Exponential Synchronization of Discrete Uncertain Spatiotemporal Networks With Topology Switching Characteristics
-
摘要: 研究了具有拓扑切换特性的离散型不确定时空网络的指数同步问题. 基于稳定性理论, 构造了具有指数形式的Lyapunov函数, 并设计了同步控制器的结构方程, 进而获得了时空网络的同步条件. 同时, 我们设计了未知参数的识别律, 有效地识别了网络中的未知参数. 最后, 选取实际的激光相位共轭波空间扩展系统作为网络节点进行仿真模拟, 验证了同步方案的可行性与控制器的有效性. 通过构造具有指数形式的Lyapunov函数, 能够有效地调节网络的同步速率. 并且获得的同步条件中不包含网络的耦合矩阵项, 消除了拓扑切换特性对同步过程的影响, 使得网络同步性能更加稳定.Abstract: The exponential synchronization problem of discrete uncertain spatiotemporal networks with topology switching characteristics is researched. Based on the stability theory, the Lyapunov function with exponential form is constructed and the structural equation of the synchronous controller is designed, and then the synchronization condition of the spatiotemporal network is obtained. Meanwhile, the unknown parameters of the network are identifled efiectively by designing the identiflcation law of the unknown parameters. Finally, we use phase-conjugate wave spatial expanded system as the nodes of the network to simulate, and the feasibility of the synchronization scheme and the efiectiveness of the controller is verifled. By constructing the Lyapunov function with exponential form, the synchronization rate of the network can be adjusted efiectively. In addition, the synchronization condition does not contain the coupling matrix term of the network, which eliminates the influence of the topology switching characteristics on the synchronization process, and makes the network synchronization performance more stable.
-
Key words:
- Topology switching /
- discrete spatiotemporal networks /
- parameter identiflcation /
- exponential synchronization
-
自然界存在着大量的复杂动态网络, 并且复杂动态网络在物理学、神经学、信息通信以及保密技术等领域中均有着重要应用. 其中, 作为复杂动态网络的一种关键性集体行为, 其同步化现象的研究也越来越受到国内外学者的广泛关注[1-5]. 长期以来, 基于稳定结构的网络同步研究一直是人们研究的主要方向[6-9]. 然而, 实际网络的拓扑结构常常是变化的. 其原因在于在实际应用中, 网络节点的连接方式经常由于连接故障或新节点的加入而改变, 进而使得网络的拓扑结构随之改变, 从而使网络同步的实现变得更加困难, 增加了网络同步状态发生崩溃的可能性. 因此, 具有拓扑切换特性的网络同步研究也就成为了目前人们研究的热点问题.
目前, 具有拓扑切换特性的复杂网络的同步研究已有文献报道. 其中典型的如Zhai[10]研究了具有拓扑切换特性的非线性系统构成的网络的同步行为. 文献[11] 研究了具有切换有向拓扑的多智能网络的一致性跟踪控制问题. 文献[12] 中, Fan等研究了具有切换拓扑特性的异构复杂网络的准同步行为. 文献[13] 报道了具有Markovian切换特性的耦合神经网络的自适应指数同步问题. 这些工作为我们进一步深入地研究具有拓扑切换特性的网络同步问题奠定了理论基础.
然而, 已报道的上述成果中其同步技术仍具有若干局限性. 例如文献[13] 中的拓扑结构切换利用的是Markovian切换, 人们很难对其进行控制. 而在实际中, 常常需要根据人们的实际需求随时进行网络拓扑结构的切换. 同时, 网络中参数的未知性也是实际网络受到外界干扰而具有的不稳定因素. 不难看出, 含有未知参数的网络同步研究十分必要. 另外, 有关连续型网络同步研究的文献也比较多见, 而离散型网络的同步研究相对较少. 其中较为典型的工作如Zhang等[14]利用线性矩阵不等式技术研究了具有时变时滞和随机扰动的离散时间网络的指数同步问题. 文献[15] 研究了一类具有不确定内耦合的离散时间随机网络的同步和状态估计. Cheng等[16]研究了具有时延耦合的离散动态网络的牵制同步, 并依据牵制控制技术得到了网络的同步判据. 但上述工作主要集中在拓扑结构固定不变的网络同步问题, 并未涉及变结构离散网络的同步问题. 同时, 时间混沌系统一直被人们用以研究网络同步的网络节点. 相比于时间网络, 时空网络因其具有独特的空间特征近年来逐渐被人们采用来研究网络同步问题, 但其同步技术依然不够成熟.
