Abstract
In this paper, we study the finite-time synchronization problem of a Kuramoto-oscillator network with a pacemaker. By constructing a cyber-physical system (CPS), the finite-time phase agreement and frequency synchronization of the network are explored for identical and non-identical oscillators, respectively. According to the Lyapunov stability analysis, sufficient conditions are deduced for ensuring the phase agreement and frequency synchronization for arbitrary initial phases and/or frequencies under distributed strategies. Furthermore, the upper bound estimations of convergence time are obtained accordingly, which is related to the initial phases and/or frequencies of oscillators. Finally, numerical examples are presented to verify the effectiveness of the theoretical results.
1 Introduction
Synchronization of complex networks has been extensively investigated by researchers due to its numerous practical applications. As one of the most celebrated periodic-oscillator models, Kuramoto model [1] and its variations have been widely used for explaining various synchronization phenomena, and they have attracted considerable attention from researchers in diverse fields ranging from biology [2, 3], mathematics [4], physics [5, 6] and engineering [7–10]. In the past decade, many progresses concerning on the synchronization of Kuramoto-oscillator networks have been made by researchers in the control community [10–20], where synchronization criteria with respect to constraints on coupling strengths and initial phases have been developed. For example, in [10], the relationship between the algebraic connectivity of a connected Kuramoto-oscillator network and critical coupling was revealed. In [11], Chopra and Spong showed that initial relative phases should be confined to π/2 and a critical coupling strength should be satisfied, which guaranteed the frequency synchronization of an all-to-all connected Kuramoto network.
In [14, 15, 17, 20], researchers have taken the pacemaker (i.e. the so-called leader) into consideration, where synchronization criteria were related to not only the constraint on coupling strengths and initial phases, but also the selection of direct controlled oscillators. Since the interactions between oscillators are usually in the sinusoidal form of phase differences, the theoretical results mentioned above were based on the requirements of initial phases, and only local stability analyses were provided. Based on the framework of cyber-physical systems [21, 22], distributed linear controllers have been adopted to synchronize Kuramoto-oscillator networks in [23, 24], where the derived stability conditions were independent of the initial phases such that the global synchronization was achieved. In [24], sufficient criteria were established for the Kuramoto-oscillator network with a pacemaker under distributed linear control.
The results aforementioned merely focused on the asymptotical synchronization, which indicated that synchronization was realized when t → ∞. Recently, in [18, 19, 25–27], more researchers have focused on the finite-time synchronization of Kuramoto-oscillator networks, which is also of significance in practical applications. For example, power girds need to get rid of local power failures as soon as possible in order to avoid the cascading failure. In [27], Wu and Li investigated the finite-time and fixed-time synchronization of Kuramoto-oscillator networks by employing a novel multiplex control. However, the finite-time synchronization of Kuramoto-oscillator network in present of a pacemaker has not been investigated so far.
Inspired by the above literatures, it is worth investigating the finite-time synchronization of Kuramoto-oscillator network with a pacemaker. In this paper, we aim to explore finite-time synchronization criteria of such network by adopting distributed schemes based on CPS. The main contributions of this paper are summarized as follows: Firstly, effective criteria are established to deal with finite-time phase agreement and frequency synchronization for Kuramoto-oscillator network with a pacemaker, and the upper bound of synchronization time is also provided; Secondly, synchronization can be achieved for arbitrary initial phases, which only influence the upper bound of synchronization time; Finally, the requirement on the connectivity of physical system is relaxed, even if it is an unconnected network.
The remainder of this paper is organized as follows. In Section 2, the framework of CPS is constructed, which consists of the physical Kuramoto-oscillator network system and the cyber (controlling) system. Furthermore, two definitions and some necessary mathematical preliminaries are encompassed in Section 2. Finite-time phase agreement in an identical Kuramoto-oscillator network and frequency synchronization in a non-identical Kuramoto-oscillator network cover the heart body of Section 3 and Section 4, respectively. Section 5 presents the numerical simulation results, and Section 6 concludes the whole paper.
2 Model and preliminaries
In the framework of CPS, a Kuramoto-oscillator network consisting of N oscillators with control input ui can be described aswhere , θi and ωi are the phase and natural frequency of the ith oscillator, respectively. denotes the adjacency matrix of an undirected network, where aij = aji (i ≠ j) > 0 iff there is an edge between oscillator i and oscillator j; otherwise, aij = 0. Let be the Laplacian matrix associated with the adjacency matrix A, where DA ∈ RN×N is a diagonal matrix with . The network associated with the adjacency matrix A is called physical network.
Assume that there is a pacemaker with dynamicswhere θ0 and ω0 are the phase and natural frequency of the pacemaker, respectively.
In this paper, we concern phase agreement and frequency synchronization with respect to the pacemaker in finite time.
