Adaptation and Synchronization over a Network : Asymptotic Error - PowerPoint PPT Presentation

Adaptation and Synchronization over a Network : Asymptotic Error Convergence and Pinning Travis E. Gibson teg807@harvard.edu Harvard Medical School 55 th Conference on Decision and Control December 12-14, 2016

Outline • Graph Notation, Balancing • Problem Statement • Undirected Graphs • Directed Graphs 2 / 16

Graph Notation and Consensus 1 4 Graph : G ( V , E ) Vertex Set : V = { 1 , 2 , . . . , n } Edge Set : ( i, j ) ∈ E ⊂ V × V 2 3 � 1 if ( j, i ) ∈ E Adjacency Matrix : [ A ] ij = 0 otherwise In-degree Laplacian : L ( G ) = D ( G ) − A ( G ) In-degree of Node i : [ D ] ii Consensus Problem � Σ i : x i = − ˙ ( x i − x j ) j ∈N in ( i ) Using Graph Notation x = [ x 1 , x 2 , . . . , x n ] T ˙ Σ : x = −L x , 3 / 16

Graph Balancing (Output Balancing) Strongly Connected (SC) there is a walk between any two vertices in G . • λ 1 ( L ( G )) = 0 > λ 2 ( L ( G )) ≥ · · · ≥ λ n ( L ( G )) Balanced in-degree = out-degree. unbalanced nodes ( � in � = � out ) 1 4 1 4 1 1 1 2 1 1 1 1 2 3 2 3 1 1 • Output Balancing : diagonal matrix D ≻ 0 s.t. weighted graph ˜ G ( V , E , D A ) is balanced. • Example: D = diag ([1 , 1 , 2 , 1] T ) • G SC = ⇒ L ( G ) 1 = 0 (. . . holds for any G not just SC G ) ⇒ 1 T ˜ • ˜ L ( ˜ G ) = 0 T G SC & balanced = ⇒ 1 T D L ( G ) = 0 T G strongly connected ∃ balancing D = 4 / 16

Incomplete Literature on Balancing and Consensus [Tsitsiklis, Phd Thesis ’84] [Jadbabaie, Lin & Morse, TAC ’03] • Focused on discrete time consensus [Olfati-Saber & Murray TAC ’04] • Focussed on balanced graphs 3 Papers by Chai Wah Wu all in 2005 (IBM Research NY) • Focussed on directed graphs that were not balanced • Synchronization of nonlinear systems • Used the graph balancing matrix in his proofs [Makhdoumi & Ozdaglar CDC ’15] • A graph balancing itself through neighbor consensus (distributed subgradient method) There are lots and lots of papers on this topic, however the existence of graph balancing matrices has not been fully exploited in the adaptive consensus literature. 5 / 16

Problem at Hand Scalar dynamics on a graph G ( V , E ) Communication Topology target node x i ( t ) = a i ˙ x i ( t ) + u i ( t ) , i ∈ V x m ���� unknown Control u i ( t ) = ˆ k i ( t ) x i ( t ) + ˆ r i ( t ) ���� ���� adaptive adaptive G Target set T ⊂ V , nodes that Scalar reference model receive information from the x m ( t ) = a m x m ( t ) + r ˙ reference model. Goal : Design adaptive laws so that x i → x m while only communicating over G • But wait ... don’t we need to add consensus and pinning to the input � � u i = ˆ k i ( t ) x i + ˆ r i ( t ) − ( x i − x j ) − ( x i − x m ) ���� ���� j ∈N in ( i ) i ∈T adaptive adaptive � �� � � �� � pinning consensus • ... NOT necessarily 6 / 16

Compact Representation of Dynamics target node x m Error e i = x i − x m ˜ e = A m ˙ e + K x + ˜ r ���� ���� diag (˜ diag ( 1 a m ) k ) G Target set T ⊂ V Locally computable errors � � Model local error = ( x i − x j ) − ( x i − x m ) x i = a i x i + u i ˙ i ∈T j ∈N in ( i ) � �� � u i = ˆ � �� � k i ( t ) x i + ˆ r i ( t ) pinning consensus x m = a m x m + r ˙ e β = ( L + M ) e � �� � =: B Global Parameters � 1 if i ∈ T x = [ x 1 , x 2 , . . . , x n ] T [ M ] ii = 0 otherwise ˆ k = [ˆ k 1 , ˆ k 2 , . . . ˆ k n ] T , ˆ r . . . ˜ k = ˆ k − k ∗ , ˜ r . . . NOTE : B is full rank & if D is a balancing of the graph, then B T D + D B ≻ 0 x m = 1 x m 7 / 16

Symmetric Graph G Error Dynamics & Local Error Target set T + Target matrix M e = A m e + ˜ ˙ Kx + ˜ r � 1 if i ∈ T [ M ] ii = e β = ( L + M ) e 0 otherwise � �� � =: B Adaptive Updates ˙ ˆ k = − diag ( x ) e β ˙ ˆ r = − e β Theorem For the error dynamics and update laws above, if G is a strongly connected symmetric graph and there is at least one target node , then all signals are uniformly bounded and e ( t ) → 0 as t → ∞ V ( e β , ˜ β B − 1 e β + ˜ k T ˜ r ) = e T r T ˜ k , ˜ k + ˜ r . . . V = 2 a m e T ˙ β B − 1 e β . NOTE: a m < 0 8 / 16

