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• 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|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
$$A=\left[\begin{array}{ccc} -(25 \gamma+10) & (25 \gamma+10) & 0 \ (28-35 \gamma) & (29 \gamma-1) & 0 \ 0 & 0 & -\frac{(\gamma+8)}{3} \end{array}\right], \quad B=\left[\begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & -1 \ 0 & 1 & 0 \end{array}\right]$$
$\beta=[1,0,0], \gamma \in[0,1]$ 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 $t$.

## 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|Model formulation

$$\dot{x} i(t)=A x i(t)+\beta x_{i}(t) B x_{i}(t)+\alpha \sum_{j=1}^{N} a_{i j}(t) H\left(x_{j}(t)-x_{i}(t)\right),$$

$$A=\left[\begin{array}{lllllllll} -(25 \gamma+10) & (25 \gamma+10) & 0(28-35 \gamma) & (29 \gamma-1) & 0 & 0 & 0 & -\frac{(\gamma+8)}{3} \end{array}\right], \quad B=\left[\begin{array}{lllllllll} 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 \end{array}\right]$$
$\beta=[1,0,0], \gamma \in[0,1]$ 是一个允数， $\alpha>0$ 表示代理之间的耦合强度， $\mathcal{A}(t)=\left[a_{i j}(t)\right] N \times N$ 是通信拓扑在时间的邻拉矩阵 $t$ ，和 $H \in \mathbb{R}^{3 \times 3}$ 是正定内链接矩阵， $i=1, \ldots, N$. 请注意，系统 (5.33) 将成为耦合 Lorenz、Chen 和 Lü 系统，如果 $\gamma=0,1$ ，和 $0.8$ ，分 别。根据图的拉普拉斯矩阵的定义，它䕊循(5.33)那
$$\dot{x} i(t)=A x_{i}(t)+\beta x_{i}(t) B x_{i}(t)-\alpha \sum_{j=1}^{N} l_{i j}(t) H x_{j}(t)$$

$$\dot{s}(t)=A s(t)+\beta s(t) B s(t),$$

$$\dot{x} i(t)=A x i(t)+\beta x_{i}(t) B x_{i}(t) \quad-\alpha \sum_{j=1}^{N} l_{i j}(t) H x_{j}(t)-\alpha c_{i}(t) H\left(x_{i}(t)-s(t)\right)$$

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

(1) 检龺是否存在至少一个代理 $n_{k}$ 属于 $W_{h}$ 可以从代理访问 $n_{g}$ 属于 $W_{j}, j=1, \ldots, \omega, j \neq h$. 如果成立，则进行步骤（2) ；如果不成立， 转至步骤 (3)。
(2) 检亱下列条件是否成立: $h<\omega$. 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 $W_{h}$ 并固定，让 $m_{h}=1$; 检查以下条件是否成立: $h<\omega$. 如果它成立，让 $h=h+1$ 并重新执行步㞡（1) ；否则 停止。

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师