. This is because there exist an infinite sequence of contiguous, nonempty and uniformly bounded time-intervals [tj , tj+1 ), j = 0, 1, 2, · · ·, such that across each time interval
p˜N
Consider the Lyapunov function candidate 1 1 Q(˜ q , p˜) = ω 2 q˜T q˜ + p˜T p˜, (26) 2 2 which is positive definite function and radially unbounded with respect to p˜ and q˜. The derivative of Q along the trajectories of the agents is given by,
1 p¯(0) sin(ωt) ω
and
as t → ∞.
(24)
Theorem 2: Consider a system of N mobile agents with dynamics (1) and a leader with dynamic (2). Let each follower be steered by the control input (24). Then, 1 qi (t) → qγ (0) cos(ωt) + pγ (0) sin(ωt) ω and pi (t) → −ωqγ (0) sin(ωt) + pγ (0) cos(ωt),
It follows from (1), (5) and the symmetry of A(t) that N i=1 p˙ i = −ω 2 q¯, p¯˙ = N which in turn implies that q¯˙ = p¯, (21) p¯˙ = −ω 2 q¯. The solution of (21) can be obtained as 1 q¯(t) = q¯(0) cos(ωt) + p¯(0) sin(ωt), ω and p¯(t) = −ω q¯(0) sin(ωt) + p¯(0) cos(ωt).
i = 1, 2, · · · , N.
Proof. Denote the position difference and the velocity difference between agent i and the leader as q˜i = qi − qγ and p˜i = pi − pγ , respectively. Then
Then, from (1) and (5) q1 = q2 = · · · = qN .
each pair of agents in the group must naturally collide, i.e., they exchange velocity information with each other when the relative distances between them are within the influencing/sensing radius r across each time interval.
q) 0 ··· 0 a1γ (˜ 0 a2γ (˜ q) · · · 0 .. .. .. .. . . . . q) 0 0 · · · aN γ (˜
(27) ⎤ ⎥ ⎥ ⎥. ⎦
It then follows from the LaSalle Invariance Principle [20] that all trajectories of the agents converge to the largest invariant set inside the region T S = [˜ q T , p˜T ] ∈ R2N : Q˙ = 0 .
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ThBIn2.7 We now consider the trajectories that lie entirely in S. Suppose that G(t) consists of n(t), 1 ≤ n(t) ≤ N , connected subgraphs and there are N k (t) agents in the kth, 1 ≤ k ≤ n(t), connected subgraph at time t. Similar to the analysis in the proof of Theorem 1, we have
T T T T q ) + H(˜ q ))˜ p = − p˘1 · · · p˘k · · · p˘n(t) −˜ pT (L(˜ ⎡ ⎤ ⎤⎡ p˘1 L1 (˜ q) · · · 0 · · · 0 ⎢ .. ⎥⎢ .. ⎥ .. .. .. .. ⎢ . ⎥⎢ . ⎥ . . . . ⎢ ⎥ ⎥⎢ ⎥⎢ p˘k ⎥ (˜ q ) · · · 0 0 · · · L ×⎢ k ⎢ ⎥ ⎥⎢ ⎢ .. ⎥⎢ .. ⎥ .. .. .. .. ⎣ ⎣ . ⎦ . . ⎦ . . . n(t) q) 0 · · · 0 · · · Ln(t) (˜ p˘
T T T T − p˘1 · · · p˘k · · · p˘n(t) ⎤ ⎤⎡ ⎡ p˘1 H1 (˜ q) · · · 0 ··· 0 ⎥⎢ .. ⎥ ⎢ .. .. .. .. .. ⎥⎢ . ⎥ ⎢ . . . . . ⎥ ⎥⎢ ⎢ ⎥⎢ p˘k ⎥ , ⎢ q) · · · 0 · · · Hk (˜ (28) ×⎢ 0 ⎥ ⎥⎢ ⎢ ⎥ ⎥ ⎢ .. . . .. .. . .. .. ⎦⎣ .. ⎦ ⎣ . . . 0 ··· 0 · · · Hn(t) (˜ q) p˘n(t) where p˘k is the velocity difference within the kth connected subgraph. Clearly, −˜ p T (L(˜ q ) + H(˜ q ))˜ p = 0 if and only if k T p˘ (Lk (˜ q ) + Hk (˜ q ))˘ pk = 0, k = 1, 2, · · · , n(t), (29) which implies that if one agent in the subgroup k is the neighbor of the leader, then the velocity difference between the agents in this subgroup and the virtual leader is zero, otherwise, the velocity difference between the agents in this subgroup is synchronized but may not be zero. As in the proof of Theorem 1, we can show that n(t) = 1, and the leader is a neighbor of the agents of the connected group. From (29), (30) p˜1 = p˜2 = · · · = p˜N = 0, from which it follows that p˜˙ 1 = p˜˙ 2 = · · · = p˜˙ N = 0. Then, from (25) it follows that q˜1 = q˜2 = · · · = q˜N = 0.
