循环切换网络的固定时同步
Fix-Time Synchronization of Cyclic Switched Networks

作者: 曹 娟 :南京航空航天大学理学院,江苏 南京;

关键词: 固定时同步反馈控制循环切换复杂网络Fix-Time Synchronization Feedback Control Cyclic Switched Complex Networks

摘要:
由于其在现实生活中的广泛应用,复杂网络的同步已经成为了一个热门的话题。通过加反馈控制器,本文研究了一类循环切换复杂网络的固定时同步问题。基于李雅普诺夫稳定性理论以及反馈控制技巧,我们详细证明了在所给的充分条件下,所得固定时同步结论的正确性。此外,我们给出停时的表达式。最后,通过一个数值实例,我们证明了理论结果的正确性。本文的结果可以适用于具体的循环切换复杂网络。

Abstract: Due to its wide application in real life, the synchronization of complex networks has become a hot topic. In this article, we investigate the fix-time synchronization for cyclic switched complex networks by feedback controllers. Based on the Lyapunov stability theory and the feedback control technique, we prove in detail the correctness of the conclusion on fix-time synchronization under the given sufficient conditions. In addition, the expression of settling time is estimated. Last but not the least, a numerical example is presented to illustrate the validity of theoretical results. The conclusions of this paper are applicable to the concrete cyclic switched complex networks.

1. 引言

同步是自然界和社会中普遍存在着的一种现象,例如钟摆的同步摆动、萤火虫的同步发光以及剧场中人们掌声的同步等。复杂网络同步 [1] 的研究已经引起了各个领域研究者们的广泛兴趣,例如通信、工程、物理学、数学、和社会学等。耦合振荡器的同步现象不仅可以解释实际生活中的许多现象 [2],而且具有许多实际应用,例如图像处理 [3]、保密通信 [4] 等。在过去几十年,各种同步种类被广泛研究,例如完全同步、投影同步、相同步、滞后同步、反同步、聚类同步等 [5] - [10]。许多研究者讨论了切换网络的同步问题,其中含有渐近同步和指数同步的研究 [11] [12]。然而,在实际工程中,人们总是希望使切换复杂系统在有限时间或者固定时间内达到同步,理由之一是想避免一些例如信息遗漏的损失。因此,越来越多的人研究切换复杂网络的有限时同步或者固定时同步。例如,文献 [13] 中,作者建立了一个有限时同步准则,该准则可以用于多权重的切换复杂动态网络。另一个例子是,李等人研究了切换网络的有限时同步问题 [14]。

值得注意的是,之前的工作中已经提出了许多种控制策略,例如牵制控制、基于观测者控制、样本数据控制、反馈控制、自适应控制、间歇控制、脉冲控制等 [15] - [21]。反馈控制是一种常用的控制方法,它很容易在工程中得以使用。因此,我们利用反馈控制技巧来设计控制器。循环切换网络是一种特殊的切换网络,这意味着切换序列具有循环的结构。对比于之前部分研究考虑的切换复杂网络的渐近同步和指数同步,有限时同步不仅具有更快的收敛速度,而且更符合实际应用的需求。而固定时比有限时具有更强的收敛性、更快的收敛速度,是一种特殊的有限时同步。研究固定时同步可以使系统具有更好的抗干扰性以及鲁棒性。固定时同步有助于确定网络在一个固定的时间段确保系统已经实现同步,这意味着通过设计适当的控制器,受控系统的轨道可以在固定的时间内趋于平衡状态。基于之前的相关文献,本文研究反馈控制下循环切换网络的固定时同步问题。

2. 初步

2.1. 模型描述

考虑如下的切换复杂网络模型:

x ˙ i ( t ) = f σ ( t ) ( x i ( t ) ) + c 1 j = 1 N a i j σ ( t ) Γ σ ( t ) x j ( t ) , (1)

