Proceedings of the 7th Asian Control Conference,Hong Kong, China, August 27-29,2009

ThB7.1

Distributed Consensus of Multi-Agent Systems with Finite-LevelQuantization

Tao Li, Minyue Fu, Lihua Xie and Ji-Feng Zhang

Abstract- This paper is concerned with consensus control ofundirected networks of discrete-time first-order agents underquantized communication. A distributed protocol is proposedbased on dynamic encoding and decoding with finite level uni­form quantizers. It is shown that under the protocol designed,for a connected network, average-consensus can be achievedwith an exponential convergence rate based on a single-bitinformation exchange between each pair of adjacent nodes ateach time step. As the number of agents increases, the explicitform of the asymptotic convergence rate is given in relation tothe number of nodes, the number of the quantization levels andthe ratio between the algebraic connectivity and the spectralradius of the Laplacian of the communication graph.

I. INTRODUCTION

Recently distributed consensus and average-consensusproblems have been paid much attention to by the controlcommunity ([1]-[6]). Many effective distributed control andestimation algorithms are proposed based on consensus al­gorithms ([7]-[8]). However, most of the works in the aboveliterature use an ideal communication model between agents.When agents have real-valued states, this assumption isequivalent to the requirement that the communication chan­nels between agents have unlimited capacity (bandwidth). Itis well known that in real digital networks, communicationchannels have a finite channel capacity, and quantizationplays an important role. Therefore, consensus problems un­der quantized communication become interesting and moremeaningful.

In [9]-[11], average-consensus algorithms were designedwith each agent having an integer-valued state. These algo­rithms can drive each agent to some interger approximationof the average of the initial states. In [12]-[14], quantizedaverage-consensus problems were studied with real-valuedstates. In [12]-[13], algorithms with an uniform quantizerof infinite levels were proposed to ensure the boundnessof the consensus error. Furthermore, an algorithm based ondynamic quantization was proposed in [14]. The number ofquantization levels, however, will diverge as the number of

This work was supported by Agency for Science, Technology andResearch under Grant SERC 052 101 0037, NSFC of China under Grants60828006, 60674038 and NSFC-Guangdong Joint Foundation (U0735003).

T. Li is with Key Laboratory of Systems and Control, Institute of SystemsScience, Academy of Mathematics and Systems Science, Chinese Academyof Sciences. Email: litao@amss.ae . en

M. Fu is with School of EE&CS, University of Newcastle, Australia.Email: minyue.fu@neweastle.edu.au

L. Xie is with the School of Electrical and Electronic Engi­neering, Nanyang Technological University, Singapore 639798. Email:elhxie@ntu.edu.sg

J. F. Zhang is with Key Laboratory of Systems and Control, Institute ofSystems Science, Academy of Mathematics and Systems Science, ChineseAcademy of Sciences, Beijing 100190, China. Email: jif@iss.ae . en

agents increases. In [15], random dither was used to makethe quantization error a "white" noise. Then the distributedstochastic approximation method ([16]-[18]) was applied toachieve approximate average-consensus.

It is well known that for feedback stabilization of lineartime-invariant systems with communication constraints, theminimal bit rate (channel capacity) was given in [19] and[20] and the case with logarithmic quantizers was consideredin [21] and [22]. Naturally, one may ask, for distributedcooperative control problems of multiagent systems withfinite communication data rate, how many bits of informationdoes each pair of adjacent agents need to exchange at eachtime step to achieve consensus of the whole network?

In this paper, we consider the average-consensus controlfor discrete-time first-order undirected networks with a finitecommunication date rate. Each agent has a real-valued statebut can only exchange symbolic data with its neighbors.The communication between agents is based on dynamicencoding and decoding with finite-level quantization. Wedesign a distributed protocol with error compensation. Theprotocol is characterized by three parameters: the controlgain, the scaling function, and the number of quantizationlevels.

