Leaderless Coordination through Time-Dependent Communication Methods

1. Leaderless coordination via bidirectional and unidirectional time-dependent communication[Luc Moreau, 2003] Andreas Goebels Paderborn, 12-11-2003

2. Outline • Introduction • Definitions and Notations • Proof • (Demonstration)

3. ti tj < Idea

4. Example

5. a b Definitions I • Heading update: with Ni(t) visible neighbours of xi • Sequence of directed graphs:

6. Definitions II • ω-limit set • „Cluster points“ of the trajectory • y єω-limit set iff {nk}kє N s.t. hnk(t) -> y as k -> ∞ • y єω-limit iff infinite sequence of values of h converge to y

7. Theorem Sequence of directed Graphs (V, G(t)) Assume that: For all t each agent is linked to each other agent across [t, ∞) There is a T such that for all t and all ka, kb є V we have that if (ka, kb) є G(t) then kb is linked to ka across [t, t+T] Then all n components of h(t) will converge to a common value as t -> ∞ with

8. Proof I mmax = limt->∞max{h1(t), h2(t), …, hn(t)} non increasing mmin = limt->∞min{h1(t), h2(t), …, hn(t)} non decreasing => mmax,mmin exist and mmin ≤mmax It remains to prove that mmin =mmax Every z in the ω–limit set of h satisfies: max{z1, …, zn} = mmax, min{z1, …, zn} = mmin (Claim 1)

9. Proof II #(z) number of components of z that equal mmax z = (z1, …, zi, …, zn ): #(z) = i <=> z1, …, zi = mmax Prove by contradiction: Assume mmin <mmax If there exist z with #(z) > 0 then there exist zc with #(zc) < #(z) • contradiction to max{z1, …, zn} = mmax (Claim 1) because #(zc) = 0

10. Proof III • choose arbitrary z: p = #(z) (first p components)(n-p) components < mmax • ω-limit set: there exists ∞ sequence of times ti s.t. h(ti) -> z as i -> ∞ • Definition of t‘i (t‘i ≥ti): t‘i first p agents (G1) remaining n-p agents (G2)

11. Proof IV • h(t‘i) converges to z‘ as i -> ∞ => z‘ єω-limit set • #(z‘) = #(z) = p • For each t‘i: T+1-tuple of graphs • finite number of tuples, infinite number of t‘is: (G(t‘i), G(t‘i+1), …, G(t‘i+T)) (G0, G1, …, GT) occurs infinitely many times sub-sequence t‘‘i of t‘i after that this tuple occurs • h(t‘‘i + r) = A(Gr-1)…A(G0)h(t‘‘i) converge to z‘‘r r є {1, 2, …, T+1}

12. Proof V z‘‘r єω-limit set => z‘‘T+1 єω-limit set G1 will not increase (taking averages) there is an edge between G1 and G2 (T) => at least one mmax heading will be decreased #(z‘‘T+1) < #(z‘) #(zc) := = #(z) (yippieh!)

13. Sketch • To show: mmin = mmax as t -> ∞ • By definition: mmin≤ mmaxand they exist • If mmin< mmax There is a z with #(z) > 0 and a #(z‘) s.t. #(z‘) < #(z) Proof: create different converging time sequences • Contradiction to max{z1, …, zn} = mmax • mmin≥ mmax • mmin= mmax

14. Leaderless Coordination

15. This is the end Thank you for your attention