其中 x i ( t ) = ( x i 1 ( t ) , x i 2 ( t ) , , x i n ( t ) ) T R n 是系统第i个节点的状态向量, c 1 c 2 是耦合强度, σ ( t ) : [ 0 , ) M = { 1 , 2 , , m } 是系统的切换信号,它是一个分段右连续的常函数。对于 σ ( t ) = r M Γ r = ( Γ r 1 , Γ r 2 , , Γ r n ) T R n × n > 0 是对角内部耦合矩阵。 A r R N × N 是外部耦合结构矩阵。激活函数 f : R n R n 连续, a i j r 0 代表耦合权重,当且仅当图中节点i和节点j之间无连接时 a i j r = 0 ,并且 A r 满足行和为零的扩散耦合条件。系统(1)的初始条件设为 x i ( t ) = φ i ( t ) 。本文我们总假设系统(1)存在唯一解。下面我们主要目标是利用反馈控制实现循环切换复杂网络的固定时同步。

我们把上述的系统(1)看作驱动系统,参照系统(1),可以给出对应的响应系统为

y ˙ i ( t ) = f σ ( t ) ( y i ( t ) ) + c 1 j = 1 N a i j σ ( t ) Γ σ ( t ) y j ( t ) + u σ ( t ) , i ( t ) , (2)

其中 y i ( t ) = ( y i 1 ( t ) , y i 2 ( t ) , , y i n ( t ) ) T R n 是响应系统第i个节点的状态向量, u σ ( t ) , i ( t ) R n 指响应系统第i个节点的连续控制输入。响应系统的初值设为 y i ( t ) = Ψ i ( t )

将同步误差向量记为 e i ( t ) = y i ( t ) x i ( t ) , i = 1 , 2 , , N ,误差系统的初始条件为 e i ( t ) = Ψ i ( t ) φ i ( t ) i = 1 , 2 , , N 。根据驱动系统(1)以及相应的响应系统(2),我们有如下的误差系统

e ˙ i ( t ) = F σ ( t ) ( e i ( t ) ) + c 1 j = 1 N a i j σ ( t ) Γ e j ( t ) + u σ ( t ) , i ( t ) ,

其中, F ( e i ( t ) ) = f ( y i ( t ) ) f ( x i ( t ) ) ,f、u都连续。我们设计如下的反馈控制器:

u σ ( t ) , i ( t ) = α σ ( t ) , i e i ( t ) β σ ( t ) , i ( 1 2 λ max ( p σ ( t ) ) ) k + 1 2 λ min ( p σ ( t ) ) s i g n ( e i ( t ) ) | e i ( t ) | k , (3)

其中, α σ ( t ) , i β σ ( t ) , i 都为正数, i = 1 , 2 , , N 0 < k < 1 s i g n ( ) 是符号函数,并且 s i g n ( e i ( t ) ) = [ s i g n ( e i 1 ( t ) ) , s i g n ( e i 2 ( t ) ) , , s i g n ( e i n ( t ) ) ] T 。本文我们所设计的该切换不连续反馈控制器含

有网络的切换序列、设计的李雅普诺夫矩阵以及右端不连续的符号函数。它的优势不仅在于可以减小网络周期循环的影响,而且可以结合所设计的李雅普诺夫函数,帮助误差系统在固定时稳定,从而实现驱动响应系统的固定时同步。本文我们仅仅考虑没有芝诺现象的情况。

2.2. 数学准备

在这一部分,我们给出一个假设,三个定义以及两个引理,以帮助获得本文的主要结果。

假设1 假设存在正定对角矩阵 P r Θ r ,使得 ( y x ) T P r [ f r ( y ) f r ( x ) Θ r ( y x ) ] η r ( y x ) T ( y x ) ;对某个 η > 0 x , y R n r M

注1 许多著名的混沌系统,包括蔡氏振荡器、Rossler系统、Lorenz系统、Chen系统和Lu系统,都满足假设1 [22]。

定义1 切换驱动系统(1)和切换响应系统(2)称为实现固定时同步,如果对于一个恰当反馈控制器(3),存在常数 T max T > 0 ,使得 lim t T e ( t ) = 0 并且当 t > T 时,有 e ( t ) = 0

定义2 对于满足假设1的所有切换序列 σ ( t ) ,定义 T j = T j g T ( j 1 ) g 为第j个循环所经历的时间段。若存在 T * 使得 T * T j ,则 T * 称为切换序列 σ ( t ) 的切换停留时间。

