Leader selection in consensus networks

Transcription

1 Leader selection in consensus networks Mihailo Jovanović mihailo joint work with: Makan Fardad Fu Lin 2013 Allerton Conference

2 Consensus with stochastic disturbances 1 RELATIVE INFORMATION EXCHANGE WITH NEIGHBORS simplest distributed averaging algorithm ψ i (t) = ( ) ψ i (t) ψ j (t) j N i + w i (t) white noise

3 UNFORCED NETWORK DYNAMICS diffusion on a graph with Laplacian L = L T ψ 1 (t).. ψ n (t) = L L L e-values of L: 0 = λ 1 λ 2 λ n ψ 1 (t).. ψ n (t) Convergence 2 connected network λ 2 (L) > 0 ψ i (t) convergence to the average t ψ(t) := 1 n ψi (t)

4 NETWORK AVERAGE undergoes random walk raft Consensus with stochastic disturbances 3 λ 2 (L) > 0 { each ψi (t) fluctuates around ψ(t) deviation from average: ψi (t) := ψ i (t) ψ(t) steady-state variance: lim t E ( ψ2 i (t)) = i λ i (L)

5 TWO GROUPS OF NODES raft Followers: relative information exchange ψ i = j N i ( ψi ψ j ) + wi Leaders: perfectly follow desired trajectory Networks with leaders 4 ψ i 0 ψ i 0

6 [ ] ψ l (t) ψ f (t) raft Network performance [ ] [ ] 0 0 ψl (t) = x vector with components L x L x principal submatrix of L ψ f (t) + [ 0 w(t) { xi = 1 node i is a leader x i = 0 otherwise remove rows & columns of L corresponding to leaders ] 5 VARIANCE OF FOLLOWERS determined by { network topology locations of leaders

7 Examples 6 x = [ ] T remove first and last rows & columns from L

8 PATH GRAPH raft x = [ ] T L = x i {0, 1}; 1 leader, 0 follower L x =

9 SELECT N l LEADERS TO: raft minimize variance of followers minimize x J f (x) = trace (L 1 x ) subject to x i {0, 1}, i = 1,..., n 1 T x = N l Leader selection problem 8

10 GREEDY ALGORITHMS WITH APPROXIMATIONS Patterson and Bamieh 10 SUBMODULAR OPTIMIZATION WITH PERFORMANCE GUARANTEES Clark, Bushnell, and Poovendran 11, 12, 13 SDP FOR SENSOR SELECTION PROBLEM Joshi and Boyd 09 Related work 9 CONTROLLABILITY OF LEADER-FOLLOWER NETWORKS Tanner 04 Liu, Chu, Wang, and Xie 08 Rahmani, Ji, Mesbahi, and Egerstedt 09 Kawashima and Egerstedt 12

11 SELECTION OF NOISE-CORRUPTED LEADERS common in applications provides insight into selection of noise-free leaders easier to solve EFFICIENT ALGORITHMS FOR BOUNDS ON GLOBAL OPTIMAL VALUE This talk 10 convex relaxations: lower bounds greedy algorithms: upper bounds EXAMPLES

12 Followers: relative information exchange ψ i = ( ψi ψ j ) + wi, i {2,..., 9} Networks with noise-corrupted leaders 11 j N i Leaders: can also measure their own state ψ i = ( ) ψi ψ j + wi α ψ i, i {1, 10} j N i

13 NETWORK DYNAMICS 12 diagonally strengthened Laplacian ψ 1 (t) L. = (L + α diag (x)) ψ n (t) L ψ 1 (t). ψ n (t) + w 1 (t). w n (t) L = raft , diag (x) = x i {0, 1}; 1 leader, 0 follower

14 SELECT N l LEADERS TO: raft minimize network variance minimize x J(x) = trace ( (L + α diag (x)) 1) subject to x i {0, 1}, i = 1,..., n 1 T x = N l Noise-corrupted formulation 13 NOISE-FREE FORMULATION leaders: followers: [ Ll + α I L T 0 L 0 L x ] 1 α [ L 1 x ]

15 FEATURES raft minimize x trace ( (L + α diag (x)) 1) subject to x i {0, 1}, i = 1,..., n convex objective function 1 T x = N l Algorithms 14 nonconvex Boolean constraints APPROACH convex relaxations: lower bounds greedy algorithms: upper bounds

16 minimize x trace ( (L + α diag (x)) 1) subject to 0 x i 1, i = 1,..., n 1 T x = N l COMPLEXITY OF COMPUTING LOWER BOUND standard SDP solvers: O ( n 4) Convex relaxation 15 customized interior point method: O ( n 3)

17 Select one-leader-at-a-time ( Ls + α e i e T i ) 1 followed by swap between a leader i and a follower j COMPLEXITY ( Ls α e i e T i + α e j e T j ) 1 Greedy algorithm 16 rank-1 update: O(n 2 ) per leader single matrix inversion: O(n 3 ) rank-2 update: O(n 2 ) per swap

18 Example: a random network with 100 nodes 17

19 raft performance bounds: upper bounds lower bounds number of leaders performance gap: number of leaders 18

20 variance raft greedy algorithm degree heuristics Degree heuristics vs. greedy algorithm number of leaders

21 N l = 5 J = 27.8 N l = 5 J = N l = 40 J = 15.0 N l = 40 J = 9.5

22 partition graph and spread leaders: boundary nodes with low-degree: Few vs many leaders 21

23 minimize x Recap trace ( (L + α diag (x)) 1) subject to x i {0, 1}, i = 1,..., n 1 T x = N l CONVEX RELAXATION: LOWER BOUND standard SDP solvers: O ( n 4) customized interior point method: O ( n 3) 22 GREEDY ALGORITHM: UPPER BOUND without exploiting structure: O ( n 4 N l ) low rank updates: O ( n 3) PAPER/SOFTWARE arxiv: mihailo/software/leaders/

24 TEAM: raft Makan Fardad (Syracuse University) Fu Lin (Argonne National Lab) Acknowledgments 23 SUPPORT: NSF CAREER Award CMMI NSF Award CMMI