## cs代写|复杂网络代写complex network代考|TSKS33

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|EXTENSIONS AND APPLICATIONS OF CNSS

In the above sections, we have surveyed some recent developments in the analysis and synthesis of CNSs with switching topologies, mainly focusing on the synchronization and consensus behaviors and comparison to complex networks and MASs’ scenarios. The above survey is by no means complete. However, it depicts the whole general framework of coordination control for CNSs with dynamic communication networks and lays the fundamental basis for other exciting and yet critical issues concerning CNSs with switching topologies. These extensions still deserve further study, although a variety of efficient tools have been successfully developed to solve various challenging problems in those active research fields. Next, we elaborate on several state-of-the-art extensions and applications of CNSs with dynamic topologies.

Resilience analysis and control of complex cyber-physical networks. Most of the units in various network infrastructures are cyber-physical systems in the Internet of Things era. One of the essential and significant features of the cyber-physical system is integrating and interacting with its physical and cyber layers. As a new generation of CNS, the complex cyber-physical network has received drastic attention in recent years. Specifically, the CNSs’ paradigm provides an excellent way to model various large-scale crucial infrastructure systems, such as power grid systems, transportation systems, water supply networks, and many others [4]. These systems all capture the basic features that large numbers of interconnected individuals through wired or wireless communication links, and many essential functions of these large-scale infrastructure systems fall under the purview of coordination of CNSs. Disruption of these critical networked infrastructures could be a real-world effect across an entire country and even further, significantly impacting public health and safety and leading to massive economic losses. The alarming historical events urgently remind us to seek solutions for maintaining certain functionality of CNSs against malicious cyberattacks (i.e., resilience or cybersecurity). It is critically essential to exploit security threats during the initial design and development phase.

## cs代写|复杂网络代写complex network代考|ALGEBRAIC GRAPH THEORY

Suppose a CNS consists of $N$ nodes (agents) which interact with each other through a communication or sensing network or a combination of both. It is natural to model the interactions among the $N$ nodes (agents) by undirected or directed graphs. Without loss generality, the $N$ nodes can be labeled as node $1, \ldots, N$. Let $\mathcal{V}={1, \cdots, N}$ be the set of nodes. Then the directed graph is described by $(\mathcal{V}, \mathcal{E})$, where the set of edges $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ represent the interactions among the $N$ nodes. For notational simplicity, the graph $(\mathcal{V}, \mathcal{E})$ is denoted by $\mathcal{G}$. The edge $(j, i) \in \mathcal{E}$ if and only if node $i$ can receive the information from node $j$. When $(j, i) \in \mathcal{E}$, node $j$ is said to be a neighbor of node $i$. Denote by $\mathcal{N}i$ the set of neighbors of node $i$. If there exists a sequence of distinct nodes $i_1, \ldots, i_m$ such that $\left(i, i_1\right),\left(i_1, i_2\right), \ldots,\left(i{m-1}, i_m\right),\left(i_m, k\right) \in \mathcal{E}$, then it is said that node $i$ has a directed path to node $k$, or node $k$ is reachable from node $i . \mathcal{G}$ is strongly connected if each node has at least one directed path to any other nodes. More generally, if there exists a node, called the root, which has at least one directed path to any other nodes, $\mathcal{G}$ is said to contain a directed spanning tree. Denote by $a_{i j}$ the weight of the edge $(j, i), i, j=1, \ldots, N$. It is assumed throughout this book that $a_{i j} \geq 0$, where $a_{i j}>0$ if and only if $(j, i) \in \mathcal{E}$, and $a_{i j}=0$, otherwise. In addition, it is assumed in this book that $a_{i i}=0$, that is, self-loop is forbidden. $\mathcal{G}$ is called an undirected graph if $(i, j) \in \mathcal{E}$ whenever $(j, i) \in \mathcal{E}$ and $a_{i j}=a_{j i}$. An undirected graph is connected if there exists at least one undirected path between each pair of distinct nodes. For undirected graphs, the existence of an undirected spanning tree is equivalent to being connected. However, for directed graphs, the existence of a directed spanning tree is a weaker condition than being strongly connected. Please see Figure $2.1$ for a directed graph which is not strongly connected but contains a directed spanning tree.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## cs代写|复杂网络代写complex network代考|CS7280

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|SYNCHRONIZATION OF COMPLEX NETWORKS

In the field of complex networks’ synchronization with switching topologies, a wide range of research has been recently focused on dealing with issues related to the switchings and their effects on synchronization.

There has been increasing recognition that each topology candidate’s properties and the switching strategy for topologies play essential roles in achieving synchronization for complex networks with switching topologies. The analytical approaches for synchronization of continuous- and discrete-time complex networks with switching topologies are generally different. Mathematically, the continuous-time complex network with switching topologies is a special kind of those with time-varying topology. However, it is preliminarily assumed in some existing works on synchronization of continuous-time network systems with time-varying topology that the connection links evolve continuously over time with a known bound for the changing rate [103] or with a time-varying Laplacian matrix being simultaneously diagonalizable [11]. Thus, the techniques developed in these works to solve synchronization problem of complex networks with special time-varying topology are generally hard to apply to that with switching topologies, especially to the case with directed switching topologies.

Specifically, averaging-based approaches were developed to analyze synchronization of continuous-time complex networks with fast switching topologies $[7,140]$ while multiple Lyapunov functions (MLFs)-based approaches were developed to analyze synchronization of continuous-time complex networks with slowly switching topologies (especially for the case with directed switching topologies) [190]. Furthermore, MLFLs-based approaches were usually employed to analyze synchronization of continuous-time complex networks with switching topologies under delayed or sampled-data coupling [90, 187]. Common Lyapunov function (CLF)- and functional (CLFL)-based approaches are applicable only to some special continuous-time complex networks with switching topologies such as each possible topology candidate is undirected [222].

## cs代写|复杂网络代写complex network代考|CONSENSUS OF MASS WITH SWITCHING TOPOLOGIES

Since the pioneer works [65] in which heading consensus of the linearized Vicsek’s model was analyzed, consensus of MASs with switching topologies has attracted increasing attention from a wide range of scientific interests.

Consensus of first-order MASs with switching topologies: In the year of 2004, consensus problem of continuous time first-order (integrator-type) MASs with directed switching and balanced topology was formulated and studied in [116]. Due to the balanced property of each possible topology candidate, a common Lyapunov function was constructed in [116] for analyzing the convergence behaviors of disagreement vector. Consensus of both continuous- and discrete time first-order MASs with directed switching topologies was further studied in [128] where each possible topology candidate is not required to be balanced. By using graphical approaches, some interesting issues on consensus of a class of first-order MASs with switching topologies were further addressed in [13]. By employing a CLFL based approach, it was proven in [83] that average consensus in continuous time first-order MASs with time delayed protocol can be achieved if each topology candidate is strongly connected and balanced, and some linear matrix inequalities hold. Note that most of the aforementioned results are concerned with consensus of first-order MASs with deterministically switching topologies. However, considering the underlying topology may randomly switch among a set of topology candidates in some practical applications, there have been a number of results focusing on consensus of first-order MASs with randomly switching topologies $[54,155,156]$.

Consensus of second-order MASs with switching topologies: Based on the stability results for switched systems provided in [108], some dwell time (DT) based criteria for consensus of continuous time second-order MASs under directed switching topologies were established in [129] where it was revealed that consensus can be achieved if each topology candidate contains a directed spanning tree and the DT for switchings among different topology candidates is larger than a threshold value. When the graph describing the communication topology among followers is undirected, it was proven in [59] by constructing a CLF that leader-following consensus could be achieved if the topology jointly contains a directed spanning tree. Later, leader-following consensus problem of MASs with switching jointly reachable interconnection and transmission delays was solved in [234] by designing the switching laws among topology candidates, where the dynamics of the leader are described by first-order integrator.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## cs代写|复杂网络代写complex network代考|CS60078

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|COMPLEX NETWORK SYSTEMS

Far from being separate entities, many natural, social, and engineering systems can be considered as CNSs associated with tight interactions among neighboring entities within them $[3,10,18,29,39,50,92,120,130,141,175,194,201,211,219]$. Roughly speaking, a CNS refers to a networking system that consists of lots of interconnected agents, where each agent is an elementary element or a fundamental unit with detailed contents depending on the nature of the specific network under consideration [175]. For example, the Internet is a CNS of routers and computers connected by various physical or wireless links. The cell can be described by a CNS of chemicals connected by chemical interactions. The scientific citation network is a CNS of papers and books linked by citations among them. The WeChat social network is a CNS whose agents are users and whose edges represent the relationships among users, to name just a few.

With the aid of coordination with neighboring individuals, a CNS can exhibit fascinating cooperative behaviors far beyond the individuals’ inherent properties. Prototypical cooperative behaviors include synchronization $[38,95,101,177]$, consensus $[76,118,128]$, swarming $[48,115]$, flocking $[117,161]$. In this book, we focus on the CNSs which include complex networks and MASs as special cases. A lot of new research challenges have been raised about understanding the emergence mechanisms responsible for various collective behaviors as well as global statistical properties of CNSs $[3,15,114,178]$. Network science, as a strong interdisciplinary research field, has been established at the first several years of the 21st century [110]. It is increasingly recognized that a detailed study on cooperative dynamics of CNSs would not only help researchers understand the evolution mechanism for macroscopical cooperative behaviors, but also prompt the application of network science to solve various engineering problems, e.g., design of distributed sensor networks [135], formation control of multiple unmanned aerial vehicles [37], distributed localization [89], and load assignment of multiple energy storage units in modern power grid [191].