值得注意的是, 虽然网络同步有滑模控制同步[17-18]、驱动响应同步[19]、自适应同步[20-21]、指数同步[22]和耦合同步[23]等多种类型, 但其中的指数同步因在调节网络同步速率方面有其独特的优势, 因而受到人们的广泛青睐. 为此, 文献[24] 研究了具有混合时变时滞的神经网络的自适应指数同步问题. Ahmed等[25]通过构造Lyapunov函数研究了一类具有有向拓扑和时滞的复杂网络的指数同步问题. 文献[26] 研究了具有时变内耦合的复杂动态网络的指数同步, 并获得了确保网络指数同步的充分条件. 但据我们所知, 目前关于离散型时空网络的指数同步方案还鲜有报道.
基于上述讨论, 本文研究了具有拓扑切换特性的离散型不确定时空网络的指数同步, 并对网络中的未知参数进行了识别. 这种同步技术的优势在于网络的拓扑结构可按照人为的需要任意切换, 其同步性能不受影响. 同时, 我们构造了一个具有指数形式的特殊的Lyapunov函数, 通过调节其中的参数, 能够有效地调节网络的同步速率.
1. 时空网络与目标系统的指数同步
考虑$ N $个离散型时空混沌系统作为节点的时空网络与单一目标系统的指数同步.目标系统取如下形式:
$$ \begin{align} y(m, n+1) = &\ F(y(m, n), \alpha) = \\ &\ f(y(m, n))+g(y(m, n))\alpha \end{align} $$ (1) 其中, $ n $为离散化时间, $ m $为空间格点坐标, $ F(y(m, n), \alpha) $是目标系统的状态函数. $ f(y(m, n)) $是经过适当分离后得到的不含系统参数的部分, $ g(y(m, n))\alpha $为含有系统参数的部分, $ \alpha $为系统参数.
网络节点的动力学方程取如下形式:
$$ \begin{align} x_i(m, \, &n+1) = F(x_i(m, n), \alpha_i)\, + \\ &\varepsilon_i\sum\limits_{j = 1}^Nc_{ij}^{s(n)}x_j(m, n)+u_i(m, n) = \\ &f(x_i(m, n))+g(x_i(m, n))\alpha_i\, + \\ &\varepsilon_i\sum\limits_{j = 1}^Nc_{ij}^{s(n)}x_j(m, n)+u_i(m, n) \end{align} $$ (2) 其中, $ n $为离散化的时间, $ m $为空间格点坐标, $ F(x_{i}(m, n), \alpha_{i}) $是网络节点的状态函数, $ x_{i}(m, n) = (x_{1}(m, n), x_{2}(m, n), \cdots, x_{N}(m, n))^{\rm T}\in {\bf R}^{W} $为网络节点的状态矢量. $ \alpha_{i} $为未知参数, 其识别量为$ \hat\alpha_{i}(m, n) $. $ u_{i}(m, n) $为网络控制器. $ \varepsilon_{i} $是网络节点之间的耦合强度, $ C^{s(n)} = (c_{ij}^{s(n)})_{N\times{N}} $是与切换信号$ s(n) $对应的网络耦合矩阵, 其中切换信号$ s(n) $的每一个取值对应某段时间内网络的一种拓扑结构, 相应的耦合矩阵即为$ C^{s(n)} $, 而切换信号在不同时间段的不同取值确保了网络拓扑结构的动态变化.网络的耦合矩阵元定义如下: 如果节点$ i $与节点$ j $ $ (i\ne{j}) $之间有连接, 那么$ c_{ij}^{s(n)} = -1 $, 否则$ c_{ij}^{s(n)} = 0 $. 矩阵对角元定义为$ c_{ii}^{s(n)} = -\sum_{j = 1, j\ne{i}}^{N}c_{ij}^{s(n)} $.
