Event-based broadcasting for multi-agent average consensus.pdf

7/25/2019 Event-based broadcasting for multi-agent average consensus.pdf

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Georg S. Seyboth, Dimos V. Dimarogonas,Karl H. Johansson

Event-based broadcasting for

multi-agent average consensus


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Why do we need it?

Former research

This paper in comparison

Introduction


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Growing interest in wireless networks high flexibility

and low installation cost. Multi-vehicle coordination.

However

Control over networks with limited resources is a

challenging task. Communication is expensive shared communication

resources, potential effects of information exchangeon power consumption of remote components.

It will be shown that

Event-based scheduling is beneficial for cooperativenetworked control and can facilitate the efficientusage of the shared resources

Why do we need it?


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The cooperative control task under consideration is averageconsensus

We learned about in class (consensus protocol for single-integrator agents) and can further be read on in Olfati-Saberand Murray (2004).

We saw in our homework (consensus protocol for double-

integrator agents) and can further be read on in Ren andAtkins (2007).

In practice those control laws have to be implemented ondigital platforms

The traditional method is time-scheduled periodic sampling

measurements are taken periodically according to a constantsampling period and the controllers are updatedsynchronously. Implementation is investigated in:Xie, Liu, Wang and Jia (2009); Ren and Cao (2008)

Former research


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Event-based sampling is an alternative to traditional periodic

sampling The idea is to sample and update the controller only when

measurements cross certain thresholds.

Outperforms periodic sampling in many scenarios - strmand Bernhardsson (1999), rzn (1999), Miskowicz (2006).

Application to networks is done in Mazo and Cao (2011), Mazoand Tabuada (2011), Wang and Lemmon (2008, 2011) where atriggering mechanism based on state norm is being used.

Both Mazo and Tabuada (2011) and Wang and Lemmon (2011)

suggest that event-based scheduling reduces the number oftransmissions required in networked control systems.

Former research cont.


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Event-based scheduling seems to be suitable for

cooperative control of multi-agent systems over networkswith limited resources. However, only a few studies haveconsidered this topic

An event-based implementation of the consensus

protocol is developed in Dimarogonas, Frazzoli, andJohansson (2012). Where all agents continuouslymonitor their neighbors states.

Former research cont.


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In contrast to Dimarogonas et al. (2012), measurement

broadcasts are scheduled in an event-based fashion,such that continuous monitoring of the neighborsstates is no longer required.

In Wang and Lemmon (2011) each subsystem is aware

of the equilibrium state to be stabilized. In the presentwork, the consensus point is unknown to the agents,which makes it more challenging to find suitabletrigger conditions.

Each agent decides based on the difference of itscurrent state and its latest broadcast state, called themeasurement error, when it has to send a new value.An event is triggered whenever the norm of themeasurement error crosses a certain threshold.

This paper in comparison


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Preliminaries (ISS)

Graph theory usage

Problem statement

Background and problem

statement


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A scalar continuous function (r) defined for r [0, a]is said to belong to class K if it is strictly increasing and (0) = 0, and it is said to belong to class Kif it isdefined for all r 0 and (r) as r .

A scalar continuous function (r, s) defined for r [0,a], s [0,] is said to belong to class KL if for each

fixed s it belongs to class K and for each fixed r it isdecreasing in s and (r, s) 0 as s , cf., Khalil(2002).

A dynamical system with state x and input w is calledinput-to-state stable (ISS) if there exist a class KLfunction and a class K function such that x(t)(x(0), t) + (w[0,t]) for all t 0, see Sontag(1989). For linear systems, ISS is equivalent to globalasymptotic stability of the unforced system (Khalil,2002).

Preliminaries


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Consider a graph G = {V,} consisting of a set of

vertices (or nodes) V = {1,,N} and edges . For undirected graphs, L is symmetric and positive

semi-definite, i.e., L = LT 0. The row sums of L arezero. Thus, the vector of ones 1is an eigenvectorcorresponding to eigenvalue 1(G) = 0, i.e., L1= 0. Forconnected graphs, L has exactly one zero eigenvalue,and the eigenvalues can be listed in increasing order0 = 1(G) < 2(G) N(G). The second eigenvalue

2(G) is called the algebraic connectivity. Lemma: Suppose L is the Laplacian of an undirected,

connected graph G. Then, for all t 0 and all vectorsv RNwith 1Tv = 0, it holds that eLtv e2(G)tv.

Graph theory usage


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Assumption: Graph G is undirected and connected.

