伴随随机攻击的信息物理系统的同步控制

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高洋 ¹ , 马洋洋 ¹ , 张亮 ¹ , 王眉林 ¹ , 王卫苹 ²
1. 中国信息安全测评中心, 北京 100085;
2. 北京科技大学计算机与通信工程学院, 北京 100083

收稿日期：2017-08-14
基金项目：国家重点研发计划项目（2016YFB0800900）；国家自然科学基金面上项目（61672534）
作者简介：高洋(1981-), 女, 副研究员。E-mail:gaoy@itsec.gov.cn

摘要：信息物理系统（CPS）广泛存在于现代基础设施体系中，例如未来智能电网、智能交通网络和公众健康系统等。该系统的安全是社会正常运行的关键。该文介绍了信息物理系统的物理层网络和信息层网络，这两层网络是相互依存的。提出了在存在随机攻击条件下，相互依存的信息物理系统的数学模型，该攻击既存在于物理层网络，也存在于信息层网络。该文研究了在存在随机攻击条件下，相互依存的信息物理系统的同步问题，并提出了一种自适应非线性控制器。利用Weiner过程将这些控制器加入到物理层网络中，以实现相互依存的信息物理系统的同步。数值模拟证明了理论结果是有效的。
关键词：信息物理系统随机攻击同步控制非线性自适应控制器
Synchronization control of cyber physical systems during malicious stochastic attacks
GAO Yang¹, MA Yangyang¹, ZHANG Liang¹, WANG Meilin¹, WANG Weiping²
1.China Information Technology Security Evaluation Center, Beijing 100085, China;
2.School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing 100083, China

Abstract: Cyber physical systems (CPSs) exist in many modern infrastructure systems, such as the future smart grid, smart transportation networks, and public health systems. CPS security is critical for normal operation of our society. The physical layer networks and the cyber layer networks are all complex networks that are interdependent. A mathematical model is given for the interdependent systems during stochastic malicious attacks, which exist in both the physical layer networks and the cyber layer networks. In addition, an adaptive nonlinear controller is given to synchronize interdependent CPSs during malicious stochastic attacks. The controller uses the properties of a Weiner process for the physical layer networks to synchronize the interdependent CPSs. Numerical simulations show the effectiveness of the theoretical results.
Key words: cyber physical systemstochastic attacksynchronization controladaptive nonlinear controller
信息物理系统(cyber physical system，CPS)是集成计算、通信与控制于一体的综合性复杂系统^[1]，具有复杂性、异构性、深度融合、自组织与自适应性、实时性、海量性等特性^[2]。它考虑了信息和物理实体之间紧密的相互作用，在改变生活的各个方面上拥有巨大潜力。信息物理系统广泛应用于现代基础设施中，如智能电网以及能源系统、智能制造中的工业控制系统就是典型的信息物理系统。基于CPS与能源领域相融合的能源互联网，将重构能源产业价值链体系^[3]。同时，信息物理系统相关技术在自动驾驶、机器人应用、智能建筑等方面已有应用。信息物理系统是减少能耗以及提高性能、可靠性和效率的关键。CPS正在成为各类大规模工业系统的基础^[4]。学界一致认为，集成的信息物理系统将在21世纪成为主要工业的支柱技术^[5]。
信息物理系统主要包括两个部分，即物理过程和信息系统。通常，物理过程是由网络系统监控或控制的，这是一个由具有传感、计算和通信(通常是无线)能力的几个微小设备组成的网络系统。所涉及的物理过程可能是一种自然现象(例如休眠火山)、人造物理系统(例如工业控制系统)或两者的更为复杂的组合。如果几个物理过程连接在一起，那么它们就构成了一个系统。在本文中可以看到一个由信息系统构建的网络。一般来说，同步是对在信息和物理网络之间发生的事件的协调，这种同步称为外部同步；同步也包括内部同步，内部同步研究一个复杂动态网络各个节点的状态是否能够趋于一致^[6]。
目前绝大部分关键基础设施是基于CPS构建的，因此对CPS攻击可能会造成灾难性的后果。例如，近年来一些黑客多次入侵了美国联邦航空管理局的空中交通控制系统^[7]。一些黑客可以破坏植入人体内的无线通讯医疗设备^[8]。2010年黑客们开发出一个称为汽车鲨鱼的软件^[9]，它能远程攻击汽车引擎，关掉刹车，这样汽车就不会停下来。该软件还可以通过监测电子控制单元(ECU)之间的通信并插入假数据包进行攻击，使仪器读数错误。CPS容易发生故障，容易受到对其物理基础设施以及对其数据管理和通信层的攻击。事实上，在越来越多的CPS中，如智能电网、智能交通系统、医疗系统等，都存在着安全隐患。CPS是继互联网之后的又一场信息革命，而安全问题是决定CPS能否被广泛使用的关键因素之一^[10]。
蒋国平等^[6]以复杂动态网络及其同步控制为基础，研究了CPS中的控制问题。互联互通的发展必然需要CPS具有更高的开放性，而开放性更高则必然导致更多的潜在安全威胁^[11]。因此，针对CPS的安全性问题进行研究迫在眉睫。
目前，CPS的主流体系架构分为3个层次，即感知执行层、数据传输层和应用控制层。李钊^[12]等即从这3个方面对CPS的安全性作了论述与研究。本文首先把CPS看成是一个网络系统，由相互依存的物理层网络和信息层网络构成，然后提出了一种CPS的数学模型，并考虑了对物理层和信息层网络的随机攻击。随后，为了确保伴随随机攻击的相互依存的CPS网络的同步，本文提出一种自适应非线性控制器。最后，给出了使用不同控制器时的模拟结果来验证理论分析的有效性。
1 CPS数据模型CPS是多层系统的集成，多个CPS相互连接，控制自身并对环境以及其他系统的网络或物理层的信号做出反应。CPS的物理层和信息层之间相互作用。因此，信息层和物理层网络之间的同步是非常重要的。本文首先提出CPS的数学模型：