网络误差定义为
$$ \begin{align} e_i(m, n) = x_i(m, n)-y(m, n) \end{align} $$ (3) 通过式(3)可以进一步得到
$$ \begin{align} \, &e_i(m, n+1) = x_i(m, n+1)-y(m, n+1) = \\ &\qquad f(x_i(m, n))+g(x_i(m, n))\alpha_i\, -\\ &\qquad f(y(m, n))-g(y(m, n))\alpha\, +\\ &\qquad \varepsilon_i\sum\limits_{j = 1}^Nc_{ij}^{s(n)}x_j(m, n)+u_i(m, n) \end{align} $$ (4) 假设1. 若存在一个大于零的常数$ l_i $, 则下列不等式成立
$$ \begin{align} &\left| {F(x_i (m, n), \hat {\alpha }_i (m, n))-F(y(m, n), \alpha )} \right|\le\\ & \qquad l_i \left| {x_i (m, n)-y(m, n)} \right| \end{align} $$ (5) 定理1. 当参数识别律和网络同步控制器分别设计为下列形式, 时空网络与目标系统将达到指数同步.
$$ \begin{align} \hat {\alpha }_i (m, n+1) = \alpha _i +Q_i (m, n)(\hat {\alpha }_i (m, n)-\alpha _i ) \end{align} $$ (6) $$ \begin{align} u_i (m, n) = -[l_i +\varepsilon _i \lambda _i^{s(n)} -\exp ({\mu }'-\mu )]e_i (m, n) \end{align} $$ (7) 其中, $ \left| {Q_i (m, n)} \right| = 1-\varphi _i \left| {g(x_i (m, n)} \right|\exp (\mu (n+1)) $为配置函数, $ {\mu }' $与$ \mu $为调节参数, $ \lambda _i^{s(n)} $为网络耦合矩阵的本征值.
证明. 构造网络的Lyapunov函数
$$ \begin{align} V(m, n) = \, &\sum\limits_{i = 1}^N {\left| {e_i (m, n)\exp (\mu n)}\right|}\, +\\ &\sum\limits_{i = 1}^N {\frac{1}{\varphi _i }\left| {\hat {\alpha }_i (m, n)-\alpha _i } \right|} \end{align} $$ (8) 其中, $ \mu $和$ \varphi _i $为调节参数.
基于式(8)可以进一步得到
$$ \begin{align*} \Delta V = &\ V(m, n+1)-V(m, n) = \nonumber\\ &\sum\limits_{i = 1}^{N}\left|e_{i}(m, n+1)\exp(\mu(n+1))\right|- \nonumber\\ &\sum\limits_{i = 1}^{N}\left|e_{i}(m, n)\exp(\mu n)\right|+\nonumber\\ &\sum\limits_{i = 1}^{N}\frac{1}{\varphi_{i}}\left| \hat{\alpha}_{i}(m, n+1)-\alpha_{i}\right|-\nonumber\\ &\sum\limits_{i = 1}^{N}\frac{1}{\varphi_{i}}\left| \hat{\alpha}_{i}(m, n)-\alpha_{i}\right| = \nonumber\\ &\sum\limits_{i = 1}^{N}\left|f(x_{i}(m, n))+g(x_{i}(m, n)) \alpha_{i}\, -\right.\nonumber\\ &f(y(m, n))-g(y(m, n))\alpha\, + \nonumber\\ &\varepsilon_{i}\sum\limits_{j = 1}^{N}c_{ij}^{s(n)}x_{j} (m, n)\left.{+u_{i}(m, n)}\right|\times\nonumber\\ &\exp(\mu(n+1))-\sum\limits_{i = 1}^{N}\left|e_{i}(m, n)\right|\exp(\mu n)\, + \nonumber\\ &\sum\limits_{i = 1}^{N}\frac{1}{\varphi_{i}}\left|\hat{\alpha}_{i}(m, n+1)- \alpha_{i}\right|-\nonumber\\ &\sum\limits_{i = 1}^{N}\frac{1}{\varphi_{i}}\left|\hat{\alpha}_{i}(m, n)- \alpha_{i}\right|\le\nonumber\\ &\sum\limits_{i = 1}^{N}\left|f(x_{i}(m, n))+g(x_{i}(m, n))\hat{\alpha}_{i} (m, n)\, -\right.\nonumber\\ &f(y(m, n))-g(y(m, n))\alpha\, + \end{align*} $$ $$ \begin{align} &\varepsilon_{i}\sum\limits_{j = 1}^{N}c_{ij}^{s(n)}x_{j}(m, n)\left.