General Graphs • For a generic graph L is not symmetric = ⇒ analysis harder. • Analysis will now require the input to use consensus & pinning u = diag (ˆ k ( t )) x + ˆ r ( t ) + c B e , B = L + M , c < 0 ���� ���� consensus pinning • Most of the literature in this area has focussed on symmetric graph • Lots of work by Mario di Bernardo is related to this work A. Das and Frank L. Lewis, Automatica 46 (2010), no. 12, • An a-priori bound on the regressor is assumed ( x is a-priori bounded) • Does not exploit the fact that D is not only s.t. L T D + D L ≻ 0 but is also a graph balancing as well • Error e converges to compact set proportional to � k ∗ � • Adaptive laws use D explicitly (thus all agents must know graph structure or learn D ). • Main contribution : a different way of proving stability • e ( t ) → 0 • Projection is used ([Das and Lewis] use sigma modification) • We exploit the graph balancing condition • If time remains I will show the [Das and Lewis] proof. 9 / 16

General Graphs, Theorem Statement Slightly easier problem for space reasons u = diag (ˆ k ( t )) x + + c B e r ���� known e = A m e + ˜ ˙ Kx + c B e ˙ k = proj ∞ ( − diag ( x ) B e , ˆ ˆ k , k max ) Theorem For the error dynamics and update laws above, with G a strongly connected graph with there being at least one target node , and with c sufficiently negative , then all signals are uniformly bounded and e ( t ) → 0 as t → ∞ V = e T D e D is a graph balancing 1 T D L = 0 T Kx + x T ˜ K D e + c e T (( L T + M ) D + D ( L + M )) e V = 2 a m e T D e + e T D ˜ ˙ Kx + x T ˜ K D e + c e T (( L T + M ) D + D ( L + M )) e V = 2 a m e T D e + e T D ˜ ˙ � �� � NOTE: D is a graph balancing 10 / 16

General Graphs, Proof continued V = e T D e D is a graph balancing 1 T D L = 0 T Kx + x T ˜ K D e + c e T (( L T + M ) D + D ( L + M )) e V = 2 a m e T D e + e T D ˜ ˙ . � �� � NOTE: D is a graph balancing Recall e = x − 1 x m ) T D L = x T D L ( x − 1 x m � �� � e c e T (( L T + M ) D + D ( L + M )) e � �� � separate into two halves   ˜ 2 a m D + c 2 Q 1 D K � e � ���� ���� � x T �   bdd by Proj ˙ e T ≻ 0 V =   2 ( L T D + D L ) ˜ c D x  K  ���� bdd by Proj • For c sufficiently negative matrix becomes ≺ 0 . • Analyze in two scenarios x / ∈ R n 1 \ 0 and x ∈ R n 1 \ 0 • x can never be in R n 1 \ 0 for any finite amount of time. 11 / 16

Compare and Contrast with [Das and Lewis] Our Result ˙ k = proj ∞ ( − diag ( x ) B e , ˆ ˆ k , k max )   ˜ 2 a m D + c 2 Q 1 D K ���� � e � ���� � x T �   ˙ bdd by Proj ≻ 0 e T V =   2 ( L T D + D L ) ˜ c x D  K  ���� bdd by Proj [Das and Lewis] ˙ k = − D diag ( x ) B e − σ ˆ ˆ k ���� Sigma mod k T � � cq 11 � � e � − σ � � e � q 12 � � q 4 ˙ ˜ e T V = + ˜ ˜ q 21 − σ k k • Different structure, x replaced by ˜ k (given by sigma mod) • For c negative, signals converges to a compact set 12 / 16

General Graphs the Complete Problem Now r is unknown u = diag (ˆ k ( t )) x + ˆ r ( t ) + c B e ���� adaptive e = A m e + ˜ ˙ Kx + ˜ r + c B e ˙ k = proj ∞ ( − diag ( x ) B e , ˆ ˆ k , k max ) ˙ ˆ r = −B e − L ˆ r ���� sharing estimates Theorem For the error dynamics and update laws above, with G a strongly connected graph with there being at least one target node , and with c sufficiently negative , then all signals are uniformly bounded and e ( t ) → 0 as t → ∞ V = e T D e + ˜ r T D ˜ r . 13 / 16

Simulations Graph is Directed Cycle of 10 nodes u = diag (ˆ u = diag (ˆ k ( t )) x + ˆ r ( t ) − 5 B e k ( t )) x + ˆ r ( t ) 5 5 e i 0 e i 0 -5 -5 0 10 20 30 0 10 20 30 1 5 k i 0 k i 0 ˆ ˆ -1 -5 0 10 20 30 0 10 20 30 0.5 2 0 0 r i r i ˆ ˆ -0.5 -2 0 10 20 30 0 10 20 30 t t 14 / 16

Simulations Cont. What about desynchronous inputs? u = diag (ˆ u = diag (ˆ k ( t )) x + ˆ r ( t ) − 5 B e k ( t )) x + 5 B e + ˆ r ( t ) 5 10 e i 0 e i 0 -5 -10 0 10 20 30 0 10 20 30 1 50 k i 0 k i 0 ˆ ˆ -1 -50 0 10 20 30 0 10 20 30 0.5 5 0 0 r i r i ˆ ˆ -0.5 -5 0 10 20 30 0 10 20 30 t t 15 / 16

Summary • For symmetric graphs consensus can be achieved with only local adaptive control • For the directed case we proved asymptotic convergence e → 0 • Conjecture : for digraphs consensus can be achieved with only local adaptive control • If you are interested in understanding the desynchronous case come to my second talk in this same session Funding provided by DARPA and Gerber Lab (Georg Gerber) we are part of HMS, BWH, & Harvard-MIT HST 16 / 16