定义3 切换序列 σ ( t ) 称作是循环的,如果存在正整数g使得 σ ( t k ) = σ ( t k + g ) , k = 0 , 1 , 2 , 。由此可得在区间 [ t k , t k + 1 ) 上, σ ( t k ) 得以激活,而在 t k + 1 时, σ ( t k + 1 ) 得以激活 k = 0 , 1 , 2 ,

引理1 [23] 假设 V ( ) : R n R + { 0 } 是一个连续、径向无界函数,并且如下两个条件成立:

1) V ( e ( t ) ) = 0 e ( t ) = 0

2) 误差系统的解满足 V ˙ ( e ( t ) ) a V ξ ( e ( t ) ) b V η ( e ( t ) )

则误差系统的原点可以实现固定时稳定并且 T max = 1 c ( 1 ξ ) + 1 c ( η 1 )

引理2 [24] 若 a 1 , a 2 , , a n 0 0 < q 1 ,则 ( i = 1 N a i ) q i = 1 N ( a i ) q

3. 主要结论

在这个部分,我们给出一个充分性准则。通过施加一个反馈控制器(3)实现循环切换驱动系统(1)和切换响应系统(2)之间的固定时同步。

定理1 若假设1成立,则循环切换驱动系统(1)和相应的循环驱动响应系统(2)可以在固定时间内实现同步,若存在对角矩阵 Ξ r > 0 ,使得如下条件成立:对于 i { 1 , 2 , , N }

1) Θ r I N I n Ξ r + c Γ r A r 0 , r M

2) 循环停留时间 T 1 * 满足 { T 1 * > 2 ϕ ( 1 ϕ ) L , 2 ϕ ( 1 ϕ ) L g ρ , T 1 * g ρ , 2 ϕ ( 1 ϕ ) L < g ρ ;

其中, ϕ = ( ν g 1 ) [ V σ ( 0 ) ( 0 ) ] 1 k 2 ν = { max { p r 1 j p r 2 j , r 1 , r 2 M , j = 1 , 2 , , n } } 1 k 2 V σ ( 0 ) ( 0 ) = 1 2 i = 1 N e i T ( 0 ) P σ ( 0 ) e i ( 0 ) L = min { β r , i , i = 1 , 2 , , N , r M } ,并且停时的上界为 T max = { h T 1 * , ϕ 0 , 2 [ V σ ( 0 ) ( 0 ) ] 1 k 2 ( 1 k ) L , ϕ = 0 , h是满足 h 1 < 2 [ V σ ( 0 ) ( 0 ) ] 1 k 2 ( 1 k ) L T 1 * 2 ϕ h 的正整数。

证明 定义切换李雅普诺夫函数为 V σ ( t ) ( t ) = 1 2 i = 1 N e i T ( t ) P σ ( t ) e i ( t ) 。根据 ν 的定义可以得出

[ V r 1 ( t ) ] 1 k 2 ν [ V r 2 ( t ) ] 1 k 2 , r 1 , r 2 M ,并且

c 1 2 i = 1 N e i T ( t ) e i ( t ) V r ( t ) c 2 2 i = 1 N e i T ( t ) e i ( t ) , r M , (4)

其中 c 1 = min { λ min ( P r ) , r M } , c 2 = max { λ max ( P r ) , r M } 。进一步,我们可以有

V ˙ r ( t ) = i = 1 N e i T ( t ) P r e ˙ i ( t ) = i = 1 N e i T ( t ) P r [ F r ( e i ( t ) ) + c 1 j = 1 N a i j r Γ r e j ( t ) α r , i e i ( t ) β r , i ( 1 2 λ max ( p r ) ) k + 1 2 λ min ( p r ) s i g n ( e i ( t ) ) | e i ( t ) | k ] = i = 1 N e i T ( t ) P r [ F r ( e i ( t ) ) Θ r e i ( t ) + Θ r e i ( t ) ] + i = 1 N j = 1 N a i j r e i T ( t ) P r Γ r e j ( t ) i = 1 N α r , i e i T ( t ) P r e i ( t ) i = 1 N β r , i ( 1 2 λ max ( p r ) ) k + 1 2 λ min ( p r ) e i T ( t ) P r s i g n ( e i ( t ) ) | e i ( t ) | k (5)