Among the various cooperative behaviors of CNSs, synchronization of complex networks and consensus of MASs are the most fundamental yet most important ones. Synchronization of complex networks exhibits the cooperative behavior that the states of all entities within these networks achieve an agreement on some quantities of interest. Compared with stability analysis of an isolated control plant, synchronization behavior analysis in CNSs are much more challenging as the synchronization process is determined by the evolution of network topology as well as the inherent dynamics of individual units within these network systems $[96,102,121,198,199]$. As a topic closely related to synchronization of complex networks, the consensus of MASs has recently gained much attention from various research fields, especially the system science, control theory, and electrical engineering communities $[22,65,88,116,128]$. In the remainder of this chapter, we will review some existing results on achieving synchronization of complex networks and consensus of MASs over dynamically changing communication topologies.

## cs代写|复杂网络代写complex network代考|DEFINITIONS OF SYNCHRONIZATION AND CONSENSUS

Before moving forward, the definition of consensus of MASs is given. Moreover, the synchronization of complex networks can be defined similarly.

Consider an MAS which consists of $N$ agents. Without loss of generality, we label the $N$ agents as agents $1, \ldots, N$, respectively. The dynamics of agent $i, i=1, \ldots, N$, are represented by
$$\dot{x}_i(t)=f\left(t, x_i(t), u_i(t)\right),$$
where $x_i(t) \in \mathbb{R}^n$ and $u_i(t) \in \mathbb{R}^m$ represent, respectively, the state and the control input, $f(\cdot, \cdot \cdot):\left[t_0,+\infty\right) \times \mathbb{R}^n \times \mathbb{R}^m \mapsto \mathbb{R}^n$ represents the nonlinear dynamics of agent i. A particular case is the general linear time-invariant MASs with the dynamics of agent $i$ are described by
$$\dot{x}_i(t)=A x_i(t)+B u_i(t), i=1, \ldots, N,$$
where $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$ represent, respectively, the state matrix and control input matrix. For convenience, throughout this book, we call MAS (1.1) to represent the MAS whose dynamics are described by (1.1).

Definition $1.1$ Consensus of the MAS (1.1) is said to be achieved if for arbitrary initial conditions $x_i\left(t_0\right), i=1, \ldots, N$,
$$\lim _{t \rightarrow \infty}\left|x_i(t)-x_j(t)\right|=0, i, j=1, \ldots, N .$$
The definition of consensus for MAS (1.1) given by Eq. (1.3) does not concern about the final consensus states. However, it is sometimes important to make the states of all agents in the considered MASs to finally converge to some predesigned trajectory, especially from the viewpoint of controlling various complex engineering systems. To ensure the states of all agents in MAS (1.1) converge to some desired states, a target system (may be virtual) is introduced to the network (1.1) as
$$\dot{s}(t)=f(t, s(t))$$
for some given initial value $s\left(t_0\right) \in \mathbb{R}^n$. Under this scenario, we call agent $i$ whose dynamics are described by (1.1) the follower $i, i=1, \ldots, N$, and call the agent whose dynamics are described by (1.4) the leader.

## cs代写|复杂网络代写complex network代考|DEFINITIONS OF SYNCHRONIZATION AND CONSENSUS

$$\dot{x}i(t)=f\left(t, x_i(t), u_i(t)\right),$$ 在哪里 $x_i(t) \in \mathbb{R}^n$ 和 $u_i(t) \in \mathbb{R}^m$ 分别表示状态和控制输入， $f(\cdot, \cdot):\left[t_0,+\infty\right) \times \mathbb{R}^n \times \mathbb{R}^m \mapsto \mathbb{R}^n$ 表示代 理 $i$ 的非线性动力学。一个特例是具有代理动力学的一般线性时不变 MAS $i$ 被描述为 $$\dot{x}_i(t)=A x_i(t)+B u_i(t), i=1, \ldots, N,$$ 在哪里 $A \in \mathbb{R}^{n \times n}$ 和 $B \in \mathbb{R}^{n \times m}$ 分别表示状态矩阵和控制输入矩阵。为方便起见，在整本书中，我们称 MAS (1.1) 来表示其动力学由 (1.1) 描述的 MAS。 定义 $1.1$ 如果对于任意初始条件，则据说可以实现 MAS (1.1) 的共识 $x_i\left(t_0\right), i=1, \ldots, N$ ， $$\lim {t \rightarrow \infty}\left|x_i(t)-x_j(t)\right|=0, i, j=1, \ldots, N .$$

$$\dot{s}(t)=f(t, s(t))$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## cs代写|复杂网络代写complex network代考|TSKS33

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|Multiple Lyapunov functions

To proceed, the notion of time dependent switching is introduced.
As a special kind of hybrid dynamic system, switched system has been studied for quite some time by researchers from applied mathematics, systems and control fields. Roughly speaking, a switched system is a dynamic system that consists of a number of subsystems and a switching rule that determines switches among these subsystems. Suppose the switched system is generated by the following family of subsystems
$$\dot{x}(t)=f_{p}(t, x(t)), x(t) \in \mathbb{R}^{n}, p \in{1, \ldots, \kappa},$$
together with a switching signal $\sigma(t):\left[t_{0},+\infty\right) \mapsto{1, \ldots, \kappa}$. Note that $\sigma(t)$ is a piecewise constant function that switches at the switching time instants $t_{1}, t_{2}, \ldots$, and is constant on the time interval $\left[t_{k}, t_{k+1}\right), k=0,1, \ldots$. In this book, we assume $\sigma(t)$ is right continuous, i.e., $\sigma(t)=\lim {\iota} t \sigma(\iota)$, and $\inf {k \in \mathbb{N}}\left(t_{k+1}-t_{k}\right) \geq \tau_{m}$ for some given positive scalar $\tau_{m}$ where inf represents the infimum. Please see Figure $2.2$ for an example. Thus the switched systems with time-dependent switching signal $\sigma(t)$ can be described by the equation
$$\dot{x}(t)=f_{\sigma(t)}(t, x(t)) .$$
According to Theorem 2.1, each subsystem has a unique solution over arbitrary interval $\left[t_{k}, t_{k+1}\right), k=0,1, \ldots$, with arbitrary initial value $x\left(t_{k}\right) \in \mathbb{R}^{n}$ if the function $f_{p}$, for each $p=1, \ldots, \kappa$, is globally Lipschitz in $x(t)$ uniformly over $t$. Thus the switched system (2.10) is well defined for arbitrary switching signal $\sigma(t)$ defined above and any given initial value $x\left(t_{0}\right) \in \mathbb{R}^{n}$. Throughout this chapter, we assume that such a globally Lipschitz condition holds for the subsystems, and thus the well-definedness of the switched system is guaranteed. We further assume that $f_{p}\left(t, \mathbf{0}{n}\right)=\mathbf{0}{n}$ for each $p=1, \ldots, \kappa$. Thus, the zero vector is an equilibrium point of the switched system (2.10). Next, some stability notions for the zero equilibrium point of switched systems are introduced.

## cs代写|复杂网络代写complex network代考|CONSENSUS OF LINEAR CNSS WITH DIRECTED SWITCHING TOPOLOGIES

In the past decade, the consensus problem of general linear CNSs has received a lot of attention $[76,146,162,185,186,224]$. Specifically, the consensus problem of linear CNSs under a directed fixed communication topology has been addressed in $[76,224]$. In [162], the robust consensus of linear CNSs with additive perturbations of the transfer matrices of the nominal dynamics was studied. In [163] and a number of subsequent papers, the robust consensus was analyzed from the viewpoint of the $\mathcal{H}_{\infty}$ control theory. Among other relevant references, we mention [146] where, while assuming that the open loop systems are Lyapunov stable, the consensus problem of linear CNSs with undirected switching topologies has been investigated. In the situation where the CNS is equipped with a leader and the topology of the system belongs to the class of directed switching topologies, the consensus tracking problem has been studied in $[185,186]$. One feature of the results in these references is that the open loop agents’ dynamics do not have to be Lyapunov stable. Note that the presence of the leader in the CNSs considered in these references facilitate the derivations and the direct analyses of the consensus error system. However, when the open loop systems are not Lyapunov stable and/or there is no designated leader in the group, the consensus problem for linear CNSs with directed switching topologies remains challenging.

Motivated by the above discussion, this section aims to study the consensus problem for linear CNSs with directed switching topologies. Several aspects of the current study are worth mentioning. Firstly, some of the assumptions in the existing works are dismissed, e.g., the open loop dynamics of the agents do not have to be Lyapunov stable in this chapter. Furthermore, the CNSs under consideration are not required to have a leader. Compared with the consensus problems for linear CNSs with a designated leader, the point of difference here concerns the assumption on the system’s communication topology. In the previous work on the consensus tracking of linear CNSs such as [185], each possible augmented system graph was required to contain a directed spanning tree rooted at the leader. Compared with that work, the switching topologies in this section are allowed to have spanning trees rooted at different nodes. This is a significant relaxation of the previous conditions since it enables the system to be reconfigured if necessary (e.g., to allow different nodes to serve as the formation leader). This also has a potential to make the system more reliable.

## cs代写|复杂网络代写complex network代考|Multiple Lyapunov functions

$$\dot{x}(t)=f_{p}(t, x(t)), x(t) \in \mathbb{R}^{n}, p \in 1, \ldots, \kappa$$

$$\dot{x}(t)=f_{\sigma(t)}(t, x(t)) .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## cs代写|复杂网络代写complex network代考|CS7280

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|EXTENSIONS AND APPLICATIONS OF CNSS WITH SWITCHING TOPOLOGIES

In the above sections, we have surveyed some recent developments in the analysis and synthesis of CNSs with switching topologies, mainly focusing on the synchronization and consensus behaviors and comparison to complex networks and MASs’ scenarios. The above survey is by no means complete. However, it depicts the whole general framework of coordination control for CNSs with dynamic communication networks and lays the fundamental basis for other exciting and yet critical issues concerning CNSs with switching topologies. These extensions still deserve further study, although a variety of efficient tools have been successfully developed to solve various challenging problems in those active research fields. Next, we elaborate on several state-of-the-art extensions and applications of CNSs with dynamic topologies.