$\begin{array}{*{20}{c}}{{{\mathit{\boldsymbol{\dot x}}}_i}\left( t \right) = \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{a_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}{\mathit{\boldsymbol{x}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{y}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} ,}\\{{{\mathit{\boldsymbol{\dot y}}}_i}\left( t \right) = \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{y}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^N {{d_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_4}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _2}\left( t \right)} \right)} .}\end{array}$

(1)

其中: x_i(t)是信息层网络t时刻第i个节点的状态向量，y_i(t)是物理层网络t时刻第i个节点的状态向量。f和g是连续的非线性向量函数。Г₁是信息层网络的内部耦合矩阵，Г₂是物理层网络的内部耦合矩阵，Г₃是信息层和物理层网络之间的外部耦合矩阵，Г₄是物理层和信息层网络之间的外部耦合矩阵。τ₁(t)和τ₂(t)是随时间变化的延迟。信息层和物理层网络都有N个节点。A=(a_ij)_N×N代表网络的拓扑结构。若网络中节点i和节点j(j≠i)之间存在连接，则令a_ij=a_ji=1；否则令a_ij=a_ji=0。然后，令$ {\mathit{a}_{\mathit{ii}}} = - \sum\limits_{\mathit{j} = 1,\mathit{j} \ne i}^\mathit{N} {{\mathit{a}_{\mathit{ij}}}} $。类似地有B=(b_ij)_N×N，C= (c_ij)_N×N，D=(d_ij)_N×N。0≤τ₁(t)≤τ₁, 0≤τ₂(t)≤τ₂。
CPS可能受到许多外部因素的攻击。在此考虑信息层和物理层网络都受到随机扰动攻击，将模型(1)修改如下：

