Zeno-free, distributed event-triggered coordination for multi-agent average consensus Cameron Nowzari 1 es 2 Jorge Cort´ 1 Electrical and Systems Engineering University of Pennsylvania cnowzari@seas.upenn.edu 2 Mechanical and Aerospace Engineering University of California, San Diego American Control Conference Portland, Oregon June 5, 2014
-Multi-agent average consensus- Consider N agents with state x = ( x 1 , . . . , x N ) ∈ R N each agent i can communicate with neighbors j ∈ N i in undirected communication graph G Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 2 / 20
-Multi-agent average consensus- Consider N agents with state x = ( x 1 , . . . , x N ) ∈ R N each agent i can communicate with neighbors j ∈ N i in undirected communication graph G x i ( t ) = u i ( t ) ˙ Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 2 / 20
-Multi-agent average consensus- Consider N agents with state x = ( x 1 , . . . , x N ) ∈ R N each agent i can communicate with neighbors j ∈ N i in undirected communication graph G x i ( t ) = u i ( t ) ˙ Well known distributed solution � u i ( t ) = − ( x i ( t ) − x j ( t )) j ∈N i Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 2 / 20
-Multi-agent average consensus- Consider N agents with state x = ( x 1 , . . . , x N ) ∈ R N each agent i can communicate with neighbors j ∈ N i in undirected communication graph G x i ( t ) = u i ( t ) ˙ Well known distributed solution � u i ( t ) = − ( x i ( t ) − x j ( t )) j ∈N i Continuous local state information Continuous communication Continuous actuation Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 2 / 20
Digital controllers Consider a single plant being controlled by a microprocessor through a feedback control loop -25 x = f ( x, u ) ˙ Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 3 / 20
Digital controllers Consider a single plant being controlled by a microprocessor through a feedback control loop -25 measure x x = f ( x, u ) ˙ Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 3 / 20
Digital controllers Consider a single plant being controlled by a microprocessor through a feedback control loop -25 measure x compute k ( x ) x = f ( x, u ) ˙ Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 3 / 20
Digital controllers Consider a single plant being controlled by a microprocessor through a feedback control loop -25 measure x compute k ( x ) feed u = k ( x ) x = f ( x, u ) ˙ Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 3 / 20
Digital controllers Consider a single plant being controlled by a microprocessor through a feedback control loop -25 measure x compute k ( x ) feed u = k ( x ) x = f ( x, u ) ˙ Notice that this is different from the idealistic system x = f ( x, k ( x )) ˙ Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 3 / 20
Digital controllers -25 measure x compute k ( x ) feed u = k ( x ) � = x = f ( x, k ( x )) ˙ x = f ( x, u ) ˙ Most existing control theory was developed ignoring the implementation details Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 4 / 20
Digital controllers -25 measure x compute k ( x ) feed u = k ( x ) � = x = f ( x, k ( x )) ˙ x = f ( x, u ) ˙ Most existing control theory was developed ignoring the implementation details As long as k ( x ) is updated sufficiently fast , everything will be okay Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 4 / 20
Digital controllers -Time-triggered- control Controller is updated periodically at an a priori chosen period T Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 5 / 20
Digital controllers -Time-triggered- control Controller is updated periodically at an a priori chosen period T Benefits: simple and easy to implement does not require extra computations Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 5 / 20
Digital controllers -Time-triggered- control Controller is updated periodically at an a priori chosen period T Benefits: simple and easy to implement does not require extra computations Drawbacks: controller is often designed assuming perfect information state is sampled and controllers are updated periodically robustness analysis done a posteriori Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 5 / 20
Digital controllers -Event-triggered- control Consider a linear system x = Ax + Bu, ˙ with ideal control law u ∗ = Kx rendering the closed loop system asymptotically stable Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 6 / 20
