⇤ Johann

Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands. ⇤⇤ Department of Electrical and Electronic Engineering, College of Science and Engineering, Ritsumeikan University, Japan

MTNS Melbourne, 2012

Harry L. Trentelman and K. Takaba (Johann Bernoulli Institute for Uncertain Linear MTNS Melbourne, 2012 Robust Synchronization of Mathematics and Computer Science, University of 20 Multi-Agent Systems 1 / Gron

Outline

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Synchronization Robust synchronization Computation of robustly synchronizing protocols Guaranteed robust synchronization radius Future research

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Harry L. Trentelman and K. Takaba (Johann Bernoulli Institute for Uncertain Linear MTNS Melbourne, 2012 Robust Synchronization of Mathematics and Computer Science, University of 20 Multi-Agent Systems 2 / Gron

Synchronization of multi-agent systems

Agent dynamics

Multi-agent networks with p agents, undirected network graph, Laplacian L. Identical nominal dynamics of the agents: ˙ xi = Axi + Bui , yi = Cxi , i = 1, 2 . . . , p (A, B) is stabilizable, (C , A) is detectable. State xi 2 Rn , input ui 2 Rm , output yi 2 Rq Neighbouring set of agent i is Ni . Information of agent i about its neighbours is Âj2Ni (yi yj )

Harry L. Trentelman and K. Takaba (Johann Bernoulli Institute for Uncertain Linear MTNS Melbourne, 2012 Robust Synchronization of Mathematics and Computer Science, University of 20 Multi-Agent Systems 3 / Gron

Synchronization of multi-agent systems

Dynamic protocol

The synchronization problem is the problem of ﬁnding a protocol that makes the network synchronized. We consider dynamic protocols of the form ˙ wi = Awi + BF j2Ni

Â (wi

wj ) + G (

j2Ni

Â (yi

yj )

Cwi ), ui = Fwi .

Structure of the protocol: combination of observer for the ith relative state Âj2Ni (xi xj ) and static feedback of the estimate wi of this relative state. Note: design parameters are the gains F and G .

Harry L. Trentelman and K. Takaba (Johann Bernoulli Institute for Uncertain Linear MTNS Melbourne, 2012 Robust Synchronization of Mathematics and Computer Science, University of 20 Multi-Agent Systems 4 / Gron

Synchronization of multi-agent systems

Network dynamics

Interconnecting the agents using this protocol yields the closed loop dynamics of the overall network. Denote x = col(x1 , x2 , . . . , xp ), w = col(w1 , w2 , . . . , wp ). Network dynamics: ✓ ◆ ✓ ◆✓ ◆ ˙ x I ⌦A I ⌦ BF x = ˙ w L ⌦ GC I ⌦ (A GC ) + (L ⌦ BF ) w

Deﬁnition

The network is said to be synchronized by the dynamic protocol if for all i, j = 1, 2 . . . , p we have xi (t) xj (t) ! 0 and wi (t) wj (t) ! 0 as t ! •.

Harry L. Trentelman and K. Takaba (Johann Bernoulli Institute for Uncertain Linear MTNS Melbourne, 2012 Robust Synchronization of Mathematics and Computer Science, University of 20 Multi-Agent Systems 5 / Gron

Synchronization of multi-agent systems

Laplacian eigenvalues

Network graph connected , L has rank p 1. In that case the eigenvalue 0 has multiplicity one. Remaining p 1 eigenvalues: 0 < l2 l3 . . . lp .

Theorem

The protocol synchronizes the network if and only if the single linear system ˙ x = Ax + Bu, y = Cx is internally stabilized by all p ˙ w = Aw + Bu + G (y 1 feedback controllers Cw ), u = li Fw , i = 2, 3, . . . , p.

This holds if and only if A GC and A + li BF (i = 2, 3, . . . , p) are Hurwitz. Such F and G exist if and only if (C , A) is detectable and (A, B) is stabilizable.

Harry L. Trentelman and K. Takaba (Johann Bernoulli Institute for Uncertain Linear MTNS Melbourne, 2012 Robust Synchronization of Mathematics and Computer Science, University of 20 Multi-Agent Systems 6 / Gron

Outline

1

Synchronization Robust synchronization Computation of robustly synchronizing protocols Guaranteed robust...