Resilience analysis and control of complex cyber-physical networks. Most of the units in various network infrastructures are cyber-physical systems in the Internet of Things era. One of the essential and significant features of the cyber-physical system is integrating and interacting with its physical and cyber layers. As a new generation of CNS, the complex cyber-physical network has received drastic attention in recent years. Specifically, the CNSs’ paradigm provides an excellent way to model various large-scale crucial infrastructure systems, such as power grid systems, transportation systems, water supply networks, and many others [4]. These systems all capture the basic features that large numbers of interconnected individuals through wired or wireless communication links, and many essential functions of these large-scale infrastructure systems fall under the purview of coordination of CNSs. Disruption of these critical networked infrastructures could be a real-world effect across an entire country and even further, significantly impacting public health and safety and leading to massive economic losses. The alarming historical events urgently remind us to seek solutions for maintaining certain functionality of CNSs against malicious cyberattacks (i.e., resilience or cybersecurity). It is critically essential to exploit security threats during the initial design and development phase.

Noteworthily, any successful cyber or physical attack mentioned above on complex cyber-physical networks may introduce undesired switching dynamics (e.g., loss of links due to DoS attacks or human-made physical damages) to the operation of these networks [194]. Inspired by the pioneering work [194], [168] further investigated the distributed observer-based cyber-security control of complex dynamical networks. This work considered the scenario that the communication channels for controllers and observers might both subject to malicious cyber attacks, which aim to block the information exchanges and result in disconnected topologies of the communication networks. New security control strategies are proposed, and an algorithm to properly select the feedback gain matrices and coupling strengths has been given. The asynchronous attacks in these two communication channels were explored in [169], where the attacks can be launched independently and may occur at different time intervals. Recently, [69] studied the distributed cooperative control for DC cyber-physical microgrids under communication delays and slow switching topologies would destruct the system’s transient behaviors at the switching time instants. The average switching dwell-time-dependent control conditions were given to ensure the exponential stability of the considered cyber-physical systems. For the event-triggered communication scenario, [26] studied the distributed consensus for general linear MASs subjected to DoS attacks. By the switched and time-delay system approaches, one constraint was provided to illustrate the convergence rate of consensus errors and uniform lower bound of non-attacking intervals of DoS attacks.

## cs代写|复杂网络代写complex network代考|ALGEBRAIC GRAPH THEORY

Suppose a CNS consists of $N$ nodes (agents) which interact with each other through a communication or sensing network or a combination of both. It is natural to model the interactions among the $N$ nodes (agents) by undirected or directed graphs. Without loss generality, the $N$ nodes can be labeled as node $1, \ldots, N$. Let $\mathcal{V}={1, \cdots, N}$ be the set of nodes. Then the directed graph is described by $(\mathcal{V}, \mathcal{E})$, where the set of edges $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ represent the interactions among the $N$ nodes. For notational simplicity, the graph $(\mathcal{V}, \mathcal{E})$ is denoted by $\mathcal{G}$. The edge $(j, i) \in \mathcal{E}$ if and only if node $i$ can receive the information from node $j$. When $(j, i) \in \mathcal{E}$, node $j$ is said to be a neighbor of node $i$. Denote by $\mathcal{N}{i}$ the set of neighbors of node $i$.

If there exists a sequence of distinct nodes $i{1}, \ldots, i_{m}$ such that $\left(i, i_{1}\right),\left(i_{1}, i_{2}\right), \ldots,\left(i_{m-1}, i_{m}\right),\left(i_{m}, k\right) \in \mathcal{E}$, then it is said that node $i$ has a directed path to node $k$, or node $k$ is reachable from node $i . \mathcal{G}$ is strongly connected if each node has at least one directed path to any other nodes. More generally, if there exists a node, called the root, which has at least one directed path to any other nodes, $\mathcal{G}$ is said to contain a directed spanning tree. Denote by $a_{i j}$ the weight of the edge $(j, i), i, j=1, \ldots, N$. It is assumed throughout this book that $a_{i j} \geq 0$, where $a_{i j}>0$ if and only if $(j, i) \in \mathcal{E}$, and $a_{i j}=0$, otherwise. In addition, it is assumed in this book that $a_{i i}=0$, that is, self-loop is forbidden. $\mathcal{G}$ is called an undirected graph if $(i, j) \in \mathcal{E}$ whenever $(j, i) \in \mathcal{E}$ and $a_{i j}=a_{j i}$. An undirected graph is connected if there exists at least one undirected path between each pair of distinct nodes. For undirected graphs, the existence of an undirected spanning tree is equivalent to being connected. However, for directed graphs, the existence of a directed spanning tree is a weaker condition than being strongly connected. Please see Figure $2.1$ for a directed graph which is not strongly connected but contains a directed spanning tree.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## cs代写|复杂网络代写complex network代考|CS60078

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|DEFINITIONS OF SYNCHRONIZATION AND CONSENSUS

Before moving forward, the definition of consensus of MASs is given. Moreover, the synchronization of complex networks can be defined similarly.

Consider an MAS which consists of $N$ agents. Without loss of generality, we label the $N$ agents as agents $1, \ldots, N$, respectively. The dynamics of agent $i, i=1, \ldots, N$, are represented by
$$\dot{x}{i}(t)=f\left(t, x{i}(t), u_{i}(t)\right),$$
where $x_{i}(t) \in \mathbb{R}^{n}$ and $u_{i}(t) \in \mathbb{R}^{m}$ represent, respectively, the state and the control input, $f(\cdot, \cdot \cdot):\left[t_{0},+\infty\right) \times \mathbb{R}^{n} \times \mathbb{R}^{m} \mapsto \mathbb{R}^{n}$ represents the nonlinear dynamics of agent i. A particular case is the general linear time-invariant MASs with the dynamics of agent $i$ are described by
$$\dot{x}{i}(t)=A x{i}(t)+B u_{i}(t), i=1, \ldots, N,$$
where $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$ represent, respectively, the state matrix and control input matrix. For convenience, throughout this book, we call MAS (1.1) to represent the MAS whose dynamics are described by (1.1).

Definition $1.1$ Consensus of the MAS (1.1) is said to be achieved if for arbitrary initial conditions $x_{i}\left(t_{0}\right), i=1, \ldots, N$,
$$\lim {t \rightarrow \infty}\left|x{i}(t)-x_{j}(t)\right|=0, i, j=1, \ldots, N .$$
The definition of consensus for MAS (1.1) given by Eq. (1.3) does not concern about the final consensus states. However, it is sometimes important to make the states of all agents in the considered MASs to finally converge to some predesigned trajectory, especially from the viewpoint of controlling various complex engineering systems. To ensure the states of all agents in MAS (1.1) converge to some desired states, a target system (may be virtual) is introduced to the network (1.1) as
$$\dot{s}(t)=f(t, s(t))$$
for some given initial value $s\left(t_{0}\right) \in \mathbb{R}^{n}$. Under this scenario, we call agent $i$ whose dynamics are described by (1.1) the follower $i, i=1, \ldots, N$, and call the agent whose dynamics are described by (1.4) the leader.

## cs代写|复杂网络代写complex network代考|SYNCHRONIZATION OF COMPLEX NETWORKS WITH SWITCHING TOPOLOGIES

In the field of complex networks’ synchronization with switching topologies, a wide range of research has been recently focused on dealing with issues related to the switchings and their effects on synchronization.

There has been increasing recognition that each topology candidate’s properties and the switching strategy for topologies play essential roles in achieving synchronization for complex networks with switching topologies. The analytical approaches for synchronization of continuous- and discrete-time complex networks with switching topologies are generally different. Mathematically, the continuous-time complex network with switching topologies is a special kind of those with time-varying topology. However, it is preliminarily assumed in some existing works on synchronization of continuous-time network systems with time-varying topology that the connection links evolve continuously over time with a known bound for the changing rate [103] or with a time-varying Laplacian matrix being simultaneously diagonalizable [11]. Thus, the techniques developed in these works to solve synchronization problem of complex networks with special time-varying topology are generally hard to apply to that with switching topologies, especially to the case with directed switching topologies.

Specifically, averaging-based approaches were developed to analyze synchronization of continuous-time complex networks with fast switching topologies $[7,140]$ while multiple Lyapunov functions (MLFs)-based approaches were developed to analyze synchronization of continuous-time complex networks with slowly switching topologies (especially for the case with directed switching topologies) [190]. Furthermore, MLFLs-based approaches were usually employed to analyze synchronization of continuous-time complex networks with switching topologies under delayed or sampled-data coupling $[90,187]$. Common Lyapunov function (CLF)- and functional (CLFL)-based approaches are applicable only to some special continuous-time complex networks with switching topologies such as each possible topology candidate is undirected [222].

For discrete-time CNSs with switching topologies, global synchronization for nonautonomous linear complex networks with randomly switching topologies was studied in [200] by developing a kind of approaches from ergodicity theory for nonhomogeneous Markovian chains. A method based on the Hajnal diameter of infinite coupling matrices was proposed in [97] to analyze the local synchronizability of a class of discrete-time complex networks with directed switching topologies. Synchronization of discrete-time complex networks with undirected switching topologies and impulsive controller was studied in [73] by constructing MLFs. Globally almost sure synchronization for discrete-time complex networks with switching topologies was investigated in [51] by using the super-martingale convergence theorem. For more recent related works, one can refer to the survey.