$\begin{array}{*{20}{c}}{{{\mathit{\boldsymbol{\dot x}}}_i}\left( t \right) = \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{a_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}{\mathit{\boldsymbol{x}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{y}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} + }\\{\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right),}\\{{{\mathit{\boldsymbol{\dot y}}}_i}\left( t \right) = \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{y}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^N {{d_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_4}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _2}\left( t \right)} \right)} + }\\{\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right).}\end{array}$

(2)

其中: ω(t)是一个n维Brown运动。σ: R⁺×R^N×R^N→R^N×N是一个噪声强度函数矩阵。
为了使信息层和物理层网络同步，定义误差向量为

${\mathit{\boldsymbol{e}}_i}\left( t \right) = {\mathit{\boldsymbol{y}}_i}\left( t \right) - {\mathit{\boldsymbol{x}}_i}\left( t \right).$

(3)

然后，无随机攻击的误差系统为

$\begin{array}{*{20}{c}}{{\mathit{\boldsymbol{e}}_i} = \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{y}}_j}\left( t \right)} - }\\{\sum\limits_{j = 1}^N {{a_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}{\mathit{\boldsymbol{x}}_j}\left( t \right)} + \sum\limits_{j = 1}^N {{d_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_4}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _2}\left( t \right)} \right)} - }\\{\sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{y}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} .}\end{array}$

(4)

随机攻击下的误差系统则为

$\begin{array}{*{20}{c}}{{\rm{d}}{\mathit{\boldsymbol{e}}_i}\left( t \right) = \left[ {\mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + } \right.}\\{\sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{y}}_j}\left( t \right)} - \sum\limits_{j = 1}^N {{a_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}{\mathit{\boldsymbol{x}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^N {{d_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_4}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _2}\left( t \right)} \right)} - }\\{\left. {\sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{y}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} } \right]{\rm{d}}t + }\\{\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right) - \mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right).}\end{array}$

(5)

在这里，ω(t)=(ω₁(t)，ω₂(t)，…，ω_N(t))^T是一个n维Brown运动向量，它基于一个完整的概率空间(Ω，$ {\cal F}$，P)且满足Eω(t)=0，E[dω(t)]²=dt。对于完整的概率空间(Ω，$ {\cal F}$，P)：Ω是一个非空集合，即样本空间，Ω的元素称作样本输出，可写作ω；$ {\cal F}$是样本空间Ω的幂集的一个非空子集，(Ω,$ {\cal F}$)合起来称为可测空间。事件就是样本输出的集合，在此集合上可定义其概率。P为概率，是从集合$ {\cal F}$到实数域R的函数。
为了获得主要结果，提出以下假设：
假设1??对于所有的x, y∈R^N都存在一个非负常数μ使得

$\left\| {\mathit{\boldsymbol{g}}\left( \mathit{\boldsymbol{x}} \right) - \mathit{\boldsymbol{g}}\left( \mathit{\boldsymbol{y}} \right)} \right\| \le \mu \left\| {\mathit{\boldsymbol{x}} - \mathit{\boldsymbol{y}}} \right\|.$

(6)

假设2??σ: R⁺×R^N×R^N→R^N×N是局部Lipschitz连续的，且满足线性增长条件，可得

$\begin{array}{*{20}{c}}{{\rm{trace}}\left\{ {{{\left[ {\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_j}\left( t \right)} \right) - \mathit{\boldsymbol{\sigma }}\left( {t,\mathit{\boldsymbol{x}}\left( t \right)} \right)} \right]}^{\rm{T}}} \times } \right.}\\{\left. {\left[ {\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_j}\left( t \right)} \right) - \mathit{\boldsymbol{\sigma }}\left( {t,\mathit{\boldsymbol{x}}\left( t \right)} \right)} \right]} \right\} \times }\\{\nu \left[ {{{\left\| {{\mathit{\boldsymbol{x}}_i}\left( t \right) - \mathit{\boldsymbol{x}}\left( t \right)} \right\|}^2} + {{\left\| {{\mathit{\boldsymbol{x}}_j}\left( t \right) - \mathit{\boldsymbol{x}}\left( t \right)} \right\|}^2}} \right].}\end{array}$