Digital controllers -Event-triggered- control Consider a linear system x = Ax + Bu, ˙ with ideal control law u ∗ = Kx rendering the closed loop system asymptotically stable Since we cannot apply u ∗ continuously, we will update it at a sequence of times { t ℓ } instead. Between updates, the applied control is u ( t ) = Kx ( t ℓ ) t ∈ [ t ℓ , t ℓ +1 ) Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 6 / 20
Digital controllers -Event-triggered- control Consider a linear system x = Ax + Bu, ˙ with ideal control law u ∗ = Kx rendering the closed loop system asymptotically stable Since we cannot apply u ∗ continuously, we will update it at a sequence of times { t ℓ } instead. Between updates, the applied control is u ( t ) = Kx ( t ℓ ) t ∈ [ t ℓ , t ℓ +1 ) Defining the error in the system as e ( t ) = x ( t ) − x ( t ℓ ), the closed loop dynamics is x = Ax ( t ) + BKx ( t ℓ ) ˙ = ( A + BK ) x ( t ) + BKe ( t ) Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 6 / 20
Digital controllers -Event-triggered- control Defining the error in the system as e ( t ) = x ( t ) − x ( t ℓ ), the closed loop dynamics is x = Ax ( t ) + BKx ( t ℓ ) ˙ = ( A + BK ) x ( t ) + BKe ( t ) Since ( A + BK ) is stable, there exists a Lyapunov function V such that V ≤ − a � x � 2 + b � x �� e � ˙ Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 7 / 20
Digital controllers -Event-triggered- control Defining the error in the system as e ( t ) = x ( t ) − x ( t ℓ ), the closed loop dynamics is x = Ax ( t ) + BKx ( t ℓ ) ˙ = ( A + BK ) x ( t ) + BKe ( t ) Since ( A + BK ) is stable, there exists a Lyapunov function V such that V ≤ − a � x � 2 + b � x �� e � ˙ If we can now enforce that � e � ≤ σ a b � x � for some σ ∈ (0 , 1), then V ≤ − (1 − σ ) a � x � 2 < 0 ˙ Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 7 / 20
Digital controllers -Event-triggered- control Event-trigger is given by � e � = σ a b � x � Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 8 / 20
Digital controllers -Event-triggered- control Event-trigger is given by � e � = σ a b � x � Solves the problem of continuous actuation requirement Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 8 / 20
Digital controllers -Event-triggered- control Event-trigger is given by � e � = σ a b � x � Solves the problem of continuous actuation requirement Still requires continuous communication in a network Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 8 / 20
Outline 1 Motivation 2 Problem statement 3 Event-triggered design Simulations 4 Conclusions Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 9 / 20
Problem statement The distributed, continuous control law � u ∗ i ( t ) = − ( x i ( t ) − x j ( t )) j ∈N i is well known to have each agent state asymptotically converge to the initial average of all agent states. Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 10 / 20
Problem statement The distributed, continuous control law � u ∗ i ( t ) = − ( x i ( t ) − x j ( t )) j ∈N i is well known to have each agent state asymptotically converge to the initial average of all agent states. Instead, we will use the control law � u i ( t ) = − (ˆ x i ( t ) − ˆ x j ( t )) , j ∈N i where ˆ x i ( t ) is the last broadcast state of agent i . Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 10 / 20
Problem statement The distributed, continuous control law � u ∗ i ( t ) = − ( x i ( t ) − x j ( t )) j ∈N i is well known to have each agent state asymptotically converge to the initial average of all agent states. Instead, we will use the control law � u i ( t ) = − (ˆ x i ( t ) − ˆ x j ( t )) , j ∈N i where ˆ x i ( t ) is the last broadcast state of agent i . Problem (Multi-agent average consensus) How should agents decide to broadcast their state to ensure their state converges to the initial average of all agent states? Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 10 / 20
Lyapunov design Ideal controller Implementable controller � u ∗ � i ( t ) = − ( x i ( t ) − x j ( t )) u i ( t ) = − x i ( t ) − ˆ (ˆ x j ( t )) j ∈N i j ∈N i Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 11 / 20
Lyapunov design Ideal controller Implementable controller � u ∗ � i ( t ) = − ( x i ( t ) − x j ( t )) u i ( t ) = − x i ( t ) − ˆ (ˆ x j ( t )) j ∈N i j ∈N i u ∗ = − Lx u = − L ˆ x Cameron Nowzari (Penn) Event-triggered consensus June 5, 2014 11 / 20
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