## cs代写|复杂网络代写complex network代考|DEFINITIONS OF SYNCHRONIZATION AND CONSENSUS

$$\dot{x} i(t)=f\left(t, x i(t), u_{i}(t)\right),$$

$$\dot{x} i(t)=A x i(t)+B u_{i}(t), i=1, \ldots, N,$$

$$\lim t \rightarrow \infty\left|x i(t)-x_{j}(t)\right|=0, i, j=1, \ldots, N .$$

$$\dot{s}(t)=f(t, s(t))$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## cs代写|复杂网络代写complex network代考|Consensus tracking of CNSs with higher-order dynamics

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|directed switching topologies

This chapter studies the consensus tracking of CNSs with higher-order dynamics and directed switching topologies. This chapter begins by overviewing some previous works and by indicating our motivations. Section $6.2$ firstly studies the case with Lipschitz nonlinear dynamics and directed fixed topology. Then we extend the results to directed switching topologies with each topology contains a directed spanning tree. This section finally studies the case with directed switching topologies that frequently contain a directed spanning tree. Section $6.3$ studies the case with general linear dynamics and occasionally missing control inputs. This section presents some sufficient criteria for achieving consensus tracking. Moreover, the convergence rate is discussed. Finally, some simulations are given to validate the theoretical results.

In contrast to CNSs with first-order nonlinear dynamics, CNSs with second-order nonlinear dynamics are more interesting as it can describe a large class of real networked systems, including coupled pendulums [5] and coupled point-mass systems with or without nonlinear disturbances [165]. Leaderless consensus problem for CNSs with second-order nonlinear dynamics and a fixed weakly connected topology was investigated in [217]. In [137], the consensus tracking problem for CNSs with second-order nonlinear dynamics in the presence of a leader under an arbitrarily given directed topology was studied from pinning control approach. Furthermore, consensus tracking problem for CNSs with higher-order Lipschitz type agent dynamics and a fixed topology was studied in [79].

In the existing literature on the consensus tracking problem for CNSs with nonlinear dynamics, it is commonly assumed that the communication topology is fixed.

## cs代写|复杂网络代写complex network代考|CONSENSUS TRACKING OF CNSS WITH HIGHER-ORDER NONLINEAR DYNAMICS

Consider a CNS consisting of a leader and $N$ followers, where the leader is labelled as agent 0 and the followers are labelled as agents $1, \ldots, N$. The dynamics of agent $i, i=0,1, \ldots, N$, are given by
$$\dot{x}{i}(t)=A x{i}(t)+C f\left(x_{i}(t), t\right)+B u_{i}(t),$$
where $x_{i}(t) \in \mathbb{R}^{n}$ represent the states of agent $i, f(\cdot, \cdot): \mathbb{R}^{n} \times[0,+\infty) \mapsto \mathbb{R}^{p}$ is a continuously differentiable vector-valued function representing the intrinsic nonlinear dynamics, $u_{i}(t) \in \mathbb{R}^{m}$ is the control input to be designed, $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}$, and $C \in \mathbb{R}^{n \times p}$ are constant real matrices. It is assumed that the matrix pair $(A, B)$ is stabilizable. In this section, it is assumed that the leader will not being affected by any followers, i.e. $u_{0}(t) \equiv \mathbf{0}{m}$ in CNS (6.1). Before moving on, the following assumption is made. Assumption 6.1 There exists a nonnegative constant $\varrho$, such that $$|f(y, t)-f(z, t)| \leq \varrho|y-z|, \forall y, z \in \mathbb{R}^{n}, t \geq 0$$ To achieve consensus tracking, the following distributed consensus tracking protocol is proposed for each follower $i$ : $$u{i}(t)=\alpha F \sum_{j=0}^{N} a_{i j}(t)\left(x_{j}(t)-x_{i}(t)\right), \quad i=1, \ldots, N,$$
where $\alpha>0$ represents the coupling strength, $F \in \mathbb{R}^{m \times n}$ is the feedback gain matrix to be designed, and $\mathcal{A}(t)=\left[a_{i j}(t)\right]_{(N+1) \times(N+1)}$ is the adjacency matrix of graph $\mathcal{G}(t)$. Here, $\mathcal{G}(t)$ describes the underlying communication topology among the $N+1$ agents at time $t$.

## cs代写|复杂网络代写complex network代考|Main results for fixed topology containing a directed spanning tree

In this section, distributed consensus tracking is addressed for CNS (6.1) with a fixed communication topology containing a directed spanning tree.

Without loss of generality, let $\mathcal{G}(t)=\mathcal{G}$ for all $t \geq 0$ since the communication topology is assumed to be fixed in this subsection. To derive the main results, the following assumption is needed.

Assumption 6.2 The communication topology $\mathcal{G}$ contains a directed spanning tree with agent 0 (i.e. the leader) as the root.

Under Assumption 6.2, , the Laplacian matrix of directed graph $\mathcal{G}$ can be written as
$$\mathcal{L}=\left[\begin{array}{cc} 0 & \mathbf{0}{N}^{T} \ \mathbf{P} & \overline{\mathcal{L}} \end{array}\right], \quad \overline{\mathcal{L}}=\left[\begin{array}{cccc} \sum{j \in N_{1}} a_{1 j} & -a_{12} & \cdots & -a_{1 N} \ -a_{21} & \sum_{j \in \mathcal{N}{2}} a{2 j} & \cdots & -a_{2 N} \ \vdots & \vdots & \ddots & \vdots \ -a_{N 1} & -a_{N 2} & \cdots & \sum_{j \in \mathcal{N}{N}} a{N j} \end{array}\right]$$
where $\mathbf{P}=-\left[a_{10}, \ldots, a_{N 0}\right]^{T}$. It can be thus obtained from Lemma $2.15$ that there exists a positive definite diagonal matrix $\Phi=\operatorname{diag}\left{\phi_{1}, \ldots, \phi_{N}\right}$ such that $\overline{\mathcal{L}}^{T} \Phi+$ $\Phi \overline{\mathcal{L}}>0$. One such $\phi=\left[\phi_{1}, \ldots, \phi_{N}\right]^{T}$ can be obtained by solving the matrix equation $\overline{\mathcal{L}}^{T} \phi=\mathbf{1}{N}$. Since $u{0}(t) \equiv \mathbf{0}{m}$, one has $$\dot{x}{0}(t)=A x_{0}(t)+C f\left(x_{0}(t), t\right) .$$
Furthermore, substituting (6.2) into (6.1) gives a closed-loop system:
$$\dot{x}{i}(t)=A x{i}(t)+C f\left(x_{i}(t), t\right)+\alpha B F \sum_{j=0}^{N} a_{i j}\left(x_{j}(t)-x_{i}(t)\right), i=1, \ldots, N,$$
where $\mathcal{A}=\left[a_{i j}\right]{(N+1) \times(N+1)}$ is the adjacency matrix of graph $\mathcal{G}$. Define $e{i}(t)=x_{i}(t)-x_{0}(t), i=1, \ldots, N$, and $e(t)=\left[e_{1}^{T}(t), \ldots, e_{N}^{T}(t)\right]^{T}$. Based on the above analysis, one has the following error dynamical system:
$$\dot{e}{i}(t)=A e{i}(t)+C\left(f\left(x_{i}(t), t\right)-f\left(x_{0}(t), t\right)\right)-\alpha B F \sum_{j=1}^{N} \bar{l}{i j}(t) e{j}(t)$$
Rewriting (6.5) into a compact form, one has
$$\dot{e}(t)=\left(I_{N} \otimes A\right) e(t)+\left(I_{N} \otimes C\right) \tilde{f}(x(t), t)-\alpha(\overline{\mathcal{L}} \otimes B F) e(t)$$

where $\tilde{f}(x(t), t)=\left[\left(f\left(x_{1}(t), t\right)-f\left(x_{0}(t), t\right)\right)^{T}, \ldots,\left(f\left(x_{N}(t), t\right)-f\left(x_{0}(t), t\right)\right)^{T}\right]^{T}$ and $x(t)=\left[x_{0}^{T}(t), x_{1}^{T}(t), \ldots, x_{N}^{T}(t)\right]^{T} .$

Before moving on, a multi-step design procedure is given for selecting the control parameters in protocol (6.2) under a fixed topology $\mathcal{G}$.

## cs代写|复杂网络代写complex network代考|CONSENSUS TRACKING OF CNSS WITH HIGHER-ORDER NONLINEAR DYNAMICS

X˙一世(吨)=一个X一世(吨)+CF(X一世(吨),吨)+乙在一世(吨),

|F(是,吨)−F(和,吨)|≤ϱ|是−和|,∀是,和∈Rn,吨≥0为了实现共识跟踪，为每个追随者提出了以下分布式共识跟踪协议一世 :

## cs代写|复杂网络代写complex network代考|Main results for fixed topology containing a directed spanning tree

X˙0(吨)=一个X0(吨)+CF(X0(吨),吨).

