Coordination of Multi-Agent Systems

Coordination of Multi-Agent SystemsMark W. SpongDonald Biggar Willett ProfessorDepartment of Electrical and Computer Engineeringand The Coordinated Science LaboratoryUniversity of Illinois at Urbana-Champaign, [email protected] CONTROL AND APPLICATIONSMay 24-26, 2006, Montreal, Quebec, Canada

IntroductionThe problem of coordination of multiple agents arises in numerous applications, both in natural and in man-made systems. Examples from nature include:Flocking of BirdsSchooling of Fish

Synchronously Flashing FirefliesA Swarm of LocustsMore Examples from Nature

Examples from EngineeringAutonomous Formation Flying and UAV Networks

Examples from Social Dynamics and Engineering SystemsCrowd Dynamics and Building EgressMobile Robot Networks

Example from Bilateral TeleoperationMulti-Robot Remote Manipulation

Other ExamplesOther Examples: circadian rhythmcontraction of coronary pacemaker cellsfiring of memory neurons in the brainSuperconducting Josephson junction arraysDesign of oscillator circuitsSensor networks

Synchronization of MetronomesExample:

Fundamental QuestionsIn order to analyze such systems and design coordination strategies, several questions must be addressed:What are the dynamics of the individual agents?How do the agents exchange information?How do we couple the available outputs to achieve synchronization?

Fundamental AssumptionsIn this talk we assume:that the agents are governed by passive dynamics.that the information exchange among agents is described by a balanced graph, possibly with switching topology and time delays in communication.

Outline of Results

We present a unifying approach to:Output Synchronization of Passive SystemsCoordination of Multiple Lagrangian SystemsBilateral Teleoperation with Time DelaySynchronization of Kuramoto Oscillators

Definition of A Passive System

Examples of Passive SystemsIn much of the literature on multi-agent systems, the agents are modeled as first-order integratorsThis is a passive system with storage functionsince

Passivity of Lagrangian SystemsMore generally, an N-DOF Lagrangian system

satisfies

where H is the total energy. Therefore, the system is passivefrom input to output

Graph Theory

Examples of Communication GraphsAll-to-All Coupling(Balanced -Undirected)Directed Not BalancedBalanced-Directed

Synchronization of Multi-Agents

First ResultsSuppose the systems are coupled by the control lawTheorem: If the communication graph is weakly connected and balanced, then the system is globally stable and the agents output synchronize. where K is a positive gain and is the set of agents communicating with agent i.

Some Corollaries1) If the agents are governed by identical linear dynamicsthen, if (C,A) is observable, output synchronization implies state synchronization2) In nonlinear systems without drift, the outputs converge to a common constant value.

Some ExtensionsWe can also prove output synchronization for systems with delay and dynamically changing graph topologies, i.e.provide the graph is weakly connected pointwise in time and there is a unique path between nodes i and j.

Further ExtensionsWe can also prove output synchronization when the coupling between agents is nonlinear, where is a (passive) nonlinearity of the form

Technical DetailsThe proofs of these results rely on methods from Lyapunov stability theory, Lyapunov-Krasovski theory and passivity-based control together with graph theoretic properties of the communication topology. References: [1] Nikhil Chopra and Mark W. Spong, Output Synchronization of Networked Passive Systems, IEEE Transactions on Automatic Control, submitted, December, 2005[2] Nikhil Chopra and Mark W. Spong, Passivity-Based Control of Multi-Agent Systems, in Advances in Robot Control: From Everyday Physics to Human-Like Movement, Springer-Verlag, to appear in 2006.

Technical DetailsSince each agent is assumed to be passive, let,,be the storage functions for the N agentsand define the Lyapunov-Kraskovski functional

Nonlinear Positive-Real Lemma

Now, after some lengthy calculations, using Moylans theorem and assuming that the interconnection graph is balanced, one can show that

Barbalats Lemma can be used to show thatand, therefore,Connectivity of the graph interconnection then implies output synchronization.

Some ExamplesConsider four agents coupled in a ring topology with dynamicsSuppose there is a constant delay T in communication and let the control input be

The closed loop system is thereforeand the outputs (states) synchronize as shown

Second-Order ExampleConsider a system of four point masses with second-order dynamicsconnected in a ring topology

coupling the passive outputs leads toand the agents synchronize as shown below

Simulation Results

Example: Coupled PendulaConsider two coupled pendula with dynamicsand suppose

Kuramoto OscillatorsKuramoto Oscillators are systems of the form

is the natural frequency and is the coupling strength.

Then we can write the system asand our results immediately imply synchronization

Multi-Robot Coordination With DelayConsider a network of N Lagrangian systemsAs before, define the input torque aswhich yieldswhere

Coupling the passive outputs yieldsand one can show asymptotic state synchronization. This gives new results in bilateral teleoperation without the need for scattering or wave variables, as well as new results on multi-robot coordination.

ConclusionsThe concept of Passivity allows a number of results from the literature on multi-agent coordination, flocking, consensus, bilateral teleoperation, and Kuramoto oscillators to be treated in a unified fashion.

THANK YOU!

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