Single-integrator agents:where xi(t) R is the state and

ui(t) R its control input.

Double-integrator agents:

where

xi(t) = [i(t), i(t)]TR2.

In both cases, the agents are coordinated in adistributed fashion, i.e., ui(t) depends only oninformation from neighbors j Ni. The communicationover each edge may be subject to a constant time-delay 0.

Problem statement

, ,i it u t i V x

0

1

0 1, ,

0 0i i ix ut t t i V

x


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Each agent consists of a controller and dynamics as shown inthe figure:

The controller of agent i monitors its own state xi(t)continuously. Based on local information, it decides when tobroadcast its current state over the network.

The latest broadcast state of agent i given by

where is the sequence ofevent times of agent i. Whenever agent i transmits its state or receives a new state

value from one of its neighbors, it re-computes its control uiimmediately.

Problem statement cont.

1,

, ,t

i i i

k k ki i t tt t x x 0 1, , ...

i it t


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The problem: Find a triggering rule that

determines, based on local information, whenagent i has to trigger and broadcast a newstate value to its neighbors, such that allagents states converge to the average of their

initial conditions.

Problem statement cont.


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Construction

Theorem

Remarks

Single-integrator agents


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With the stack vectors x = [x1,,xN]Tand

u = [u1,,uN]T, the multi-agent system withsingle integrator agents can be written as

, x(0) = x0RN.

Recall that the continuous distributed controllaw globally asymptoticallysolves the average consensus problem, i.e.,

for all i V as t .

The closed-loop system can be written as.

Construction

t u tx

i

i i j

j N

x t x tu t

1 0i ii V

x N xt

t Lx t x


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An event-based implementation of the continuous

distributed control law is proposed, given byor in stack vector form

A trigger function fi() is defined for each agentwhich takes values in R and depends on local

information only, i.e., on time t and the true andbroadcast states .

An event for agent i is triggered as soon as thetrigger condition is fulfilled.

For each i V and t 0, a measurement error isdefined . With e = [e1,,eN]T, theclosed-loop system is described by

Construction cont.

.u t Lx t i

i i j

j N

x t x tu t

, i ix t x t

, , 0i i itf x t x t

i i ite x t x t

L x t e t t Lx t x


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Using the measurement error definition its proposed to usethe following trigger functions , with

thresholds of the form . Based on the original idea of event-based control in strm

and Bernhardsson (1999) - trigger an event and close thefeedback loop whenever the state deviates from theequilibrium and crosses a predefined threshold c0, say.

Let a(t) = (1/N)1T

x(t) be the average of all states. Thederivative of a(t) is (t) = (1/N)1Tu(t) = 0 sinceu(t) = L (t) has zero average, i.e., 1Tu(t) = 0T. Thus,a(t) = a(0) = a for all t 0 and the state x(t) can bedecomposed according to x(t) = a1+ (t), where (t) is thedisagreement vector of the multi-agent system, i.e.,

1

T

(t) = 0. From the Lemma presented before it follows that the rate

of convergence for single-integrator agents with continuousdistributed control law is at least 2(G), because(t)= exp(Lt)(0) exp(2(G)t)(0).

Construction cont.

,i i i it e t e t h t f 0:ih t R R

0 1 expih t c c t

ax


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Consider the multi-agent system of single-

integrator agents with controllaw . Suppose the triggerfunction is given by

with constants c0 0 and c1 0, c0+ c1> 0, and0 < < 2(G). Then, for all initial conditionsx0R

N, the closed-loop system does notexhibit Zeno behavior. Moreover, thedisagreement vector of the closed-loopsystem converges to a ball centered at theorigin with radius r = L c0/2(G).

Theorem

, ,i it u t i V x i

i i j

j N

x t x tu t

0 1, ti i it e t e t c c ef

N


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The graph G is assumed to be undirected in this paper.However, it can be shown that the theorem extends to

strongly connected and balanced directed graphs. The radius r = L c0/2(G) can be chosen arbitrarily

small since it scales with c0. Forc1= 0 the density of events is independent of c0forlarge t, but for small t the inter-event intervals are

short if c0is small. This motivates why c1> 0 might besuitable in practice. The theorem states that the radius vanishes for c0= 0.

The closed-loop system reaches average consensusasymptotically in this case. The condition < 2(G) is

intuitive, because the states should converge fasterthan the threshold decreases. However, if c0> 0, a positive lower bound on the inter-

event times exists for > 2(G) as well. Consequently,knowledge of 2(G) is not necessary if c0> 0.