X˙一世(吨)=一个X一世(吨)+CF(X一世(吨),吨)+一个乙F∑j=0ñ一个一世j(Xj(吨)−X一世(吨)),一世=1,…,ñ,

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## cs代写|复杂网络代写complex network代考|CONSENSUS TRACKING OF CNSS WITH LORENZ TYPE DYNAMICS

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## cs代写|复杂网络代写complex network代考|Model formulation

Consider a CNS with Lorenz type dynamics which are given by
$$\dot{x}{i}(t)=A x{i}(t)+\beta x_{i}(t) B x_{i}(t)+\alpha \sum_{j=1}^{N} a_{i j}(t) H\left(x_{j}(t)-x_{i}(t)\right),$$
where
$$A=\left[\begin{array}{ccc} -(25 \gamma+10) & (25 \gamma+10) & 0 \ (28-35 \gamma) & (29 \gamma-1) & 0 \ 0 & 0 & -\frac{(\gamma+8)}{3} \end{array}\right], \quad B=\left[\begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & -1 \ 0 & 1 & 0 \end{array}\right] \text {, }$$
$\beta=[1,0,0], \gamma \in[0,1]$ is a parameter, $\alpha>0$ represents the coupling strength among the agents, $\mathcal{A}(t)=\left[a_{i j}(t)\right]{N \times N}$ is the adjacency matrix of the communication topology at time $t$, and $H \in \mathbb{R}^{3 \times 3}$ is the positive definite inner linking matrix, $i=1, \ldots, N$. Note that systems (5.33) will become the coupled Lorenz, Chen and Lü systems if $\gamma=0,1$, and $0.8$, respectively. By the definition of the Laplacian matrix for a graph, it follows from (5.33) that $$\dot{x}{i}(t)=A x_{i}(t)+\beta x_{i}(t) B x_{i}(t)-\alpha \sum_{j=1}^{N} l_{i j}(t) H x_{j}(t),$$
where $L(t)=\left[l_{i j}(t)\right]{N \times N}$ is the Laplacian matrix of communication topology $\mathcal{G}(\mathcal{A}(t))$, $i=1, \ldots, N$. It is assumed in this section that $t{0}=0$.

The control goal here is to design some pinning controllers to some designed agents such that the states of all the agents in $(5.33)$ to converge to a common target trajectory $s(t)$ in the sense of $\lim {t \rightarrow \infty}\left|x{i}(t)-s(t)\right|=0$, for all $i=1, \ldots, N$, with
$$\dot{s}(t)=A s(t)+\beta s(t) B s(t),$$
with arbitrarily given initial value $s\left(t_{0}\right) \in \mathbb{R}^{3}$. Motivated by the works in $[74,136$, $205,216]$, pinning CNS (5.33) by using some linear controllers $-\alpha c_{i}(t) H\left(x_{i}(t)-s(t)\right)$ to agent $i$ leads to
\begin{aligned} \dot{x}{i}(t)=& A x{i}(t)+\beta x_{i}(t) B x_{i}(t) \ &-\alpha \sum_{j=1}^{N} l_{i j}(t) H x_{j}(t)-\alpha c_{i}(t) H\left(x_{i}(t)-s(t)\right) \end{aligned}
where $c_{i}(t) \in{0,1}$ and $c_{i}(t)=1$ if the agent $i$ of $(5.33)$ is pinned at time $t$.
Let $e_{i}(t)=x_{i}(t)-s(t), i=1, \ldots, N$, it thus follows from (5.37) that
\begin{aligned} \dot{e}{i}(t)=& A e{i}(t)+\beta x_{i}(t) B x_{i}(t)-\beta s(t) B s(t) \ &-\alpha \sum_{j=1}^{N} l_{i j}(t) H e_{j}(t)-\alpha c_{i}(t) H e_{i}(t) \end{aligned}

## cs代写|复杂网络代写complex network代考|Main results for directed fixed communication topology

In this subsection, consensus tracking of CNS (5.33) with target trajectory given in (5.36) under a fixed communication topology is studied. Without loss of generality, let $\mathcal{G}(\mathcal{A}(t))=\mathcal{G}(\mathcal{A})$ for all $t \geq 0$. And we label the target as agent 0 .

Assumption 5.2 There exists at least one directed spanning tree rooted at agent 0 (i.e., the target) in the augmented communication topology $\mathcal{G}(\widetilde{\mathcal{A}})$.

It is clearly that Assumption $5.2$ will hold if all the agents $1, \ldots, N$ are pinned, i.e., $c_{i}(t)=1$, for all $i=1, \ldots, N$ and $t \geq 0$. However, it is more interesting to study how to make Assumption $5.2$ hold if only a small fraction of the agents in $\mathcal{G}(\mathcal{A})$ could be selected and pinned. To do this, the following algorithm is proposed to determine at least how many and what kinds of agents should be pinned such that Assumption $5.2$ holds.

Algorithm 5.2 Find the strongly connected components of $\mathcal{G}(\mathcal{A})$ by employing the Tarjan’s algorithm [157]. Note that the time complexity of this operation is $O(N+E)$, where $N$ and $E$ are, respectively, the numbers of agents and links of $\mathcal{G}(\mathcal{A}) .$ Suppose that there are $\omega$ strongly connected components in $\mathcal{G}(\mathcal{A})$, labeled as $W_{1}, W_{2}, \ldots, W_{\omega}$. Set $m_{i}=0, i=1, \ldots, \omega$, and $h=1$. Then, execute the following steps
(1) Check whether there exists at least one agent $n_{k}$ belonging to $W_{h}$ which is reachable from an agent $n_{g}$ belonging to $W_{j}, j=1, \ldots, \omega, j \neq h$. If it holds, go to step (2); if it dose not hold, go to step (3).
(2) Check whether the following condition holds: $h<\omega$. If it holds, let $h=h+1$ and re-perform step (1); else stop.
(3) Arbitrarily selected one agent in $W_{h}$ and pinned, let $m_{h}=1$; Check whether the following condition holds: $h<\omega$. If it holds, let $h=h+1$ and re-perform step (1); else stop.

## cs代写|复杂网络代写complex network代考|Main results for directed switching communication topologies

The underlying topology of the CNS considered in this subsection is modeled by directed switching graphs. Let $\overline{\mathcal{G}}=\left{\mathcal{G}\left(\mathcal{A}^{1}\right), \ldots, \mathcal{G}\left(\mathcal{A}^{\kappa}\right)\right}, \kappa \geq 2$, indicate the set of all possible directed communication topologies. Suppose that there exists an infinite sequence of uniformly bounded non-overlapping time intervals $\left[t_{k}, t_{k+1}\right), k \in \mathbb{N}$, with $t_{0}=0$, over which the interaction graph is fixed. The time sequence $t_{k}, k \in \mathbb{N}$ is then called the switching sequence, at which the interaction graph changes. Furthermore, introduce a switching signal $\sigma(t):[0,+\infty) \mapsto{1, \ldots, \kappa}$. Then, let $\mathcal{G}\left(\mathcal{A}^{\sigma(t)}\right)$ be the communication topology of the CNS at time $t$. Note that $\mathcal{G}\left(\mathcal{A}^{\sigma(t)}\right) \in \overline{\mathcal{G}}$, for all $t \geq 0$. The error dynamical system (5.39) can be rewritten as
\begin{aligned} \dot{e}{i}(t)=& A e{i}(t)+\beta x_{i}(t) B e_{i}(t)+\beta e_{i}(t) B s(t)-\alpha \sum_{j=1}^{N} l_{i j}^{\sigma(t)} H e_{j}(t) \ &-\alpha c_{i}(t) H e_{i}(t), i=1, \ldots, N, \end{aligned}
where $\mathcal{L}^{\sigma(t)}=\left[l_{i j}^{\sigma(t)}\right]{N \times N}$ is the Laplacian matrix of communication topology $\mathcal{G}\left(\mathcal{A}^{\sigma(t)}\right)$. Throughout this section, the time derivatives of functions $e{i}(t)$ and $x_{i}(t)$ at any switching instant represent its right derivative.

Assumption 5.3 There exists at least one directed spanning tree rooted at agent 0 (i.e., the target) in the augmented communication topology $\mathcal{G}\left(\widetilde{\mathcal{A}}^{\sigma(t)}\right)$.

Remark 5.10 Applying Algorithm $5.2$ to each possible communication topology $\mathcal{G}\left(\mathcal{A}^{i}\right), i=1, \ldots, \kappa$, one gets that Assumption $5.3$ will hold if the selected agents are pinned.

Similar to the last subsection, the Laplacian matrix of $\mathcal{G}\left(\tilde{\mathcal{A}}^{i}\right), i=1, \ldots, \kappa$, can be written as
$$\begin{gathered} \tilde{\mathcal{L}}^{i}=\left[\begin{array}{cc} 0 & \mathbf{0}{N}^{T} \ \mathbf{P}^{i} & \overline{\mathcal{L}}^{i} \end{array}\right], \ \overline{\mathcal{L}}^{i}=\left[\begin{array}{cccc} \sum{j \in N_{1}} a_{1 j}^{i} & -a_{12}^{i} & \cdots & -a_{1 N}^{i} \ -a_{21}^{i} & \sum_{j \in \mathcal{N}{2}} a{2 j}^{i} & \cdots & -a_{2 N}^{i} \ \vdots & \vdots & \ddots & \vdots \ -a_{N 1}^{i} & -a_{N 2}^{i} & \cdots & \sum_{j \in \mathcal{N}{N}} a{N j}^{i} \end{array}\right], \end{gathered}$$
where $\mathbf{P}^{i}=-\left[a_{10}^{i}, \cdots, a_{N 0}^{i}\right]^{T}$. with $a_{j 0}^{i}=c_{j}^{i}, j=1, \ldots, N$. Under Assumption 5.3, it can be got from Lemma $2.15$ that there exists a sequence of positive definite diagonal matrices $\Phi^{i}=\operatorname{diag}\left{\phi_{1}^{i}, \ldots, \phi_{N}^{i}\right}$ such that $\left(\overline{\mathcal{L}}^{i}\right)^{T} \Phi^{i}+\Phi^{i} \overline{\mathcal{L}}^{i}>0$, where $\phi^{i}=$ $\left[\phi_{1}^{i}, \ldots, \phi_{N}^{i}\right]^{T}$ can be obtained by solving the matrix equation $\left(\overline{\mathcal{L}}^{i}\right)^{T} \phi^{i}=\mathbf{1}_{N}, i=$ $1, \ldots, \kappa$.