We show that if the network is connected, then for anygiven number of quantization levels, the control gain andthe scaling function can be chosen properly such that averageconsensus can be asymptotically achieved. In particular, thecontrol parameters can be properly chosen such that average­consensus can be achieved by using a single-bit quantizer.This indicates that no matter how large a network is, aslong as it is connected, one can always design a distributedprotocol to ensure average-consensus with merely one bitinformation exchange between each pair of adjacent agentsat each time step. We also give the relationship between theconvergence rate and the number of quantization levels. Weshow that under the protocol designed, the consensus errorvanishes exponentially, and faster convergence requires morebits for quantization.

Our proposed consensus algorithm may be applied tothe distributed estimation over large scale sensor networks,where the number of nodes is often large. This give riseto investigating the asymptotic property of the closed-loopsystem as the number N of nodes approaches infinity. Weshow that in some sense, the asymptotic optimal convergence

rate is O(exp{ - K~ t} ) when using a (2K +1)-level quan-2v N . flectitizer, where Q N, an important physical quantIty re ecting

the synchronizability of a network, is the ratio between thesecond smallest eigenvalue (algebraic connectivity) and the

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0, -1/2 < Y < 1/2,i, 2i-1 < Y < 2i+1

2 - 2'q(y) == i == 1,2, ... ,K -1, (4)

K, y > 2K-I- 2 '

-q(-y), y ~ -1/2.

In (2), the ith agent needs the perfect state information ofits neighbors. In this paper, we assume that perfect stateinformation is not available, but only symbolic data can beexchanged between agents, and the communication channelsare modeled as noiseless digital channels each with a pairof encoder and decoder. The encoder <I>j of the jth agent isgiven by

where ej (t) is the internal state of <I>j, and ~j (t), which isthe output of <I>j, is sent to the neighbors of the jth agent.Here, q(.) is a finite-level uniform quantizer, and 9 (t) > °is a scaling function.

The quantizer q(.): JR ---+ r is a map from lR to the set rof quantized levels. In this paper, we consider a finite-leveluniform quantizer with

r == {O,±i, i == 1,2, ...K}.

The number of quantization levels is 2K +1. The associatedquantizer q(.) is given by

(5)

(2)

(3)

0,9 (t - 1)~j (t) + ej (t - 1),

( Xj ( t ) - ~j ( t - l ) )q g(t-l) ,

t == 1,2, ...

0,g(t - 1)~j(t) + Xji(t - 1),t == 1,2, ...

Remark 1: The encoder <I>j is a scaled difference encoder,and ej (t) is a one-step predictor. In this difference encodingalgorithm, at each time step the "prediction error", x j (t) ­ej (t - 1) is quantized. Generally speaking, the amplitude ofprediction error is smaller than that of state x j (t) itself, soit can be represented by fewer bits.

Remark 2: If consensus is achieved, then the predictionerror x j (t) - ej (t - 1) vanishes as t ---+ 00. Therefore,intuitively the scaling function g(t) should satisfy the fol­lowing properties. On one hand, g(t) should converges tozero asymptotically make the quantizer persistently excited,such that the agents receive the their neighbors' informationcontinuously. On the other hand, g(t) should be large enoughsuch that the quantizer will not be saturated.

For each communication channel (j, i) E E, the ith agentreceives ~j (t), and then uses the following decoder \lJji toestimate x j (t ) :

i == 1,2, ... ,N.

B. Protocol design

In [4], a weighted-average protocol was proposed:

Ui(t) == L aij(Xj(t) - Xi(t)), t == 0,1, ... ,jENi

largest eigenvalue (spectral radius) of the Laplacian matrixof the topology graph. Our result shows that when thecommunication data rate is limited, the convergence rate ofdistributed consensus not only depends on the connectivitybut also the synchronizability of the communication graph.

The remainder of this paper is organized as follows. InSection II, we present the model of the network, designthe distributed protocol, and formulate the problem to beinvestigated. In Section III, we prove that under the protocoldesigned and mild conditions, average-consensus can beachieved with an exponential convergence rate. Then weanalyze the asymptotic performance as N ---+ 00 and give anexplicit form of the asymptotic convergence rate. In SectionIV, we give a numerical example to demenstrate our results.In Section V, we give some concluding remarks.

