Singularly Perturbed Algorithms for Dynamic Average Consensus - PowerPoint PPT Presentation

Singularly Perturbed Algorithms for Dynamic Average Consensus Solmaz S. Kia, Jorge Cortés, Sonia Martínez Mechanical and Aerospace Engineering Dept. University of California San Diego http://tintoretto.ucsd.edu/solmaz European Control Conference, July 18, 2013 1 / 19

Problem definition Dynamic Average Consensus Autonomous and cooperative agents u 1 ( t ) 1 x i = − c i , x i , c i ∈ R ˙ - x i : agreement state u 2 ( t ) u 3 ( t ) 2 3 - c i : driving command Design c i = f ( i , neighbors ) s.t. ∀ i ∈ { 1 , . . . , N } u 4 ( t ) u 5 ( t ) 4 5 N x i ( t ) → 1 � u j ( t ) , t → ∞ N j = 1 Applications: distributed fusion of dynamic and evolving information multi-robot coordination sensor fusion distributed tracking feature-based map merging 2 / 19

Dynamic average consensus in the literature Previous literature Focus on convergence to consensus Specific initialization conditions: Spanos et al. 05, Zhu and Martinez 10 Specific set of inputs: Spanos et al. 05, Olfati-Saber and Shamma 05 Track with s.s. error: Olfati-Saber and Shamma 05, Spanos et al. 05, Freeman et al. 06, Zhu and Martinez 10 Require knowledge of the dynamics generating inputs: Bai et al. 10 Inputs with bounded derivatives: all of them No explicit attention to limited control authority No explicit attention to rate of convergence of individual agents 3 / 19

This talk Dynamic average consensus with pre-specified rate of convergence β : N N � x i → 1 � � x i ( 0 ) → 1 � � � � � u j ( t ) u j ( 0 ) � e − β t � � k � � � � N N j = 1 j = 1 u 1 ( t ) Network of agents with limited control authority; Control over time of arrival u 2 ( t ) u 3 ( t ) Dynamic average consensus with pre-specified rate of convergence β i at each agent: u 4 ( t ) N N � x i → 1 � � x i ( 0 ) → 1 � � � � � � e − β i t u j ( t ) u j ( 0 ) � � k � � � � N N j = 1 j = 1 Network of agents with different levels of control authority; Control over time of arrival of each agent Design tool: singular perturbation theory 4 / 19

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 5 Strongly connected: i → j for any i , j Weight-balanced: N N � � a ji = a ij , i ∈ V j = 1 j = 1 5 / 19

Dynamic average consensus with pre-specified rate of convergence Goal: Dynamic average consensus with pre-specified rate of convergence β : N N � � � x i → 1 � x i ( 0 ) → 1 � � � � u j ( t ) u j ( 0 ) � e − β t � � k � � � � N N j = 1 j = 1 Design methodology � N Simplest dynamics: x i → 1 j = 1 u j ( t ) with rate β N N N � � x i − 1 + 1 x i = − β � u j � u j ˙ ˙ N N j = 1 j = 1 Requirement: � N � N j = 1 u j + 1 u j in a distributed manner ! - fast dynamics to generate β 1 j = 1 ˙ N N 6 / 19

Dynamic average consensus with pre-specified rate of convergence Goal: Dynamic average consensus with pre-specified rate of convergence β : N N � � � x i → 1 � x i ( 0 ) → 1 � � � � u j ( t ) u j ( 0 ) � e − β t � � k � � � � N N j = 1 j = 1 Design methodology � N Simplest dynamics: x i → 1 j = 1 u j ( t ) with rate β N N N � � x i − 1 + 1 x i = − β � u j � u j ˙ ˙ N N j = 1 j = 1 Requirement: � N � N j = 1 u j + 1 u j in a distributed manner ! - fast dynamics to generate β 1 j = 1 ˙ N N 6 / 19

Dynamic average consensus with pre-specified rate of convergence Design methodology x i = − β x i − ( β 1 � N j = 1 u j + 1 � N � u j ) � Desired dynamics: ˙ j = 1 ˙ N N � N j = 1 u j + 1 � N Requirement: fast distributed dynamics to generate β 1 u j j = 1 ˙ N N Two-time scale algorithm: Initialize at k = 0 , x i ( 0 ) ∈ R Fast dynamics : at each time k , obtain u i ( k ) and ˙ u i ( k ) . ∀ i ∈ { 1 , . . . , N } , run � u i ( k )) − � N i = 1 a ij ( z i − z j ) − � N z i = −( z i − β u i ( k ) − ˙ i = 1 a ij ( ν i − ν j ) ˙ ν i = � N i = 1 a ji ( z i − z j ) ˙ N N � � z i ( t , k ) → β 1 u j ( k ) + 1 u j ( k ) , exponentially as t → ∞ , (Due to [a]) ˙ N N j = 1 j = 1 Slow dynamics : x i ( k + 1 ) = x i ( k ) − ∆ t ( β x i ( k ) − z i ( k )) k ← k + 1 [a] R. Freeman et al., “Stability and convergence properties of dynamic average consensus estimators,” CDC 2006 7 / 19

