Edge-wise funnel synchronization Stephan Trenn Technomathematics - PowerPoint PPT Presentation

Edge-wise funnel synchronization Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany 2017 GAMM Annual Meeting, Weimar, Germany Tuesday, 7th March 2017

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Contents Synchronization of heterogenous agents 1 High-gain and funnel control 2 Funnel synchronization 3 Edgewise Funnel synchronization 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Problem statement Given x 2 N agents with individual scalar dynamics: x 1 x i = f i ( t , x i ) + u i ˙ x 3 x 4 undirected connected coupling-graph G = ( V , E ) local feedback u 1 = F 1 ( x 1 , x 2 , x 3 ) Desired u 2 = F 2 ( x 2 , x 1 , x 3 ) Control design for practical synchronization u 3 = F 3 ( x 3 , x 1 , x 2 ) u 4 = F 4 ( x 4 , x 3 ) x 1 ≈ x 2 ≈ . . . ≈ x n Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization A ”high-gain“ result Let N i := { j ∈ V | ( j , i ) ∈ E } and d i := |N i | and L be the Laplacian of G . Diffusive coupling � u i = − k ( x i − x j ) u = − k L x or, equivalently, j ∈N i Theorem (Practical synchronization, Kim et al. 2013) Assumptions: G connected, all solutions of average dynamics N s ( t ) = 1 � ˙ f i ( t , s ( t )) N i =1 remain bounded. Then ∀ ε > 0 ∃ K > 0 ∀ k ≥ K: Diffusive coupling results in lim sup t →∞ | x i ( t ) − x j ( t ) | < ε ∀ i , j ∈ V Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Example (taken from Kim et al. 2015) x 2 Simulations in the following for N = 5 agents with dynamics x 3 f i ( t , x i ) = ( − 1 + δ i ) x i + 10 sin t + 10 m 1 i sin(0 . 1 t + θ 1 i ) + 10 m 2 i sin(10 t + θ 2 i ) , x 1 with randomly chosen parameters δ i , m 1 i , m 1 i ∈ R and θ 1 i , θ 2 x 4 i ∈ [0 , 2 π ]. x 5 Parameters identical in all following simulations, in particular δ 2 > 1, hence agent 2 has unstable dynamics (without coupling). Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Example (taken from Kim et al. 2015) k=2 10 x 2 x 3 0 x 1 -10 x 4 x 5 -20 -30 u = − k L x 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 gray curve: k=20 10 N s ( t ) = 1 0 � ˙ f i ( t , s ( t )) N i =1 -10 N s (0) = 1 � x i (0) -20 N i =1 -30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Contents Synchronization of heterogenous agents 1 High-gain and funnel control 2 Funnel synchronization 3 Edgewise Funnel synchronization 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Reminder Funnel Controller ϕ y ( t ) = h ( t , y ( t )) + u ( t ) ˙ y ( t ) e ( t ) ϕ ( t ) ϕ t F e − ϕ ( t ) u ( t ) = − k ( t ) e ( t ) + − y ref ( t ) Theorem (Practical tracking, Ilchmann et al. 2002) Funnel Control 1 k ( t ) = ϕ ( t ) − | e ( t ) | works, in particular, errors remains within funnel for all times. Basic idea for funnel synchronization u = − k L x − → u = − k ( t ) L x Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Content Synchronization of heterogenous agents 1 High-gain and funnel control 2 Funnel synchronization 3 Edgewise Funnel synchronization 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Approach from Shim & T. 2015 Local error   � �  =: − k d i ( x i − x i ) =: − k i e i u i = − k x i − x j = − k  d i x i − x j j :( i , j ) ∈ E j :( i , j ) ∈ E Funnel synchronization feedback rule 1 u i ( t ) = − k i ( t ) e i ( t ) with k i ( t ) = ϕ ( t ) − | e i ( t ) | States Gains 10 120 5 0 100 -5 80 -10 -15 60 -20 -25 40 -30 20 -35 -40 0 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Unpredictable limit trajectory Problems Synchronization occurs as desired, but No proof available yet Non-predictable limit trajectory Laplacian feedback Non-Laplacian feedback Diffusive coupling Funnel synchronization   k 1 ( t ) k 2 ( t )   u = − k L x u = − K ( t ) L x = −  L x  ...     k N ( t ) has Laplacian feedback has non-Laplacian feedback matrix K ( t ) L , in particular [1 , 1 , . . . , 1] ⊤ is not a left-eigenvector of K ( t ) L . matrix k L Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Weakly centralized Funnel synchronization, Shim & T. 2015 Restoring Laplacian feedback structure Weakly decentralized funnel synchronization u = − k max ( t ) L x with k max ( t ) := max k i ( t ) i again has (time-varying) Laplacian feedback matrix − k max ( t ) L . States Gain 10 120 5 0 100 -5 80 -10 -15 60 -20 -25 40 -30 20 -35 -40 0 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Problem Each agent needs knowledge of gains of all other agents! Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Content Synchronization of heterogenous agents 1 High-gain and funnel control 2 Funnel synchronization 3 Edgewise Funnel synchronization 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Diffusive coupling revisited Diffusive coupling for weighted graph N N � � u i = − k α ij · ( x i − x j ) − → u i = − k ij · α ij · ( x i − x j ) i i where α ij = α ji ∈ { 0 , 1 } is the weight of edge ( i , j ) Conjecture If k ij = k ji are all sufficiently large, then practical synchronization occurs with predictable limit trajectory s . Proof technique from Kim et al. 2013 should still work in this setup. Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Adjusted proof technique of Kim et al. 2013 1 ⊤ � ξ � � � = 1 N Consider coordinate transformation x , then closed loop has the form r N R ( k ij ) ξ = 1 ˙ N 1 ⊤ N f ( t , 1 N ξ + Qr ) r = − Λ( k ij ) r + R ( k ij ) f ( t , 1 N ξ + Qr ) ˙ Show that r → 0, then ξ → s where s = 1 N 1 ⊤ ˙ N f ( t , 1 N s ) Problem Coordinate transformation depends on k ij → Approach breaks down when k ij becomes time/state-dependent Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization

Synchronization of heterogenous agents High-gain and funnel control Funnel synchronization Edgewise Funnel synchronization Edgewise Funnel synchronization Diffusive coupling → edgewise Funnel synchronization N N � � u i = − k ij · α ij · ( x i − x j ) − → u i = − k ij ( t ) · α ij · ( x i − x j ) i i Edgewise error feedback 1 e ij := x i − x j k ij ( t ) = ϕ ( t ) − | e ij | , with Properties: Decentralized, i.e. u i only depends on state of neighbors Symmetry, k ij = k ji Laplacian feedback, u = −L K ( t , x ) x Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Edge-wise funnel synchronization