Definition 1Network Eq. 1 with control input ui achieves (pacemaker-based) finite-time phase agreement, if there exists a settling time T > 0 depending on the initial states θi(0) (i ∈ {0}⋃I), such thatand θi − θ0 ≡ 0 for t ≥ T.
Definition 2Network Eq. 1 with control input ui achieves (pacemaker-based) finite-time frequency synchronization, if there exists a settling time T > 0 depending on the initial states , such thatand for t ≥ T.In order to obtain the sufficient conditions, the following Lemmas are needed.
Lemma 1[28] For an undirected graphwithNnodes,holds, whereandis the Laplacian matrix of.
Lemma 2[29] Consider the system of differential equationwhereis continuous on an open neighborhoodof the origin andf (0) = 0. A continuously differentiable functionis said to be a solution ofEq. 4on the intervalifxsatisfiesEq. 4for allt ∈ I.If there exists a continuous functionV(x):such that(1)V(x) is positive definite;(2) There exist real numbersc > 0, 0 < ρ < 1, and an open neighborhoodof the origin such that.Then, the origin is a finite-time stable equilibrium ofEq. 4and the finite settling timeTsatisfiesIf in addition, the origin is globally finite-time stable equilibrium.For the sake of convenience, let ξi = θi − θ0, then . For a real symmetric matrices , let be the minimum eigenvalue of matrix . Denote , where the signum function sign(x) is defined as
3 Finite-time phase agreement for identical Kuramoto oscillators
In this section, we first concentrate on the case of oscillators with identical natural frequency, i.e., ωi = ω0, ∀i ∈ I. Thus, network Eq. 1 with control input ui becomes
For achieving finite-time phase agreement, a distributed control strategy is constructed aswhere denotes the adjacency matrix of an undirected network with elements bij defined similar to aij, fi ≥ 0, and parameter 0 < α < 1. The network associated with the connections between the oscillators in the controller Eq. 6 is called cyber network. Let be the Laplacian matrix associated with the adjacency matrix B, where its elements are defined similar to .
By transforming θi into ξi, network Eq. 5 with distributed control strategy Eq. 6 becomes
Network Eq. 1 with identical oscillator under distributed control strategy Eq. 6 achieves finite-time phase agreement with the settling time bounded byifwhere fmin = min{f1, …, fN}, and γ ∈ (π, 2π) satisfies tan γ = γ.
Proof 1Consider the following Lyapunov functional candidateThe derivation ofV1along trajectoriesEq. 7givesAccording toLemma 1and the fact, we can obtainAnd,CombiningEqs. 10–12, Eq. 9yieldsIf, we getByLemma 2andDefinition 1, networkEq. 1with identical oscillator under distributed control strategyEq. 6achieves finite-time phase agreement with the settling time bounded byThis completes the proof.
Remark 1According to (8), we find that the upper bound of synchronization time is proportionate to initial state ‖ξ(0)‖, and is inversely proportional tofmin. According toTheorem 1, it is sufficient to achieve finite-time phase agreement if. Therefore, even if the physical network is not connected, phase agreement could be also achieved with the help of cyber network, which relaxes the requirement on the connectivity of the physical network.
4 Finite-time frequency synchroniztion for non-identical Kuramoto oscillators
Now we further concentrate on the case of oscillators with non-identical natural frequencies, i.e., there exists some i ∈ I such that ωi ≠ ω0. For achieving finite-time frequency synchronization, a distributed control strategy ui is designed aswhere , fi ≥ 0, parameter 0 < α < 1, and bij denotes the same as that in Eq. 6. Let be the Laplacian matrix associated with the adjacency matrix B, where its elements are defined similar to .
Network Eq. 1 with non-identical oscillators under distributed control strategy Eq. 13 achieves frequency synchronization with the settling time bounded byifwhere .
Proof 2By taking the derivation ofEq. 1, we obtainConsider the following Lyapunov functional candidateThe derivation ofV2along trajectoriesEq. 15givesAccording toLemma 1and the fact, we can obtainAnd,CombiningEqs. 17–19, Eq. 16yieldsIf, we getByLemma 2andDefinition 2, networkEq. 1with non-identical oscillators under distributed control strategyEq. 13achieves finite-time frequency synchronization with the settling time bounded byThis completes the proof.
Remark 2According toEq. 14, we find that the upper bound of synchronization time is proportionate to initial state, and is inversely to thefmin. According toTheorem 2, it is sufficient to achieve finite-time frequency synchronization if. Therefore, even if the physical network is not connected, frequency synchronization could be also achieved with the help of cyber network, which relaxes the requirement on the connectivity of the physical network.
5 Numerical simulation
In this section, we assume networks associated with adjacency matrices A and B as shown in Figures 1A,B, respectively.
