arXiv:1501.06678v2 [cs.SY] 29 Jan 2016 measurements. [email protected] ink ag e.+86-0731-84576455. Tel. Wang. Xiangke e words: Key Fina analysis. rate. theoretical convergence the the verify to of and given estimates interval are the quantization provide the also of and relationship the i equ out agreement and desired figure the neighborhood to agreement exponentially for converge While the agents interval. quantization of the radius with increases the qu radius uniform of for bound Particularly, before. logarit upper addressed and uniform been not both o for has connection properties mathematical convergence explicit the the and pro reveal the also of stability but the law, guarantee log seco only and o not uniform of do based both We problem which considered. dynamics in agreement studied, agreement is edge edge system quantized multi-agent the The of subgraph. model tree reduced a obtain ⋆ the about concept important an second-ord a edge introduce of the we problem Under measurements. agreement quantized under edge system agent the explores work This Abstract rpitsbitdt Elsevier to submitted Preprint deAreeto ut-gn ytmwith System Multi-agent of Agreement Edge mi addresses: Email hspprwsntpeetda n etn rjunl Corre journal. or meeting any at presented not was paper This unie esrmnsvateDirected the via Measurements Quantized a olg fMcarnc n uoain ainlUniversit National Automation, and Mechatronics of College hwnZeng Zhiwen deareet deLpain ut-gn ytm quant system, multi-agent Laplacian, edge agreement, Edge [email protected] XageWang), (Xiangke a deLaplacian Edge ink Wang Xiangke , ehooy 103 China 410073, Technology, [email protected] seta deLaplacian edge essential a hqagZheng Zhiqiang , Zie Zeng), (Zhiwen oaihi unies the quantizers, logarithmic lbim nadto,we addition, In ilibrium. nies epoiethe provide we antizers, h unie interval quantized the f oe unie control quantized posed mcqatzr,which quantizers, hmic rtmcqatzr are quantizers arithmic h ovrec speed convergence the l,smlto results simulation lly, reetframework, greement rnniermulti- nonlinear er Ziin Zheng). (Zhiqiang dodrnonlinear nd-order fDefense of y dct htthe that ndicate pnigauthor sponding h spanning the n 1Jn 2021 June 11 a ized n also and 1 Introduction

Graph theory contributes significantly in the analysis and synthesis of multi- agent systems, since it provides natural abstractions for how information are shared among agents in a network [1,2,3,4]. Pioneering researches on edge agreement [5,6] not only provide totally new insights that how the spanning trees and cycles effect the performance of the agreement protocol, but also set up a novel systematic framework for analysing multi-agent systems from the edge perspective. In our previous work [7], the concept of edge Laplacian was extended to more general directed graphs and the classical input-to-state nonlinear control methods together with the recently developed cyclic-small- gain theorem were successfully implemented to drive multi-agent system to reach robust consensus.

Early efforts on multi-agent systems mainly focuses on the study with high accurate data exchanging among agents. However, it is hard to be guaranteed in the real digital networks when considering that communication channel has a limited bandwidth, and energy used for transmission is generally re- strained. Frankly speaking, constraints on communication have a considerable impact on the performance of multi-agent system. To cope with the limita- tions, the measurement data are always processed by quantizers. In practice, to realize the quantized communication scheme, a encoder-decoder pair is em- ployed. Generally, the quantized data is always encoded by the sender side before transmitting and dynamically decoded at the receiver side. Recently, the gossiping algorithms [8], the coding/decoding schemes [9] and nonsmooth analysis [10] have been proposed to solve the coordination control problem of first-order multi-agent systems with quantized information. However, as known that second-order multi-agent systems have significantly different co- ordination behaviour even if agents are coupled through similar topology. To the best of authors’ knowledge, there are still little works explore the quan- tization effects on second-order dynamics. Considering different quantizers, [11] studies the synchronization behaviour of mobile agents with second-order dynamics under an undirected graph topology. The authors also point out that the quantization effects may cause undesirable oscillating behaviour un- der directed topology. To more recent literature [12], the authors address the quantized consensus problem of second-order multi-agent systems via sampled data under directed topology. Considering the fact that quantization intro- duces strong nonlinear characteristics such as discontinuity and saturation to the system, the control law designed for the ideal case may lead to instability. The research on second-order multi-agent systems in the presence of quantized measurements under directed topology is still open.

