A. S. Morse Yale University

IMA Short Course Distributed Optimization and Control Flocking Asynchronously in Continuous Time A. S. Morse Yale University University of Minnesota June 2, 2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAA

qi i = heading s = speed s 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 Each agent’s heading is updated at the same time as the rest using a local rule based on the average of its own current heading plus the headings of its “neighbors.” Suppose agent’s clocks are not synchronized – what happens?

i i l l d h i d i h h E t t t t t t v e n m e s n o n e c e s s a r y e v e n y s p a c e o r s y n c r o n z e w o e r a g e n s ( ] ( ) ( ) 0 0 d d i h d i i l l f µ t t t t i h i d l f i d i µ t d t i f t t A A U R A H t t t t t t t t i t t t k t t i i 0 t t t a n u p a e s s e a n g m o n o o n c a y o n r o m o : a g e n c o m p u e s s w a y - p o n g p e n a e s u e v e e n o r m g e e s n s e a n g a s s u m e o s a s y w = k ( ) i k i k i i i k + k i 1 i 2 i i 1 i ; ; ; ; : : : i t t e v e n m e s . Indices of neighbors of agent i at time tik

waypoint i ti4 ti1 ti5 ti2 ti3

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. T = {0, t1, t2, ... } ordered set of event times of all n agents. To state the convergence result, we stipulate that each agent ihas only itself as a neighbor at each time in T which is not at event time of agent i. This has no effect on the update rules. Ni (t) = {i} for any t2T which is not an event time of agent i. Thus each Ni (t) is well defined for all t2T Extended neighbor graph E(t) is the neighbor graph of index sets N1(t), N2(t), ..., Nn(t) t2T.

1 3 4 2 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.

CONVERGENCE Synchronous Case: For any trajectory of the synchronous system 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. Asynchronous Case: For any trajectory on T of the asynchronoussystem i2 {1, 2, … ,n} along which the sequence of extended neighbor graphs E(0), E(t1), …. is repeatedly jointly rooted, there is a constant ss to which each agent’s heading iconverges exponentially fast. How can one prove this?

First develop a more explicit model i ti4 ti1 ti5 ti2 ti3 0 1 i

But to use this formula we need to know values of the jat agent i’s event times Can combine agent i’s two update equations to get the familiar update equation This formula tells how ievolves only on agent i’s event time set. 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, …}

t21 t22 t11 t23 t14 t27 t12 t26 t25 t15 t24 t16 t13 agent 2 agent 1 interacting

t1 t21 t22 t2 t11 t3 t4 t23 t8 t14 t27 t12 t5 t12 t11 t26 t25 t9 t15 t10 t6 t24 t16 t13 t13 t7 agent 2 agent 1 T =

Merge all event time sequences into a single ordered sequence T. Define the “synchronized state” of Pi at event times t2to be the original unsynchronized state of Piat these times plus possibly some additional variables. At times in T between two successive event times in Ti, define the state of Pi to be constant at the same value as at the first of these two event times. Analyze the synchronous system S comprised of the n synchronized Pi Analytic Synchronization The n mutually unsynchronized processes below, P1, P2, …Pn together constitute the asynchronous system to be analyzed via “analytic synchronization.” i2 {1, 2, … ,n}

Synchronizing Pi For all times tk2 T = {t0 , t1, .... } between agent i’s qth and (q +1)th event times tiq and ti(q+1) respectively, including timetiq,define Can show that these variable evolve on all of T as where Ti is the set of event times of agenti Can you do this? i2 {1, 2, … ,n}

stochastic matrix Defining the Synchronous System S Comprised of the n Synchronized Pi Asynchronous flocking matrix S

Asynchronous Flocking Matrices R = set of all lists of n real numbers r = {r1, r2, …., rn} where ri2 [0, 1] B = set of all lists of n integers b = {b1, b2, …., bn} where bi2 {0, 1} Gsa = set of all self arced directed graphs with n vertices It is possible to construct a function, F : Gsa£R£B! set of all 2n £ 2n stochastic matrices which is continuous on R, such that where Note that the set of all asynchronous flocking matrices, namely image of F, is compact because R is closed.

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

1 & 4: 2 & 3: µ1 µ2µ3 µ4w1w2w3w4

= ° (F) Not necessarily rooted Vertices without self arcs

Summary For all times tk2 T = {t0 , t1, .... } between agent i’s qth and (q +1)th event times tiq and ti(q+1) respectively, including timetiq, 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 xiin x converge to ss. Check this!

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 extended neighborgraphs, not sequences of asynchronous flocking matrix graphs. To prove that all i converge to a common heading ss can be shown to be equivalent to proving that call 2n entries xiin x converge to ss.

A. S. Morse Yale University