Dynamic average consensus with pre-specified rate of convergence Design methodology x i = − β x i − ( β 1 � N j = 1 u j + 1 � N � u j ) � Desired dynamics: ˙ j = 1 ˙ N N � N j = 1 u j + 1 � N Requirement: fast distributed dynamics to generate β 1 u j j = 1 ˙ N N Two-time scale algorithm: Initialize at k = 0 , x i ( 0 ) ∈ R Fast dynamics : at each time k , obtain u i ( k ) and ˙ u i ( k ) . ∀ i ∈ { 1 , . . . , N } , run � u i ( k )) − � N i = 1 a ij ( z i − z j ) − � N z i = −( z i − β u i ( k ) − ˙ i = 1 a ij ( ν i − ν j ) ˙ ν i = � N i = 1 a ji ( z i − z j ) ˙ N N � � z i ( t , k ) → β 1 u j ( k ) + 1 u j ( k ) , exponentially as t → ∞ , (Due to [a]) ˙ N N j = 1 j = 1 Slow dynamics : x i ( k + 1 ) = x i ( k ) − ∆ t ( β x i ( k ) − z i ( k )) k ← k + 1 [a] R. Freeman et al., “Stability and convergence properties of dynamic average consensus estimators,” CDC 2006 7 / 19

A dynamic average consensus with pre-specified rate of convergence An average consensus with pre-specified rate of coverage Fast dynamics : at each time k , ∀ i ∈ { 1 , . . . , N } � u i ( k )) − � N i = 1 a ij ( z i − z j ) − � N z i = −( z i − β u i ( k ) − ˙ i = 1 a ij ( ν i − ν j ) ˙ ν i = � N i = 1 a ji ( z i − z j ) ˙ Slow dynamics : x i ( k + 1 ) = x i ( k ) − ∆ t ( β x i ( k ) − z i ( k )) Innovation Combine the fast and slow dynamics in one continuous-time algorithm No need to wait for fast dynamics to converge to take steps in the slow dynamics Design is based on singularly perturbed systems 8 / 19

Singular perturbation � ˙ � ˙ x = f ( t , x , z ) , x ∈ R n x = f ( t , x , z ) , x ∈ R n ǫ = 0 − − − − → z = g ( t , x , z ) , z ∈ R m , ǫ ˙ 0 = g ( t , x , z ) Slow dynamics: g ( t , x , z ) = 0 ⇒ z = h ( t , x ) x = f ( t , x , h ( t , x )) ˙ Fast dynamics: fixed ( t , x ( t )) and τ = t /ǫ dz d τ = g ( t , x , z ) 9 / 19

Singular perturbation � ˙ � ˙ x = f ( t , x , z ) , x ∈ R n x = f ( t , x , z ) , x ∈ R n ǫ = 0 − − − − → z = g ( t , x , z ) , z ∈ R m , ǫ ˙ 0 = g ( t , x , z ) Slow dynamics: g ( t , x , z ) = 0 ⇒ z = h ( t , x ) x = f ( t , x , h ( t , x )) ˙ Fast dynamics: fixed ( t , x ( t )) and τ = t /ǫ dz d τ = g ( t , x , z ) 9 / 19

Singular perturbation � ˙ � ˙ x = f ( t , x , z ) , x ∈ R n x = f ( t , x , z ) , x ∈ R n ǫ = 0 − − − − → z = g ( t , x , z ) , z ∈ R m , ǫ ˙ 0 = g ( t , x , z ) Slow dynamics: g ( t , x , z ) = 0 ⇒ z = h ( t , x ) x = f ( t , x , h ( t , x )) ˙ Fast dynamics: fixed ( t , x ( t )) and τ = t /ǫ dz d τ = g ( t , x , z ) 9 / 19

Singular perturbation � ˙ � ˙ x = f ( t , x , z ) , x ∈ R n x = f ( t , x , z ) , x ∈ R n ǫ = 0 − − − − → z = g ( t , x , z ) , z ∈ R m , ǫ ˙ 0 = g ( t , x , z ) Slow dynamics: g ( t , x , z ) = 0 ⇒ z = h ( t , x ) x = f ( t , x , h ( t , x )) ˙ Fast dynamics: fixed ( t , x ( t )) and τ = t /ǫ dz d τ = g ( t , x , z ) 9 / 19

Singular perturbation � ˙ � ˙ x = f ( t , x , z ) , x ∈ R n x = f ( t , x , z ) , x ∈ R n ǫ = 0 − − − − → z = g ( t , x , z ) , z ∈ R m , ǫ ˙ 0 = g ( t , x , z ) Slow dynamics: g ( t , x , z ) = 0 ⇒ z = h ( t , x ) x = f ( t , x , h ( t , x )) ˙ Fast dynamics: fixed ( t , x ( t )) and τ = t /ǫ dz d τ = g ( t , x , z ) 9 / 19

Singular perturbation on infinite intervals Full dynamics: Slow and fast dynamics � � x = f ( t , x , z ) , ˙ ˙ x = f ( t , x , h ( t , x )) dz ǫ ˙ z = g ( t , x , z ) d τ = g ( t , x , z ) Solution : x ( t , ǫ ) Solution : ¯ x ( t ) Theorem For [ t , x , z − h ( t , x ) , ǫ ] ∈ [ 0 , ∞ ) × D x × D y × [ 0 , ǫ 0 ) On any compact subset of D x × D y : continuous and bounded: f , g , ∂ f | ∂ x , ∂ z , ∂ǫ , ∂ g | ∂ x , ∂ z , ∂ǫ , ∂ t bounded partial derivative w.r.t arg: h ( t , x ) , ∂ g ( t , x , z , 0 ) /∂ z ∂ f ( t , x , h ( t , x ) , 0 ) /∂ x is Lipschitz in x , uniformly in t , The slow dynamics is exponentially stable The fast dynamics is exponentially stable For any t 0 � 0 , ∃ ǫ ⋆ s.t. for 0 < ǫ � ǫ ⋆ we have x ( t , ǫ ) − ¯ x ( t ) ∈ O ( ǫ ) , t ∈ [ t 0 , ∞ ) 10 / 19