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A preliminary result on synchronization of heterogeneous agents via funnel control Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany joint work with Hyungbo Shim (Seoul National University, Korea) 54th IEEE Conference on Decision and Control, CDC 2015, Osaka, Japan Wednesday, 16th December 2015, WeA07.6, 11:40

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Contents Synchronization of heterogenous agents 1 High-gain and funnel control 2 Simulations 3 Weakly centralized Funnel synchronization 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Problem statement x 2 Given 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 ) x 1 := 1 agents know average of neighbor states 2( x 2 + x 3 ) x 2 := 1 Desired 2( x 1 + x 3 ) Control design for practical synchronization x 3 := 1 3( x 1 + x 2 + x 4 ) x 1 ≈ x 2 ≈ . . . ≈ x n x 4 := x 3 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization A ”high-gain“ result Let N i := { j ∈ V | ( j , i ) ∈ E } and d i := |N i | . Diffusive coupling � u i = − k ( x i − x j ) = − kd i ( x i − x i ) 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, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Contents Synchronization of heterogenous agents 1 High-gain and funnel control 2 Simulations 3 Weakly centralized Funnel synchronization 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized 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. Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Funnel synchronization Reminder diffusive coupling: u i = − k i e i with e i = x i − x i . Combine diffusive coupling with Funnel Controller 1 u i ( t ) = − k i ( t ) e i ( t ) with k i ( t ) = ϕ ( t ) − | e i ( t ) | Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Content Synchronization of heterogenous agents 1 High-gain and funnel control 2 Simulations 3 Weakly centralized Funnel synchronization 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Example (taken from Kim et al. 2015) Simulations in the following for N = 5 agents with dynamics 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 ) , with randomly chosen parameters δ i , m 1 i , m 1 i ∈ R and θ 1 i , θ 2 i ∈ [0 , 2 π ]. 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, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Simulation with constant k black curve: x 2 N x 3 s ( t ) = 1 � ˙ f i ( t , s ( t )) N u i = − k e i with k = 10 x 1 i =1 x 4 N s (0) = 1 � x 5 x i (0) N i =1 10 0 -10 -20 -30 0 5 10 15 20 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Funnel synchronization 10 0 x 2 x 3 -10 -20 x 1 -30 x 4 x 5 -40 0 5 10 15 20 u i ( t ) = − k i ( t ) e i ( t ) 120 1 100 k i ( t ) = ϕ ( t ) − | e i ( t ) | 80 60 ϕ ( t ) = ϕ + ( ϕ − ϕ ) e − λ t 40 20 ϕ = 20, ϕ = 1, λ = 1 0 0 5 10 15 20 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Observations for funnel synchronization from simulations Funnel synchronization seems to work errors remain within funnel practical synchronizations is achieved limit trajectory does not coincide with solution s ( · ) of N N s ( t ) = 1 s (0) = 1 � � ˙ f i ( t , s ( t )) , x i . N N i =1 i =1 What determines the new limiting trajectory? Coupling graph? Funnel shape? Gain function? Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Funnel synchronization, directed graph 10 x 2 0 x 3 -10 x 1 -20 -30 x 4 x 5 -40 0 5 10 15 20 u i ( t ) = − k i ( t ) e i ( t ) 120 1 100 k i ( t ) = 80 ϕ ( t ) − | e i ( t ) | 60 ϕ ( t ) = ϕ + ( ϕ − ϕ ) e − λ t 40 20 ϕ = 20, ϕ = 1, λ = 1 0 0 5 10 15 20 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Funnel synchronization, complete graph 10 0 x 2 x 3 -10 -20 x 1 -30 x 4 x 5 -40 0 5 10 15 20 u i ( t ) = − k i ( t ) e i ( t ) 120 1 100 k i ( t ) = ϕ ( t ) − | e i ( t ) | 80 60 ϕ ( t ) = ϕ + ( ϕ − ϕ ) e − λ t 40 20 ϕ = 20, ϕ = 1, λ = 1 0 0 5 10 15 20 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control

Synchronization of heterogenous agents High-gain and funnel control Simulations Weakly centralized Funnel synchronization Funnel synchronization with bigger funnel 10 0 x 2 x 3 -10 -20 x 1 -30 x 4 x 5 -40 0 5 10 15 20 u i ( t ) = − k i ( t ) e i ( t ) 120 1 100 k i ( t ) = ϕ ( t ) − | e i ( t ) | 80 60 ϕ ( t ) = ϕ + ( ϕ − ϕ ) e − λ t 40 20 ϕ = 30, ϕ = 2, λ = 0 . 3 0 0 5 10 15 20 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germanyjoint work with Hyungbo Shim (Seoul National University, Korea) A preliminary result on synchronization of heterogeneous agents via funnel control