(7)

其中：ν是个正常数，i, j=1, 2, …, N。
2 CPS的同步控制2.1 无随机攻击的CPS的同步控制当CPS没有受到随机攻击时，可以利用非线性自适应控制器来控制相互依存的信息层和物理层网络以实现同步。
控制器u_i(t)下的误差系统(4)可以作如下描述：

$\begin{array}{*{20}{c}}{{{\mathit{\boldsymbol{\dot e}}}_i}\left( t \right) = \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + }\\{\sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{y}}_j}\left( t \right)} - \sum\limits_{j = 1}^N {{a_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}{\mathit{\boldsymbol{x}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^N {{d_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_4}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _2}\left( t \right)} \right)} - }\\{\sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{y}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} + {\mathit{\boldsymbol{u}}_i}\left( t \right).}\end{array}$

(8)

定理1??在假设1和2下，信息层和无攻击的物理层网络在以下自适应非线性控制器下全局渐近同步：

$\left\{ \begin{array}{l}{\mathit{\boldsymbol{u}}_i}\left( t \right) = \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + \\\;\;\;\;\;\;\;\sum\limits_{j = 1}^N {{a_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}{\mathit{\boldsymbol{x}}_j}\left( t \right)} - \sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{x}}_j}\left( t \right)} + \\\;\;\;\;\;\;\;\sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} - \\\;\;\;\;\;\;\;\sum\limits_{j = 1}^N {{d_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_4}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _2}\left( t \right)} \right)} - {k_i}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right),\\{{\dot k}_i}\left( t \right) = {l_i}{\left\| {{\mathit{\boldsymbol{e}}_i}\left( t \right)} \right\|^2}.\end{array} \right.$

(9)

其中：l_i是正常数，i=1, 2, …, N。
证明：
控制器(9)下的误差系统(4)可以描述为

建立以下的Lyapunov函数：

$\begin{array}{*{20}{c}}{V\left( t \right) = \frac{1}{2}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} + \frac{1}{2}\sum\limits_{i = 1}^N {\frac{{{{\left( {{k_i}\left( t \right) - h} \right)}^2}}}{{{l_i}}}} + }\\{\frac{1}{2}\sum\limits_{i = 1}^N {\frac{1}{{1 - {\tau _1}}}\int_{t - {\tau _1}\left( t \right)}^t {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( s \right){\mathit{\boldsymbol{e}}_i}\left( s \right){\rm{d}}s} } .}\end{array}$

其中h是正常数。利用上述两式可得

$\begin{array}{*{20}{c}}{\dot V\left( t \right) = \sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right){{\mathit{\boldsymbol{\dot e}}}_i}\left( t \right)} + \sum\limits_{i = 1}^N {\frac{{\left( {{k_i}\left( t \right) - h} \right)}}{{{l_i}}}{{\dot k}_i}\left( t \right)} + }\\{\frac{1}{{2\left( {1 - {\tau _1}} \right)}}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} - }\\{\frac{{1 - {{\dot \tau }_1}\left( t \right)}}{{2\left( {1 - {\tau _1}} \right)}}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( {t - {\tau _1}\left( t \right)} \right){\mathit{\boldsymbol{e}}_i}\left( {t - {\tau _1}\left( t \right)} \right)} = }\\{\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right)\left[ {\mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) - {k_i}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} \right]} + }\\{\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right)\left[ {\sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{e}}_j}\left( t \right)} - \sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{e}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} } \right]} + }\\{\sum\limits_{i = 1}^N {\left( {\frac{1}{{2\left( {1 - {\tau _1}} \right)}} + {k_i}\left( t \right) - h} \right)\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} - }\\{\frac{{1 - {{\dot \tau }_1}\left( t \right)}}{{2\left( {1 - {\tau _1}} \right)}}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( {t - {\tau _1}\left( t \right)} \right){\mathit{\boldsymbol{e}}_i}\left( {t - {\tau _1}\left( t \right)} \right)} .}\end{array}$