While the analysis of the node agreement (consensus problem) has matured, work related to the edge agreement has not been deeply studied yet. In this

2 paper, we are going to explore the quantization effects on the edge agreement problem of second-order nonlinear multi-agent systems. The main contribu- tions are twofold. First, by introducing the essential edge Laplacian, we high- light the role of the spanning tree subgraph and then we can obtain a reduced model of second-order edge agreement dynamics across the spanning tree un- der the edge agreement framework. Second, we propose a general analysis of the convergence properties for second-order nonlinear multi-agent systems under quantized measurements. Unlike previous works [11,12], for uniform quantizers, we provide the explicit upper bound of the radius of the agreement neighborhood and also indicate that the radius increases with the quantization interval. While for logarithmic quantizers, the agents converge exponentially to the desired agreement equilibrium. Moreover, we also provide the estimates of the convergence rate as well as pointing out that the coarser the quantizer is, the slower the convergence speed.

The rest of the paper is organized as follows: preliminaries are proposed in Section 2. The quantized edge agreement with second-order nonlinear dynam- ics under directed graph is studied in Section 3. The simulation results are provided in Section 4 while the last section draws the conclusions.

2 Basic Notions and Preliminary Results

The null space of matrix A is denoted by (A). Denote by In the iden- n×Nn tity matrix and by 0n the zero matrix in R . Let 0 be the column vector with all zero entries. Let = ( , ) be a digraph of order N specified by a node set and an edgeG set V E with size L. The set of neigh- V E ⊆V×V bors of node i is denoted by i = j : ek =(j, i) . The adjacency matrix N N{×N ∈ E} of is defined as AG = [aij] R with nonnegative adjacency elements G ∈ aij > 0 (j, i) ε. The degree matrix ∆G = [∆ij] is a diagonal matrix ⇔ N ∈ with [∆ii]= aij, i =1, 2, , N, and the graph Laplacian of is defined j=1 ··· G by LG ( ) = ∆G AG. Denote by ( ) the L L diagonal matrix of wk, G P − W G × for k = 1, 2 , L, where wk = aij represents the weight of ek = (j, i) . ··· × ∈ E The incidence matrix E ( ) RN L for a directed graph is a 0, 1 -matrix with rows and columns indexedG ∈ by nodes and edges of respectively.{ ± } For edge G ek =(j, i) ,[E ( )]jk = +1, [E ( )]ik = 1 and [E ( )]lk = 0 if l = i, j. The ∈E G N×LG − G 6 in-incidence matrix E⊙ ( ) R is a 0, 1 matrix and for ek =(j, i) , G ∈ { − } ∈E [E⊙ ( )]lk = 1 for l = i,[E⊙ ( )]lk = 0 otherwise. The weighted in-incidence G w − wG matrix E⊙( ) is defined as E⊙( )= E⊙ ( ) ( ). As thus, the graph Lapla- G G G W G w T cian of has the following expression [7]: LG ( )= E⊙( )E( ) . The weighted edge LaplacianG of a directed graph can beG defined asG [7] G G T w Le( ) := E( ) E ( ). (1) G G ⊙ G

3 A spanning tree T = ( , 1) of a directed graph = ( , ) is a directed tree formed by graphG edgesV E that connect all the nodesG ofV theE graph; a co- spanning tree C = ( , 1) of T is the subgraph of having all the vertices and exactlyG thoseV E−E edges of G that are not in . GraphG is called G GT G quasi-strongly connected if and only if it has a directed spanning tree [13]. A quasi-strongly connected directed graph can be rewritten as a union form: = . In addition, according toG certain permutations, the incidence G GT ∪ GC matrix E( ) can always be rewritten as E( )= E ( ) E ( ) as well. Since G G T G C G the co-spanning tree edges can be constructed from the spannin g tree edges via a linear transformation [5], such that

E ( ) T ( )= E ( ) (2) T G G C G −1 with T ( )= E ( )T E ( ) E ( )T E ( ) and rank(E ( )) = N 1 from G T G T G T G C G G − [13]. We define 

R ( )= I T ( ) (3) G  G  and then we have

E ( )= E ( ) R ( ) . (4) G T G G The column space of E( )T is known as the cut space of and the null space of E( ) is called as theGflow space of E( ). Additionally,G the rows of R ( ) G G G form a basis of the cut space of and the rows of T ( )T I form a basis of − G  the flow space, respectively [13].

