Int. Workshop on Complex Eng. Systems and Design, Hong Kong, 22 May 2009 Recent Advances in Consensus of Multi-Agent Systems Jinhu Lü Academy of Mathematics and Systems Science Chinese Academy of Sciences 1
With Thanks To City University of Hong Kong Prof. Guanrong Chen The Hong Kong Polytechnic University Prof. C. K. Michael Tse RMIT University, Australia Prof. Xinghuo Yu Princeton University, USA Prof. Simon A. Levin and Prof. Iain D. Couzin University of Virginia, USA Prof. Zongli Lin AMSS, Chinese Academy of Sciences Yao Chen 2
Outline Introduction Consensus of Multi-Agent Systems CASE I: Nonlinear Local Rules CASE II: Leader and Asymmetric Matrix CASE III: Informed-Uninformed Local Rules Conclusions 3
Part I: Introduction Some Examples Several Representative Models 4
Some Examples: Snow Geese Flock 5
Some Examples: Bird Flock 6
Some Examples: Fish Flock 7
Some Examples: Fish Swarm Barracuda Tuna 8
Some Examples: Rotating Ants Mill 9
Some Examples: Bacteria Group 10
Formation Control with Distance Constraints The problem: Maintaining a formation in 2D Autonomous agent 11
Multi-Agent Systems (MAS) Agents Insect, bird, fish, people, robot, node, individual, particle, Local Rules 12
A Unifying Framework of MAS Many Individuals Many Interactions Emergent Behaviors Decentralization Simple Local Rules? Collective Behaviors: Modeling, Analysis and Control 13
Several Representative Models: Boids Flocking Model (1987) Three Basic Local Rules: (The First Model) Alignment: Steer to move toward the average heading of local flockmates Separation: Steer to avoid crowding local flockmates Cohesion: Steer to move toward the average position of local flockmates Reynolds, Flocks, herd, and schools: A distributed behavioral model, Computer Graphics, 1987, 21(4): 15-24. 14
Several Representative Models: Vicsek Particles Model (1995) One Basic Local Rule: Alignment (The Simplest Model) Position: Heading: Vicsek, et al., Novel type of phase transition in a system of self lf-driven particles, Phys. Rev. Lett., 1995, 75 (6): 1226. 15
Several Representative Models: Vicsek Particles Model (1995) (a) Initial: Random positions/velocities (b) Low density/noise: grouped, random (c) High density/noise: correlated, random (d) High density / Low noise ordered motion Vicsek, et al., Novel type of phase transition in a system of self lf-driven particles, Phys. Rev. Lett., 1995, 75 (6): 1226. 16
Several Representative Models: Couzin-Levin Model (2005) Local Rules: Separation, Cohesion, Alignment Updating rules: (Information) Couzin, I. D., Krause, J., Franks, N. R. & Levin, S. M. Effective leadership and decision-making in animal groups on the Move, Nature, 2005, 433: 513-516. 17
Part II: Consensus of Multi-Agent Systems CASE I: Nonlinear Local Rules CASE II: Leader and Asymmetric Matrix CASE III: Informed-Uninformed Local Rules 18
CASE I: Nonlinear Local Rules Y. Chen, J. Lü, Z. Lin, Consensus of discrete-time multi-agent systems with nonlinear local rules and time-varying delays, CDC, 2009. 19
Background: Linear Local Rules x () t i x () t j 1 xi( t + 1) = x j( t) N i ( t) j N i ( t) i V = {1,2,..., n} 20
Some Further Results The linear local rules have been widely investigated over the last ten years. n i i + = ij j τ j j = 1 x ( t 1) a ( t) x ( t ( t)) 21
Nonlinear Local Rules There are few results reported on nonlinear local rules. Vicsek model (PRL, 1995): i sin( θ j( t τ j( t))) j Ni () t θi ( t + 1) = arctan i cos( θ j( t τ j( t))) j Ni () t x ( t+ 1) = x ( t) + vcos( θ ( t+ 1)) i i i y ( t+ 1) = y ( t) + vsin( θ ( t+ 1)) i i i Nonlinear L. Moreau, Stability of multiagent systems with time-dependent communication links, IEEE TAC, 2005, 50(2): 169-182. L. Fang, P.J. Antsaklis, Asynchronous consensus protocols using nonlinear paracontractions theory, IEEE TAC, 2008, 53(10): 2351-2355. 22
Nonlinear Local Rules x () t i y () t i j x () t j i i y () t = f( x ( t τ ())) t j j j 1 i xi( t + 1) = F( y j( t)) ni () t j Ni () t 23