+u_{i} (m, n)\right|\times\\ &\exp(\mu(n+1))-\sum\limits_{i = 1}^{N}\left|e_{i}(m, n)\right|\exp(\mu n)\, + \\ &\sum\limits_{i = 1}^{N}\left|g(x_{i}(m, n))\right|\left| \hat\alpha_{i}(m, n)-\alpha_{i}\right|\times\\ &\exp(\mu(n+1))\, + \\ &\sum\limits_{i = 1}^{N}\frac{1}{\varphi_{i}}\left|\hat\alpha_{i}(m, n+1)- \alpha_{i}\right|-\\ &\sum\limits_{i = 1}^{N}\frac{1}{\varphi_{i}}\left|\hat\alpha_{i}(m, n)- \alpha_{i}\right| \end{align} $$ (E9) 利用假设1和定理1, 并考虑到网络耦合矩阵$ C^{s(n)} $满足耗散条件, 因此$ \varepsilon _i \sum_{j = 1}^N {c_{ij}^{s(n)} y(m, n)} = 0 $, 则式(9)可以改写为
$$ \begin{align*} \Delta V\le&\sum\limits_{i = 1}^{N}\left|(l_{i}+\varepsilon_{i}\lambda_{i}^{s(n)})e_{i}(m, n)\, +\right.\nonumber\\ &\left.u_{i}(m, n)\right|\exp(\mu(n+1))\, - \nonumber\\ &\sum\limits_{i = 1}^{N}\left|e_{i}(m, n)\right|\exp(\mu n)\, + \nonumber\\ &\sum\limits_{i = 1}^{N}\left|g(x_{i}(m, n))\right|\left|\hat{\alpha}_{i}(m, n)-\alpha_{i}\right|\times\nonumber\\ &\exp(\mu(n+1))\, + \nonumber\\ &\sum\limits_{i = 1}^{N}\frac{1}{\varphi_{i}}\left|\hat{\alpha}_{i}(m, n+1)-\alpha_{i}\right|-\nonumber\\ &\sum\limits_{i = 1}^{N}\frac{1}{\varphi_{i}}\left|\hat{\alpha}_{i}(m, n)-\alpha_{i}\right| = \nonumber\\ &\sum\limits_{i = 1}^{N}\left|\exp({\mu}'-\mu)e_{i}(m, n)\right|\exp(\mu(n+1))\, -\nonumber\\ &\sum\limits_{i = 1}^{N}\left|e_{i}(m, n)\right|\exp(\mu n)\, + \nonumber\\ &\sum\limits_{i = 1}^{N}[\left|g(x_{i}(m, n))\right|\exp(\mu(n+1))-\frac{1}{\varphi_{i}}]\, \times \nonumber\\ &\left|\hat{\alpha}_{i}(m, n)- \alpha_{i}\right|+\nonumber \end{align*} $$ $$ \begin{align} &\sum\limits_{i = 1}^{N}\frac{1}{\varphi_{i}}\left|\hat{\alpha}_{i}(m, n+1)-\alpha_{i}\right| = \\ &\sum\limits_{i = 1}^{N}\left|\exp({\mu}'-\mu)e_{i}(m, n)\right| \exp(\mu(n+1))\, -\\ &\sum\limits_{i = 1}^{N}\left|e_{i}(m, n)\right|\exp(\mu n) = \\ &\sum\limits_{i = 1}^{N}\left|e_{i}(m, n)\right|\exp(\mu n)(\exp({\mu}')-1) \end{align} $$ (10) 显然, 若$ \exp ({\mu }')-1<0 $, 即$ {\mu }'<0 $, 必有$ \Delta V $ $ < $ $ 0 $. 基于Lyapunov稳定性理论, 上述网络与目标系统达到指数同步.
2. 仿真模拟与分析
为了验证同步方案, 我们选取具有时空混沌行为的激光相位共轭波空间拓展系统[27-28]作为目标系统和时空网络的节点进行仿真模拟.
激光相位共轭波空间拓展系统的状态方程为
$$ \begin{align} x(m, n+1) = &\ (1-\varepsilon )\omega J_0^2 [(x(m, n)-B)^{\frac{1}{2}}]\, + \\ &\ \varepsilon \omega J_0^2 [(x(m-1, n)-B)^{\frac{1}{2}}] \end{align} $$ (11) 其中, 系统参数$ \omega = 28 $, $ B = 1 $, $ \varepsilon = 0.1 $, 而$ J_0 $表示零级Bessel函数.
针对上述方程, 取周期性边界条件, 并且初始值在$ (1.8, 2) $区间内随机取值, 我们仿真模拟了激光相位共轭波空间拓展系统状态变量随时空的演化如图 1所示.明显地, 该系统处于时空混沌态.