Lemma 1 ([7]) For a quasi-strongly connected graph , the graph Laplacian G LG( ) and the edge Laplacian Le( ) have the same N 1 nonzero eigenvalues, whichG are all in the open right-halfG plane. −

Lemma 2 ([7]) For a general quasi-strongly connected graph = , G GT ∪ GC Le( ) contains L N +1 zero eigenvalues. Moreover, if the edge set of is G − GC not empty, then zero is a simple root of the minimal polynomial of Le( ). G

3 Quantized Edge Agreement with Second-order Nonlinear Dy- namics under Directed Graph

In this section, the edge agreement of second-order nonlinear multi-agent sys- tems under quantized measurements is studied. To ease the notation, we sim- w w ply use E, E and Le instead of E( ), E ( ) and Le( ). ⊙ G ⊙ G G

4 3.1 Problem Formulation

We consider a group of N networked agents and the dynamics of the i-th agent is represented by

x˙ i(t)= vi(t) (5)

v˙i(t)= f (xi(t), vi(t), t)+ ui(t) (6)

n n n where xi(t) R is the position, vi(t) R is the velocity and ui(t) R ∈ ∈ n n ∈ n is the control input. The nonlinear term f (xi(t), vi(t), t): R R R is × → unknown and satisfies the following assumption:

Assumption 3 For a nonlinear function f, there exists nonnegative con- stants ξ1 and ξ2 such that

f (x, v, t) f (y,z,t) ξ x y + ξ v z , | − | ≤ 1 | − | 2 | − | x,v,y,z Rn; t 0. ∀ ∈ ∀ ≥

The goal for designing distributed control law ui(t) is to synchronize velocities and positions of the N networked agents.

The generally studied second-order consensus protocol proposed in [14] is de- N N scribed as follows: ui(t)= α aij (xj (t) xi (t)) + β aij (vj (t) vi (t)), j∈Ni − j∈Ni − for i = 1, 2 , N, where αP > 0 and β > 0 are theP coupling strengths. As in [15], we··· assume that each agent i has only quantized measurements of relative position Q (xi xj) and velocity information Q (vi vj), where − − Q (.): Rn Rn denotes the quantization function. Therefore, the protocol can be modified→ as

N N ui(t)=α aijQ (xj (t) xi (t)) + β aijQ (vj (t) vi (t)) (7) ∈N − ∈N − jXi jXi for i = 1, 2 , N. In this paper, two typical quantization operators are con- ··· sidered: uniform and logarithmic quantizer. For a given δu > 0, a uniform quantizer qu : R R satisfies qu (a) a δu, a R; for a given δl > 0, → | − | ≤ ∀ ∈ a logarithmic quantizer ql : R R satisfies ql (a) a δl a , a R. The → | − | ≤ | | ∀ ∈ positive constants δu and δl are known as quantization interval. For a vector T n ∆ T ν = [ν1, ν2, , νn] R , Let Qu (ν) = [qu (ν1) , qu (ν2) , , qu (νn)] and ∆ ··· ⊂ T ··· Ql (ν) = [ql (ν ) , ql (ν ) , , ql (νn)] . Then we obtain the following bounds: 1 2 ··· Qu (ν) ν √nδu, Ql (ν) ν δl ν . | − | ≤ | − | ≤ | |