Remarks

N


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The class of time-dependent trigger functions can be extended tofi(t, ei(t)) = |ei(t)| hi(t) with clexp(t) hi(t) cuexp(t),

i V, where 0 < cl< cuand 0 < < 2(G). Then, for all initial conditions x0R

N, the closed-loop systemdoes not exhibit Zeno behavior. Moreover, the disagreement vector of the closed-loop system converges to the origin asymptotically.

This enlarges the class of suitable trigger functions compared tothe theorem and shows that the agents do not need to share the

same trigger function. However, in this case or when c0= 0, in order to choose an

appropriate trigger function, each agent has to be aware of 2(G). This assumption can partly be avoided by using the lower bound

on 2(G) in terms of N and diameter d given by:2(G) 4/(Nd) 4/(N(N 1)). Mohar (1991).

If is chosen smaller than this bound, then obviously < 2(G).Therefore it is sufficient that each agent is aware of N or an upperbound thereof.

A more advanced method to resolve this issue is throughdistributed estimation of 2(G).

Remarks cont.


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Construction

Theorem

Remarks

Delayed communication


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From a practical point of view, the effect of

communication delays needs to be accounted for. The disagreement dynamics are asymptotically

stable for e(t) = 0 and therefore ISS with respect toe(t). This is consistent with the shown theorem,which shows that (t)is bounded for boundede(t)and converges to zero asymptotically if e(t)vanishes.

The ISS property is exploited in the analysis of thedelayed case.

Assuming a delay of 0 in all channels, thecontrol law is u(t) = Lx(t ) and ( t) = L(t ) Le(t )

Construction


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Consider the multi-agent system of single-

integrator agents with controllaw . Suppose the triggerfunction is given by

with constants c0 0 and c1 0, c0+ c1> 0, and0 < < 2(G). Then, for all initial conditionsx0R

N, the closed-loop system does notexhibit Zeno behavior. Moreover, thedisagreement vector of the closed-loopsystem converges to a ball centered at theorigin with radius r = L c0/2(G).

Theorem

, ,i it u t i V x i

i i j

j N

x t x tu t

0 1, ti i it e t e t c c ef

N


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The graph G is assumed to be undirected in this paper.However, it can be shown that the theorem extends to

strongly connected and balanced directed graphs. The radius r = L c0/2(G) can be chosen arbitrarily

small since it scales with c0. Forc1= 0 the density of events is independent of c0forlarge t, but for small t the inter-event intervals are

short if c0is small. This motivates why c1> 0 might besuitable in practice. The theorem states that the radius vanishes for c0= 0.

The closed-loop system reaches average consensusasymptotically in this case. The condition < 2(G) is

intuitive, because the states should converge fasterthan the threshold decreases. However, if c0> 0, a positive lower bound on the inter-

event times exists for > 2(G) as well. Consequently,knowledge of 2(G) is not necessary if c0> 0.

Remarks

N


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Simulation network

Single-Integrator agents

Double-Integrator agents

Simulations


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A network of five agents with G as shown here is

considered.

The theoretical results are illustrated through

simulations and the event-based control strategy iscompared to traditional time-scheduled control.

Simulation network


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considered.



Single-integrator agents

S


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considered.



Single-integrator agents cont.

D bl i


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considered.



Double-integrator agents


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???

Summary and future work


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Construction

Theorem

Double-integrator agents

C t ti


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Consider the multi-agent system (1) with

control law (4). Suppose the trigger function isgiven by fi(t, ei(t)) = |ei(t)| c0 + c1e t(7) with constants c0 0 and c1 0, c0 + c1 >0, and 0 < < 2(G). Then, for all initial

conditions x0 R N , the closed-loop systemdoes not exhibit Zeno behavior. Moreover, thedisagreement vector of the closed-loopsystem converges to a ball centered at theorigin with radius r = L Nc0/2(G).

Construction

Th


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Consider the multi-agent system (2) with

control law (18). Suppose the trigger functionis given by fi(t, e ,i(t), e ,i(t)) = e,i(t) e ,i(t) c0 + c1e t (22)with constants c0 0 and c1 0, c0 + c1 > 0,

and 0 < < |Re(3( ))|. Then, for all initialconditions 0, 0 R N , the closed-loopsystem does not exhibit Zeno behavior.Moreover, the disagreement vector of theclosed-loop system converges to a ballcentered at the origin with radius r = c0cV L 2N/|Re(3( ))|. (23)

Theorem

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Event-based broadcasting for multi-agent average consensus.pdf