Based on the above analysis, one may get the following theorem which is the main result of this subsection.

## cs代写|复杂网络代写complex network代考|Model formulation

X˙一世(吨)=一个X一世(吨)+bX一世(吨)乙X一世(吨)+一个∑j=1ñ一个一世j(吨)H(Xj(吨)−X一世(吨)),

b=[1,0,0],C∈[0,1]是一个参数，一个>0表示代理之间的耦合强度，一个(吨)=[一个一世j(吨)]ñ×ñ是通信拓扑在时间的邻接矩阵吨， 和H∈R3×3是正定内链接矩阵，一世=1,…,ñ. 请注意，系统 (5.33) 将成为耦合 Lorenz、Chen 和 Lü 系统，如果C=0,1， 和0.8， 分别。根据图的拉普拉斯矩阵的定义，从 (5.33) 可以得出

X˙一世(吨)=一个X一世(吨)+bX一世(吨)乙X一世(吨)−一个∑j=1ñl一世j(吨)HXj(吨),

s˙(吨)=一个s(吨)+bs(吨)乙s(吨),

X˙一世(吨)=一个X一世(吨)+bX一世(吨)乙X一世(吨) −一个∑j=1ñl一世j(吨)HXj(吨)−一个C一世(吨)H(X一世(吨)−s(吨))

## cs代写|复杂网络代写complex network代考|Main results for directed fixed communication topology

(1) 检查是否存在至少一个代理nķ属于在H可以从代理访问nG属于在j,j=1,…,ω,j≠H. 如果成立，则进行步骤（2）；如果不成立，转至步骤（3）。
(2) 检查下列条件是否成立：H<ω. 如果它成立，让H=H+1并重新执行步骤（1）；否则停止。
(3) 任意选择一名代理人在H并固定，让米H=1; 检查以下条件是否成立：H<ω. 如果它成立，让H=H+1并重新执行步骤（1）；否则停止。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## cs代写|复杂网络代写complex network代考|Consensus tracking of CNSs with first-order nonlinear

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|dynamics and directed switching topologies

This chapter studies the consensus tracking of CNSs with first-order nonlinear dynamics and directed switching topologies. This chapter begins by overviewing some previous works and indicating our motivations. Section $5.2$ studies the case with Lipschitz type nonlinear dynamics without assuming that each possible network topology has a directed spanning tree. Specifically, this section proposes an algorithm for selecting the pinned nodes such that the graph contains a directed spanning tree. Section $5.3$ studies the case with Lorenz type nonlinear dynamics under directed fixed topology as well as directed switching topologies, where the Lorenz systems include the Chen and Lü systems. Finally, some simulations are given to validate the obtained theoretical results.

According to whether the final synchronization states depend on the initial value or not, synchronization in CNSs can be generally categorized into local synchronization [98, 102] and global synchronization [96]. Compared with the local synchronization, the global synchronization means that the network synchronization can be achieved under any given initial conditions, thus is more favorable in practical applications. In [96], a distance between the nodes’ states and the synchronization manifold was introduced, based on which a new methodology was proposed to investigate the global synchronization of coupled systems. Later, general algebraic connectivity was proposed in [218] to study the global synchronization as well as local synchronization problems in strongly connected networks. Global synchronization for coupled linear systems via state or output feedback control was studied in [224]. In $[179,204]$, global synchronization for a class of CNSs with sampling-data coupling was

addressed. For the case that the considered networks are not strongly connected or even do not contain any directed spanning tree, the pinning synchronization problem arises $[74,176,178,216]$. Pinning synchronization in scale-free and small-world complex networks were addressed in [178] and [176], respectively. Later, local and global pinning synchronization in random and scale-free networks were studied in [74]. It is worth noting that global synchronization is actually consensus tracking by regarding the target system in the network as a leader. In [216], pinning synchronization of undirected CNSs was further addressed. Without assuming the network topology is undirected or strongly connected, it was proved in [20] that a single controller can pin a coupled CNS to its homogeneous trajectory under some suitable conditions. Global pinning synchronization for a class of CNSs has been investigated in [66] under a $V$-stability framework. However, it is previously assumed in the aforementioned literature that each possible network topology contains a directed spanning tree with the leader being the root node. This indicates that each agent in the considered network can be influenced by the leader directly or indirectly. In some real cases, the aforementioned condition is hard or even impossible to verify.

Motivated by the aforementioned works on consensus tracking (i.e., global pinning synchronization) of CNSs, this chapter aims to solve the consensus tracking problem for a class of switched CNSs where some possible network topologies may not contain any directed spanning tree. By using a combined tool from $M$-matrix theory and stability analysis of switched systems, a new kind of topology-dependent MLFs for the switched networks is constructed. Theoretical analysis indicates that the consensus tracking in such a CNS can be achieved if some carefully selected nodes are pinned such that the network topology contains a directed spanning tree rooted at the target node frequently enough as the network evolves with time. Without causing any confusion, global pinning synchronization is referred as consensus tracking in the subsequent analysis in this chapter.

## cs代写|复杂网络代写complex network代考|Model formulation

Suppose that the considered CNS consists of $N$ nodes, the dynamics of agent $i$ are given by
$$\dot{x}{i}(t)=f\left(x{i}(t), t\right)+\alpha \sum_{j=1}^{N} a_{i j}(t)\left(x_{j}(t)-x_{i}(t)\right)$$
where $x_{i}(t)=\left[x_{i 1}(t), \ldots, x_{i n}(t)\right]^{T} \in \mathbb{R}^{n}$ for $i=1, \ldots, N$ represent the states of agent $i, \alpha>0$ is the coupling strength, and $\mathcal{A}(t)=\left[a_{i j}(t)\right]{N \times N}$ is the adjacency matrix of graph $\mathcal{G}(\mathcal{A}(t))$ at time $t$. Throughout this chapter, the derivatives of all functions at switching time points should be considered as their right-hand derivatives. According to the definition of Laplacian matrix for a graph, it follows from (5.1) that $$\dot{x}{i}(t)=f\left(x_{i}(t), t\right)-\alpha \sum_{j=1}^{N} l_{i j}(t) x_{j}(t),$$
where $\mathcal{L}(t)=\left[l_{i j}(t)\right]_{N \times N}$ is the Laplacian matrix of graph $\mathcal{G}(\mathcal{A}(t))$.

The control goal in this section is to design pinning controllers for some appropriately selected agents in (5.2) such that the states of each agent in the considered network will approach $s(t)$ when $t$ approaches $+\infty$, i.e., $\lim {t \rightarrow \infty}\left|x{i}(t)-s(t)\right|=0$, for all $i=1, \ldots, N$ and arbitrarily given initial conditions, where
$$\dot{s}(t)=f(s(t), t) .$$
Here, $s(t)$ may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Motivated by the works in [74], pinning network (5.2) by using linear controllers $-\alpha c_{i}(t)\left(x_{i}(t)-s(t)\right)$ to agent $i$ leads to
$$\dot{x}{i}(t)=f\left(x{i}(t), t\right)-\alpha \sum_{j=1}^{N} l_{i j}(t) x_{j}(t)-\alpha c_{i}(t)\left(x_{i}(t)-s(t)\right)$$
where $c_{i}(t) \in{0,1}$ and $c_{i}(t)=1$ if and only if agent $i$ of (5.2) is pinned at time $t$.
Let $e_{i}(t)=x_{i}(t)-s(t), i=1, \ldots, N$. It thus follows from (5.4) that
$$\dot{e}{i}(t)=f\left(x{i}(t), t\right)-f(s(t), t)-\alpha \sum_{j=1}^{N} l_{i j}(t) e_{j}(t)-\alpha c_{i}(t) e_{i}(t)$$
By taking the target system (5.3) as a virtual leader of the CNS under consideration, one may get the augmented network topology $\mathcal{G}(\widetilde{\mathcal{A}}(t))$ consisting of $N+1$ agents. Labeling the index of the virtual agent as 0 , the Laplacian matrix $\widetilde{L}(t)$ of the augmented network topology $\mathcal{G}(\widetilde{\mathcal{A}}(t))$ can be written as:
$$\tilde{\mathcal{L}}(t)=\left[\begin{array}{cc} 0 & \mathbf{0}{N}^{T} \ \mathbf{P}(t) & \overline{\mathcal{L}}(t) \end{array}\right] \in \mathbb{R}^{(N+1) \times(N+1)}$$ $$\overline{\mathcal{L}}(t)=\left[\begin{array}{cccc} \sum{j \in \mathcal{N}{1}} a{1 j}(t) & -a_{12}(t) & \ldots & -a_{1 N}(t) \ -a_{21}(t) & \sum_{j \in \mathcal{N}{2}} a{2 j}(t) & \ldots & -a_{2 N}(t) \ \vdots & \vdots & \ddots & \vdots \ -a_{N 1}(t) & -a_{N 2}(t) & \ldots & \sum_{j \in \mathcal{N}{N}} a{N j}(t) \end{array}\right]$$