One Open Problem What kinds of functions f, F and G(t) can guarantee the consensus of MAS? 1 i xi( t+ 1) = F( f( xj( t τ j( t)))) ni () t j Ni () t (1) Consensus of MAS: lim xi() t xj() t = 0 i, j t V *G(t)=(V, E(t)) is the graph corresponding to the relation among V.. 24
Assumptions (I) Let g = F 1 (A1): f and g are both continuous functions defined on [a, b] and f ([ ab, ]) g([ ab., ]) (A2): f is monotonous increasing and g is strict monotonous increasing. (A3): f () x g() x for x [ ab,.] Γ (A4): There exists an integer Γ > 0 such that s= 1 Gt ( + s) is strongly connected for any t 0. (A5) There exists an integer B > 0 satisfying 0 τ i j ( t) < Bfor any i j. i (A6): τ () t = 0for t 0. i 25
Let G( ) = lim i k G( i). k (A3 ): f ( x) g( x) for x [ ab, ]. (A4 ): Gt () Assumptions (II) is directed, but G( ) is undirected and strongly connected. (A4 ): There exists some integer when (A4 ): Γ 0 (, i j) G( ) and (, i j) G() t there is some t ' with t t' t+γ satisfying ( j, i) G( t'). () is directed, but G is strongly connected. Gt ( ) (A4 ): There exists some integer Γ>0 such that the in-degree of each node in Gk = Γ s= 1 G( k+ s) equal to its corresponding out-degree. 26
Main Results (I) Theorem 1: Suppose that (A1)-(A6) hold for a given MAS (V, G(t), (1)). For any given initial states, the states of all agents can reach consensus. 27
Main Results (II) Theorem 2: Suppose (A1)-(A3), (A4 ), (A4 ), (A5), (A6) hold for a given MAS (V, G(t), (1)). Then this MAS can reach consensus. 28
Main Results (III) Theorem 3: Suppose (A1)-(A3), (A4 ), ( A4 ), (A5), (A6) hold for a given MAS (V, G(t), (1)). Then this MAS can reach consensus. 29
Main Results (IV) Theorem 4: Suppose that Assumptions (A1)-(A3), (A5) and (A6) hold for a given MAS (V, G(t), (1)). If there exists a sequence { t } and an integer λ m > 0 k such that there exists a directed path from Ωm( tk) to Ωm( tk) and 0 < tk+ 1 tk < λm for k > 0. Then the MAS (V, G(t), (3)) can reach consensus. Ω () t = {: i m () t = min m ()}, t m i j j {1,2,..., n} Ω () t = {: i m () t = max m ()}. t m i j j {1,2,..., n} 30
Application To Vicsek Model { i i i x (), t y (), t θ ( t) } i sin( θ j( t τ j( t))) j Ni () t θi ( t + 1) = arctan i cos( θ j( t τ j( t))) j Ni () t x ( t+ 1) = x ( t) + vcos( θ ( t+ 1)) i i i y ( t+ 1) = y ( t) + vsin( θ ( t+ 1)) i i i (2) { } 2 2 N () t = j V : ( x () t x ()) t + ( y () t y ()) t < r, r > 0 i i j i j 31
Application To Vicsek Model Theorem 5: Suppose that Assumptions (A5) and (A6) hold for the Vicsek model with time-varying delays in the updating rules (2). Also, assume that the graph π π G( ) is connected. If the initial headings θi () t (, ), 2 2 then the headings of agents can reach consensus. a ij () t = x () t = tan θ () t i i cos θ ( t τ ( t)) j N () t i i j i cos θ ( t τ ( t)) j j j x ( t+ 1) = a () t x ( t τ ()) t i i ij j j j N() t i *Theorem 5 is a special case of Theorem 2. 32
Numerical Simulations Let x+ 1 0 x 1 2 x 1 x 2 f = x+ 2 2 x 3 1 x+ 4 3 x 6 3 1 x 0 x 4, F( x) = 2 2x 6 4 x 6. B = 2 The phase portrait of f(x) and F(x) S = { x: f( x) = g( x)} = { x:1 x 2, x= 6} a 33
Numerical Simulations (I) S1 = { x:1 x 2} Phase trajectories from the initial states T x( 1) = (1.12,1.52,0.58,1.65,0.17,0.17,0.76,2.06), T x(0) = (0.60,0.98,0.16, 2.08,0.00,1.59,0.65, 2.39). 34
Numerical Simulations (II) S 2 = {6} Phase trajectories from the initial states T x( 1) = (0.30,5.87,0.71,3.70,0.94,0.05,4.45,4.66), T x(0) = (4.82, 0.04, 0.59, 4.91,1.17,1.75,5.63, 0.82). 35
CASE II: Leader and Asymmetric Matrix W. Guo, S. Chen, J. Lü, X. Yu, Consensus of multi-agent systems with an active leader and asymmetric adjacency matrix, CDC, 2009. 36
Problem Formulation The state vectors of all agents of MAS are described by m x = u R, i = 1,, n. i i The underlying dynamics of leader is described by x 0 = v0 v 0 = a() t = a0() t + δ () t y = x0 m x, v, δ R 0 where, y() t = x () t 0 0 is the measured output, u i are the control input, and at () is the (acceleration) input. 37
