Saturation-tolerant average consensus with controllable rates of - PowerPoint PPT Presentation

Saturation-tolerant average consensus with controllable rates of convergence Solmaz S. Kia, Jorge Cortés, Sonia Martínez Mechanical and Aerospace Engineering Dept. University of California San Diego http://tintoretto.ucsd.edu/solmaz SIAM Conference on Control and Its Applications July 9, 2013 1 / 21

Problem definition Static Average Consensus Autonomous and cooperative agents u 1 1 x i = − c i , x i , c i ∈ R ˙ u 2 u 3 3 2 - x i : agreement state - c i : driving command Design c i = f ( i , neighbors ) s.t. ∀ i ∈ { 1 , . . . , N } u 4 u 5 4 5 N x i ( t ) → 1 � u j , t → ∞ N j = 1 Applications: coordination and information fusion multi-robot coordination distributed fusion in sensor networks distributed optimization smart meters 2 / 21

Static average consensus in the literature Static average consensus is one of the most studied problems in networked systems Inspired by analysis of group behavior (flocking) in nature: Vicsek 95 , Reynolds 87 , Toner and Tu 98 Mathematical models of static consensus and averaging: Jadbabaie et al. 03 , Olfati Saber and Murray 03 and 04 , Boyd et al. 05 Previous literature: Focus on convergence to consensus: time delay, switching, noisy links Focus on increase rate of convergence, No explicit attention to rate of convergence of individual agents No explicit attention to limited control authority 3 / 21

Problems considered in this talk u 1 1 x i = − c i , x i , c i ∈ R ˙ - x i : Agreement state - c i : Driving command u 2 u 3 2 3 Design c i = f ( i , neighbors ) s.t. u 4 u 5 5 4 � N x i → 1 j = 1 u j , t → ∞ , with rate β i 1 N Agents with limited control authority opt for slower rate Consistent response over different communication topologies Control over time of arrival � N x i → 1 x i = − sat ¯ j = 1 u j , t → ∞ , even though ˙ c i ( c i ) 2 N Average consensus is achieved despite limited control authority 4 / 21

Network model Communication topology: weighted digraph G ( V , E , A ) Node set: V = { 1 , · · · , N } 1 Edge set: E ⊆ V × V 2 3 Weights (for i , j ∈ { 1 , . . . , N } ) a ij > 0 if ( i , j ) ∈ E , a ij = 0 if ( i , j ) / ∈ E 4 Strongly connected: i → j for any i , j Weight-balanced: N N � � a ji = a ij , i ∈ V j = 1 j = 1 Laplacian matrix: L = D out − A N � D : out degree, D out A : Adjacency matrix; = a ij , i ∈ V ii j = 1 5 / 21

Laplacian static average consensus Laplacian algorithm: a solution by R. Olfati-Saber and R. Murray 2003, 2004 x i = − c i , x i , c i ∈ R ˙ u 1 1 N � c i = a ij ( x i − x j ) , x i ( 0 ) = u i u 2 u 3 2 3 j = 1 Unbounded c i u 4 u 5 4 5 Weight-balanced Strongly connected � N � N j = 1 u j as t → ∞ x i → 1 j = 1 x j ( 0 ) = 1 N N Exponential convergence with rate ˆ λ 2 = min { λ ( 1 2 ( L + L ⊤ )) > 0 } N N N � x i ( t ) − 1 � � x ( t ) − 1 � � x ( 0 ) − 1 � � u j � � � � � � e − ˆ u j 1 N u j 1 N λ 2 t , � � � � t � 0 � � � � � � N N N j = 1 j = 1 j = 1 6 / 21

Laplacian static average consensus Laplacian algorithm: a solution by R. Olfati-Saber and R. Murray 2003, 2004 � u 1 x = − Lx , x i ( 0 ) = u i 1 ˙ x = ( x 1 , · · · , x N ) u 2 u 3 2 3 Unbounded c i Weight-balanced u 4 u 5 4 5 Strongly connected � N � N x i → 1 j = 1 u j as t → ∞ j = 1 x j ( 0 ) = 1 N N Exponential convergence with rate ˆ λ 2 = min { λ ( 1 2 ( L + L ⊤ )) > 0 } N N N � � � � x i ( t ) − 1 � x ( t ) − 1 � x ( 0 ) − 1 � u j � � � � � � e − ˆ u j 1 N u j 1 N λ 2 t , � � � � t � 0 � � � � � � N N N j = 1 j = 1 j = 1 6 / 21

Laplacian static average consensus Laplacian algorithm: a solution by R. Olfati-Saber and R. Murray 2003, 2004 � u 1 x = − Lx , x i ( 0 ) = u i 1 ˙ x = ( x 1 , · · · , x N ) u 2 u 3 2 3 Unbounded c i Weight-balanced u 4 u 5 4 5 Strongly connected � N � N x i → 1 j = 1 u j as t → ∞ j = 1 x j ( 0 ) = 1 N N Exponential convergence with rate ˆ λ 2 = min { λ ( 1 2 ( L + L ⊤ )) > 0 } N N N � � � � x i ( t ) − 1 � x ( t ) − 1 � x ( 0 ) − 1 � u j � � � � � � e − ˆ u j 1 N u j 1 N λ 2 t , � � � � t � 0 � � � � � � N N N j = 1 j = 1 j = 1 6 / 21

Laplacian static average consensus: example Response of Laplacian algorithm for two different graph topologies 1 2 5 3 4 ˆ λ 2 = 1 . 38 1 2 5 3 4 ˆ λ 2 = 0 . 5 7 / 21

Static average consensus: controllable rate of convergence at each agent Think about physical processes x i = − c i × ˙ Accommodate agents with limited control authority Consistent transient across all communication topologies Control over time of arrival Every agent controls its own convergence rate 8 / 21