The following notation will be used throughout this paper:1 denotes a column vector with all ones. I denotes theidentity matrix with an appropriate size. For a given set 5, thenumber of its elements is denoted by 151 . For a given vectoror matrix A, its transpose is denoted by AT, its Euclideannorm is denoted by IIAII2. For a given positive number x, themaximum integer less than or equal to x is denoted by lxJ;the minimum integer greater than or equal to x is denoted

by rxl·II. PROBLEM FORMULATION

A. Average-consensus problem

In this paper, the dynamics of each agent is modeled as adiscrete-time first-order integrator:

Xi(t + 1) == Xi(t) + hUi(t), t == 0,1, ... , i == 1,2, ... , N, (1)

where Xi (t) E lR is the ith agent's state, Ui (t) E lR is theith agent's control input, and h is the control gain. Theinformation flow among agents are modeled as an undirectedgraph g == {V, E, A}, where V == {I, 2, ..., N} is the set ofnodes with i representing the ith agent, E is the set of edgesand A==[aij]ElRN X N is the weighted adjacency matrix ofg. An edge denoted by the unordered pair (j, i) representsa communication channel from j to i. Note that A is asymmetric matrix. For any i, j E V, aij==aji 2:: 0, and

aij > °if and only if j E Ni. Also, deg, == L~l aijis called the degree of i, and d* ==maxi deg, is called thedegree of g. The Laplacian matrix of g is defined as E ==V - A, where V == diag(deg., ... , deg N ) . A sequence ofedges (iI, i2), (i 2, i3), ..., (ik-l, ik) is called a path fromnode i l to node ike The graph g is called a connected graphif for any i, j E V, there is a path from i to j.

The dynamic system (1) together with the communicationgraph g is usually called a dynamic network ([4]). A groupof controls U == {Ui, i == 1, 2... , N} is called a distributedprotocol if for all i, Ui(t) only depends on Xi(S) and Xj(s),j E Ns, S ~ t. The average-consensus control is to design adistributed protocol for the dynamic network, such that forany initial values Xl (0), ..., XN(O), all the agents asymptoti­cally reach an agreement with -k L~l Xj(O) when t ---+ 00.

That is, -k L~l Xj(O) can be computed asymptotically ina distributed manner.

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is a one-bit quantizer.

We propose a distributed protocol as

(12)X(t + 1) == (I - h£')X(t) - hAe(t).

The last term in (11), which we call an error-compensationterm, plays an important role in our protocol. Substituting theprotocol (3), (5) and (7) into the system (1) leads to

{X(t + 1) == (I - h£')X(t) + h£'e(t),X(t+ 1) = g(t)Q[XCt+;{tjXCt)] + X(t),

where Q([Y1, ...,YN]T) == [q(Y1), ...,q(YN)]T. From theabove, noting that J N Lc == 0, we have

1 N 1 NNLXj(t+1)= NLXj(t), t=O,l, ...

j=l j=l

It can be seen that the closed-loop system preserves the av­erage state under the protocol (7). If the error-compensationterm is removed, then by (11), the protocol (7) reduces tothe protocol (9), and the closed-loop system becomes

(7)

(6)

i == 1,2, ...N.

Ui(t) == L aij(Xji(t) - ei(t)), t == 0,1, ... ,jENi

where Xji(t) is the output of Wji.

Remark 3: When the output ~j (t) of the quantizer is zero,the jth agent does not send any information, so for a (2K +1)-level quantizer q(.), the communication channel (j, i), i E

N, is required to be capable of transmitting pog2(2K)l bitswithout error at each time step. In particular, the quantizerq(.) given by

{

0, -1/2<y<1/2,q(y) == 1, Y 2: 1/2,

-1, y::; 1/2

where A2 (£,) is called the algebraic connectivity of g. Inparticular, if g is connected, then A2 (£,) > 0.