在仿真模拟中, 目标系统取单一的激光相位共轭波空间拓展系统
$$ \begin{align} y(m, n+1) = &\ (1-\varepsilon )\omega J_0^2 [(y(m, n)-B)^{\frac{1}{2}}]\, + \end{align} $$ $$ \begin{align} &\varepsilon \omega J_0^2 [(y(m-1, n)-B)^{\frac{1}{2}}] \end{align} $$ (12) 任选网络节点数$ N = 7 $, 基于式(2)构造网络节点的动力学方程
$$ \begin{align} x_i (m, n+1) = &\ (1-\varepsilon )\omega J_0^2 [(x_i (m, n)-B)^{\frac{1}{2}}]\, + \\ &\ \varepsilon \omega J_0^2 [(x_i (m-1, n)-B)^{\frac{1}{2}}]\, + \\ &\ \varepsilon _i \sum\limits_{j = 1}^7 c_{ij}^{s(n)} x_j(m, n)+u_i(m, n) \end{align} $$ (13) 其中, 参数$ \varepsilon $在模拟过程中假设为未知量.
选取时间段$ [0, 250] $, 并假设相应的切换信号$ s(n) $如图 2所示.我们选取4种时空网络的拓扑结构与切换信号$ s(n) $对应, 如图 3所示.具体来说, 例如当时间$ n = [0, 50) $时, 对应切换信号$ s(n) = 1 $, 即此时的时空网络的拓扑结构为图 3(a), 对应的耦合矩阵为$ C^1 $; 当时间$ n = [50, 80) $时, 切换信号$ s(n) $ $ = $ $ 3 $, 即此时的时空网络的拓扑结构为图 3(c), 对应的耦合矩阵为$ C^3 $. 显然, 不同时间段的切换信号$ s(n) $都有一一对应的拓扑结构, 且与拓扑结构对应的耦合矩阵为
$$ \begin{align} &C^1 = \begin{bmatrix} \; \; \; 3 & -1 & \; \; \; 0 & -1 & \; \; \; 0 & -1 &\; \; \; 0 \\ \; \; \; 0 & \; \; \; 1 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 &\; \; \; 0 & -1 \\ -1 & \; \; \; 0 & \; \; \; 1 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 \\ \; \; \; 0 & \; \; \; 0 & -1 & \; \; \; 1 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 \\ -1& \; \; \; 0 & \; \; \; 0 & \; \; \; 0 & \; \; \; 1 & \; \; \; 0 & \; \; \; 0 \\ \; \; \; 0 & \; \; \; 0 &\; \; \; 0 & \; \; \; 0 & -1 & \; \; \; 1 & \; \; \; 0 \\ -1 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 & \; \; \; 1 \\ \end{bmatrix} \\ &C^2 = \begin{bmatrix} \; \; \; 3 & -1 & \; \; \; 0 & -1 &\; \; \; 0 & -1 & \; \; \; 0 \\ \; \; \; 0 &\; \; \; 1 & \; \; \; 0 &\; \; \; 0 & \; \; \; 0 & \; \; \; 0 & -1 \\ \; \; \; 0 & \; \; \; 0 &\; \; \; 0 & \; \; \; 0 & \; \; \; 0 &\; \; \; 0 &\; \; \; 0 \\ \; \; \; 0 & \; \; \; 0 & -1 & \; \; \; 1 & \; \; \; 0 &\; \; \; 0 & \; \; \; 0 \\ -1 &\; \; \; 0 &\; \; \; 0 & \; \; \; 0 & \; \; \; 1 &\; \; \; 0 &\; \; \; 0 \\ \; \; \; 0 &\; \; \; 0 & \; \; \; 0 & \; \; \; 0 & -1 &\; \; \; 1 & \; \; \; 0 \\ \; \; \; 0 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 & 0 & \; \; \; 0 &\; \; \; 0 \\ \end{bmatrix} \\ &C^3 = \begin{bmatrix} \; \; \; 2 & -1 & \; \; \; 0 & -1 &\; \; \; 0 & \; \; \; 0 & \; \; \; 0 \\ \; \; \; 0 &\; \; \; 1 &\; \; \; 0 &\; \; \; 0 &\; \; \; 0 &\; \; \; 0 & -1 \\ \; \; \; 0 & -1 &\; \; \; 1 & \; \; \; 0 & \; \; \; 0 &\; \; \; 