FIGURE 1
(A) Network associated with adjacency matrix A. (B) Network associated with adjacency matrix B.
We first verify Theorem 1. Obviously, . For simplicity, set ωi = 0 (i = 0, 1, 2, 3, 4, 5), α = 0.5, fi = 2 (i = 1, 2, 3, 4, 5), and (θ0 (0), θ1 (0), θ2 (0), θ3 (0), θ4 (0), θ5 (0)) = (0.25,−0.1028,28.8866,−10.0289,5.3575,−17.3534)T. In Figure 2A, phase differences θi − θ0 converge to zero, which means finite-time phase agreement is achieved. Besides, it also shows phase agreement is achieved about 2.2s, which is less than the upper bound of settling time T1 = 5.9600s. Time evolutions of the controller Eq. 6 of each oscillator are showed in Figure 2B.
FIGURE 2
(A) Time evolutions of phase differences θi − θ0 (i = 1, 2, 3, 4, 5) under distributed control strategy Eq. 6. (B) Time evolutions of the distributed control strategy Eq. 6.
Secondly, we verify Theorem 2. Obviously, . Set α = 0.5, fi = 2 (i = 1, 2, 3, 4, 5), , (ω1, ω2, ω3, ω4, ω5) = (−10,−4,0,4,10)T, and . In Figure 3Afrequency differences converge to zero, which means finite-time frequency synchronization is achieved. Besides, it also shows frequency synchronization is achieved about 0.825s, which is less than the upper bound of settling time bound T2 = 2.1147s. Time evolutions of the controller Eq. 13 of each oscillator are showed in Figure 3B.
FIGURE 3
(A) Time evolutions of frequencies differences under distributed control strategy Eq. 13. (B) Time evolutions of the distributed control strategy Eq. 13.
Finally, we move to see the influence of parameter α on synchronization time. In the simulations, we set α = 0.1, 0.3, 0.5, 0.7, 0.9. In Figure 4, it is showed that the synchronization time decreases as α grows.
6 Conclusion
In this paper, the problems of finite-time phase agreement and frequency synchronization of Kuramoto-oscillator networks with a pacemaker have been investigated. Two distributed control strategies, based on the CPS, have been designed to drive the Kuramoto-oscillator networks. In the light of finite-time stability theory, the sufficient criteria have been derived for guaranteeing the phase agreement and frequency synchronization of identical and non-identical Kuramoto-oscillator networks with a pacemaker. At the same time, the upper bounds estimation of convergence time of Kuramoto-oscillator networks have been given accordingly. Numerical examples have validated the effectiveness of the derived theoretical results.
However, the convergence time estimations of this paper are heavily related to initial phases and/or frequencies of oscillators. Therefore, it is urgent to explore the fixed-time synchronization of Kuramoto model with a pacemaker in the future.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
This work is supported by National Natural Science Foundation of China (Grant No. 61903142), the Science and Technology Project of Jiangxi Education Department (Grant No. GJJ180356).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
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References
1.
KuramotoY. Chemical oscillators, waves, and turbulence. Berlin, Germany: Springer (1984).
2.
LiuCWeaverDRStrogatzSHReppertSM. Cellular construction of a circadian clock: period determination in the suprachiasmatic nuclei. Cell (1997) 91:855–60. 10.1016/s0092-8674(00)80473-0
3.
ChenCLiuSShiX-QChatéHWuYL. Weak synchronization and large-scale collective oscillation in dense bacterial suspensions. Nature (2017) 542:210–4. 10.1038/nature20817
4.
KimJ-HParkJ-H. Exponential synchronization of Kuramoto oscillators using spatially local coupling. Physica D: Nonlinear Phenomena (2014) 277:40–7. 10.1016/j.physd.2014.03.006
5.
WiesenfeldKColetPStrogatzSH. Frequency locking in josephson arrays: connection with the Kuramoto model. Phys Rev E (1998) 57:1563–9. 10.1103/physreve.57.1563
6.
DaidoH. Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions. Phys Rev Lett (1992) 68:1073–6. 10.1103/physrevlett.68.1073
7.
SunWLüJHYuXHChenYChenSH. Cooperation of multiagent systems with mismatch parameters: a viewpoint of power systems. IEEE Trans Circuits Syst (2016) 63:693–7. 10.1109/tcsii.2016.2530178
8.
SeybothGSWuJBQinJHYuCBAllgöwerF. Collective circular motion of unicycle type vehicles with non-identical constant velocities. IEEE Trans Control Netw Syst (2014) 1:167–76. 10.1109/tcns.2014.2316995
9.
SepulchreRPaleyDALeonardNE. Stabilization of planar collective motion: all-to-all communication. IEEE Trans Automat Contr (2007) 52:811–24. 10.1109/tac.2007.898077
10.