5 Considering the dynamics of the networked agents described in (5) and (6), by directly applying the quantized protocol (7), we obtain

x˙ i (t)= vi (t)  N v˙i (t)= f (xi (t) , vi (t) , t)+ α aijQ (xj (t) xi (t))  j∈Ni −   N P + β aijQ (vj (t) vi (t))  j∈Ni −   P  To ease the difficulty of the analysis, we technically chose α = σ2 and β = σ3 (σ > 0) as in [16]. The biggest advantage to using this trick is that we can easily construct a positive definite matrix which will be used in the proof of the main results. As thus, the system can be collected as

x˙ (t)= v (t) 2 w T v˙ (t)= F (x (t) , v (t) , t) σ (E In)Qˆ (E In)x (t) (8)  − ⊙ ⊗ ⊗  3 w T  σ (E In)Qˆ (E In)v (t)   − ⊙ ⊗ ⊗    

with x(t), v(t) and F (x(t), v(t), t) denoting the column stack vector of xi(t), vi(t) and f (xi(t), vi(t), t); and Qˆ represents the vector form of the quantization function Q.

T T Define xe = (E In)x and ve = (E In)v, which denote the difference ⊗ ⊗ of position and velocity of two neighbouring nodes respectively. We suppose ˆ ˆ exe = Q (xe) xe and eve = Q (ve) ve as in [15]. Then by left-multiplying T − − E In of both sides of (8), we have ⊗

x˙ e (t)= ve (t) T 2 3 v˙e (t)=(E In)F σ (Le In)xe σ (Le In)ve (9)  ⊗ − ⊗ − ⊗  2 3 σ (Le In)exe σ (Le In)eve . − ⊗ − ⊗   The edge agreement dynamics (9) describes the evolution of the edge states T T T z = xe ve , which depends on its current state and its adjacent edges’   states. In comparison to the node agreement (consensus), the edge agreement, rather than requiring the convergence to the agreement subspace, expects the edge dynamics (9) to converge to the origin, i.e., limt→∞ xe (t) = 0 and | | limt→∞ ve (t) = 0. | |

6 3.2 Main Results

For the quasi-strongly connected graph , the incidence matrix can be written T G as E = . Let z = T T denotes the states across the spanning ET EC T xT vT     T T T T T tree T with xT =(E In)x, vT =(E In)v and zC = x v denotes the G T ⊗ T ⊗ C C T   T states across the cos-spanning tree C with xC =(EC In)x, vC =(EC In)v, respectively. Notice that E T ( )G = E as mentioned⊗ in (2); therefore⊗ the T G C co-spanning tree states can be reconstructed through matrix T , i.e., xC (t) = T T (T ( ) In)xT (t) and vC (t)=(T ( ) In)vT (t). Moreover, based on the ob- G ⊗ G ⊗ T servation that E = ET R ( ) form (4), then we can obtain xe = R ( ) InxT , T G G ⊗ ve = R ( ) Inv and G ⊗ T

T R ( ) In 0¯ z = G ⊗ zT (10)  T  0¯ R ( ) In  G ⊗    with zero matrix 0¯ of compatible dimension.

To simplify the subsequent analysis, the essential edge Laplacian will be em- ployed, which helps us to obtain a reduced model of the closed-loop multi-agent system based on the spanning tree subgraph . GT Before moving on, we introduce the following transformation matrix:

T −1 T −1 R ( ) R( ) R ( ) Se ( )= R( ) θ ( ) Se( ) = G G G G e G  T   G G  θe ( )  G    where R ( ) is defined via (3) and θe ( ) denotes the orthonormal basis of the G G flow space, i.e., Eθe ( ) = 0. Since rank(E) = N 1, one can obtain that G T − dim(θe ( )) = (E) and θe ( ) θe ( )= IL−N . Applying the above similar G N G G +1 transformation lead to

ˆ T w −1 Le ET E⊙θe ( ) Se( ) LeSe ( )=  G  (11) G G 0¯ 0¯     T w T where Lˆe = E E R( ) is referred to as the essential edge Laplacian. For the T ⊙ G essential edge Laplacian, we have the following lemma.

Lemma 4 The essential edge Laplacian Lˆe contains exactly N 1 nonzero − eigenvalues of Le.

−1 PROOF. Based on the similar transformation Se ( ) and Se( ) , the eigen- G G

7 values of the block matrix (11) are the solution of

(L−N+1) λ det λI Lˆe =0 −   which shows that Lˆe contains exactly all the nonzero eigenvalues of Le from Lemma 1 and 2.