0 &\; \; \; 0 \\ \; \; \; 0 &\; \; \; 0 & -1 & \; \; \; 1 & \; \; \; 0 &\; \; \; 0 & \; \; \; 0 \\ -1 &\; \; \; 0 &\; \; \; 0 & \; \; \; 0 &\; \; \; 1 &\; \; \; 0 &\; \; \; 0 \\ \; \; \; 0 &\; \; \; 0 &\; \; \; 0 & \; \; \; 0 & -1 &\; \; \; 1 &\; \; \; 0 \\ \; \; \; 0 &\; \; \; 0 &\; \; \; 0 & \; \; \; 0 & \; \; \; 0 &\; \; \; 0 & \; \; \; 0\\ \end{bmatrix} \end{align} $$ $$ \begin{align} &C^4 = \begin{bmatrix} \; \; \; 1 & \; \; \; 0 &\; \; \; 0 & -1 &\; \; \; 0 & \; \; \; 0 & \; \; \; 0 \\ \; \; \; 0 & \; \; \; 1 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 & -1 \\ \; \; \; 0 & -1 & \; \; \; 1 & \; \; \; 0 & \; \; \; 0 &\; \; \; 0 &\; \; \; 0 \\ \; \; \; 0 &\; \; \; 0 & -1 & \; \; \; 1 & \; \; \; 0 & \; \; \; 0 & \; \; \; 0 \\ -1 &\; \; \; 0 &\; \; \; 0 & \; \; \; 0 & \; \; \; 2 &\; \; \; 0 & -1 \\ \; \; \; 0 &\; \; \; 0 & \; \; \; 0 & \; \; \; 0 & -1 &\; \; \; 1 &\; \; \; 0 \\ \; \; \; 0 &\; \; \; 0 & \; \; \; 0 & \; \; \; 0 &\; \; \; 0 & \; \; \; 0 &\; \; \; 0\\ \end{bmatrix} \end{align} $$ 基于式(6)和式(7), 我们可以获得对应的网络控制器及其参数识别律, 其中相关参数取为$ {\mu }' = $ $ -0.09 $, $ \mu = -0.01 $, $ \varphi _i = 0.012 $. 取周期性边界条件, 网络节点状态变量的初始值仍随机选取, 我们模拟网络误差随时空的演化如图 4~10所示, 参数识别量随时空的演化如图 11~17所示.
图 11 未知参数$\varepsilon_1(m, n)$随时空的演化Fig. 11 Spatiotemporal evolution of unknown parameter $\varepsilon_1(m, n)$
图 12 未知参数$\varepsilon_2(m, n)$随时空的演化Fig. 12 Spatiotemporal evolution of unknown parameter $\varepsilon_2(m, n)$
图 13 未知参数$\varepsilon_3\, (m, n)$随时空的演化Fig. 13 Spatiotemporal evolution of unknown parameter $\varepsilon_3\, (m, n)$
图 14 未知参数$\varepsilon_4(m, n)$随时空的演化Fig. 14 Spatiotemporal evolution of unknown parameter $\varepsilon_4(m, n)$
图 15 未知参数$\varepsilon_5(m, n)$随时空的演化Fig. 15 Spatiotemporal evolution of unknown parameter $\varepsilon_5(m, n)$
图 16 未知参数$\varepsilon_6(m, n)$随时空的演化Fig. 16 Spatiotemporal evolution of unknown parameter $\varepsilon_6(m, n)$
图 17 未知参数$\varepsilon_7(m, n)$随时空的演化Fig. 17 Spatiotemporal evolution of unknown parameter $\varepsilon_7(m, n)$由图 4~10可以发现网络误差$ e_i (m, n) $在初始阶段存在一定振荡但经过较短时间后振荡明显减弱并且当$ n = 60 $时同步误差迅速收敛于零此时网络达到了指数同步状态.同时, 保持其他参数不变, 调节参数$ \mu $取不同的数值, 网络的同步误差趋于零的快慢程度是不同的, 可见指数同步在控制同步速率方面有较大的优势.另外, 从图中也可以看到, 在整个时间序列[0, 250]上, 即使网络拓扑结构随着切换信号任意变化, 但网络误差收敛于零后一直保持稳定状态, 说明网络拓扑结构的切换并不影响网络的同步性能.图 11~17显示, 由于初始值不同, 未知参数$ \varepsilon _i(m, n) $的识别图像在$ n = 60 $之前有明显的波动.但在此之后逐渐趋于定值0.1, 与参数的实际值一致, 表明未知参数的识别效果明显.