Static average consensus: controllable rate of convergence at each agent Think about physical processes x i = − c i × ˙ Accommodate agents with limited control authority Consistent transient across all communication topologies Control over time of arrival Every agent controls its own convergence rate 8 / 21

Static average consensus: controllable rate of convergence at each agent Problem Definition x i = − c i , x i , c i ∈ R ˙ u 1 1 - x i : Agreement state - c i : Driving command Design c i = f ( i , neighbors ) s.t. u 2 u 3 2 3 N x i → 1 � u j , t → ∞ with rate β i , i.e. N u 4 u 5 j = 1 4 5 N N � x i ( t ) − 1 � � x i ( 0 ) − 1 � � u j � � u j � � e − β i t � � κ � � � � N N j = 1 j = 1 9 / 21

Static average consensus: controllable rate of convergence at each agent Design methodology � N Simplest dynamics: x i → 1 j = 1 u j with rate β i N N x i = − β i ( x i − 1 � u j ) ˙ N j = 1 � N j = 1 u j in a distributed manner ! Requirement: fast dynamics to generate 1 N Two-time scales: � N z = − Lz , z i ( 0 ) = u i : z i → 1 j = 1 u j Fast dynamics : ˙ N � N x i = − β i ( x i − 1 j = 1 u j ) Slow dynamics : ˙ N 10 / 21

Static average consensus: controllable rate of convergence at each agent Design methodology � N Simplest dynamics: x i → 1 j = 1 u j with rate β i N N x i = − β i ( x i − 1 � u j ) ˙ N j = 1 � N j = 1 u j in a distributed manner ! Requirement: fast dynamics to generate 1 N Two-time scales: � N z = − Lz , z i ( 0 ) = u i : z i → 1 j = 1 u j Fast dynamics : ˙ N � N x i = − β i ( x i − 1 j = 1 u j ) Slow dynamics : ˙ N 10 / 21

Static average consensus: controllable rate of convergence at each agent Design methodology � N Simplest dynamics: x i → 1 j = 1 u j with rate β i N N x i = − β i ( x i − 1 � u j ) ˙ N j = 1 � N j = 1 u j in a distributed manner ! Requirement: fast dynamics to generate 1 N Two-time scales: � N z = − Lz , z i ( 0 ) = u i : z i → 1 j = 1 u j Fast dynamics : ˙ N � N x i = − β i ( x i − 1 j = 1 u j ) Slow dynamics : ˙ N 10 / 21

Static average consensus: controllable rate of convergence at each agent Proposed solution � z i = � N j = 1 a ij ( z i − z j ) , z i ( 0 ) = u i , ǫ ˙ i ∈ { 1 , . . . , N } x i = − β i ( x i − z i ) , x i ( 0 ) ∈ R , ˙ Lemma For strongly connected and weight-balanced digraphs, ∀ ǫ , β i > 0 , N � x i ( t ) → 1 u j , as t → ∞ , i ∈ { 1 , . . . , N } , N j = 1 exponentially fast, with a rate min { β i , ǫ − 1 ˆ λ 2 } . ˆ λ 2 = min { λ ( 1 2 ( L + L T )) > 0 } 11 / 21

Static average consensus: controllable rate of convergence at each agent Sketch of the proof: z = − ǫ − 1 Lz , z i ( 0 ) = u i ∈ R , ˙ x i = − β i ( x i − z i ) , x i ( 0 ) ∈ R . ˙ Laplacian algorithm : N N � � � z i ( t ) − 1 � z ( 0 ) − ( 1 � u j � � � � e − ǫ − 1 ˆ u j ) 1 N λ 2 t , � � t � 0 � � � � N N j = 1 j = 1 Solution of the agreement dynamics: � t x i ( t ) = x i ( 0 ) e − β i t + β i e − β i ( t − τ ) z i ( τ ) d τ 0 For β i = ǫ − 1 ˆ λ 2 : N N | x i ( t ) − 1 � u j | � | x i ( 0 ) − 1 � u j | e − β i t + t β i κ z e − β i t ; N N j = 1 j = 1 For β i � = ǫ − 1 ˆ λ 2 : N β i κ z � | x i ( t ) − 1 u j | � κ x e − β i t + λ 2 t − e − β i t ) . ( e − ǫ − 1 ˆ β i − ǫ − 1 ˆ N λ 2 j = 1 ˆ λ 2 = min { λ ( 1 2 ( L + L T )) > 0 } 12 / 21

Static average consensus: controllable rate of convergence at each agent Sketch of the proof: z = − ǫ − 1 Lz , z i ( 0 ) = u i ∈ R , ˙ x i = − β i ( x i − z i ) , x i ( 0 ) ∈ R . ˙ Laplacian algorithm : N N � � � z i ( t ) − 1 � z ( 0 ) − ( 1 � u j � � � � e − ǫ − 1 ˆ u j ) 1 N λ 2 t , � � t � 0 � � � � N N j = 1 j = 1 Solution of the agreement dynamics: � t x i ( t ) = x i ( 0 ) e − β i t + β i e − β i ( t − τ ) z i ( τ ) d τ 0 For β i = ǫ − 1 ˆ λ 2 : N N | x i ( t ) − 1 � u j | � | x i ( 0 ) − 1 � u j | e − β i t + t β i κ z e − β i t ; N N j = 1 j = 1 For β i � = ǫ − 1 ˆ λ 2 : N β i κ z � | x i ( t ) − 1 u j | � κ x e − β i t + λ 2 t − e − β i t ) . ( e − ǫ − 1 ˆ β i − ǫ − 1 ˆ N λ 2 j = 1 ˆ λ 2 = min { λ ( 1 2 ( L + L T )) > 0 } 12 / 21