(31)
Finally, from (2), (30) and (31), we have qi (t) → qγ (0) cos(ωt) +
1 pγ (0) sin(ωt), ω
[tj , tj+1 ), j = 0, 1, 2, · · ·, such that during each time interval all agents in the group and the leader must naturally collide, i.e., each agent in the group exchange velocity information with the leader when the relative distance between them are below the influencing/sensing radius r across each time interval. IV. S IMULATION S TUDY All the simulations are performed with 10 agents whose initial positions and velocities are randomly chosen within [−4, 4] and [−2, 2], respectively. In the simulation, we choose the influencing/sensing radius r = 0.5, the parameter ω = 3.162, and ε = 0.1 for the σ-norm. Fig. 1 shows the synchronization of coupled harmonic oscillators under control input (5). In particular, Plots (a) and (b) are respectively the evolutions of the position and velocity of the ten agents. It is obvious from these plots that the control input (5) is capable of achieving stable synchronization motion in the absence of any network connectivity assumption. Fig. 2 shows the synchronization with a leader of coupled harmonic oscillators under control input (24). In this figure, initial position and velocity of the leader are set to be qγ (0) = 1 and pγ (0) = 1. Shown in Plots (a) and (b) are respectively the evolution of the position difference and velocity differences between the ten agents and the leader. These simulation results show that all agents asymptotically attain the position and velocity of the leader in the absence of any connectivity assumption on the followers. V. C ONCLUSIONS In this paper, we have investigated synchronization of coupled harmonic oscillators, both without and with a leader. By exploiting the inherent dynamical properties of the uncoupled systems, i.e., the coupled harmonic oscillators will naturally exchange information with each other from time to time through the coupling, we were able to design control laws that achieve synchronization without requiring any connectivity assumption on the agents. This is in contrast with existing algorithms for the same problem where certain network connectivity assumptions are needed. In the situation when a leader is present, all agents in the group will track the leader in the absence of any connectivity assumption on the followers. R EFERENCES
and pi (t) → −ωqγ (0) sin(ωt) + pγ (0) cos(ωt), as t → ∞.
Remark 2: Unlike many existing algorithms with a leader, Theorem 2 shows that all harmonic oscillators can asymptotically attain the position and velocity of the leader even without any connectivity assumption on the followers. This is because there exists an infinite sequence of contiguous, nonempty and uniformly bounded time-intervals
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[1] X. Wang, “Complex networks: topology, dynamics, and synchronization,” International Journal of Bifurcation and Chaos, Vol. 12, No. 5, pp. 885-916, 2002. [2] C. W. Wu, Synchronization in Coupled Chaotic Circuits and Systems, World Scientific, Singapore, 2002. [3] H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,” Progress of Theoretical Physics, Vol. 69, No. 1, pp. 32-47, 1983. [4] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, Vol. 64, No. 8, pp. 821-824, 1990. [5] X. Wang and G. Chen, “Synchronization in scale-free dynamical networks: robustness and fragility,” IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, Vol. 49, No. 1, pp. 54-62, 2002.
ThBIn2.7 Position Convergence
Velocity Convergence
3
8
6 2 4
1
2
p
q
0 0
−2
−1
−4
−6 −2 −8
−3
0
5
10
−10
15
0
5
t
10
15
10
15
t
(a)
(b)
Fig. 1.
Synchronization of 10 agents under control input (5).
Position Difference
Velocity Difference
2
8
1.5
6
1 4 0.5 2 p
q
0
−0.5
0
−1 −2 −1.5 −4
−2
−2.5
0
5
10
15
t
0
5 t
(a)
Fig. 2.
−6
(b)
Synchronization with a leader of 10 agents under control input (24) with a leader.
[6] X. Wang and G. Chen, “Complex networks: small-world, scale-free and beyond,” IEEE Circuits and Systems Magzine, Vol. 3, No. 1, pp. 6-20, 2003. [7] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, 1984. [8] A. Jadbabaie, N. Motee and M. Barahona, “On the stability of the Kuramoto model of coupled nonlinear oscillators,” In Proceedings of the American Control Conference, pp. 4296-4301, 2004. [9] A. Papachristodoulou and A. Jadbabaie, “Synchronization in oscillator networks: switching topologies and presence of nonhomogeneous delays,” Proc. of the 44th IEEE Conference on Decision and Control, pp. 5692-5697, 2005. [10] W. Ren, “Synchronization of coupled harmonic oscillators with local interaction,” Automatica, Vol. 44, No. 12, pp. 3195-3200, 2008. [11] W. Ren and E. Atkins, “Distributed multi-vehicle coordinated control via local information exchange,” International Journal of Robust and Nonlinear Control, Vol. 17, No. 10, pp. 1002-1033, 2007. [12] G. Xie and L. Wang, “Consensus control for a class of networks of dynamic agents,” Internatonal Journal of Robust and Nonlinear Control, Vol. 17, No. 10-11, pp. 941-959, 2007. [13] W. Ren, “On consensus algorithms for double-integrator dynamics,” IEEE Transactions on Automatic Control, Vol. 53, No. 6, pp. 1503-
4990
1509, 2008. [14] R. Olfati-Saber, “Flocking for multi-agent dynamic systems: algorithms and theory,” IEEE Transactions on Automatic Control, Vol. 51, No. 3, pp. 401-420, 2006. [15] H. G. Tanner, A. Jadbabaie and G. J. Pappas, “Stable flocking of mobile agents, part I: fixed topology,” Proc. the 42nd IEEE Conference on Decision and Control, pp. 2010-2015, December 2003. [16] H. G. Tanner, A. Jadbabaie and G. J. Pappas, “Stable flocking of mobile agents, part II: dynamic topology,” Proc. of the 42nd IEEE Conference on Decision and Control, pp. 2016-2021, December 2003. [17] H. Su, X. Wang and Z. Lin, “Flocking of multi-agents with a virtual leader,” IEEE Transactions on Automatic Control, Vol. 54, No. 2, pp. 293-307, 2009. [18] C. W. Reynolds, “Flocks, herds, and schools: a distributed behavioral model,” Computer Graphics, ACM SIGGRAPH 87 Conference Proceedings, Vol. 21, No. 4, pp. 25-34, 1987. [19] C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, Vol. 207, Springer, New York, 2001. [20] H. K. Khalil, Nonlinear Systems, Third Edition, Prentice Hall, Upper Saddle River, New Jersey, 2002.
