Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif – sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS - 1987 BOIDSBOID neighborhood Flocking Rules separation alignment cohesion Demetri Terzopoulos Flocking Rules separation alignment cohesion Motivated by simulation results reported in V i c s e k et al . si m u l a t e d a ° o c k of n ag e n t s f p a r t i c l e s g al l mo v i n g in t h e pl a n e a t t h e sa m e sp ee d s, bu t wi t h di ® e re n t he a di n g s µ 1 ; µ 2 ; : : : ; µ n . s = sp e e d s µ i µi = he a di n g Each agent’s heading is updated at the same time as the rest using a local rule based on the average of its own current heading plus the headings of its “neighbors.” Vicsek’s simulations demonstrated that these nearest neighbor rules can cause all agents to eventually move in the same direction despite 1. the absence of a leader and/or centralized coordination 2. the fact that each agent’s set of neighbors changes with time. Vicsek Model ri neighbors of agent i agent i Each agent is a neighbor of itself Each agent has its own sensing radius ri So neighbor relations are not symmetric HEADING UPDATE EQUATIONS s = sp e e d s µ i µi = he a di n g 0 N i(t) = set of indices of agent i s neighbors at time t ni(t) = number of indices in N i(t) Average at time t of headings of neighbors of agent i. Another rule: Vicsek Flocking Problem: Under what conditions do all n headings converge to a common value? Convex combination {Requires collaboration!} Neighbor Graph N of Index Sets N 1, N 2 ,…., N n G = all directed graphs with vertex set V = {1,2,…,n} N = graph in G with an arc from j to i whenever j 2 N i, i 2 {1,2,…,n} j is a ne i g h b or of i i j A self-arced graph = any graph G with self-arcs at all vertices 1 (1,2) Ne i g h b or gr a p h s = se lf -a r c e d gr a ph s 2 3 4 5 7 6 State Space Model ‘ ’ Adjacency Matrix AG of a graph G 2 G: An n£ n matrix of 0 s and 1 s with aij = 1 whenever there is an arc in G from i to j. _ 4 (1,2) 1 In-degree = 4, out-degree = 1 2 3 4 In-degree of vertex i = number of arcs entering vertex i 5 7 6 Out-degree of vertex i = number of arcs leaving vertex i State Space Model ‘ ’ Adjacency Matrix AG of a graph G 2 G: An n£ n matrix of 0 s and 1 s with aij = 1 whenever there is an arc in G from i to j. Flocking Matrix FN of a neighbor graph N 2 G: bijection where f g D N = di a go n a l d1 ; d2 ; : : : ; dn an d Xn ni = di = in - d e g r e e of v er t e x i = aj i j = 1 Update Eqns: State Model: Vicsek flocking problem: Under what conditions do all n headings converge to a common value? A switched linear system No common quadratic Lyapunov function exists B u t t h e no n - n e ga t i v e f un c t i o n f g ¡ f g V ( µ ) = ma x µ i mi n µ i i i Verify this! is at least non-increasing along trajectories But it takes much more to conclude that V ! 0 µ ( t + 1) = FN ( t ) µ ( t ) Vicsek flocking problem: Under what conditions do all n headings converge to a common value? µ ( t + 1) = FN ( t ) µ ( t ) Problem reduces to determining conditions on the sequence N(0), N(1), ... under which where For if this is so, then where and so {Right} Stochastic Matrices Sn£ n= stochastic if 1. it ha s on l y no n - n e g a t i v e en t r i e s 2. its row sums all equal 1 Stochastic matrices closed under multiplication – flocking matrices are not Flocking matrices are stochastic Therefore it is sufficient to determine conditions on an infinite sequence of n£ n stochastic matrices S1, S2, .... so that This is a well studied problem in the theory of non-homogeneous Markov chains If S is a compact set of n£ n stochastic matrices whose members each have at least one positive column, then for each sequence of matrices S1, S2, … from S, and this limit is approached exponentially fast. Why is this true? Induced Norms and Semi-Norms n£ n n£ n For M 2 R and p > 0, let ||M||p denote the induced matrix p norm on R . We will be interested primarily in the cases p = 1, 2, 1 : For any such p, define 1. Nonnegative: |M|p ¸ 0 2. Homogeneous: |rM|p = r|M|p 3. Triangle inequality: |M + M | · |M | + |M | 1 2 p 1 p 2 p verify! These three properties mean that |¢ |p is a semi-norm {If |M|p = 0 were to imply M = 0, then |¢|p would be a norm.