DörflerFBulloF. Synchronization in complex networks of phase oscillators: a survey. Automatica (2014) 50:1539–64. 10.1016/j.automatica.2014.04.012
11.
ChopraNSpongMW. On exponential synchronization of Kuramoto oscillators. IEEE Trans Automat Contr (2009) 54:353–7. 10.1109/tac.2008.2007884
12.
HaS-YLiZCXueXP. Formation of phase-locked states in a population of locally interacting Kuramoto oscillators. J Differential Equations (2013) 255:3053–70. 10.1016/j.jde.2013.07.013
13.
LiHQChenGLiaoXFHuangTW. Attraction region seeking for power grids. IEEE Trans Circuits Syst (2017) 64:201–5. 10.1109/tcsii.2016.2561410
14.
LiXRaoPC. Synchronizing a weighted and weakly-connected Kuramoto oscillator digraph with a pacemaker. IEEE Trans Circuits Syst (2015) 62:899–905. 10.1109/tcsi.2014.2382193
15.
RaoPCLiXOgorzalekMJ. Stability of synchronous solutions in a directed Kuramoto-Oscillator network with a pacemaker. IEEE Trans Circuits Syst (2017) 64:1222–6. 10.1109/tcsii.2017.2679216
16.
JadbabaieAMoteeNBarahonaM. On the stability of the Kuramoto model of coupled nonlinear oscillators. Proc Amer Control Conf (2004) 5.
17.
TongDBRaoPCChenQYOgorzalekMJLiX. Exponential synchronization and phase locking of a multilayer Kuramoto-oscillator system with a pacemaker. Neurocomputing (2018) 308:129–37. 10.1016/j.neucom.2018.04.067
18.
WuJLiuMQWangXFMaR-R. Achieving fixed-time synchronization of the Kuramoto model via improving control techniques. J Korean Phys Soc (2021) 79:998–1006. 10.1007/s40042-021-00302-z
19.
WuJLiX. Global stochastic synchronization of Kuramoto-oscillator networks with distributed control. IEEE Trans Cybern (2021) 51:5825–35. 10.1109/tcyb.2019.2959854
20.
WangYQDoyleFJ. Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker. IEEE Trans Automat Contr (2013) 58:989–94. 10.1109/tac.2012.2215772
21.
DerlerPLeeEAVincentelliAS. Modeling cyber-physical systems. Proc IEEE (2012) 100:13–28. 10.1109/jproc.2011.2160929
22.
LeeEA. Computing foundations and practice for cyber-physical systems: a preliminary report. California: University of California (2007).
23.
ZhangW-YYangCGuanZ-HLiuZ-WChiMZhengG-L. Bounded synchronization of coupled Kuramoto oscillators with phase lags via distributed impulsive control. Neurocomputing (2016) 218:216–22. 10.1016/j.neucom.2016.08.054
24.
RaoPCLiX. Pacemaker-based global synchronization of Kuramoto oscillators via distributed control. IEEE Trans Circuits Syst (2018) 65:1768–72. 10.1109/tcsii.2017.2763184
25.
DongJ-GXueXP. Finite-time synchronization of Kuramoto-type oscillators. Nonlinear Anal Real World Appl (2015) 26:133–49. 10.1016/j.nonrwa.2015.05.006
26.
ZhangXXSunZYYuCB. Finite-time synchronization of networked Kuramoto-like oscillators. In: Proceedings of Australian Control Conference; 3-4 Nov; Newcastle (2016).
27.
WuJLiX. Finite-time and fixed-time synchronization of kuramoto-oscillator network with multiplex control. IEEE Trans Control Netw Syst (2019) 6:863–73. 10.1109/tcns.2018.2880299
28.
Olfati-SaberRMurrayRM. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Automat Contr (2004) 49:1520–33. 10.1109/tac.2004.834113
29.
BhatSPBernsteinDS. Finite-time stability of continuous autonomous systems. SIAM J Control Optim (2000) 38:751–66. 10.1137/s0363012997321358
Summary
Keywords
finite-time, synchronization, Kuramoto-oscillator, pacemaker, cyber-physical system (CPS)
Citation
Rao P and Guo X (2022) Finite-time synchronization of Kuramoto-oscillator networks with a pacemaker based on cyber-physical system. Front. Phys. 10:1077045. doi: 10.3389/fphy.2022.1077045
Received
22 October 2022
Accepted
09 November 2022
Published
24 November 2022
Volume
10 - 2022
Edited by
Jianbo Wang, Southwest Petroleum University, China
Reviewed by
Zhao-Long Hu, Zhejiang Normal University, China
Rongzhong Yu, Jiujiang University, China
Updates
Copyright
© 2022 Rao and Guo.
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*Correspondence: Pengchun Rao, pcrao@ecjtu.edu.cn
This article was submitted to Social Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.