Meanwhile, we can construct the following Lyapunov equation as

T HLˆe + Lˆ H = IN− (12) e − 1 where H is positive definite.

Next, we will provide a reduced multi-agent system model in terms of the corresponding dynamics across T . For edge dynamics (9), we make use of the following transformation G

−1 xT −1 vT (Se In)xe =   (Se In)ve =   . ⊗ 0 ⊗ 0         ET T w −1 T T −1 Since E = ET R ( ) and Le = E E⊙, we have Se E =   and Se Le = G 0     T w T E E⊙ T T T ˆ T w  . Then we define ω = exe eve and let LO = ET E⊙. By using 0     the similar transformation (11), system (9) finally can be recast into a compact matrix form as follows:

z˙ = +( In)z +( In)ω (13) T FT LT ⊗ T LT 1 ⊗

0N−1 IN−1 0N−1×L 0N−1×L 0 with T =  , T 1 =   and T =  . L 2 ˆ 3 ˆ L 2 ˆ 3 ˆ F T σ Le σ Le σ LO σ LO (E In)F − −  − −   T ⊗        Remark 5 The decomposition of the spanning tree and co-spanning tree sub- graph has been wildly applied to solve many magnetostatic problems, such as tree-cotree gauging [17], finite element analysis [18]. As is well known, the spanning tree plays a vital role in the stability analysis of networked multi- agent system. Under the edge agreement framework, we reveal the connection of the algebraic properties and the graph structure and highlight the role of the spanning tree subgraph by introducing the essential edge Laplacian.

8 To further look at the relation between the quantization interval and the edge agreement, we propose the following theorem.

Theorem 6 Considering the quasi-strongly connected directed graph as- T G sociated with the edge Laplacian Le , suppose = + with Q − PLT LT P   σH H = , where H is obtained by (12). If σ > λmax(H) +1 and P   2 H σH q   λ ( ) 2max (ξ ,ξ ) > 0. Then, under the quantized protocol (7), min Q − 1 2 kPk system (13) has the following convergence properties:

(1): With uniform quantizers, the agents converge to a ball of radius

2√2nLδu z kPLT 1 k (14) T ≤ λ ( ) 2max (ξ ,ξ ) min 1 2 Q − kPk which is centred at the agreement equilibrium;

(2): With logarithmic quantizers, the agents converge exponentially to the de- sired agreement equilibrium, provided that δl satisfies

λmin ( ) 2max (ξ1,ξ2) δl < Q − kPk. (15) 2 RT kPLT 1 kk k The estimated trajectories of the edge Laplacian dynamics (13) is as

λmax( ) − π t λmax(P) zT (t) P e zT (0) for t 0 | | ≤ λmin( ) | | ≥ P with

T π = λ ( ) 2max (ξ ,ξ ) 2δl R . min Q − 1 2 kPk − kPLT 1 k

PROOF. For the edge Laplacian dynamics (13), we choose the following Lyapunov function candidate:

T V (z )= z ( In)z (16) T T P ⊗ T

σH H in which =   , where H can be obtained from (12) and is positive P H σH P   definite while choosingσ> 1.

9 By taking the derivative of (16) along the trajectories of (13), we have

T T T V˙ (z )=z In + In z +2z ( In) T T PLT ⊗ LT P ⊗ T T P ⊗ FT  T  +2zT ( T 1 In)ω T PL ⊗ T T = z ( In)z +2z ( In) +2z ( In)ω − T Q ⊗ T T P ⊗ FT T PLT 1 ⊗ in which

2 3 σ IN− σ IN− σH T 1 1 − = T + T =   . Q − PL L P 3 4   σ IN−1 σH σ IN−1 2H  − −   

1 2 2 3 Let = Q Q  with 1 = σ IN−1, 2 = σ IN−1 σH and 3 = Q T Q Q − Q 2 3 4 Q Q  σ IN− 2H . According to Schur complements theorem [14], by selecting 1 − λ (H) σ > max +1 s 2

T −1 2 then we have > 0 and = H (2(σ 1) IN− H) > 0, so Q1 Q3 − Q2 Q1 Q2 − 1 − that is positive definite. Q In the meanwhile, we notice that