令E=B?Γ₂，F=C?Γ₃，有

$\begin{array}{*{20}{c}}{\dot V\left( t \right) \le \sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( {\mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} - h} \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} + }\\{\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right)\mathit{\boldsymbol{E}}{\mathit{\boldsymbol{e}}_j}\left( t \right)} } - }\\{\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right)\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{e}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} } - }\\{\frac{{1 - {{\dot \tau }}\left( t \right)}}{{2\left( {1 - {\tau _1}} \right)}}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( {t - {\tau _1}\left( t \right)} \right){\mathit{\boldsymbol{e}}_i}\left( {t - {\tau _1}\left( t \right)} \right)} .}\end{array}$

令e(t)=(e₁(t), e₂(t), …, e_N(t))，可得

$\begin{array}{*{20}{c}}{\dot V\left( t \right) \le \left( {\mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} - h} \right){\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{e}}\left( t \right) + }\\{{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\frac{{{\mathit{\boldsymbol{E}}^{\rm{T}}} + \mathit{\boldsymbol{E}}}}{2}\mathit{\boldsymbol{e}}\left( t \right) + {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{Fe}}\left( {t - {\tau _1}\left( t \right)} \right) - }\\{\frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {t - {\tau _1}\left( t \right)} \right)\mathit{\boldsymbol{e}}\left( {t - {\tau _1}\left( t \right)} \right).}\end{array}$

因为

$\begin{array}{*{20}{c}}{{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{Fe}}\left( {t - {\tau _1}\left( t \right)} \right) \le \frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{e}}\left( t \right) + }\\{\frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {t - {\tau _1}\left( t \right)} \right)\mathit{\boldsymbol{e}}\left( {t - {\tau _1}\left( t \right)} \right),}\end{array}$

所以

$\begin{array}{*{20}{c}}{\dot V\left( t \right) \le \left( {\mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} - h} \right){\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{e}}\left( t \right) + }\\{{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\frac{{{\mathit{\boldsymbol{E}}^{\rm{T}}} + \mathit{\boldsymbol{E}}}}{2}\mathit{\boldsymbol{e}}\left( t \right) + \frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{e}}\left( t \right) \le }\\{\left[ {\mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} - h + } \right.}\\{\left. {{\lambda _{\max }}\frac{{{\mathit{\boldsymbol{E}}^{\rm{T}}} + \mathit{\boldsymbol{E}}}}{2} + {\lambda _{\max }}\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{F}}^{\rm{T}}}} \right]{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{e}}\left( t \right).}\end{array}$

λ_max为最大特征值。如果存在足够大的正常数h，并且它满足

$\begin{array}{*{20}{c}}{h \ge \mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} + }\\{{\lambda _{\max }}\frac{{{\mathit{\boldsymbol{E}}^{\rm{T}}} + \mathit{\boldsymbol{E}}}}{2} + {\lambda _{\max }}\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{F}}^{\rm{T}}} + 1,}\end{array}$

则有$ \mathit{\dot V}\left( \mathit{t} \right) \le - {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{e}}\left( \mathit{t} \right) \le 0$，因此误差是收敛的。证明完成。
2.2 具有随机攻击的网络物理系统的同步控制CPS常常会受到外部攻击的影响，且这种攻击在很多情况下是不确定的，可以将它认定为随机扰动。因此，研究带有随机扰动的CPS同步性问题具有重要意义。
控制器u_i(t)下的误差系统可以描述为