3. 结论
在时空网络的拓扑结构随着切换信号的变换而变化的情况下, 通过设计有效的同步控制器和含有指数形式的Lyapunov函数, 我们实现了含有未知参数的离散型时空网络与单一目标的指数同步, 并且未知参数也得到了有效的识别.通过仿真模拟, 我们进一步验证了设计的同步原理的有效性.模拟结果表明, 网络误差$ e_i(m, n) $在较短的时间内趋于零, 未知参数的识别曲线经过短时间波动后也趋于定值0.1.我们发现, 变化的网络拓扑结构并没有对网络的同步性能产生影响, 并且指数同步策略能够有效地控制网络的同步速率.
-
图 11 未知参数$\varepsilon_1(m, n)$随时空的演化
Fig. 11 Spatiotemporal evolution of unknown parameter $\varepsilon_1(m, n)$
图 12 未知参数$\varepsilon_2(m, n)$随时空的演化
Fig. 12 Spatiotemporal evolution of unknown parameter $\varepsilon_2(m, n)$
图 13 未知参数$\varepsilon_3\, (m, n)$随时空的演化
Fig. 13 Spatiotemporal evolution of unknown parameter $\varepsilon_3\, (m, n)$
图 14 未知参数$\varepsilon_4(m, n)$随时空的演化
Fig. 14 Spatiotemporal evolution of unknown parameter $\varepsilon_4(m, n)$
图 15 未知参数$\varepsilon_5(m, n)$随时空的演化
Fig. 15 Spatiotemporal evolution of unknown parameter $\varepsilon_5(m, n)$
图 16 未知参数$\varepsilon_6(m, n)$随时空的演化
Fig. 16 Spatiotemporal evolution of unknown parameter $\varepsilon_6(m, n)$
-
[1] Xu X F, Zong G D, Hou L L. Passivity-based stabilization and passive synchronization of complex nonlinear networks. Neurocomputing, 2016, 175: 101-109 doi: 10.1016/j.neucom.2015.10.040 [2] Wang F, Yang Y Q, Hu M F, Xu X Y. Projective cluster synchronization of fractional-order coupled-delay complex network via adaptive pinning control. Physica A, 2015, 434: 134-143 doi: 10.1016/j.physa.2015.03.089 [3] Xu Y H, Zhou W N, Fang J A, Xie C R, Tong D B. Finite-time synchronization of the complex dynamical network with non-derivative and derivative coupling. Neurocomputing, 2016, 173: 1356-1361 doi: 10.1016/j.neucom.2015.09.008 [4] Wu H Q, Wang L F, Niu P F, Wang Y. Global projective synchronization in finite time of nonidentical fractionalorder neural networks based on sliding mode control strategy. Neurocomputing, 2017, 235: 264-273 doi: 10.1016/j.neucom.2017.01.022 [5] Anbuvithya R, Mathiyalagan K, Sakthivel R, PrakashP. Non-fragile synchronization of memristive BAM networks with random feedback gain fluctuations. Communications in Nonlinear Science and Numerical Simulation, 2015, 29: 427-440 doi: 10.1016/j.cnsns.2015.05.020 [6] Yang Y, Wang Y, Li T Z. Outer synchronization of fractional-order complex dynamical networks. Optik, 2016, 127: 7395-7407 doi: 10.1016/j.ijleo.2016.05.029 [7] Wang S G, Zheng S, Zhang B W, Cao H T. Modified function projective lag synchronization of uncertaincomplex networks with time-varying coupling strength. Optik, 2016, 127: 4716-4725 doi: 10.1016/j.ijleo.2016.01.085 [8] Chandrasekar A, Rakkiyappan R. Impulsive controller design for exponential synchronization of delayed stochastic memristor-based recurrent neural networks. Neurocomputing, 2016, 173: 1348-1355 doi: 10.1016/j.neucom.2015.08.088 [9] Srinivasan K, Chandrasekar V K, Gladwin P R, Murali K, Lakshmanan M. Different types of synchronization in coupled network based chaotic circuits. Communications in Nonlinear Science and Numerical Simulation, 2016, 39: 156-168 doi: 10.1016/j.cnsns.2016.03.002 [10] Zhai S D. Modulus synchronization in a network of nonlinear systems with antagonistic interactions and switching topologies. Communications in Nonlinear Science and Numerical Simulation, 2016, 33: 184-193 doi: 10.1016/j.cnsns.2015.09.010 [11] Wang X, Yang G H. Distributed H$_\infty$ consensus tracking control for multi-agent networks with switching directed topologies. Neurocomputing, 2016, 207: 693-699 doi: 10.1016/j.neucom.2016.05.052 [12] Fan J B, Wang Z X, Jiang G P. Quasi-synchronization of heterogeneous complex networks with switching sequentially disconnected topology. Neurocomputing, 