T = (E In) max (ξ ,ξ ) z . (17) |FT | T ⊗ F ≤ 1 2 | T |

For uniform quantizers, we can calculate the upper bound of the quantization error as

ω √2nLδu. (18) | | ≤ By combining (17) and (18), one can obtain

2 V˙ (z ) λ ( ) z +2max (ξ ,ξ ) z 2 T ≤ − min Q T 1 2 kPk| T |

+2√2nLδu z kPL T 1 k| T | = z ( λ ( )+2max (ξ ,ξ ) ) z T − min Q 1 2 kPk T  √ +2 2 nLδu T 1 . kPL k Clearly, the edge agreement can be reached and the radius of the agreement neighbourhood is as (14).

For logarithmic quantizers, according to Ql (a) a δl a and equation (10) | − | ≤ | |

10 , we have

T ω δl z δl R z . (19) | | ≤ | | ≤ | T |

Combining (17) and (19), we have

2 V˙ (z ) λ ( ) z +2max (ξ ,ξ ) z 2 T ≤ − min Q T 1 2 kPk| T | T 2 +2δl R z kPLT 1 k | T | 2 = π z . − T

Obviously, while the condition (15) is satisfied, the edge Laplacian dynamics (13) converges exponentially to the desired agreement equilibrium.

Moreover, one can obtain that

2 π V˙ (z (t)) π z V (z (t)) . T ≤ − T ≤ −λ ( ) T max P By applying the Comparison Lemma [19], we have

− π t V (z (t)) e λmax(P) V (z (0)) . T ≤ T Then we can provide the following estimates of the convergence rate for the closed-loop multi-agent system

λmax( ) − π t λmax(P) zT (t) P e zT (0) for t 0. (20) | | ≤ λmin( ) | | ≥ P Remark 7 From equation (14), one can see that the radius of the conver- gence neighbourhood trends to zero as the quantization interval δu decreases. Additionally, for logarithmic quantizers, the corresponding convergence rate of the quantized system depend on δl, i.e., the coarser the logarithmic quantizer is, more time it takes to converge. As the convergence time can be used to mea- sure the performance of the quantized control law, we provide the estimation of the upper bound of the convergence time based on (20) as follows: λ ( ) λ ( ) r T = max P ln min P − π λ ( ) z (0) max P | T | where r > 0 denotes the expected radius of the agreement error.

4 Simulation

Consider multi-agent system consisting of a group of 5 agents associated with a quasi-strongly connected graph as shown in Fig 1, where e , e , e , e 1 2 3 4 ⊂ GT

11 and e . 5 ⊂ GC 2 e e 1 2 1 3 e e e 3 5 4 4 5

Fig. 1. A quasi-strongly connected graph of 5 agents.

The dynamics of the i-th agent is described as (5) and (6) with xi (t) , vi (t) 3 ui (t) R . Let x(m, :) and v(m, :) denote the column vector of the m-th (m ∈ =1,2,3) variable of x (t) , v (t) respectively. The inherent nonlinear dynamics 3 3 f (xi(t), vi(t), t): R R R is described by Chua’s circuit × →

f (xi(t), vi(t), t)=(ζ ( vi1(t)+ vi2(t) l (vi1(t))) , − − T τ(vi (t) vi (t)+ vi (t)), χvi (t)) 1 − 2 3 − 2 where l (vi (t)) = bvi (t)+0.5(a b)( vi (t)+1 vi (t) 1 ). The system is 1 1 − | 1 |−| 1 − | chaotic when ζ =0.01, τ =0.001, χ =0.018, a = 4/3 and b = 3/4. In view − − −3 of Assumption 3, simple calculation leads to ξ1 = 0 and ξ2 = 4.3871 10 [14]. ×

Suppose that the weighted diagonal matrix is = diag 0.12, 0.24, 0.44, 0.43, 0.09 . By choosing σ =1.64, we have W { }

0.21 0.09 0.00 0.09 0.12 0.00 0.00 0.00 −0.09 ˆ −0.12 0.24 0.00 0.00 ˆ −0.12 0.24 0.00 0.00 0.00 Le = 0.00 −0.24 0.44 0.00 , LO = 0.00 −0.24 0.44 0.00 −0.00 . 0.00 −0.24 0.00 0.43! 0.00 −0.24 0.00 0.43 0.00 !