## cs代写|复杂网络代写complex network代考|Main results

Based on the analysis in the last section, one has that, for each $s \in \mathcal{P}, \mathcal{G}\left(\widetilde{\mathcal{A}}^{s}\right)$ contains a directed spanning tree rooted at agent 0 . Denote the Laplacian matrix of $\mathcal{G}\left(\widetilde{\mathcal{A}}^{s}\right)$ by $\widetilde{\mathcal{L}}^{s}$. Without loss of generality, let
$$\begin{gathered} \tilde{\mathcal{L}}^{s}=\left[\begin{array}{cc} 0 & \mathbf{0}{N}^{T} \ \mathbf{P}^{s} & \overline{\mathcal{L}}^{s} \end{array}\right] \in \mathbb{R}^{(N+1) \times(N+1)}, \ \overline{\mathcal{L}}^{s}=\left[\begin{array}{cccc} \sum{j \in \mathcal{N}{1}} a{1 j}^{s} & -a_{12}^{s} & \cdots & -a_{1 N}^{s} \ -a_{21}^{s} & \sum_{j \in \mathcal{N}{2}} a{2 j}^{s} & \cdots & -a_{2 N}^{s} \ \vdots & \vdots & \ddots & \vdots \ -a_{N 1}^{s} & -a_{N 2}^{s} & \cdots & \sum_{j \in \mathcal{N}{N}} a{N j}^{s} \end{array}\right], \end{gathered}$$
where $\mathbf{P}^{s}=-\left[a_{10}^{s}, \ldots, a_{N 0}^{s}\right]^{T}, a_{i 0}^{s}=c_{i}^{s}$, and $a_{i 0}^{s}=1$ if agent $i$ in graph $\mathcal{G}\left(\mathcal{A}^{s}\right)$ is pinned, $i=1, \ldots, N$. According to the condition that, for each $s \in \mathcal{P}, \mathcal{G}\left(\widetilde{\mathcal{A}}^{s}\right)$ contains a directed spanning tree, then $\overline{\mathcal{L}}^{s}$ is a nonsingular $M$-matrix. Then, by using Lemma $2.15$, we can get some positive definite matrices $\left(\bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)} \overline{\mathcal{L}}^{\sigma\left(\bar{t}{\rho}\right)}+\left(\overline{\mathcal{L}}^{\sigma\left(\bar{t}{\rho}\right)}\right)^{T} \bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)}\right)$ by letting $\bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)}=\operatorname{diag}\left{\phi{1}^{\sigma\left(\bar{t}{\rho}\right)}, \ldots, \phi{N}^{\sigma\left(\bar{t}{\rho}\right)}\right}$ with $\phi^{\sigma\left(\bar{t}{\rho}\right)}=\left[\phi_{1}^{\sigma\left(\bar{t}{\rho}\right)}, \ldots, \phi{N}^{\sigma\left(\bar{t}{\rho}\right)}\right]^{T}$ satisfies $\left(\overline{\mathcal{L}}^{\sigma\left(\bar{t}{\rho}\right)}\right)^{T} \phi^{\sigma\left(\bar{t}{\mu}\right)}=\mathbf{1}{N} .$

For notational brevity, one may let
where $\mathcal{Q}{\text {sub }}^{t{\min }^{p}}=\left{\sigma(t): t \in\left[t_{\min }^{\rho}, \bar{t}{\rho+1}\right)\right}, \tilde{\lambda}{0}^{i}$ is the smallest eigenvalue of $\overline{\mathcal{L}}^{i}+$ $\left(\bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)}\right)^{-1}\left(\overline{\mathcal{L}}^{i}\right)^{T} \bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)}$. Furthermore, let
$$\mu=\max {i \neq j, i, j \in \mathcal{P}} \frac{\phi{\max }^{i}}{\phi_{\min }^{j}},$$
where $\phi_{\min }^{s}=\min {r=1, \ldots, N} \phi{r}^{s}, \phi_{\max }^{s}=\max {r=1, \ldots, N} \phi{r}^{s}$, for each $s \in \mathcal{P}$.
Based on the above analysis, one may get the following theorem which summarizes the main results of this section.

## cs代写|复杂网络代写complex network代考|Model formulation

X˙一世(吨)=F(X一世(吨),吨)+一个∑j=1ñ一个一世j(吨)(Xj(吨)−X一世(吨))

X˙一世(吨)=F(X一世(吨),吨)−一个∑j=1ñl一世j(吨)Xj(吨),

The control goal in this section is to design pinning controllers for some appropriately selected agents in (5.2) such that the states of each agent in the considered network will approachs(吨)什么时候吨方法+∞， IE，林吨→∞|X一世(吨)−s(吨)|=0， 对全部一世=1,…,ñ并且任意给定初始条件，其中

s˙(吨)=F(s(吨),吨).

X˙一世(吨)=F(X一世(吨),吨)−一个∑j=1ñl一世j(吨)Xj(吨)−一个C一世(吨)(X一世(吨)−s(吨))

## cs代写|复杂网络代写complex network代考|Main results

where\mathcal{Q}{\text {sub }}^{t{\min }^{p}}=\left{\sigma(t): t \in\left[t_{\min }^{\rho} , \bar{t}{\rho+1}\right)\right}, \波浪号{\lambda}{0}^{i}\mathcal{Q}{\text {sub }}^{t{\min }^{p}}=\left{\sigma(t): t \in\left[t_{\min }^{\rho} , \bar{t}{\rho+1}\right)\right}, \波浪号{\lambda}{0}^{i}是的最小特征值大号¯一世+ (披¯σ(吨¯ρ))−1(大号¯一世)吨披¯σ(吨¯ρ). 此外，让

μ=最大限度一世≠j,一世,j∈磷φ最大限度一世φ分钟j,

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## cs代写|复杂网络代写complex network代考|Cooperative Control of Complex Network Systems with Dynamic Topologies

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|CNSS WITH STATIC COUPLING AND SWITCHING TOPOLOGIES

This section studies the consensus disturbance rejection problem for CNSs under directed switching topologies. Before moving forward, the definition of consensus disturbance rejection is given.

Definition 4.1 The consensus disturbance rejection of CNSs (4.1) and (4.3) with disturbances generated by (4.2) is said to be achieved if
$$\ lim {t \rightarrow \infty}\left|x{i}(t)-x_{0}(t)\right|=0, \lim {t \rightarrow \infty}\left|\hat{d}{i}(t)-d_{i}(t)\right|=0,$$
hold for arbitrary initial values $x_{i}\left(t_{0}\right), x_{0}\left(t_{0}\right), \hat{d}{i}\left(t{0}\right), d_{i}\left(t_{0}\right), i=1, \ldots, N$.
Theorem 4.2 Suppose Assumptions 4.1-4.3 hold. If the ADT $\tau_{a}>\ln \nu$, then the consensus disturbance rejection of CNSs (4.1) and (4.3) with the disturbances generated by (4.2) can be achieved by adopting the consensus error estimator (4.5), the state estimator (4.9), and the disturbance observer (4.10) based controller (4.11) with $K=-B^{T} P^{-1}, Q=\mu R^{-1} D^{T}, \rho \geq 4 \alpha / \lambda_{0}, \mu \geq 4 / \lambda_{0}$, where $\alpha$ is a positive constant, $\lambda_{0}$ is given by (4.4), $P>0$ and $R>0$ are, respectively, obtained by solving the LMIs (4.18) and (4.19),
$$\begin{gathered} A P+P A^{T}-\alpha B B^{T}+P<0, \ W^{T} R+R W-D^{T} D+2 R<0 . \end{gathered}$$
Proof $4.2$ For any $t \in\left[t_{j}, t_{j+1}\right), j=0,1,2, \ldots$, we construct the following $M L F s$
$$V_{1}(t)=V_{11}(t)+V_{12}(t)+V_{13}(t)+V_{14}(t),$$
where
\begin{aligned} &V_{11}(t)=\zeta^{T}(t)\left(I_{N} \otimes P^{-1}\right) \zeta(t), \ &V_{12}(t)=\frac{\gamma_{1}}{2} \sum_{i=1}^{N} \phi_{i}^{\sigma(t)}\left(2+\varrho_{i}(t)\right) \varrho_{i}(t), \ &V_{13}(t)=\gamma_{1} \gamma_{2} \tilde{d}^{T}(t)\left(\Phi^{\sigma(t)} \otimes R\right) \tilde{d}(t), \ &V_{14}(t)=\gamma_{1} \gamma_{3} \tilde{\delta}^{T}(t)\left(\Phi^{\sigma(t)} \otimes S\right) \tilde{\delta}(t), \end{aligned}

