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Coordination of Multi-Agent Systems Mark W. Spong Donald Biggar Willett Professor Department of Electrical and Computer Engineering and The Coordinated Science Laboratory University of Illinois at Urbana-Champaign, USA mspong@uiuc.edu IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Introduction The 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

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Synchronously Flashing Fireflies A Swarm of Locusts More Examples from Nature

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Autonomous Formation Flying and UAV Networks Examples from Engineering

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Crowd Dynamics and Building Egress Mobile Robot Networks Examples from Social Dynamics and Engineering Systems

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Example from Bilateral Teleoperation Multi-Robot Remote Manipulation

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Other Examples Other Examples: circadian rhythm contraction of coronary pacemaker cells firing of memory neurons in the brain Superconducting Josephson junction arrays Design of oscillator circuits Sensor networks

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Synchronization of Metronomes Example :

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Fundamental Questions In order to analyze such systems and design coordination strategies, several questions must be addressed: 1.What are the dynamics of the individual agents? 2.How do the agents exchange information? 3.How do we couple the available outputs to achieve synchronization?

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Fundamental Assumptions In this talk we assume: 1.that the agents are governed by passive dynamics. 2.that the information exchange among agents is described by a balanced graph, possibly with switching topology and time delays in communication.

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Outline of Results We present a unifying approach to: Output Synchronization of Passive Systems Coordination of Multiple Lagrangian Systems Bilateral Teleoperation with Time Delay Synchronization of Kuramoto Oscillators

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Definition of A Passive System

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Examples of Passive Systems In much of the literature on multi-agent systems, the agents are modeled as first-order integrators This is a passive system with storage function since

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Passivity of Lagrangian Systems More generally, an N-DOF Lagrangian system satisfies where H is the total energy. Therefore, the system is passive from input to output

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Graph Theory 3 2 14

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada 3 2 14 4 3 2 1 5 3 2 14 Examples of Communication Graphs All-to-All Coupling (Balanced -Undirected) Directed – Not Balanced Balanced-Directed

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Synchronization of Multi-Agents

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada First Results Suppose the systems are coupled by the control law Theorem: 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.

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Some Corollaries 1) If the agents are governed by identical linear dynamics then, if (C,A) is observable, output synchronization implies state synchronization 2) In nonlinear systems without drift, the outputs converge to a common constant value.

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Some Extensions We 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.

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada where is a (passive) nonlinearity of the form Further Extensions We can also prove output synchronization when the coupling between agents is nonlinear,

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Technical Details The 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.

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Technical Details Since each agent is assumed to be passive, let,…,be the storage functions for the N agents and define the Lyapunov-Kraskovski functional

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Nonlinear Positive-Real Lemma

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Now, after some lengthy calculations, using Moylan’s theorem and assuming that the interconnection graph is balanced, one can show that

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Barbalat’s Lemma can be used to show that and, therefore, Connectivity of the graph interconnection then implies output synchronization.

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Some Examples 3 2 14 Consider four agents coupled in a ring topology with dynamics Suppose there is a constant delay T in communication and let the control input be

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada The closed loop system is therefore and the outputs (states) synchronize as shown

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Second-Order Example Consider a system of four point masses with second- order dynamics connected in a ring topology 3 2 14

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada where which is passive from The key here is to define ``the right’’ passive output. Define a preliminary feedback so that the dynamic equations become to

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada coupling the passive outputs leads to and the agents synchronize as shown below

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Simulation Results

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Example: Coupled Pendula Consider two coupled pendula with dynamics and suppose

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada is the natural frequency and is the coupling strength. Kuramoto Oscillators Kuramoto Oscillators are systems of the form where is the phase of the i-th oscillator,

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Suppose that the oscillators all have the same natural frequency and define Then we can write the system as and our results immediately imply synchronization

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Multi-Robot Coordination With Delay Consider a network of N Lagrangian systems As before, define the input torque as which yields where

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Coupling the passive outputs yields and 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.

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada Conclusions The 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.

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IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada THANK YOU! QUESTIONS?

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