η r i = 1 N e i T ( t ) e i ( t ) + i = 1 N e i T ( t ) ( P r Θ r α r , i P r ) e i ( t ) + i = 1 N j = 1 N a i j r e i T ( t ) P r Γ r e j ( t ) L i = 1 N ( 1 2 λ max ( p r ) ) k + 1 2 | e i ( t ) | k + 1 j = 1 N P r j e ˜ j T ( t ) γ r j A r e ˜ j ( t ) + j = 1 N P r j e ˜ j T ( t ) ( θ r j I N Ξ r ) e ˜ j ( t ) L i = 1 N ( 1 2 λ max ( p r ) e i T ( t ) e i ( t ) ) k + 1 2 ,

其中 e ˜ j ( t ) = [ e 1 j , e 2 j , , e N j ] T ,而 Θ r , I n , Γ r 为对角矩阵,由定理的条件(1)可知如下的不等式成立:

θ r j I N Ξ r + c γ r j A r 0 , j = 1 , 2 , , n (6)

根据引理2可得

L i = 1 N [ 1 2 e i T ( t ) p r e i ( t ) ] k + 1 2 L [ i = 1 N 1 2 e i T ( t ) p r e i ( t ) ] k + 1 2 = L [ V r ( t ) ] k + 1 2 . (7)

将(6)和(7)式代入(5)式,可以得出 V ˙ ( t ) L [ V r ( t ) ] k + 1 2

考虑到网络的周期循环性,由引理1,

[ V σ ( 0 ) ( T 1 ) ] 1 k 2 [ V σ ( 0 ) ( 0 ) ] 1 k 2 ( ν g 1 ) [ V σ ( 0 ) ( 0 ) ] 1 k 2 1 k 2 ν L T 1 . (8)

由定理中循环停留时间 T 1 * 的假设, ϕ < 1 k 2 ν L T 1 * 。结合上式我们有,若 t [ 0 , t g ] , i = 1 N | e i ( t ) | 0 则在第一个循环之后, [ V σ ( t ) ( t ) ] 1 k 2 不小于 ϕ 1 k 2 ν L T 1 * 。通过数学归纳,我们可以得出,对于第 j ( j = 2 , 3 , ) 个循环,若 t [ 0 , T j ] , i = 1 N | e i ( t ) | 0 ,则

[ V σ ( 0 ) ( T j ) ] 1 k 2 [ V σ ( 0 ) ( T j 1 ) ] 1 k 2 ϕ 1 k 2 ν L T 1 * . (9)

V σ ( t ) ( t ) = 0 之后,不等式(8)和(9)总会成立。若 ϕ 0 ( u > 1 ) ,则存在一个正整数h满足 h 1 < 2 [ V σ ( 0 ) ( 0 ) ] 1 k 2 ( 1 k ) L T 1 * 2 ϕ h ,在第h个循环之后, V σ ( t ) ( h T 1 * ) = 0 ,由不等书(4)可得 e i ( h T 1 * ) = 0 ,其中 i = 1 , 2 , , N 。由此可得收敛时间 t 1 * h T 1 * 。若 ϕ = 0 ( u = 1 ) ,则 t 1 * 2 [ V σ ( 0 ) ( 0 ) ] 1 k 2 ( 1 k ) L 。综合以上分析,停时的上界为 T max = { h T 1 * , ϕ 0 , 2 [ V σ ( 0 ) ( 0 ) ] 1 k 2 ( 1 k ) L , ϕ = 0 , 其中h是满足 h 1 < 2 [ V σ ( 0 ) ( 0 ) ] 1 k 2 ( 1 k ) L T 1 * 2 ϕ h 的正整数。因此,根据定义1,驱动系统(1)和响应系统(2)可以实现固定时同步。这样就完成了证明。