We need the following lemmas.Lemma 3.2: If Assumption AI) holds and h < ANC£) ,

then Ph < 1, where

Generally speaking, the closed-loop system (12) does notpreserve the average state, and worse still, it can be shownthat the closed-loop system (12) may be divergent if e(t) is abounded white noise ([23]). That is why we use the protocol(7) rather than the protocol (9). This type of protocols whichcan preserve the state average have been proposed in [12] andsome similar methodology of error compensation has beenaddressed in [24].

III. FINITE-LEVEL QUANTIZED CONSENSUS

For the protocol designed and the resulting closed-loopsystem (12), We are concerned about whether the networkcan achieve consensus with finite-level quantized commu­nication. If so, how many bits are necessary for each pairof adjacent agents to exchange at each time step? Can wegive a quantitative description of the relationship between theconvergence rate and the control parameters? In this section,we will answer the above questions.

To get the main results, we need the following assump­tions:

AI) o is connected.A2) maxi IXi(O)1 < c; maxi 18i(0)1 ::; C8 , where c, and

C8 are known nonnegative constants.Below is a basic result on Laplacian matrices:Lemma 3.1: ([25]) If g == {V, e, A} is an undirected

graph, then the Laplacian £, is a symmetric matrix, and hasN real eigenvalues, in an ascending order:

(13)

and

Denote

i == 1,2, ...N. (9)

X(t) == [X1(t), ..., XN(t)]T, X(t) == [e1(t), ...,eN(t)]T,e(t) == X(t) - X(t), 8(t) == X(t) - JNX(t), (8)

It can be seen that, in our protocol (7), the control inputof the ith agent consists of three terms. The first term,LjENi aij[Xj(t) - Xi(t)], which is just the control inputof the protocol (2), plays the main role. The second term,- LjENi aij(Xj(t) - Xji(t)), represents the weighted sumof estimation errors for the neighbors' states. The last termLjENi aji (Xi (t) - Xij(t)) is the weighted sum of estimationerrors for Xi(t) by the neighbors.

where J N == -Kr 11 T.

Remark 4: If ei(t) is replaced by Xi(t), then the protocol(7) becomes

Ui(t) == L aij(Xji(t) - Xi(t)), t == 0,1, ...jENi

The protocol (9) is a natural extension of the protocol (2) tothe case with quantized communications and it has somecomputational advantage over the protocol (7). One maywonder why we use the protocol (7) rather than the protocol(9). We give some explanations below.

From (3) and (5), it follows that

Xji(t) == ej(t), t == 0,1, ... , i E N j , j == 1,2, ... , N. (10)

Thus, the internal state ~j (t) of encoder <I> j is equal to theestimates of x j (t) by its neighbors. By the symmetry of Aand (10), the protocol (7) can be rewritten as

Ui(t) == L aij[Xj(t) - Xi(t) - (Xj(t) - Xji(t))jENi+(Xi(t) - ei(t))]L aij[Xj(t) - Xi(t)] - L aij(Xj(t) - Xji(t))jENi jENi+ L aji(xi(t) - Xij(t)). (11)

jENi

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Furthermore, if h < A2(£)~AN(£)' then Ph == 1 - hA2(£).Lemma 3.3: Suppose Assumptions Al)-A2) hold. For

any given h E (0, AN(£)) and, E (Ph, 1), let

1K 1(h,,) lM1(h,, ) - 2J+ 1, (14)

VNh2AJv(£) 1 + hAN(£)M1(h,,) + (15)

2,(,- Ph) 2,'

and

{ Cx 2(,- Ph)(C8 , + hCxAN(£))} (16)90 > max K + ~ , hAN(£) .

Then for any given K 2: K 1(h,,), under the protocol (3),(5) and (7) with the (2K +1)-level uniform quantizer (4) andthe scaling function g(t) == gO,t, the closed-loop system (12)satisfies

1 Nlim Xi(t) == N """" Xj(0), i == 1,2 ..., N.

t---+oo L..Jj=l

The convergence rate of average-consensus ([26]) is de­fined as

For the convergence rate of our algorithm, we have thefollowing theorem.

