Adaptation and Synchronization over a Network: stabilization without a reference model

Transcription

1 Adapaion and Synchronizaion over a Nework: sabilizaion wihou a reference model Travis E. Gibson (gibson@mi.edu) Harvard Medical School Deparmen of Pahology, Brigham and Women s Hospial 55 h Conference on Decision and Conrol December 12-14, 216

2 Problem Saemen Adapive Sysems Nework Consensus Sysem Error Generaion Parameer Adapaion Consensus Error How do we achieve consensus and learning wihou a reference model? Can synchronous inpus enhance adapaion? 2/22

3 Inroducion and Ouline Synchronizaion can hur learning Example of wo unsable sysems (builds on Narendra s recen work) Synchronizaion and Learning in Neworks Resuls Using Graph Theory Concree connecions o classic adapive conrol (if ime allows) 3/22

4 Synchronizaion vs. Learning: Tradeoffs 4/22

5 Two sysems sabilizing each oher Consider wo unsable sysems [Narendra and Harshangi (214)] Updae laws wih e = x 1 x 2. Σ 1 : ẋ 1 () = a 1 ()x 1 ()+u 1 () Σ 2 : ẋ 2 () = a 2 ()x 2 ()+u 2 () ȧ 1 () = x 1 ()e() a 1 () > ȧ 2 () = x 2 ()e() a 2 () > No Inpu u 1 = u 2 = x1,x x1 x2 a1,a a1 a /22

6 Synchronizaion Hurs Learning Synchronous Inpu u 1 = e u 2 = +e e = x 1 x 2 x1,x x1 1.8 a1 x2 a a1,a2 Desynchronous Inpu u 1 = +e u 2 = e x1,x x1 x2 a1,a a1 a2 6/22

7 Sabiliy Resuls for Synchronous and Desynchronous Inpus Σ 1 : ẋ 1 () = a 1 ()x 1 ()+u 1 () Σ 2 : ẋ 2 () = a 2 ()x 2 ()+u 2 () ȧ 1 () = x 1 ()e() ȧ 2 () = x 2 ()e() Theorem: Synchronous Inpus The dynamics above wih synchronous inpus have a se of iniial condiions wih non-zero measure for which x 1 and x 2 end o infiniy while e L 2 L and e as. Furhermore, his se of iniial condiions ha are unsable is also unbounded. Theorem: Desynchronous Inpus The dynamics above wih desynchronous inpus are sable for all a 1 () a 2 () 7/22

8 Synchronizaion and learning in neworks 8/22

9 Graph Noaion and Consensus Graph : G(V,E) Verex Se : V = {v 1,v 2,...,v n } Edge Se : (v i,v j ) E V V v 1 v 4 Adjacency Marix : [A] ij = v 2 v 3 { 1 if (v j,v i ) E oherwise In-degree Laplacian : L(G) = D(G) A(G) In-degree of Node i : [D] ii Consensus Problem Σ i : ẋ i = Using Graph Noaion j N in(i) (x i x j ) Σ : ẋ = Lx, x = [x 1, x 2,..., x n ] T 9/22

10 Review: Sufficien Condiion for Consensus Σ : ẋ = Lx Theorem: (Olfai-Saber and Murray, 24) For he dynamics above wih G srongly conneced i follows ha lim x() = ζ1, for some finie ζ R. If G is also balanced hen ζ = 1 n n i=1 x i(), i.e. average consensus is reached. srongly conneced here is a walk beween any wo verices in he nework. balanced if he in-degree of each node is equal o is ou-degree. Any srongly conneced digraph can be balanced (Marshall and Olkin, 1968). Disribued algorihms exis o balance a digraph (Dominguez-Garcia and Hadjicosis, 213). 1/22

11 Reurn o Adapive Sabilizaion Consider a se of n possibly unsable sysems Σ i ẋ i () = a i x i +θ i ()x Updae Law θ i = x i j N in(i) (x i x j ) Compac form Σ : ẋ = Ax+diag(θ)x [A] ii = a i θ = x Lx θ = [θ 1, θ 2,..., θ n ] T 11/22

