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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

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 Birds

Schooling of Fish

Examples from Engineering

Autonomous Formation Flying and UAV Networks

Examples from Social Dynamics and Engineering Systems

Mobile Robot Networks

Crowd Dynamics and Building Egress

Example from Bilateral Teleoperation

Multi-Robot Remote Manipulation

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

Synchronization of Metronomes

Example:

Fundamental Questions

In 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 Assumptions

In 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 Systems
- Coordination of Multiple Lagrangian Systems
- Bilateral Teleoperation with Time Delay
- Synchronization of Kuramoto Oscillators

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

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

3

2

3

2

4

1

5

1

4

3

2

1

4

Examples of Communication GraphsBalanced-Directed

All-to-All Coupling

(Balanced -Undirected)

Directed – Not Balanced

First Results

Suppose the systems are coupled by the control law

where K is a positive gain and is the set of agents communicating with agent i.

Theorem: If the communication graph is weakly connected and balanced, then the system is globally stable and the agents output synchronize.

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.

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.

Further Extensions

We can also prove output synchronization when the coupling between agents is nonlinear,

where is a (passive) nonlinearity of the form

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.

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

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

Barbalat’s Lemma can be used to show that

and, therefore,

Connectivity of the graph interconnection then implies output synchronization.

3

2

1

4

Some ExamplesConsider four agents coupled in a ring topology with dynamics

Suppose there is a constant delay T in communication and let the control input be

The closed loop system is therefore

and the outputs (states) synchronize as shown

3

2

1

4

Second-Order ExampleConsider a system of four point masses with second-order dynamics

connected in a ring topology

The key here is to define ``the right’’ passive output. Define a preliminary feedback

so that the dynamic equations become

where which is passive from

to

coupling the passive outputs leads to

and the agents synchronize as shown below

is the phase of the i-th oscillator,

where

Kuramoto OscillatorsKuramoto Oscillators are systems of the form

is the natural frequency and is the coupling strength.

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

Multi-Robot Coordination With Delay

Consider a network of N Lagrangian systems

As before, define the input torque as

which yields

where

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.

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.

THANK YOU!

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