## cs代写|复杂网络代写complex network代考|CNSS WITH DYNAMIC COUPLING AND FIXED TOPOLOGY

We could learn from Theorem $4.2$ that the coupling strength $\rho$ depends on the smallest eigenvalue $\lambda_{0}$ which is a global information associated with all the possible communication graphs. Consequently, the controller (4.11) can not be implemented in a distributed way. Motivated by this observation, we give a new state estimator with dynamic coupling strengths upon which a fully distributed controller can be reconstructed. While, unlike the last subsection, the directed topology of the CNSs considered in this subsection is assumed to be fixed. The state estimator is given as follows.
\begin{aligned} \dot{\hat{\xi}}{i}(t) &=A \hat{\xi}{i}(t)+\alpha B K \hat{\xi}{i}(t)+\left(\rho{i}+\varrho_{i}\right) B K\left(\hat{\zeta}{i}(t)-\hat{\delta}{i}(t)\right) \ \dot{\rho}{i} &=\left(\hat{\zeta}{i}(t)-\hat{\delta}{i}(t)\right)^{T} \Theta\left(\hat{\zeta}{i}(t)-\hat{\delta}{i}(t)\right) \ \varrho{i} &=\left(\hat{\zeta}{i}(t)-\hat{\delta}{i}(t)\right)^{T} P^{-1}\left(\hat{\zeta}{i}(t)-\hat{\delta}{i}(t)\right) \end{aligned}
where $\hat{\zeta}{i}(t)=\sum{j=1}^{N} a_{i j}\left(\hat{\xi}{i}(t)-\hat{\xi}{j}(t)\right)+a_{i 0} \hat{\xi}{i}(t), \Theta=P^{-1} B B^{T} P^{-1}, P>0$ will be given later, and the initial value $\rho{i}\left(t_{0}\right)>0$. Based on the estimator (4.34), the disturbance observer and the controller are then given by (4.35) and (4.36), respectively.
$$\begin{gathered} \hat{d}{i}(t)=z{i}(t)+Q \hat{\delta}{i}(t) \ \dot{z}{i}(t)=W z_{i}(t)+(W Q-Q A) \hat{\delta}{i}(t)-\alpha Q B K \hat{\zeta}{i}(t) \ u_{i}(t)=\alpha K \hat{\xi}{i}(t)-E \hat{d}{i}(t) \end{gathered}$$
By using the same analyses to those presented in Section $4.2$, we get
\begin{aligned} \dot{\hat{\delta}}(t)=&\left(I_{N} \otimes A\right) \hat{\delta}(t)+\alpha\left(I_{N} \otimes B K\right) \hat{\zeta}(t) \ &-(\overline{\mathcal{L}} \otimes D) \tilde{d}(t)+\left[I_{N} \otimes(G A-F C-A)\right] \tilde{\delta}(t) \end{aligned}$\dot{\tilde{\delta}}(t)=\left[I_{N} \otimes(G A-F C)\right] \tilde{\delta}(t)$, $\dot{\tilde{d}}(t)=\left(I_{N} \otimes W\right) \tilde{d}(t)-(\overline{\mathcal{L}} \otimes Q D) \tilde{d}(t)+\left[I_{N} \otimes Q(G A-F C-A)\right] \tilde{\delta}(t)$ $\dot{\hat{\zeta}}(t)=\left[I_{N} \otimes(A+\alpha B K)\right] \hat{\zeta}(t)+\overline{\mathcal{L}}(\rho+\varrho) \otimes B K)$ where $\hat{\zeta}(t)=\left[\hat{\zeta}{1}^{T}(t), \ldots, \hat{\zeta}{2}^{T}(t)\right]^{T}, \rho=\operatorname{diag}\left{\rho_{1}, \ldots, \rho_{N}\right}$, and the other symbols are the same as those defined in Section 4.2. the same as those defined in Section 4.2.

## cs代写|复杂网络代写complex network代考|NUMERICAL SIMULATIONS

We perform two examples to validate Theorems $4.2$ and $4.3$, respectively. The CNSs under consideration consist of five YF-22 research UAVs [34] whose longitudinal

dynamics satisfy (4.1) with
$$A=\left[\begin{array}{cccc} -0.284 & -23.096 & 2.420 & 9.913 \ 0 & -4.117 & 0.843 & 0.272 \ 0 & -33.884 & -8.263 & -19.543 \ 0 & 0 & 1 & 0 \end{array}\right]$$
$$B=\left[\begin{array}{c} 20.168 \ 0.544 \ -39.085 \ 0 \end{array}\right], D=B\left[\begin{array}{ll} 1 & 0 \end{array}\right], C=\left[\begin{array}{llll} 1 & 1 & 0 & 0 \ 0 & 0 & 1 & 1 \end{array}\right] \text {, }$$
where $x_{i}(t)=\left[x_{i 1}(t), x_{i 2}(t), x_{i 3}(t), x_{i 4}(t)\right]^{T}$ and $x_{i 1}(t), x_{i 2}(t), x_{i 3}(t), x_{i 4}(t)$ represent, respectively, the speed, the attack angle, the pitch rate, and the pitch angle, $i=0,1, \ldots, 4$. The harmonic disturbances are generated by (4.2) with $d_{i}(t)=$ $\left[d_{i 1}(t), d_{i 2}(t)\right]^{T}$ and
$$W=\left[\begin{array}{cc} 0 & 1.5 \ -1.5 & 0 \end{array}\right]$$
It is not difficult to verify that Assumptions 4.1, 4.2 and the conditions (1) and (2) in Remark $4.2$ hold. Then, we get
\begin{aligned} H &=\left[\begin{array}{cccc} -0.2135 & -0.0058 & 0.4137 & 0 \ 0.4029 & 0.0109 & -0.7808 & 0 \end{array}\right]^{T}, \ G &=\left[\begin{array}{cccc} 0.7865 & -0.2135 & 0.4029 & 0.4029 \ -0.0058 & 0.9942 & 0.0109 & 0.0109 \ 0.4137 & 0.4137 & 0.2192 & -0.7808 \ 0 & 0 & 0 & 1 \end{array}\right] . \end{aligned}
Solving the LMI (4.8) gives that
$$F=\left[\begin{array}{rrrr} 11.3220 & 1.5912 & 5.9677 & 0.0839 \ 5.2910 & 0.7606 & 3.1369 & 0.3498 \end{array}\right]^{T} .$$

## cs代写|复杂网络代写complex network代考|CNSS WITH STATIC COUPLING AND SWITCHING TOPOLOGIES

$$\ lim {t \rightarrow \infty}\left|x{i}(t) – x_{0}(t)\right|=0, \lim {t \rightarrow \infty}\left|\hat{d}{i}(t)-d_{i}(t)\right|=0, H○ldF○r一个rb一世吨r一个r是一世n一世吨一世一个l在一个l在和sX一世(吨0),X0(吨0),d^一世(吨0),d一世(吨0),一世=1,…,ñ.吨H和○r和米4.2小号在pp○s和一个ss在米p吨一世○ns4.1−4.3H○ld.我F吨H和一个D吨τ一个>ln⁡ν,吨H和n吨H和C○ns和ns在sd一世s吨在rb一个nC和r和j和C吨一世○n○FCñ小号s(4.1)一个nd(4.3)在一世吨H吨H和d一世s吨在rb一个nC和sG和n和r一个吨和db是(4.2)C一个nb和一个CH一世和在和db是一个d○p吨一世nG吨H和C○ns和ns在s和rr○r和s吨一世米一个吨○r(4.5),吨H和s吨一个吨和和s吨一世米一个吨○r(4.9),一个nd吨H和d一世s吨在rb一个nC和○bs和r在和r(4.10)b一个s和dC○n吨r○ll和r(4.11)在一世吨Hķ=−乙吨磷−1,问=μR−1D吨,ρ≥4一个/λ0,μ≥4/λ0,在H和r和一个一世s一个p○s一世吨一世在和C○ns吨一个n吨,λ0一世sG一世在和nb是(4.4),磷>0一个ndR>0一个r和,r和sp和C吨一世在和l是,○b吨一个一世n和db是s○l在一世nG吨H和大号米我s(4.18)一个nd(4.19), 一个磷+磷一个吨−一个乙乙吨+磷<0, 在吨R+R在−D吨D+2R<0. 磷r○○F4.2F○r一个n是吨∈[吨j,吨j+1),j=0,1,2,…,在和C○ns吨r在C吨吨H和F○ll○在一世nG米大号Fs V_{1}(t)=V_{11}(t)+V_{12}(t)+V_{13}(t)+V_{14}(t)， 在H和r和 在11(吨)=G吨(吨)(我ñ⊗磷−1)G(吨), 在12(吨)=C12∑一世=1ñφ一世σ(吨)(2+ϱ一世(吨))ϱ一世(吨), 在13(吨)=C1C2d~吨(吨)(披σ(吨)⊗R)d~(吨), 在14(吨)=C1C3d~吨(吨)(披σ(吨)⊗小号)d~(吨),$$

## cs代写|复杂网络代写complex network代考|CNSS WITH DYNAMIC COUPLING AND FIXED TOPOLOGY

X^˙一世(吨)=一个X^一世(吨)+一个乙ķX^一世(吨)+(ρ一世+ϱ一世)乙ķ(G^一世(吨)−d^一世(吨)) ρ˙一世=(G^一世(吨)−d^一世(吨))吨θ(G^一世(吨)−d^一世(吨)) ϱ一世=(G^一世(吨)−d^一世(吨))吨磷−1(G^一世(吨)−d^一世(吨))

d^一世(吨)=和一世(吨)+问d^一世(吨) 和˙一世(吨)=在和一世(吨)+(在问−问一个)d^一世(吨)−一个问乙ķG^一世(吨) 在一世(吨)=一个ķX^一世(吨)−和d^一世(吨)

d^˙(吨)=(我ñ⊗一个)d^(吨)+一个(我ñ⊗乙ķ)G^(吨) −(大号¯⊗D)d~(吨)+[我ñ⊗(G一个−FC−一个)]d~(吨)d~˙(吨)=[我ñ⊗(G一个−FC)]d~(吨), d~˙(吨)=(我ñ⊗在)d~(吨)−(大号¯⊗问D)d~(吨)+[我ñ⊗问(G一个−FC−一个)]d~(吨)$\dot{\hat{\zeta}}(t)=\left[I_{N} \otimes(A+\alpha BK)\right] \hat{\zeta}(t)+ \overline{\mathcal{L }}(\rho+\varrho) \otimes BK )在H和r和\hat{\zeta}(t)=\left[\hat{\zeta}{1}^{T}(t), \ldots, \hat{\zeta}{2}^{T}(t)\ right]^{T}、\rho=\operatorname{diag}\left{\rho_{1}、\ldots、\rho_{N}\right}$，其他符号同4.2节定义. 与第 4.2 节中定义的相同。

## cs代写|复杂网络代写complex network代考|NUMERICAL SIMULATIONS

H=[−0.2135−0.00580.41370 0.40290.0109−0.78080]吨, G=[0.7865−0.21350.40290.4029 −0.00580.99420.01090.0109 0.41370.41370.2192−0.7808 0001].

F=[11.32201.59125.96770.0839 5.29100.76063.13690.3498]吨.

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。