12 Sabilizaion over Srongly Conneced Graphs Theorem ẋ = Ax+diag(θ)x θ = x Lx For he dynamics above wih G a srongly conneced digraph, and all he a i +θ i() no idenical i follows ha lim x() =. G is srongly conneced = λ i(l) closed righ-half plane of C. L is Mezler = Diagonal D > s.. L T D DL. Non-increasing funcion n n [D] iiθ i() = x T DLxd+ [D] iiθ i() i=1 = 1 2 i=1 x T (DL+L T D)xd+ n [D] iiθ i(). L1 = = 1 T (DL+L T D)1 =. κ λ 2(DL+L T D) > = i [D]iiθi() κ x T xd+ i θi() when x / span(1). i=1 12/22

13 Sabilizaion over Conneced Graphs Any conneced digraph can be pariioned ino disjoin subses called Srongly Conneced Componens (SCCs) where each subses is a maximal srongly conneced subgraph Graph G Condensed Graph G SCC roo condensed nodes in red For any conneced G he corresponding G SCC is a Direced Acyclic Graph (DAG) Every conneced DAG conains a roo node (no unique). 13/22

14 Sabilizaion over Conneced Graphs Con. Theorem ẋ = Ax+diag(θ)x θ = x Lx For he dynamics above wih he adapaion occurring over a conneced graph G such ha a roo can be chosen in G SCC ha is a condensed node, hen lim x() = The roo is a srongly conneced subgraph (hus sabilizes iself) All informaion flowing over G decimaes from a sable SCC. Sabiliy of each SCC hen follows from he hierarchical srucure of he DAG. G SCC 14/22

15 Sabilizaion over Conneced Graphs: Example of Necessiy This node can never sabilize if iniially unsable 15/22

16 Consensus and Leaning Bring everyhing ogeher as a layered archiecure The communicaion graph is G G a is he adapaion graph and is consrained by he communicaion in G G s is he synchronizaion graph and is similarly consrained G G a G s (Doyle and Csee, 211), (Alderson and Doyle, 21) 16/22

17 Adapive Sabilizaion over a Nework Σ : ẋ = Ax+diag(θ)x θ = x L a x G a = xi.5 θi /22

18 Adapive Sabilizaion and Desynchronous Inpu Σ : ẋ = Ax+L s x+diag(θ)x θ = Γx L a x G a = G s = 1 2 xi θi /22

19 Summary Borrowing from Narendra, Murray, and My Thesis, we have Found ha synchronizaion can hur learning. As always conex is imporan Wha abou oher learning paradigms, i.e. Jadbabaie s work or he broader Machine Learning lieraure Bibliography Alderson, D. L and J. C Doyle. 21. Conrasing views of complexiy and heir implicaions for nework-cenric infrasrucures, Sysems, Man and Cyberneics, Par A: Sysems and Humans, IEEE Transacions on 4, no. 4, Dominguez-Garcia, A. D. and C. N. Hadjicosis Disribued marix scaling and applicaion o average consensus in direced graphs, Auomaic Conrol, IEEE Transacions on 58, no. 3, Doyle, J. C. and M. Csee Archiecure, consrains, and behavior, Proceedings of he Naional Academy of Sciences 18, no. Supplemen 3, Marshall, A. W and I. Olkin Scaling of marices o achieve specified row and column sums, Numerische Mahemaik 12, no. 1, Narendra, K. S. and P. Harshangi Unsable sysems sabilizing each oher hrough adapaion, American Conrol Conference, pp Olfai-Saber, R. and R. M Murray. 24. Consensus problems in neworks of agens wih swiching opology and ime-delays, Auomaic Conrol, IEEE Transacions on 49, no. 9, /22

20 Closed-loop Reference Model (CRM) feedback gain l r Reference Model Plan x m x - e θ() γ learning rae 2/22

21 How does CRM Help? Classic Open-loop Reference Model (ORM) Adapive (l = ) The reference model does no adjus o any ouside facors reference: x m plan: x Closed-loop Reference Model (CRM) Adapive The reference model adjuss o rapidly reduce he model following error e = x x m reference: x m plan: x 21/22

22 γ = 1 l = γ = 1 l = 1 CRM Simulaion Examples Sae Sae xm 1 θ xp.5 k Parameer x o m 1 θ xm.5 k 1.5 xp Parameer γ = 1 l = 1 Sae x o m 1 θ xm.5 k 1.5 xp How do you choose γ and l? Parameer 22/22