Consensus and Synchronization of Multi-Agent Systems

Transcription

1 Consensus and Synchronization of Multi-Agent Systems (Combinatorial) optimization on matrix semigroups Raphaël Jungers UCLouvain L aquila, April 2016

2 Smart cities

3 Consensus of multiagent systems

4 Gossip algorithms S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):

5 Gossip algorithms S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):

6 Gossip algorithms S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):

7 Gossip algorithms 0.35 ( )/2 = S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):

8 Gossip algorithms S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):

9 Gossip algorithms

10 Outline Consensus Primitivity of semigroups of matrices Open problem: Cerny s conjecture Discussion

11 Outline Consensus Primitivity of semigroups of matrices Open problem: Cerny s conjecture Discussion

12 Classical assumptions: A folklore theorem Theorem: (Tsitsiklis 86, Hendrickx et al. 2005) Under the assumptions above, and if there exists a bound B such that for all t, A(t)+ +A(t+B) >0, then consensus is achieved 0.35 Other variants where symmetry, delays, Or other nonidealities/properties are taken into account ( )/2 =

13 Consensus of multi-agent systems We are given a set of stochastic matrices, representing several connectivity topologies Problem 1: Do products of these matrices always converge to consensus? Problem 2: Is there a product which converges to consensus?

14 Properties of stochastic matrices Property 1 with Any consensus state is an equilibrium Property 2 If then is an invariant polyhedron

15 Faces of a polyhedron Proposition [Jadbabaie 03] : After projection along the (1,1,,1) vector, convergence to consensus is equivalent to convergence to zero

16 Faces of a polyhedron Proposition [Jadbabaie 03] : After projection along the (1,1,,1) vector, convergence to consensus is equivalent to convergence to zero Do all the products converge to zero? Is there one product converging to zero? These are classical switching systems problems! Theorem (Lagarias and Wang, 92): If a set of matrices has an invariant polytope, then the image of any open face is included in an open face All the points of an open face are essentially the same We can represent the whole dynamics on a finite graph

17 The graph of faces Nodes: One for int(p), and one for each open face Edges: From F to G if there is a matrix A which maps F into G Example: matrix set

18 The graph of faces (2) Nodes: One for int(p), and one for each open face Edges: From F to G if there is a matrix A which maps F into G

19 Consensus is decidable Theorem: Stability (i.e. all products lead to consensus) iff the graph is acyclic Theorem: Stabilizability (i.e. there exists a converging sequence) iff int(p) is reachable from every vertex

20 Consensus is decidable Complexity: there are matrices nodes and m

21 An explicit upperbound on the length of the period of a periodic contracting product Further work The face lattice is actually a ranked poset A cyclic product avoiding consensus must remain at the boundary of the polytope it is an antichain in the poset The face lattice of the consensus polytope has the Sperner property : (but some polytopes do not) The longest antichain is simply a particular level of the poset

22 Conclusions

23 Perspective (sensor) networks [Furstenberg Kesten, 1960] [Gurvits, 1995] [Kozyakin, 1990] Wireless control Bisimulation design [Rota, Strang, 1960] [Blondel Tsitsiklis, 98+] [Johansson Rantzer 98] consensus problems Social/big data control 60s 70s 90s 2000s now Mathematical properties TCS inspired Negative Complexity results Lyapunov/LMI Techniques (S-procedure) CPS applic. Ad hoc techniques

24 Open problems

25 Questions? More on: