# Synchronization of Coupled Harmonic Oscillators Using Quantized Sampled Position Data

Authors: Xinjing Wang; Peipei He
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

For coupled harmonic oscillators (CHO), the paper studies the synchronization problem by using the quantized current sampled position data. The logarithmic quantizer is adopted here to quantize the information transmitted; thus the quantization error of the sampled position data can be illustrated as the uncertainty according to the sector bound property. On the basis of that, the synchronization problem is converted into the asymptotical stability of the subsystems and even the solving problem of characteristic equation. Some sufficient conditions ensuring the synchronization of CHO are obtained relating to coupling strength, sampling period, and quantizer parameter. The usefulness of the theoretical result is shown by an example at the end.


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## Body

## 1. Introduction

Recently, as a special multiagent system, CHO has gotten much attention. By assuming that the undirected network is not connectivity, literature [1] shows that the CHO is also able to achieve synchronization. When the CHO is affected by controller loss and communication time-delays, an algorithm is proposed in [2] to ensure the synchronization of the system and a brief procedure of the convergence analysis for such algorithm is given. By using some special analysis tools, literature [3] shows that the CHO can be synchronized under the conditions of mild connectivity for the directed graph. A frequency dependent topology condition is given in [4] to synchronize the CHO. With the measurement noise, literature [5] gives an effective algorithm and the convergence analysis of the CHO for the directed network topologies with or without leaders. A sufficient condition for the synchronized oscillatory motions of CHO is given by the influence of noise for the strongly connected directed graph.The control algorithms proposed in the above literature use both velocity information and position information. In some cases, each oscillator exchanges just the velocity information or the position information with its neighbors. The relating papers which design control algorithms just use velocity information or position information include [6–9]. Assuming that the network is connected, the sufficient conditions are given for the synchronized oscillatory motions by just using velocity information in brief [6]. The results of [6] are generalized to the sampled-data CHO affected by input signals loss in [7]. Literature [8] proposes a distributed control method just using the outdated position states. For both positive and negative coupling strengths, it derives some necessary and sufficient conditions (NSCs) for the synchronization of CHO. In [9], by adopting current and outdated position data, respectively, two distributed protocols are adopted for the synchronization of CHO.On the other hand, as an important issue, data quantization has received extensive attention in multiagent systems. Theoretical results in the past two years include [10–16]. The consensus of multiagent systems is studied according to the robust learning control approach in [10]. For the undirected connected graphs, literature [11] obtains some strong results for the problem of expected time to convergence for quantized consensus. Under the influence of quantized input and disturbances, the distributed quantized H∞ consensus is investigated in [12]. For the high-order multiagent systems, the quantized consensus is studied in [13], in which it assumed that only the first state can be measured. By adopting both uniform and logarithmic quantizers, some quantized consensus results are achieved for heterogeneous systems in [14]. By using probabilistic versus deterministic quantizers, the quantized consensus problem for multiagent systems is studied in [15]. Moreover, event-triggered consensus [16], averaging consensus [17], consensus tracking [18], metropolis consensus [19], containment control [20], and other consensus problems [21–23] affected by data quantization are discussed in recent papers.Summarized above, the synchronization of CHO and quantized consensus for the general multiagent systems have received enough attention, respectively. However, there are few results relating to the synchronization of CHO affected by data quantization, which is the problem studied in our paper. In fact, this paper’s main contribution is to extend the results of literature [9] to the synchronization problem for CHO by using the quantized position data. By adopting the logarithmic quantizer, the quantization error of the sampled position data can be illustrated as the uncertainty according to the sector bound property. With this setup, some sufficient conditions relating to coupling strength, sampling period, and quantizer parameter are obtained for synchronized CHO.The organizational structure of the paper is as follows. Section2 demonstrates the problem discussed and the CHO network; distributed protocol and logarithmic quantizer are illustrated there in detail. Some sufficient conditions for synchronization of CHO and the detailed proving process are shown in Section 3. The usefulness of the theoretical results is illustrated by a simulation example in Section 4. Section 5 gives some conclusions.Notation. The n-dimensional Euclidean space is denoted by Rn. N (R+) indicates the positive integers (real numbers) set. We denote by c the module of complex number c. A-1 and AT are the inverse and the transpose of matrix A, respectively. 0n and 1n denote the vectors with dimension n×1 whose elements are all zeros or ones, respectively. For a multiagent system, especially for CHO, information exchange between agents can be modeled by a network or graph [24]. Let graph G be with node set Ω=1,…,M and edge set Υ⊆Ω×Ω. If there exists at least one node having a directed path to every other node, the digraph G is called containing a directed spanning tree (DST). The adjacency matrix A=aijM×M satisfies aij>0 if j,i∈Υ and aij=0 if j,i∉Υ. Mi=j∣aij>0,j∉i,∀i∈1,…,M is a set including all the neighbor nodes of node i. We use L=lijM×M to denote the Laplacian matrix, whose elements satisfy lij=-aiji≠j and lii=∑k=1,k≠iMaik [25]. 0=η1≤η2≤⋯≤ηM represent the eigenvalues of L when the digraph G contains a DST. diaga1,…,aM indicates the diagonal matrix with diagonal elements a1,…,aM.


## 2. Problem Formulation

For continuous-time CHO, we illustrate the synchronization problem and protocol by using the quantized sampled position data in this section.For a positive constanth, the synchronization problem of the following CHO network composed by M nodes is discussed:(1)x˙it=vit,v˙it=-hxit+uit,i∈1,…,Mwith position states xit, velocity states vit, and control input uit for node i.Assuming that the information between nodes is transmitted over the network, then it must be quantized before transmission. A protocol using the quantized value of the current relative sampled position data is adopted in this paper:(2)uit=γ∑j∈Miaijqxitk-qxjtk,t∈tk,tk+1,i∈1,…,M,k∈N,in which coupling strength γ>0 is to be designed and xitk,i∈1,…,M, denote the sampled position at time instants tk=kT,k∈N, with t0=0. The quantizer used here is a general logarithmic one defined as [26, 27](3)qχ=κwifχ∈11+δκw,11-δκw,χ>00ifχ=0-q-χifχ<0 for any χ∈R, with w∈N∪0,δ∈0,1, κw+1=ρκw,v0>0, and ρ=1-δ/1+δ∈0,1.The aim of our paper is designing suitableγ and T such that the CHO (1) under the control input (2) achieves synchronization; that is,(4)limt→∞⁡xit-xjt=0,limt→∞⁡vit-vjt=0for any i,j∈1,…,Mi≠j.


## 3. The Sufficient Conditions for Synchronization

For CHO (1) with protocol (2), the synchronization problem is studied in this section. The sufficient conditions for synchronization are pursued on the basis of the following two lemmas.Lemma 1 (see [28]).
The fact that the graphG contains a DST is equal to the fact that the matrix L has a simple zero eigenvalue and other eigenvalues having positive real parts. Furthermore, 1M is the right eigenvector relating to zero eigenvalue of L. Moreover, we can always get a suitable ϕ, which is left eigenvector relating to zero eigenvalue of L, satisfying ϕTL=0 and ϕT1M=1.Lemma 2 (see [29]).
Letps=s2+ϖ1+iσ1s+ϖ0+iσ0 be a complex-coefficient polynomial with real constants ϖ0,σ0,ϖ1, and σ1, and then the asymptotical stability of ps is equivalent to ϖ1>0 and ϖ1σ1σ0+ϖ12ϖ0-σ02>0.By using the above lemmas, the following theorem is given to discuss the synchronization of CHO (1) under protocol (2), which is the main result of this paper.Theorem 3.
If the graphG contains a DST and the initial states satisfy ϕTx0=ϕTv0=0 with ϕ defined in Lemma 1, then network (1) under protocol (2) can achieve synchronization if γ and T are selected satisfying(5a)γ∉hmaxi∈2,…,M⁡Re⁡ξ~i,hmini∈2,…,M⁡Re⁡ξ-i,(5b)T∉lπh∣l∈N,(5c)γ<hRe⁡ξ~imaxξ~imax2,(5d)T∈ΓTmaxi∈2,…,M,l∈N⁡arctan⁡eil,where ξ~i and ξ¯i,i∈2,…,M are, respectively, the nonzero eigenvalues of LIM+Λ¯ and LIM-Λ¯ with Λ¯=δIM, imax=argmaxi∈2,…,Mξ~i, ΓTφ≜T∣tan2⁡hT/2>tan2⁡φ, for any φ∈0,π/2, and (6)eil=h-γξil2bil2ailbil2+ail2cilfor any i∈2,…,M,l∈N with ail=hγReξil-γ2ξil2,bil=hγImξil, and cil=h2-hγReξil, in which ξil denote possible values for ξitk,k∈N, defined by (10). Moreover, let xt=x1t,…,xMtT and vt=v1t,…,vMtT, and we obtain(7)limt→∞⁡xit=cos⁡htϕTx0+1hsin⁡htϕTv0=0,limt→∞⁡vit=-hsin⁡htϕTx0+cos⁡htϕTv0=0for all i∈1,…,M.Proof.
By the property of logarithmic quantizer, we getqxitk=1+εitkxitk with εitk≤δ, ∀k∈N∪0,i∈1,…,M. For any t∈R+∪0, let zit=xit,vitT,i∈1,…,M and zt=z1Tt,…,zMTtT; then network (1) under protocol (2) can be rewritten as(8)z˙t=IM⊗Azt+γL⊗Bqx1tk0qx2tk0⋮qxMtk0=IM⊗Azt+γL⊗B10⋱101+ε1tkx1tk1+ε1tkv1tk⋮1+εMtkxMtk1+εMtkvMtk=IM⊗Azt+γL⊗BIM⊗DIM+Λtk⊗I2ztk=IM⊗Azt+γLIM+Λtk⊗Bztk,t∈tk,tk+1,k∈N∪0,where A=01-h0,B=0010,D=1000, and Λtk=diagε1tk,…,εMtk.
For the matrixLIM+Λtk,∀k∈N∪0, the following claim holds by Lemma 1.
Claim. If the graph G contains a DST, then the matrix LIM+Λtk,∀k∈N∪0, has a simple zero eigenvalue and other eigenvalues having positive real parts. Furthermore, 1/1+ε1tk,1/1+ε1tk,…,1/1+εMtkT and ϕ defined in Lemma 1 are the right eigenvector and left eigenvector, respectively, relating to zero eigenvalue of LIM+Λtk.
Based on the above claim, for anyk∈N∪0, there must exist matrices Qtk=p1tk,…,pMtk∈RM×M and Q-1tk=q1tk,…,qMtkT∈RM×M with p1tk=1/1+ε1tk,1/1+ε1tk,…,1/1+εMtkT and q1tk=ϕ such that(9)Jtk=Q-1tkLIM+ΛtkQtk=00M-1T0M-1L^tkwith Jordan canonical form Jtk and block upper triangular matrix L^tk represented as(10)L^tk=ξ2tkτ0⋯00ξ3tkτ⋯0⋱000⋯ξMtkin which ξitk,i∈2,…,M satisfying Reξ2tk≤⋯≤ReξMtk are the nonzero eigenvalues of LIM+Λtk, and τ is equal to 1 or 0.
Letmt=m1Tt,…,mMTtT=Q-1tk⊗I2zt,∀t∈tk,tk+1 and mit∈R2, and then (8) gives that(11)m˙t=Q-1tk⊗I2IM⊗AQtk⊗I2mt+γQ-1tk⊗I2LIM+Λtk⊗BQtk⊗I2mtk=IM⊗Amt+γJtk⊗Bmtk,t∈tk,tk+1,k∈N∪0indicating(12a)m˙1t=Am1t,(12b)M˙2t=IM-1⊗AM2t+γL^tk⊗BM2tk,for any t∈tk,tk+1,k∈N∪0, with M2t=m2Tt,…,mMTtT.
Next, we will verify that the asymptotic stability of subsystem (12b) is equivalent to the synchronization of (8) if the initial states satisfy ϕTx0=ϕTv0=0. On one hand, if limt→∞⁡mit=02,i∈2,…,M, combining with zt=Qtk⊗I2mt,t∈tk,tk+1 and p1tk=1/1+ε1tk,1/1+ε1tk,…,1/1+εMtkT gives zit→1/1+εitkm1t,t∈tk,tk+1,i∈1,…,M. Moreover, by defining (13)Et=cos⁡ht1hsin⁡ht-hsin⁡htcos⁡ht,the equalities m˙1t=Am1t and ϕTx0=ϕTv0=0 tell us that(14)m1t=eAtm0=EtϕT⊗I2z0=cos⁡htϕTx0+1hsin⁡htϕTv0-hsin⁡htϕTx0+cos⁡htϕTv0=02for any t∈R+∪0. Thus we get limt→∞⁡zit=02,i∈1,…,M which means that the synchronization of (8) is guaranteed with synchronization state 02. On the other hand, if the synchronization of (8) is ensured, by letting the synchronization state as z∗t=x∗t,v∗tT∈R2, we will show x∗t,v∗tT=02. In fact, it holds that m1t=ϕT⊗I2zt=∑i=1Mϕizit‍ according to mt=Q-1⊗I2zt. Based on ϕT1M=1, we get z∗t=m1t. Combined with (14), it gives x∗t=v∗t=0, and thus limt→∞⁡zt=02M. Taking mt=Q-1tk⊗I2zt,∀t∈tk,tk+1, into consideration, we get limt→∞⁡mit=02,i∈1,…,M, which ensures the asymptotic stability of subsystem (12b).
By the claim above and (10), we know that the asymptotic stability of (12b) is equal to that of the following subsystems: (15)m˙it=Amit+γξitkBmitk,i∈2,…,M, the solution of which is as(16)mit=eAt-tkmitk+γξitk∫tkteAt-sdsBmitk=eAt-tkmitk+γξitk∫0t-tkeAsdsBmitk=Eit-tkmitk+Fiξitk,t-tkmitk for any t∈tk,tk+1,k∈N∪0,i∈1,…,M, where (17)Eit-tk=cos⁡ht-tk1hsin⁡ht-tk-hsin⁡ht-tkcos⁡ht-tkFiξitk,t-tk=γξitk1-cos⁡ht-tkh0γξitksin⁡ht-tkh0. Let Siξitk,t-tk=Eit-tk+Fiξitk,t-tk; then equality (16) gives (18)mit=Siξitk,t-tkSiξitk-1,TSiξitk-2,T,…,Siξi0,Tmi0,i∈2,…,M for any t∈tk,tk+1,k∈N∪0.
The boundedness ofSiξitk,t-tk results in that mit tends to 02 if ρSiξitk,T<1,k∈N∪0,i∈2,…,M. Denoting the possible values for ξitk,k∈N∪0,i∈2,…,M by ξil,l∈N, then ρSiξitk,T<1 is ensured by ρSiξil,T<1 which is equal to the fact that the following equation’s solutions satisfy λ<1:(19)filλ,T=det⁡λI2-Siξil,T=λ2-2cos⁡hT+1-cos⁡hThγξilλ+1+cos⁡hT-1hγξil=0.If we set λ=s+1/s-1, characteristic equation (19) is rewritten as(20)21-cos⁡hT1-γξilhs2+21-cos⁡hTγξilhs+21+cos⁡hT=0for any i∈2,…,M, l∈N. According to the property of bilinear transformation, λ<1 achieves if and only if Res<0. Moreover, if we define Λ¯=δIM and denote the nonzero eigenvalues of LIM+Λ¯ and LIM-Λ¯ as ξ~i and ξ¯i,i∈2,…,M, respectively, according to the relationship between the eigenvalues and the parameters εi,i∈1,…,M, we get (21)mini∈2,…,M⁡Reξ¯i≤Reξil≤maxi∈2,…,M⁡Reξ~i which ensures that h≠γξil,i∈2,…,M,l∈N, if (5a) is satisfied. Combining with (5b) gives that equality (20) can be transferred into, ∀i∈2,…,M,l∈N, (22)s2+γξilh-γξils+1+cos⁡hT1-cos⁡hThh-γξil=0. Considering 1+cos⁡hT/1-cos⁡hT>0, Lemma 2 tells us that the roots of (22) located in the left-half plane are equal to(23)ailh-γξil2>0,(24)ailbil2+ail2cilh-γξil2-1+cos⁡hT1-cos⁡hTbil2>0, where(25)ail=hγReξil-γ2ξil2,bil=hγImξil,cil=h2-hγReξilfor all i∈2,…,M, l∈N. Recalling (5a), we have that condition (23) is equivalent to (26)γ<hReξilξil2,i∈2,…,M,l∈N, which is guaranteed by (27)γ<mini∈2,…,M,l∈N⁡hReξilξil2. Based on the relationship between the eigenvalues and the parameters εi,i∈1,…,M and inequality (21), we know that(28)mini∈2,…,M,l∈N⁡hReξilξil2=hReξ~imaxξ~imax2 with imax=argmaxi∈2,…,M⁡ξ~i.
Moreover, (26) gives γ<h/Reξil indicating ail>0 and cil>0. By defining (29)eil=h-γξil2bil2ailbil2+ail2cil, we get that (24) holds if(30)T∈ΓTmaxi∈2,…,M,l∈N⁡arctan⁡eil with ΓTφ,φ∈0,π/2 defined by ΓTφ=T∣tan2⁡hT/2>tan2⁡φ. Therefore, if (5a) and (5b) are satisfied, we obtain that the roots of (20) hold negative real part if (5c) and (5d) hold which means that λ<1, where λ denotes the eigenvalues of Siξil,T. Thus ρSiξil,T<1 is guaranteed which gives limt→∞⁡mit=02 indicating limt→∞⁡zt=02M. Up to now, it shows that the synchronization of network (1) under protocol (2) is obtained when (5a), (5b), (5c), and (5d) are satisfied. This completes the proof.Remark 4.
In this paper, the constraint on the initial state,ϕTx0=ϕTv0=0, is a necessary condition to ensure the synchronization of (8) according to zit→1/1+εitkm1t,t∈tk,tk+1,i∈1,…,M, in which εi and εj,i≠j are different from each other. How to relax the constraint on the initial state is one of our further research directions, which will need completely different proof line.Remark 5.
When we design parameters based on the above theorem, it is obvious that the difficulty lies in ascertaining the value ofφ=maxi∈2,…,M,l∈N⁡arctan⁡eil. In fact, it can always increase or decrease the value of εi,i∈1,…,M to study the proportional relationship between εi and eil and thus determine the value of φ and the feasible interval for T.Remark 6.
It is worth mentioning that the synchronization expression of CHO (1) under protocol (2) is the same as the one under protocol [9](31)uit=γ∑l∈Miailxitk-xjtk,t∈tk,tk+1,i∈1,…,M,k∈N, where the current relative sampled position data has not been quantized. This is due to the fact that Et defined in (13), which determines the synchronization state, is not influenced by quantization.


## 4. Simulation

A simulation example is given in this section to show the usefulness of the result. The CHO network composed of four nodes and the adjacency matrixA is given as(32)A=0101100111000110. Select parameters as h=0.8,δ=0.1, and γ satisfying (33)γ=0.2∉hmaxi∈2,…,M⁡Reξ~i,hmini∈2,…,M⁡Reξ¯i=0.2424,0.3556,γ=0.2<hReξ~imaxξ~imax2=0.2424. Based on φ=maxi∈2,…,M,l∈N⁡arctan⁡eil=0.6255 and [9](34)ΓTφ=T∣tan2⁡hT2>tan2⁡φ=⋃l=0∞2ςπ+φh,2ς+1πh∪⋃l=1∞2ς-1πh,2ςπ-φhwe know T∈1.3987,3.5124 by setting ς=0. Selecting T=3∈ΓTφ and the initial states as z0=-1,0,2,0,-1,2,-1,-1T satisfying ϕTx0=ϕTv0=0 with ϕ=5/21,7/21,3/21,6/21T, the synchronization states can be shown in Figure 1, from which we see that the CHO network reaches synchronization. If the initial state is set as z0=1,0,2,0,-1,2,-1,1T which violates ϕTx0=ϕTv0=0, Figure 2 shows that the synchronization errors do not tend to zeros, which means that the synchronization for CHO network is not reached.Figure 1
Synchronization with initial statesz0=-1,0,2,0,-1,2,-1,-1T.Figure 2
Synchronization errors with initial statesz0=1,0,2,0,-1,2,-1,1T.


## 5. Conclusion

By using the quantized current sampled position data, the sufficient conditions for synchronization of CHO are given according to coupling strength, sampling period, and quantizer parameter. It is worth mentioning that the results here need some restrictions on initial states. How to obtain the synchronization for CHO network without these restrictions holds some challenges, and it is one of our further research directions.



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