Theorem 3.1: Suppose the conditions of Lemma 3.3hold, then under the protocol (3), (5) and (7) with the(2K + l)-level uniform quantizer (4), the closed-loop system(12) satisfies

and rasym ~ " where 8(t) is the consensus error defined by(8).

Due to space limit, the proofs of Lemmas 3.2, 3.3 andTheorem 3.1 are omitted here.

Remark 5: Lemma 3.3 says that by using a scalingfunction decaying exponentially and a pog2 (2K1(h, ,))l-bituniform quantizer, the protocol (3), (5) and (7) can ensureaverage-consensus to be achieved asymptotically. It is worthpointing out that for any given hand " the bit numberpog2 (2K1(h, ,))l is a conservative estimate, and in practice,fewer bits may be required. However, the number K1(h,,)

gives us some intuitive clues on the relationship betweenthe number of bits required and the control gain h and thescaling factor ,.

Remark 6: Theorem 3.1 gives an estimate for the conver­gence rate of the consensus. The smaller the" the faster theconsensus error converges to zero. Note that, can be madearbitrarily close to Ph, which is the convergence rate for thecase with perfect communication ([26]). From Lemma 3.3, itis shown that a smaller " namely a faster convergence rate,requires more bits to be communicated, and when , ~ Ph,the required number of bits goes to infinity.

ThB7.1

From Lemma 3.3 and Theorem 3.1, it can be seen that ifthe convergence rate , is fixed (i.e., independnent of N ofagents), then the number of quantization levels, 2K1(h, ,) +1, will increase to infinity as N ~ 00. However, in manycases, we do not expect that the number of bits is too large.To satisfy this requirement, we can use a fixed number ofquantization levels at the cost of slower convergence. Wehave the following result.

Theorem 3.2: Suppose Assumptions Al)-A2) hold. Forany given K 2: 1, let

2OK = {(a, (8)1 a E (0, A2(£) + AN(£))',8 E (p,., 1),

1M 1(a , {3 ) < K + 2}' (17)

where Pa is defined by (13) and M 1(a , {3 ) is defined by(15). Then, (i) OK is nonempty. (ii) For any (h,,) E OK,under the protocol (3), (5) and (7) with g(t) == gO,t and the(2K +1)-level uniform quantizer (4), the closed-loop system(12) satisfies

1 N

lim Xi(t) == N """" Xj(0), i == 1,2 ..., N,t---+oo L..J

j=l

where go is a constant satisfying (16).Proof: (i) Noting that

lim[VNaA~(£) + l+aAN(£)j = ~o--e-O 2A2(£) 2 2'

we know that for any given K 2: 1, there exists a* E

(0, A2(£)~AN (£)) such that

VNa*A~(£) l+a*AN(£) K ~ (18)2A2(£) + 2 < + 2·

By Lemma 3.2, it is known that Pa* == 1 - o." A2(£) < 1.By this and (15), we get

1. M (* ) _ VNa* AJv(£) 1 + a* AN(£)1m 1 a " - , (I") + .

"(---+1 2/\2 J..., 2

Then by (18), we know that there exists ,* E (Pa*, 1), suchthat

M 1(a*,I' *) < K +~.

Therefore (a*,,*) E OK, that is, OK is nonempty.(ii) For any (h,,) E OK, by (17), we know that h E

(0, AN(£))' , E (Ph, 1), and ~ < M1(h,,) < K +~. Thus,by (14), one gets K1(h,,) ~ K, which together with Lemma3.3 leads to the conclusion of the theorem.

Remark 7: From Theorem 3.2, it is shown that as long asthe network is connected, we can always design a distributedprotocol to ensure average-consensus with each agent send­ing merely one bit of information to its neighbors at eachtime step.

The set OK is a plane point set described by threenonlinear inequalities. Generally speaking, it is difficult toget an explicit solution of these inequalities. However, by

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introducing a free parameter to E (0,1), we can get a simpl ealgorithm to choose (h , 'Y) from OK for any given K 2: 1.