注2 由定理中的循环停留时间以及固定时停时可以看出该反馈控制器的精度和准确性。

4. 结论

本文详细地研究了一类循环切换复杂网络的固定时同步问题。我们考虑无芝诺现象的情形。基于李雅普诺夫稳定性理论和固定时控制技巧,我们设计出一种反馈控制器来帮助该网络实现固定时同步。同时,我们分析出循环停留时间满足具体条件时,停时的上界。我们考虑的切换模式是时间依赖的情形,进一步,我们还可以研究采取状态依赖模式的循坏切换网络,如何通过反馈控制来实现该网络的固定时同步。

文章引用: 曹 娟 (2021) 循环切换网络的固定时同步。 动力系统与控制, 10, 70-76. doi: 10.12677/DSC.2021.101008

参考文献

[1] Wang, Z.X., Jiang, G.P., Yu, W.W., He, W.L., Cao, J.D. and Xiao, M. (2017) Synchronization of Coupled Heterogeneous Complex Networks. Journal of the Franklin Institute, 354, 4102-4125.

[2] Mirollo, R.E. and Strogatz, S.H. (1990) Synchronization of Pulse-Coupled Biological Oscillators. SIAM Journal on Applied Mathematics, 50, 1645-1662.
https://doi.org/10.1038/nature10680

[3] Wei, G.W. and Jia, Y.Q. (2002) Synchronization-Based Image Edge Detection. Europhysics Letters, 59, 814-819.
https://doi.org/10.1038/nmat1967

[4] Nechifora, A., Albub, M., Hairc, R. and Terzijaa, V. (2015) A Flexible Platform for Synchronized Measurements, Data Aggregation and Information Retrieval. Electric Power Systems Research, 120, 20-31.
https://doi.org/10.1021/nn1015874

[5] Liu, J., Li, L.L. and Fardoun, H.M. (2020) Complete Synchronization of Coupled Boolean Networks with Arbitrary Finite Delays. Frontiers of Information Technology & Electronic Engineering, 21, 281-293.
https://doi.org/10.1021/nl102069z

[6] Ren, L. and Zhang, G.S. (2019) Adaptive Projective Synchronization for a Class of Switched Chaotic Systems. Mathematical Methods in the Applied Sciences, 42, 6192-6204.
https://doi.org/10.1063/1.4818458

[7] Aguirre, L.A. and Freitas, L. (2018) Control and Observability Aspects of Phase Synchronization. Nonlinear Dynamics, 91, 2203-2217.
https://doi.org/10.1021/nl103079j

[8] Cai, S.M., Zhou, F.L. and He, Q.B. (2019) Fixed-Time Cluster Lag Synchronization in Directed Heterogeneous Community Networks. Physica A, 525, 128-142.
https://doi.org/10.1063/1.3536529

[9] Zhang, C.L., Deng, F.Q., Zhang, W.F., Hou, T. and Yang, Z.W. (2019) Anti-Synchronization and Synchronization of Coupled Chaotic System with Ring Connection and Stochastic Perturbations. IEEE Access, 7, 76902-76909.
https://doi.org/10.1021/acs.nanolett.5b00939

[10] Gan, Q., Xiao, F., Qin, Y. and Yang, J. (2019) Fixed-Time Cluster Synchronization of Discontinuous Directed Community Networks via Periodically or Aperiodically Switching Control. IEEE Access, 7, 83306-83318.
https://doi.org/10.1038/nnano.2013.46

[11] Liu, X.Y., Cao, J.D., Yu, W.W. and Song, Q. (2016) Nonsmooth Finite-Time Synchronization of Switched Coupled Neural Networks. IEEE Transactions on Cybernetics, 46, 2360-2371.
https://doi.org/10.1088/0953-8984/22/33/334204

[12] Wang, Z.X., He, H.B., Jiang, G.-P. and Cao, J.D. (2020) Distributed Tracking in Heterogeneous Networks with Asynchronous Sampled-Data Control. IEEE Transactions on Industrial Informatics, 16, 7381-7391.
https://doi.org/10.1098/rsta.2010.0213

[13] Yang, D., Li, X.D. and Song, S.J. (2020) Design of State-Dependent Switching Laws for Stability of Switched Stochastic Neural Networks with Time-Delays. IEEE Transactions on Neural Networks & Learning Systems, 31, 1808-1819.
https://doi.org/10.1080/00018732.2011.582251