} |M|p = 0 ; M = 0 Additional Properties of · · 1. |M|p 1 if ||M||p 1 Because |M|p · ||M||p M is semi - contractive in the p semi-norm if |M|p < 1 n£ n 2. Sub-multiplicative: Suppose M is a subset of R such that M1 = 1 for all M 2 M . Then Proof : Let c0 ,c1 and c2 denote values of c which minimize ||M2M1 - 1c||p, ||M1-1c||p, and ||M2-1c||p respectively. 1 = M21 Suppose M is a subset of Rn£ n such that M1 = 1 for all M 2 M . Let p be fixed and let C be a compact set of semi - contractive matrices in M . Let Then for each infinite sequence of matrices M1 , M2, ... in C, the matrix product converges as i ! 1 as fast as ¸ i converges to zero, to a rank one matrix of the form 1c. Proof: See board We want to use this fact to prove that: If S is a compact set of n£ n stochastic matrices whose members each have at least one positive column, then for each sequence of matrices S1, S2, … from S, and this limit is approached exponentially fast. To do this , it is enough to show that: A stochastic matrix S is semi-contractive in the semi-norm | ¢ |1 if S has a positive column. Any stochastic matrix S can be written as S = 1c + T where c is the largest row vector for which S - 1c is nonnegative and T = S – 1c T1 = S1 – 1c1 = (1 - c1)1 so all row sums of T = (1 - c1) ¸ 0 because T ¸ 0 Moreover c ≠ 0 if and only if S has a positive column. verify! Therefore (1 – c1) < 1 if and only if S has a positive column 0 jSj1 = m in jjS¡ 1d jj1 · jjS¡ 1cjj1 = jjT jj1 = ( 1¡ c1) d A stochastic matrix S is semi-contractive in the semi-norm | ¢ |1 if S has a positive column. Transitioning from Matrices to Graphs For a nonnegative matrix Mn£ n, °(M) is that graph whose adjacency matrix is the transpose of the matrix which results when each non-zero entries in M is replaced by a 1. In other words, for a nonnegative matrix M, °(M) is that graph which has an arc (i, j) from i to j whenever mj,i ≠ 0. Transitioning from Matrices to Graphs For a nonnegative matrix Mn£ n, °(M) is that graph whose adjacency matrix is the transpose of the matrix which results when each non-zero entries in M is replaced by a 1. 0 °( FN ) = °( A N ) = N A graph is strongly rooted if at least one vertex is adjacent to every vertex in the graph strongly rooted graph Motivation for strongly rooted: For any nonnegative matrix M, °(M) has an arc (i, j) whenever mj,i ≠ 0. °(M) is strongly rooted , M has a positive column Transitioning from Matrices to Graphs If S is a compact set of n£ n stochastic matrices whose members each have at least one positive column, then for each sequence of matrices S1, S2, … from S, and this limit is approached exponentially fast. If S is a compact set of n£ n stochastic matrices whose members each have a strongly rooted graph, then for each sequence of matrices S1, S2, … from S, and this limit is approached exponentially fast. Transitioning from Matrices to Graphs When does . T Tq T2 T1 If then Thus establishing convergence to 1c of an infinite product of stochastic matrices boils down to determining when the graph of a product of stochastic matrices is strongly rooted. Transitioning from Matrices to Graphs As before G = set of all directed graphs with vertex set {1,2,....,n}. By the composition of graph G2 2 G with graph G1 2 G , written G2 ± G1, is that directed graph in G which has an arc (i, j) from i to j whenever there is an integer k such that (i, k) is an arc in G1 and (k, j) is an arc in G2.