Local Control Scheme A neighbor-based feedback law ui = k aij()( t xi xj) + bi()( t xi x0 ) + vi, k > 0, j Ni A dynamic neighbor-based system to estimate v i = a0 γk aij()( t xi xj) + bi()( t xi x0),0< γ < 1, j Ni where N is the neighbor set of this agent and i = 1,, n. i Y. Hong, J. Hu,, L. Gao, Automatica,, 2006, 42: 1177-1182 1182 v 0 38
The Closed-Loop System The closed-loop system is described by x = u = k( L+ B) Imx+ kb1 x0 + v v = 1 a γ k( L + B) I x + kb1 x 0 m 0 Where is Kronecker product and 1 = (1,1,,1) T 39
One Open Problem When the adjacency matrix is asymmetric, what kind of conditions can guarantee the consensus of MAS? 40
Case A: Strongly Connected Digraph Theorem 1: For any given 0< γ < 1, select a constant k > i 2 2 (1 ) s where λ is the minimal eigenvalue of M. If the interconnection graph G keeps strongly connected or L is irreducible, then there exists some constant C satisfying lim t Moreover, if at () is known, that is, a(t)=a 0 (t) or. lim Then one has ω() t = 0. t maxξ γ γ λ ω() t Cδ δ = 0 41
Case B: Non-Strongly Connected Digraph If the interconnection graph G is not strongly connected, then its Laplacian L is reducible. L= ( l ij ) n n L 11 0 0 0 0 L 0 0 qq = Lq+ 1,1 Lq+ 1, q Lq+ 1, q+ 1 0 L L L L p1 pq p, q+ 1 pp 42
Case B: Non-Strongly Connected Digraph Theorem 2: If for i = 1,, q, b b i ( i + 1) ( s + 1) + + S > 0 then one can select some large enough k satisfying that each agent of MAS can follow the leader or the tracking errors can be estimated. 43
Some Remarks Each agent of MAS can follow the active leader if the input of the active leader is known beforehand The tracking error of MAS can be estimated if the input of the active leader is unknown beforehand 44
Numerical Simulations A MAS consists of 6 three-dimensional agents and a leader with velocity v 0. When δ = 0, assume that v 0 satisfies the Lorenz system x 0 = v0 v 01 = a( v02 v01) + δ1( t) v 02 = cv01 v01v03 v02 + δ2() t v 03 = v01v02 bv03 + δ3() t 45
Case A: Strongly Connected Digraph The adjacency matrices and parameters are given as follows: 0 1 2 1 1 1 2 0 2 1 0 1 1 3 0 0 1 2 A = 2 1 1 0 2 3 1 2 1 2 0 1 2 1 2 1 1 0 B= diag{0,5,0,0,0,0}, γ = 0.8, k = 5 46
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Case B: No-Strongly Connected Digraph The adjacency matrices and parameters are given as follows: 0 0 0 0 0 0 0 0 2 0 0 0 0 3 0 0 0 0 A = 2 1 1 0 2 3 1 2 1 2 0 1 2 1 2 1 1 0 B= diag{3,5,0,0,0,0}, γ = 0.8, k = 5 49
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CASE III: Informed-Uninformed Local Rules J. Lü, I. D. Couzin, J. Liu, S. A. Levin, Emerging collective decision from conflict in animal groups, WCICA, 2008. 52
Problem Formulation Information Source (point or direction) MAS Information Source (point or direction) The sketch map of MAS and the information sources, where the golden ball is a MAS, the red and blue balls are two different information sources, respectively. 53
Updating Local Rules Each agent attempts to balance the influence of its reciprocal action with that of its alignment. The desired travel direction of agent i at time is given by Reciprocal Action Alignment Ratio Alignment 54
A Simple Model With Information Consider the information sources, the final travel direction of agent i at time is described by Desired Unit Vector Weight Preferred Unit Direction Vector 55
Numerical Simulations (1) The influence of the proportion of informed agents on the consensus of MAS. 56
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Numerical Simulations (2) Changed information MAS Initial information The influence of the variation of the information sources on the consensus of MAS. 58
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Part III: Conclusions Consensus of MAS with nonlinear local rules Consensus of MAS with an active leader and asymmetric matrix Consensus of MAS with Information 60
URL: http://lsc.amss.ac.cn/~ljh Email: jhlu@iss.ac.cn 61