A. S. Morse

Yale University

University of Minnesota June 2, 2014

IMA Short Course

Distributed Optimization and Control

Flocking Asynchronouslyin Continuous Time

Each agent’s heading is updated at the same time as the rest using a local rulebased on the average of its own current heading plus the headings of its“neighbors.”

i = headingqi

s = speeds

Vicsek et al. simulated a flock of n agents {particles} all moving in the plane at the same speed s, but with different headings 1, 2, …. n

at the same time as the rest

Suppose agent’s clocks are not synchronized – what happens?

and updates its heading monotonically on (tik; ti(k+1)] from µi(tik) to wi(tik):

At tik agent i computes its kth way-point

Agent i0s event times = 0;ti1; ti2; : : : assumed to satisfy

Update Rule for Agent i0s Heading µi

Event times not necessarily evenly spaced or synchronized with other agentsevent times.

Indices of neighbors of agent i at time tik

i

ti1 ti2 ti3ti4 ti5

waypoint

T = {0, t1 , t2, ... } ordered set of event times of all n agents.

Ni (tik) = set of labels of agent i’s neighbors at time tik

Thus each agent’s neighbors are defined at all of its own event times.

To state the convergence result, we stipulate that each agent i has only itself as a neighbor at each time in T which is not at event time of agent i.

Ni (t) = {i} for any t 2 T which is not an event time of agent i.

Thus each Ni (t) is well defined for all t 2 T

Extended neighbor graph E(t) is the neighbor graph of index sets N1(t), N2(t), ..., Nn(t) t 2 T.

This has no effect on the update rules.

1

3

42

Extended neighbor graph E(t) at a time t which is an event time of only agents 1 and 3.

Note that agents 2 and 4 have only themselves as neighbors.

along which the sequence of neighbor graphs N(0), N(1), …. is repeatedly jointly rooted,there is a constant ss to which each agent’s heading i converges exponentially fast.

For any trajectory of the synchronous system Synchronous Case:

CONVERGENCE

Asynchronous Case:

How can one prove this?

along which the sequence of extended neighbor graphs E(0), E(t

1), …. is repeatedly

jointly rooted, there is a constant ss to which each agent’s heading i convergesexponentially fast.

For any trajectory on T of the asynchronous system

i 2 {1, 2, … ,n}

i

ti1 ti2 ti3ti4 ti5

0

1i

First develop a more explicit model

i

ti1 ti2 ti3ti4 ti5

0

1i

First develop a more explicit model

i

ti1 ti2 ti3ti4 ti5

0

1i

First develop a more explicit model

Can combine agent i’s two update equations to get the familiar update equation

This formula tells how i evolves only on agent i’s event time set.

But to use this formula we need to know values of the j at agent i’s event times

In the synchronous case where event times are the same for all agents,the tik are independent of i, and the preceding update equations are sufficient.

For the asynchronous case a common time scale is needed …..

A Common Time Scale

T = set of all event times tik of all n agents

Re-label the elements of T as t0, t1, t2, … where t0 = 0 and t < t +1 for 2 {0, 1, 2, …}

t11 t12 t15 t16t13 t14

t26t25t24t21 t27t23t22

agent 2

agent 1

interacting

agent 2

agent 1

t11 t12 t15 t16t13 t14 t26t25t24t21 t27t23t22 t3 t5 t10 t13t7 t8 t11t9t6t1 t12t4t2 T =

The n mutually unsynchronized processes below, P1, P2, …Pn together constitutethe asynchronous system to be analyzed via “analytic synchronization.”

Merge all event time sequences into a single ordered sequence T.

Analytic Synchronization

At times in T between two successive event times in Ti, define the stateof Pi to be constant at the same value as at the first of these two event times.

Define the “synchronized state” of Pi at event times t 2 to be the originalunsynchronized state of Pi at these times plus possibly some additional variables.

Analyze the synchronous system S comprised of the n synchronized Pi

i 2 {1, 2, … ,n}

Synchronizing Pi

Can show that these variable evolve on all of T as

i 2 {1, 2, … ,n}

For all times tk 2 T = {t0 , t1, .... } between agent i’s qth and (q +1)th event times tiq and ti(q+1) respectively, including time tiq, define

Can you do this?where Ti is the set of event times of agent i

Defining the Synchronous System S Comprised of the n Synchronized Pi

stochastic matrix

S

Asynchronous flocking matrix

R = set of all lists of n real numbers r = {r1, r2, …., rn} where ri 2 [0, 1]

B = set of all lists of n integers b = {b1, b2, …., bn} where bi 2 {0, 1}

Asynchronous Flocking Matrices

Note that the set of all asynchronous flocking matrices, namely image of F,

is compact because R is closed.

Gsa = set of all self arced directed graphs with n vertices

F : Gsa £ R £ B ! set of all 2n £ 2n stochastic matrices

where

It is possible to construct a function,

which is continuous on R, such that

Example

Suppose tk = T is an event time of agents 2 and 3 in a 4 agent network

Suppose the extended neighbor graph E(T) is

µ1 µ2 µ3 µ4 w1 w2 w3 w4

1 & 4:

2 & 3:

Not necessarily rooted

Vertices without self arcs

1

3

54

6 8

7

2

= ° (F)

For all times tk 2 T = {t0 , t1, .... } between agent i’s qth and (q +1)th event times tiq and ti(q+1) respectively, including time tiq, we defined

Then we defined

and asserted that

To prove that all i converge to a common heading ss can be shown to be equivalent to proving that call 2n entries xi in x converge to ss.Check this!

Summary

To prove that all i converge to a common heading ss can be shown to be equivalent to proving that call 2n entries xi in x converge to ss.

Thus as before the problem reduces to determining conditions under which

But the graphs of the F(k) do not necessarily have self arcs at all vertices.

So the preceding facts about compositions of self-arced graphs do not apply

Moreover the convergence condition is stated in terms of sequences of extendedneighbor graphs, not sequences of asynchronous flocking matrix graphs.