$\begin{array}{*{20}{c}}{{\rm{d}}{\mathit{\boldsymbol{e}}_i}\left( t \right) = \left[ {\mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{y}}_j}\left( t \right)} - } \right.}\\{\sum\limits_{j = 1}^N {{a_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}{\mathit{\boldsymbol{x}}_j}\left( t \right)} + \sum\limits_{j = 1}^N {{d_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_4}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _2}\left( t \right)} \right)} - }\\{\left. {\sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{y}}_i}\left( {t - {\tau _1}\left( t \right)} \right)} + {\mathit{\boldsymbol{u}}_i}\left( t \right)} \right]{\rm{d}}t + }\\{\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right) - \mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right).}\end{array}$

(10)

定理2??在假设1和2下，如果存在一个足够大的正常数h满足

$\begin{array}{*{20}{c}}{h \ge \mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} + {\lambda _{\max }}\frac{{{\mathit{\boldsymbol{E}}^{\rm{T}}} + \mathit{\boldsymbol{E}}}}{2} + }\\{{\lambda _{\max }}\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{F}}^{\rm{T}}} + 1 + \nu ,}\end{array}$

则网络(5)通过自适应非线性控制器(9)全局渐近同步。
证明??控制器(9)下的误差系统(5)为

$\begin{array}{*{20}{c}}{{\rm{d}}{\mathit{\boldsymbol{e}}_i}\left( t \right) = \left[ {\mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{e}}_j}\left( t \right)} - } \right.}\\{\left. {\sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{e}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} - {k_i}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} \right]{\rm{d}}t + }\\{\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right) - \mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right).}\end{array}$

构造下面的Lyapunov函数：

$\begin{array}{*{20}{c}}{V\left( t \right) = \frac{1}{2}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} + \frac{1}{2}\sum\limits_{i = 1}^N {\frac{{{{\left( {{k_i}\left( t \right) - h} \right)}^2}}}{{{l_i}}}} + }\\{\frac{1}{2}\sum\limits_{i = 1}^N {\frac{1}{{1 - {\tau _1}}}\left( {\int_{t - {\tau _1}\left( t \right)}^t {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( s \right){\mathit{\boldsymbol{e}}_i}\left( s \right){\rm{d}}s} } \right)} .}\end{array}$

式中h是正常数，然后有

$\begin{array}{*{20}{c}}{{\cal L}V\left( t \right) = \sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right){{\mathit{\boldsymbol{\dot e}}}_i}\left( t \right)} + \sum\limits_{i = 1}^N {\frac{{\left( {{k_i}\left( t \right) - h} \right)}}{{{l_i}}}{{\dot k}_i}\left( t \right)} + }\\{\frac{1}{{2\left( {1 - {\tau _1}} \right)}}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} - }\\{\frac{{1 - \dot \tau \left( t \right)}}{{2\left( {1 - {\tau _1}} \right)}}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( {t - {\tau _1}\left( t \right)} \right){\mathit{\boldsymbol{e}}_i}\left( {t - {\tau _1}\left( t \right)} \right)} = }\\{\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( \mathit{t} \right)\left[ {\mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) - {k_i}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} \right]} + }\\{\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right)\left[ {\sum\limits_{j = 1}^N {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{e}}_j}\left( t \right)} - \sum\limits_{j = 1}^N {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{e}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} } \right]} + }\\{\sum\limits_{i = 1}^N {\left( {\frac{1}{{2\left( {1 - {\tau _1}} \right)}} + {k_i}\left( t \right) - h} \right)\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} - }\\{\frac{{1 - \dot {\tau_1} \left( t \right)}}{{2\left( {1 - {\tau _1}} \right)}}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( {t - {\tau _1}\left( t \right)} \right){\mathit{\boldsymbol{e}}_i}\left( {t - {\tau _1}\left( t \right)} \right)} + }\\{\frac{1}{2}\sum\limits_{i = 1}^N {{\rm{trace}}\left\{ {{{\left[ {\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right)} \right]}^{\rm{T}}} \times } \right.} }\\{\left. {\left[ {\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right)} \right]} \right\}.}\end{array}$