2017, 237: 342-349 doi: 10.1016/j.neucom.2017.01.025 [13] Dai A D, Zhou W N, Xu Y H, Xiao C. Adaptive exponential synchronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays by pinning control. Neurocomputing, 2016, 173: 809-818 doi: 10.1016/j.neucom.2015.08.034 [14] Zhang Q J, Chen G R, Wan L. Exponential synchronization of discrete-time impulsive dynamical networks with time-varying delays and stochastic disturbances. Neurocomputing, 2018, 309: 62-69 doi: 10.1016/j.neucom.2018.04.070 [15] Sakthivel R, Sathishkumar M, Kaviarasan B, Marshal Anthoni S. Synchronization and state estimation for stochastic complex networks with uncertain inner coupling. Neurocomputing, 2017, 238: 44-45 doi: 10.1016/j.neucom.2017.01.035 [16] Cheng R R, Peng M S, Zuo J. Pinning synchronization of discrete dynamical networks with delay coupling. Physica A, 2016, 450: 444-453 doi: 10.1016/j.physa.2016.01.004 [17] Mohammadzadeh A, Ghaemi S. A modified sliding mode approach for synchronization of fractional-order chaotic/hype- rchaotic systems by using new self-structuring hierarchical type-2 fuzzy neural network. Neurocomputing, 2016, 191: 200-213 doi: 10.1016/j.neucom.2015.12.098 [18] Zhao L, Jia Y M. Neural network-based distributed adaptive attitude synchronization control of spacecraft formation under modified fast terminal sliding mode. Neurocomputing, 2016, 171: 230-241 doi: 10.1016/j.neucom.2015.06.063 [19] Yang L X, Jiang J, Liu X J. Synchronization of fractional-order colored dynamical networks via open-plus-closed-loop control. Physica A, 2016, 443: 200-211 doi: 10.1016/j.physa.2015.09.062 [20] Fan Y Q, Xing K Y, Wang Y H, Wang L Y. Projective synchronization adaptive control for different chaoticneural networks with mixed time delays. Optik, 2016, 127: 2551-2557 doi: 10.1016/j.ijleo.2015.11.227 [21] Li J M, He C, Zhang W Y, Chen M L. Adaptive synchronization of delayed reaction-diffusion neural networks with unknown non-identical time-varying coupling strengths. Neurocomputing, 2017, 219: 144-153 doi: 10.1016/j.neucom.2016.09.006 [22] Ahmed M A A, Liu Y R, Zhang W B, Alsaadi F E. Exponential synchronization via pinning adaptive control for complex networks of networks with time delays. Neurocomputing, 2017, 225: 198-204 doi: 10.1016/j.neucom.2016.11.022 [23] Xu Q, Zhuang S X, Liu S J, Xiao J. Decentralized adaptive coupling synchronization of fractional-order complex-vari- able dynamical networks. Neurocomputing, 2016, 186: 119-126 doi: 10.1016/j.neucom.2015.12.072 [24] Han X M, Wu H Q, Fang B L. Adaptive exponential synchronization of memristive neural networks with mixed time-varying delays. Neurocomputing, 2016, 201: 40-50 doi: 10.1016/j.neucom.2015.11.103 [25] Ahmed M A A, Liu Y R, Zhang W B, Alsaedi A, Hayat T. Exponential synchronization for a class of complex networks of networks with directed topology and time delay. Neurocomputing, 2017, 266: 274-283 doi: 10.1016/j.neucom.2017.05.039 [26] Dai H, Chen W S, Jia J P, Liu J Y, Zhang Z Q. Exponential synchronization of complex dynamical networks with time-varying inner coupling via event-triggered communication. Robotics and Autonomous Systems, 2017, 245: 124-132 [27] Beli M R, Stojkov P. Chaos in phase-conjugate resonators as a multimodal mapping. Optical and Quantum Electronics, 1990, 22: 157-165 doi: 10.1007/BF02189951 [28] Kaneko K. Spatial period-doubling in open flow. Physics Letters A, 1980, 111: 321-325 http://www.sciencedirect.com/science/article/pii/0375960185903597 期刊类型引用(0)
其他类型引用(2)
-
下载:
下载:
计量
- 文章访问数: 831
- HTML全文浏览量: 212
- PDF下载量: 640
- 被引次数: 2