4.1 Uniform Quantizer

First, we consider the quantized protocol (7) with the following uniform quan- tizer as the one used in [11],

x 1 qu (x)= δu + . (21)  δu  2

The simulation results with δu = 1 are shown in Fig. 2, from which we can see that xe(t) and ve(t) indeed converge to a small neighbourhood near the equilibrium points. To show the effect of δu on the agreement error ess , we | | further take δu =0.01, 0.1, 2 and 3 to run the simulation. The results in Tab.1 shows that error trends to zero when δu 0, as shown in Theorem 6. →

12 40 25

30 20

15 20 x(2,:)

10 10 v(1,:) x(3,:) 5 v(2,:)

x 0 v 0 −10 x(1,:) −5 v(3,:)

−20 −10

−30 −15

−40 −20 0 5 10 15 20 25 30 0 5 10 15 20 25 30 t(seconds) t(seconds)

60 40

40 30

20 20

0

e e 10 x v −20 0.5 e 0 x 0 −0.5 −40 19.998 20 20.002 0.05 e 0

t(seconds) v −10 −0.05 −60 19.98 20 20.02 t(seconds) −80 −20 0 5 10 15 20 25 30 0 5 10 15 20 25 30 t(seconds) t(seconds)

Fig. 2. Edge agreement under uniform quantizer with δu = 1. 4.2 Logarithmic Quantizer

The logarithmic quantizer we apply to (7) is an odd map ql : R R [11], → equ(ln x) when x> 0  ql = 0 when x =0 (22)   equ(ln(−x)) when x< 0  −   −δu where qu is defined as (21) and the parameter δl = 1 e . To satisfy the − stability constraints (15), we require δl < 0.0301. The simulation results with −0.01 δu =0.01 and δl =1 e =0.01 are shown in Fig.3, from which we can see − that the agreement is indeed achieved as well as xi(t) and vi(t) converge to the desired agreementπ values. The estimation of the convergence rate is given λmax(P) − t as ψ = e λmax(P) z (0) for t 0 with π = 0.5387. From Fig.4, one | | λmin(P) | T | ≥ Table 1 The effect of δu on the agreement error.

δu 0.01 0.1 1 2 3 error 0.005 0.05 0.62 1.98 4.45

13 can see that z exponentially converge to the origin. | T |

40 25

30 20

x(2,:) 15 20 10 v(1,:) 10 x(3,:) 5 v(2,:)

x 0 v 0 −10 x(1,:) −5 v(3,:)

−20 −10

−30 −15

−40 −20 0 5 10 15 20 25 30 0 5 10 15 20 25 30 t(seconds) t(seconds)

60 40

40 30

20 20

0

e e 10 x v −20 0.05 e 0 x −0.05 0 −40 19.99 20 20.01 0.02

t(seconds) e 0 −10 v −60 −0.02 19.995 20 20.005 t(seconds) −80 −20 0 5 10 15 20 25 30 0 5 10 15 20 25 30 t(seconds) t(seconds)

Fig. 3. Edge agreement under logarithmic quantizer with δl = 0.01.

350 |z | T |ψ| 300

250

200 | T |z 150

100

50

0 0 5 10 15 20 25 30 t(seconds)

Fig. 4. The convergence estimation of z (t) . | T |

14 5 Conclusions

In this paper, we presented an important concept about the essential edge Laplacian and also obtained a reduced model of the second-order edge agree- ment dynamics across the spanning tree subgraph. Under the edge agreement framework, the synchronization of second-order nonlinear multi-agent systems under quantized measurements was studied. We revealed the explicit mathe- matical connection of the quantized interval and the convergence properties for both uniform and logarithmic quantizers. Specifically, we obtained the upper bound of the radius of the agreement neighbourhood for uniform quantizers, which indicates that the radius increases with the quantization interval. While for logarithmic quantizers, we pointed out that the agents converge exponen- tially to the desired agreement equilibrium. In addition, we also provided the estimates of the convergence rate.

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