[14] Lee, T.C. and Jiang, Z.P. (2008) Uniform Asymptotic Stability of Nonlinear Switched Systems with an Application to Mobile Robots. IEEE Transactions on Automatic Control, 53, 1235-1252.
https://doi.org/10.1103/PhysRevB.91.045414

[15] Zhou, B., Zheng, W.X. and Duan, G.-R. (2011) Stability and Stabilization of Discrete-Time Periodic Linear Systems with Actuator Saturation. Automatica, 47, 1813-1820.
https://doi.org/10.1063/1.4959880

[16] Branicky, M.S. (1998) Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems. IEEE Transactions on Automatic Control, 43, 475-482.
https://doi.org/10.1063/1.4978312

[17] Ding, S.B. and Wang, Z.S. (2020) Event-Triggered Synchronization of Discrete-Time Neural Networks: A Switching Approach. Neural Networks: The Official Journal of the International Neural Network Society, 125, 31-40.
https://doi.org/10.1063/1.5111354

[18] Zhou, L., Ding, H. and Xiao, X.Q. (2020) Input-to-State Stability of Discrete-Time Switched Nonlinear Systems with Generalized Switching Signals. Applied Mathematics and Computation, 392, Article ID: 125727.
https://doi.org/10.1103/PhysRevB.81.153401

[19] Huang, Z.G., Xia, J.W., Wang, J., Wang, J. and Shen, H. (2020) Observer-Based Finite-Time Bounded Analysis for Switched Inertial Recurrent Neural Networks under the PDT Switching Law. Physica A, 538, Article ID: 122699.
https://doi.org/10.1016/j.carbon.2009.12.057

[20] Liu, Y., Chen, F., Yang, B., Wang, X. and Wang, W.M. (2020) Finite-Time Synchronization for a Class of Multiweighted Complex Networks with Markovian Switching and Time-Varying Delay. Complexity, 1-25.
https://doi.org/10.1103/PhysRevB.82.125429

[21] Yang, F., Gu, Z., Cheng, J. and Liu, J. (2019) Event-Driven Finite-Time Control for Continuous-Time Networked Switched Systems under Cyber Attacks. Journal of the Franklin Institute, 357, 11690-11709.

[22] Ren, H.L., Zong, G.D. and Li, T.S. (2018) Event-Triggered Finite-Time Control for Networked Switched Linear Systems with Asynchronous Switching. IEEE Transactions on Systems, Man & Cybernetics: Systems, 48, 1874-1884.
https://doi.org/10.1038/nnano.2008.67

[23] Liu, X.K., Shi, X.R. and Li, Y. (2019) Neural Networks-Based Adaptive Finite-Time Control of Switched Nonlinear Systems under Time-Varying Actuator Failures. Advances in Difference Equations, 2019, Article No. 482.
https://doi.org/10.1016/j.physrep.2009.02.003

[24] Chen, C.Y., Zhu, S., Wang, M., Yang, C.Y. and Zeng, Z.G. (2020) Finite-Time Stabilization and Energy Consumption Estimation for Delayed Neural Networks with Bounded Activation Function. Neural Networks, 131, 163-171.
https://doi.org/10.1063/1.3518979

[25] Abdurahman, A., Jiang, H.J. and Teng, Z.D. (2015) Finite-Time Synchronization for Memristor-Based Neural Networks with Time-Varying Delays. Neural Networks, 69, 20-28.
https://doi.org/10.1088/1367-2630/13/2/025008

[26] Li, J.H., Dong, H.L., Wang, Z.D. and Zhang, W.D. (2018) Protocol-Based State Estimation for Delayed Markovian Jumping Neural Networks. Neural Networks, 108, 355-364.
https://doi.org/10.1016/j.carbon.2013.07.088

[27] Goudreau, M.W. and Giles, C.L. (1995) Using Recurrent Neural Networks to Learn the Structure of Interconnection Networks. Neural Networks, 8, 793-804.
https://doi.org/10.1155/2013/101765

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