根据假设2，可以得到

$\begin{array}{*{20}{c}}{\frac{1}{2}\sum\limits_{i = 1}^N {{\rm{trace}}\left\{ {{{\left[ {\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{y}}_i}\left( t \right) - \mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right)} \right.} \right]}^{\rm{T}}} \times } \right.} }\\{\left. {\left[ {\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right)} \right]} \right\} \le }\\{\frac{1}{2}\sum\limits_{i = 1}^N {\left[ {\nu {{\left\| {{\mathit{\boldsymbol{y}}_i}\left( t \right) - {\mathit{\boldsymbol{x}}_i}\left( t \right)} \right\|}^2} + } \right.} }\\{\left. {\nu {{\left\| {{\mathit{\boldsymbol{y}}_i}\left( t \right) - {\mathit{\boldsymbol{x}}_i}\left( t \right)} \right\|}^2}} \right] = }\\{\sum\limits_{i = 1}^N {\nu {{\left\| {{\mathit{\boldsymbol{e}}_i}\left( t \right)} \right\|}^2}} = \sum\limits_{i = 1}^N {\nu \mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} .}\end{array}$

令E=B?Γ₂, F=C?Γ₃，有

$\begin{array}{*{20}{c}}{{\cal L}V\left( t \right) \le \sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right)\left( {\mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} - h + \nu } \right){\mathit{\boldsymbol{e}}_i}\left( t \right)} + }\\{\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right)\mathit{\boldsymbol{E}}{\mathit{\boldsymbol{e}}_j}\left( t \right)} } - }\\{\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( t \right)\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{e}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} } - }\\{\frac{{1 - {{\dot \tau }_1}\left( t \right)}}{{2\left( {1 - {\tau _1}} \right)}}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{e}}_i^{\rm{T}}\left( {t - {\tau _1}\left( t \right)} \right){\mathit{\boldsymbol{e}}_i}\left( {t - {\tau _1}\left( t \right)} \right)} .}\end{array}$

令e(t)=(e₁(t), e₂(t), …, e_N(t))，可得

$\begin{array}{*{20}{c}}{{\cal L}V\left( t \right) \le \left( {\mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} - h + \nu } \right){\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{e}}\left( t \right) + }\\{{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\frac{{{\mathit{\boldsymbol{E}}^{\rm{T}}} + \mathit{\boldsymbol{E}}}}{2}\mathit{\boldsymbol{e}}\left( t \right) + {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{Fe}}\left( {t - {\tau _1}\left( t \right)} \right) - }\\{\frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {t - {\tau _1}\left( t \right)} \right)\mathit{\boldsymbol{e}}\left( {t - {\tau _1}\left( t \right)} \right).}\end{array}$

由

得

$\begin{array}{*{20}{c}}{{\cal L}V\left( t \right) \le \left( {\mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} - h + \nu } \right){\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{e}}\left( t \right) + }\\{{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\frac{{{\mathit{\boldsymbol{E}}^{\rm{T}}} + \mathit{\boldsymbol{E}}}}{2}\mathit{\boldsymbol{e}}\left( t \right) + \frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{e}}\left( t \right) \le }\\{\left( {\mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} - h + \nu + } \right.}\\{\left. {{\lambda _{\max }}\frac{{{\mathit{\boldsymbol{E}}^{\rm{T}}} + \mathit{\boldsymbol{E}}}}{2} + {\lambda _{\max }}\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{F}}^{\rm{T}}}} \right){\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{e}}\left( t \right).}\end{array}$

如果存在足够大的正常数h，并且它满足

$\begin{array}{*{20}{c}}{h \ge \mu + \frac{1}{{2\left( {1 - {\tau _1}} \right)}} + \nu + {\lambda _{\max }}\frac{{{\mathit{\boldsymbol{E}}^{\rm{T}}} + \mathit{\boldsymbol{E}}}}{2} + }\\{{\lambda _{\max }}\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{F}}^{\rm{T}}} + 1,}\end{array}$

