Team-triggered coordination of networked systems Cameron Nowzari Jorge Cort´ es Mechanical and Aerospace Engineering University of California, San Diego cnowzari@ucsd.edu American Control Conference Washington D.C. June 18, 2013
-Coordination of networked systems- Each individual subsystem senses immediate environment communicates with others wirelessly processes gathered information takes action in response Multiple subsystems provide inherent robustness adaptive behavior enable tasks beyond individuals’ capabilities Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 2 / 23
Review of real-time implementation strategies Given desired task, design − − − − → agent plans information If continuous feedback and control is possible, ideal execution − − − − − − − − − → ideal performance agent plans Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23
Review of real-time implementation strategies Given desired task, design − − − − → agent plans information If continuous feedback and control is possible, ideal execution − − − − − − − − − → ideal performance agent plans Instead, given desired performance, we can use Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23
Review of real-time implementation strategies Given desired task, design − − − − → agent plans information If continuous feedback and control is possible, ideal execution − − − − − − − − − → ideal performance agent plans Instead, given desired performance, we can use Time-triggered control: periodic periodic execution information − − − − − → agent plans − − − − − − − − − − − → desired performance Find suitable period T Simple Wasteful Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23
Review of real-time implementation strategies Given desired task, design − − − − → agent plans information If continuous feedback and control is possible, ideal execution − − − − − − − − − → ideal performance agent plans Instead, given desired performance, we can use Event-triggered control: triggered execution continuous information − − − − − − − → agent plans − − − − − − − − − − − − → desired performance Design suitable event-trigger Only necessary updates Requires continuous information Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23
Review of real-time implementation strategies Given desired task, design − − − − → agent plans information If continuous feedback and control is possible, ideal execution − − − − − − − − − → ideal performance agent plans Instead, given desired performance, we can use Self-triggered control: design execute information? ← − − − − agent plans? ← − − − − − desired performance What is necessary information to achieve a desired level of performance ? Implementable, asynchronous Conservative Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 3 / 23
-Team-triggered- control Objective: Combine best properties of event- and self-triggered strategies into a unified, implementable approach How? Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 4 / 23
-Team-triggered- control Objective: Combine best properties of event- and self-triggered strategies into a unified, implementable approach How? Agents make promises to neighbors about their future states Agents warn each other when promises need to be broken Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 4 / 23
Outline 1 Motivation 2 Problem Formulation problem statement review of existing strategies 3 Team-Triggered Coordination promises control policies self-triggered updates broken promises (event-triggered updates) 4 Algorithm Certification performance guarantees simulations Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 5 / 23
Problem statement Coordination task: Drive N agents with linear dynamics x i = A i x i + B i u i , ˙ x i ∈ X i , u i ∈ U i , to a set D . Agents can communicate with other agents through a graph with a fixed topology Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 6 / 23
Problem statement Coordination task: Drive N agents with linear dynamics x i = A i x i + B i u i , ˙ x i ∈ X i , u i ∈ U i , to a set D . Agents can communicate with other agents through a graph with a fixed topology Given a Lyapunov function V with a distributed gradient N d � ∇ i V ( x i dtV ( x ) = N ) ˙ x i , i =1 and a distributed continuous control law u ∗ that monotonically optimizes V while x is not in D , how can we design a real-time implementation? Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 6 / 23
Time-triggered coordination Communication and actuator updates happen at a fixed period T T t 0 t 1 t 2 t 3 d T must be small enough such that dt V ( x ) < 0 at all times Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 7 / 23
Time-triggered coordination Communication and actuator updates happen at a fixed period T T t 0 t 1 t 2 t 3 d T must be small enough such that dt V ( x ) < 0 at all times Drawbacks: Requires global information to compute T T can be quite conservative Assumes all agents to be synchronized Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 7 / 23
Event-triggered coordination Given an update at time t last , the next update time t next is decided by the occurrence of an event d dtV ( x ( t next ) = 0 Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23
Event-triggered coordination Given an update at time t last , the next update time t next is decided by the occurrence of an event N d � N ( t next ))( A i x i + B i u ∗ i ( x N ∇ i V ( x i dtV ( x ( t next ) = i ( t last )) = 0 i =1 Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23
Event-triggered coordination Given an update at time t last , the next update time t next is decided by the occurrence of an event N d � N ( t next ))( A i x i + B i u ∗ i ( x N ∇ i V ( x i dtV ( x ( t next ) = i ( t last )) = 0 i =1 Requires global information. Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23
Event-triggered coordination Given an update at time t last , the next update time t next is decided by the occurrence of an event N d � N ( t next ))( A i x i + B i u ∗ i ( x N ∇ i V ( x i dtV ( x ( t next ) = i ( t last )) = 0 i =1 Requires global information. Instead, use local trigger ∇ i V ( x i N ( t i )( A i x i ( t i next ) + B i u ∗ i ( x N i ( t i next ) last )) = 0 � �� � exact Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23
Event-triggered coordination Given an update at time t last , the next update time t next is decided by the occurrence of an event N d � N ( t next ))( A i x i + B i u ∗ i ( x N ∇ i V ( x i dtV ( x ( t next ) = i ( t last )) = 0 i =1 Requires global information. Instead, use local trigger ∇ i V ( x i N ( t i )( A i x i ( t i next ) + B i u ∗ i ( x N i ( t i next ) last )) = 0 � �� � exact Drawback: Requires continuous information from neighbors to check trigger Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 8 / 23
Self-triggered coordination Agent i receives information at time t i last , constructs guaranteed sets X i j ( t ) = R j ( t − t last , x j ( t last )) ⊂ X j Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 9 / 23
Self-triggered coordination Agent i receives information at time t i last , constructs guaranteed sets X i j ( t ) = R j ( t − t last , x j ( t last )) ⊂ X j Recall the local trigger next ) + B i u ∗ i ( x N ∇ i V ( x i N ( t i next ))( A i x i ( t i i ( t i last )) Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 9 / 23
Self-triggered coordination Agent i receives information at time t i last , constructs guaranteed sets X i j ( t ) = R j ( t − t last , x j ( t last )) ⊂ X j Recall the local trigger next ) + B i u ∗ i ( x N ∇ i V ( x i N ( t i next ))( A i x i ( t i i ( t i last )) next ) + B i u ∗ i ( x N ∇ i V ( y N )( A i x i ( t i i ( t i ≤ sup last )) y N ∈ X i N ( t i next ) Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 9 / 23
Self-triggered coordination Agent i receives information at time t i last , constructs guaranteed sets X i j ( t ) = R j ( t − t last , x j ( t last )) ⊂ X j Recall the local trigger next ) + B i u ∗ i ( x N ∇ i V ( x i N ( t i next ))( A i x i ( t i i ( t i last )) next ) + B i u ∗ i ( x N ∇ i V ( y N )( A i x i ( t i i ( t i ≤ sup last )) y N ∈ X i N ( t i next ) Drawback: Conservative updates Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 9 / 23
Outline 1 Motivation 2 Problem Formulation problem statement review of existing strategies 3 Team-Triggered Coordination promises control policies self-triggered updates broken promises (event-triggered updates) 4 Algorithm Certification performance guarantees simulations Cameron Nowzari (UCSD) Team-triggered coordination June 18, 2013 10 / 23
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