Algorithm 1:(i) Choose a constant to E (0,1).(ii) Choo se the control gain h E (0, hi« to)), where

hi« to) = min{ A2 (£) ~AN (£)'2KeoA2(£) }

25

100

50 100 150Tlmelstep]

50

0'0.'-----~---~-----

0'0'-----~---~-----,50

Fig. 3. Curves of states with K = 2, '"Y = 0.975.

25

IK=2 ,h=O.1 ,y=O.975,90=20 I

30' rr----~----~---~

Fig. 2. Curves of states with K = 1. 'Y = 0.9 5.

30' rr----~----~---~

I K=1 ,h=O.1,Y=O.95 ,90=20 1

Tlmelstep]

Fig. I. Network topology of Example I. The edges of the graphare randomly generated according to P {(i , j ) E [ g} = 0.2, for

any unordered pair ii; j).

V. CONCL UDING REMARKS

In this paper, the average-consensus control problem hasbeen considered for undire cted networks of discrete-timefirst-order agents under finite bit-rate communication. Basedon scaled uniform quantization, a dynamic difference en­coding and decoding scheme is used for the communicationbetween each pair of agents. A distributed protocol has beenproposed, where the control input of each agent is a weightedsum of the difference between the estimate of its neighbor'sstate and the internal state of its own encoder. This typeof protocol is equivalent to adding an error compensationterm to the original weighted average type protocol. It isshown that for a connected undirected dynamic network withfirst-order agents, no matter how many agents there are,we can always design a distributed protocol to ensure thataverage-consensus is achieved asymptotically with as few as

Then we have OK = U eoE (O,l) OK,eo'

In many cases, the number N of the network nodes islarge and we are concerned about the asymptotic property asN approaches infinity. In the following, we investigate theasymptotic performance of the closed-loop system. It can beseen that the asymptotic value of 'Y has a very compendiousexpression.

Theorem 3.4: Suppose Assumption AI) holds. Then forany given K 2: 1,

r inf (h ,'Y)EllK 'Y 1rm KQ2 = ,

N v-scx: exp{-~}2VN

h Q - A2(£)were N - AN(£)'

Remark 8: QN is an important physical factor. It is shownthat a network exhibits better synchronizability if QN islarge ([27]) . Theorem 3.4 shows that in some sense, the

best convergence rate we can achieve is O(exp{- ~~ t}).Therefore, the best convergence rate is closely related tothe number of the quanti zation levels, the scale and thesynchronizability of the network.

Due to space limit, the proofs of Theorem 3.3 and 3.4 areomitted here.

OK,eo = {( a , 13)1a E (0, hi« to)) ,f3 = 1 - (1 - to )a ).,2(£ )} '

IV. N UMERI CAL E XAMPLE

In this section, we use an example to demonstrate thevalidity of the proposed consensus protocol and Theorem3.1.

Example 1: We consider a network with 30 nodes and0 - 1 weights 1 shown in Fig. 1. The initial states are chosenas Xi(O) = i, i = 1, ..., 30. The one-bit quantizer is used. Thecurve s of states with h = 0.1, g(t) = 20(0 .95) t are shownin Fig. 2. Then the 5-level quantizer is used. The curves ofstates with h = 0.1, g(t ) = 20(0 .975)t are shown in Fig.3. It can be seen that average-consensus is achieved with anexponential rate in both cases and a smaller 'Y leads to afaster convergence.

(iii) Let 'Y = 1 - (1 - to) h ).,2 (£ ).The result below show that any pair (h , 'Y) generated by

Algorithm 1 belongs to OK and any point in OK can begenerated by Algorithm 1.

Theorem 3.3: For any given K 2: 1, and to E (0,1), let

1 0 - I weights means that aij = 1, if (i, j) E E; otherwise , aij = o.

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7th ASCC, Hong Kong, China, Aug. 27-29,2009

one bit information exchange between each pair of adjacentagents at each time step. It is shown that the convergencerate is closely related to the number of network nodes, thenumber of quantization levels and the synchronizability ofthe network.

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