则$ \mathit{\dot V}\left( \mathit{t} \right) \le - {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( \mathit{t} \right)\mathit{\boldsymbol{e}}\left( \mathit{t} \right) \le 0$，因此误差是收敛的，证明完成。
3 示例本节提供了两个例子来说明本文获得的定理的有效性。
使用Lorenz系统描述信息层网络和物理层网络，并考虑由3个节点组成的网络。没有攻击的相互依存的网络可以描述如下：

$\begin{array}{*{20}{c}}{\mathit{\boldsymbol{\dot x}}\left( t \right) = \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^3 {{a_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}{\mathit{\boldsymbol{x}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^3 {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{y}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} ,}\\{{{\mathit{\boldsymbol{\dot y}}}_i}\left( t \right) = \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^3 {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{y}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^3 {{d_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_4}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _2}\left( t \right)} \right)} .}\end{array}$

其中：

$\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) = \left( {10{x_{i2}} - 10{x_{i1}},28{x_{i1}} - {x_{i1}}{x_{i3}} - {x_{i2}},{x_{i1}}{x_{i2}} - \frac{8}{3}{x_{i3}}} \right),}\\{\mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) = \left( {10{y_{i2}} - 10{y_{i1}},28{y_{i1}} - {y_{i1}}{y_{i3}} - {y_{i2}},{y_{i1}}{y_{i2}} - \frac{8}{3}{y_{i3}}} \right),}\end{array}}&{1 \le i \le 3}\end{array}.$

Г₁=Г₂=Г₃=Г₄=diag(1, 1, …, 1)。A、B、C、D为全连接矩阵，它们的权重都是1。
令l_i=1，然后得到图 1所示的控制器(9)下的误差曲线。根据定理1的条件，验证了定理1。

图 1 无攻击的信息层网络和物理层网络的同步误差

图选项

接下来为了验证定理2，可以将具有攻击的相互依存的网络描述如下：

$\begin{array}{*{20}{c}}{\mathit{\boldsymbol{\dot x}}\left( t \right) = \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^3 {{a_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}{\mathit{\boldsymbol{x}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^3 {{c_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}{\mathit{\boldsymbol{y}}_j}\left( {t - {\tau _1}\left( t \right)} \right)} + }\\{\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right),}\\{{{\mathit{\boldsymbol{\dot y}}}_i}\left( t \right) = \mathit{\boldsymbol{g}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^3 {{b_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}{\mathit{\boldsymbol{y}}_j}\left( t \right)} + }\\{\sum\limits_{j = 1}^3 {{d_{ij}}{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_4}{\mathit{\boldsymbol{x}}_j}\left( {t - {\tau _2}\left( t \right)} \right)} + }\\{\mathit{\boldsymbol{\sigma }}\left( {t,{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right).}\end{array}$

本文使用Lorenz系统。Brown运动满足Eω(t)=0，Dω(t)=1。
令l_i=2，得到图 2所示控制器(9)下的误差曲线，从而验证了定理2。

图 2 伴随攻击的信息层网络和物理层网络的同步误差(Brown运动满足Eω(t)=0，Dω(t)=1)

图选项

两次验证所需的数值均为随机给出。
4 结论在CPS中，信息层网络和物理层网络是相互依存的复杂网络。本文首先提出了伴随随机攻击的相互依存的CPS的数学模型，并在物理层网络中加入了自适应非线性控制器，以实现物理层和信息层网络在随机攻击下的同步。最后，本文以Lorenz系统为例进行了仿真。仿真结果证明了理论结果是有效的。本文提出的模型与设计原理对于分析和控制现实生活中的针对CPS网络的随机攻击的动态行为具有积极作用。

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