Getting multi-agent systems to cooperate Luca Schenato University - PowerPoint PPT Presentation

Getting multi-agent systems to cooperate Luca Schenato University of Padova OptHySys 2017

Outline � Motivations and target applications � Challenges � The consensus algorithm � Application of consensus � Conclusions and open vistas

Outline � Motivations and target applications � Challenges � The consensus algorithm � Application of consensus � Conclusions and open vistas

Networked Control Systems INTELLIGENT TRAFFIC SWARM SYSTEMS ROBOTICS SMART GRIDS Physically distributed dynamical systems interconnected by a communication network WIRELESS SENSOR NETWORKS SMART CITIES SMART BUILDINGS

Target applications: MAgIC Lab. at University of Padova Wireless Sensor Actuator Networks Smart Camera Networks Robotic Networks Smart Energy Grids

Joint work with Colleagues at Univ. of Padova Gianluigi Pillonetto Alessandro Chiuso Sandro Zampieri Ruggero Carli Angelo Cenedese International collaborators: Former/current students: Fabio Fagnani Lara Brinon-Arranz Sinan Yildirim Simone Del Favero Andrea Carron Marco Todescato Turin Politech, Italy IST, Portugal Ege Univ., Turkey Damiano Varagnolo Saverio Bolognani Filippo Zanella Alexandra von Meier Reza Argandeh Kameshwar Poolla Univ. of Lulea, Swedenn MIT, USA Sellf Inc. UC. Berkeley, USA CIEE, Berkeley, USA UC. Berkeley, USA

Outline � Motivations and target applications � Challenges � The consensus algorithm � Application of consensus � Conclusions and open vistas

Challenges � Unreliable (wireless) communication: � Random delay, packet loss, limited communication range � Scalability: � Complexity (CPU, memory, communication) per agent must be constant � Robustness: � Mild performance degradation when local failures � Architecture: � Centralized vs hierarchical vs distributed vs decentralized � Cooperative vs competitive

Challenges: a personal experience � Prototyping time � Leader-based/hierarchical algorithms too complex to code � Debugging time � Few LEDs for visual inspection � Ex-post analysis of dozens of agent data logs after a failure � Rapid peer-to-peer communication � Wi-Fi, bluetooth, zigbee not suitable for peer-to-peer Courtesy of Antonio Franchi, CNRS, Toulouse Need for simple asynchronous peer-to-peer algorithms

Some working complex systems INTERNET Cell phones networks

A leading paradigm: ISO layers with few primitives Application layer Communication layers

Multi-agent systems: an ISO-like paradigm ? Smart Power Grids Intelligent transportation � What should be the right ISO-model ? Need to seamlessly integrate: � Communication network(s) � Sensing and control � Physical constraints (conservation mass/energy) � Markets

ISO for multi-agent systems Application Sensor Map Time-synch ??? layer calibration building Communication ??? Point-to-point Broadcast Multi-cast layer

Consensus algorithm: a primitive for cooperation Application Sensor Map Time-synch ??? layer calibration building Cooperation Average Consensus ??? layer consensus Communication ??? Point-to-point Broadcast Multi-cast layer

Outline � Motivations and target applications � Challenges � The consensus algorithm � Application of consensus � Conclusions and open vistas

The consensus problem � Main idea � Having a set of agents to agree upon a certain value (usually global function) using only local information exchange (local interaction) � Also known as: � Agreement problem (economics, social networks) � Load balancing (Computer Science & communications) � Synchronization (statistical mechanics) � Rendezvous and flocking (robotics) � Old problem: Markov Chains (60’s), Load balancing (70’s), Distributed decision making (80’s), flocking(00’s)

Multi-agent modeling � Network of � N agents � Communication graph: � i-th node neighbors: � Local variable: node i store

Recursive Distributed Algorithms DEFINITION: Recursive Distributed Algorithm consistent with the graph G : Any recursive algorithm where the i-th node’s update law depends only on the local variables of i and its neighbors

Consensus definitions DEFINITION: 10 A Recursive Distributed 8 Algorithm consistent with the 6 x i graph G is said to asymptotically 4 achieve consensus if 2 0 0 10 20 30 40 Consensus Iteration DEFINITION: 10 A Recursive Distributed 8 Algorithm consistent with the 6 graph G is said to asymptotically x i 4 achieve average consensus if 2 0 0 10 20 30 40 Consensus Iteration

A robotics example: the rendezvous problem 2 Receiving node: 1 Other nodes: 3 4

A robotics example: the rendezvous problem 2 Receiving node: 1 Other nodes: 4 3

A robotics example: the rendezvous problem Receiving node: 1 2 4 3

The linear consensus algorithm PROPERTIES OF P(t) (Stochastic Matrix) � Consistent with the graph: � Component-wise non-negative: � Row-sum unitary: � P(t) doubly stochastic if also column-sum unitary:

Constant matrix P Synchronous communication: At each time all nodes communicate according to the communication graph and update their local variables (Laplacian weigths)

Time varying P(t): broadcast Broadcast communication: At each time one node wakes up and broadcasts its information to all its neighbors

Time varying P(t): symmetric gossip Symmetric gossip communication: At each time one node wakes up and choses one of its neighbors. These two nodes exchange their local variables

Asynchronous consensus: convergence � Standard Consensus (Broadcast) � Graph rooted on average � Self-loops, i.e. P(t) with positive diagonal � P(t) row-stochastic � Average Consensus (Gossip) � Graph connected on average � Self-loops, i.e. P(t) with positive diagonal � P(t) doubly stochastic

Convergence for time-varying communication union

Asynchronous consensus: communication burden � Broadcast-based Consensus � Achieves consensus � updates per 1 sent message � No ACK message required � Gossip-based Consensus � Achieves average consensus � 2 updates per (at least) 3 sent messages � Non-trivial communication protocol

Average consensus: the (broadcast) ratio consensus Standard Ratio Transmitter node Transmitter node Receiver nodes Receiver nodes Other nodes: Other nodes: Column stochastic Row stochastic

Average consensus: the ratio consensus Ratio Standard • D. Kempe, A. Dobra, and J. Gehrke, 2003 • M. Alighanbari and J. How, 2008 • F. Benezit, V. Blondel, P. Thiran, J. Tsitsiklis, M. Vetterli, 2010

Realistic scenarios Ideal scenario Collisions Packet losses

Packet loss: Broadcast consensus Ratio Standard Transmitter node Transmitter node Receiver nodes Receiver nodes Other nodes: Other nodes: Row stochastic Column sub-stochastic

Packet losses: symmetric gossip consensus Gossip nodes Other nodes Row stochastic

Asynchronous consensus: packet loss and random delay � Standard Consensus (broadcast) � Guaranteed (slower) convergence � Average Consensus (gossip) � Guaranteed (slower) convergence, but loss of average � Under randomized communication: (Fagnani-Zampieri 2009, Frasca-Hendrickx 2013) � Ratio Consensus (broadcast) � No convergence � Robust Ratio Consensus (broadcast) � Guaranteed average consensus � Additional local variables required (Dominguez Garcia-Hadjicostis-Vaidya, 2014)

Consensus algorithm: a primitive for multi-agent systems Application layer Sensor Distributed Time-synch ??? calibration optimization Cooperation layer Robust asynchronous broadcast-based and Average Consensus ??? consensus relatively simple implementations Communication layer available ??? Point-to-point Broadcast Multi-cast

Outline � Motivations and target applications � Challenges � The consensus algorithm � Application of consensus � Conclusions and open vistas

Consensus-based applications Wireless Sensor Actuator Networks Smart Camera Networks Robotic Networks Smart Energy Sensor Calibration Grids � RF indoor tracking � Perimeter patrolling Clock Synchronization � � Cardinality estimation � Rendez-vous � Map building � Localization Multi-area state � � Source-seeking estimation �

Sensor calibration issues in RF-based localization i Systematic calibration errors d i k d k d j j

WSN sensor calibration Ideally: Calibrated measurement � Estimate : � Use to compensate the offset: What we propose is: All nodes overestimate or underestimate the distance similarly. The errors, in the triangulation process, cancel out partially.

Calibration as consensus problem Define we want Recalling: update equation Steady state

Experimental Testbed 25 Tmote-Sky nodes with Chipcon CC2420 RF transceiver randomly placed inside a single conference room 8 Network topology and 7 6 nodes displacement: 5 Edge if packet loss x [m] 4 3 probability <25% 2 1 0 0 2 4 6 8 10 12 y [m]