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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|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 $4.2$.

## 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
$$\dot{e}{i}(t)=f\left(x{i}(t), t\right)-f(s(t), t)-\alpha \sum_{j=1}^{N} l_{i j}(t) e_{j}(t)-\alpha c_{i}(t) e_{i}(t) .$$

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

$$\dot{\hat{\xi}} i(t)=A \hat{\xi} i(t)+\alpha B K \hat{\xi} i(t)+\left(\rho i+\varrho_{i}\right) B K(\hat{\zeta} i(t)-\hat{\delta} i(t)) \dot{\rho} i \quad=(\hat{\zeta} i(t)-\hat{\delta} i(t))^{T} \Theta(\hat{\zeta} i(t)-\hat{\delta} i(t)) \varrho i=(\hat{\zeta} i(t)-\hat{\delta} i(t))^{T} P^{-1}(\hat{\zeta} i(t)-\hat{\delta} i(t)$$

$$\hat{d} i(t)=z i(t)+Q \hat{\delta} i(t) \dot{z} i(t)=W z_{i}(t)+(W Q-Q A) \hat{\delta} i(t)-\alpha Q B K \hat{\zeta} i(t) u_{i}(t)=\alpha K \hat{\xi} i(t)-E \hat{d} i(t)$$

$\dot{\hat{\delta}}(t)=\left(I_{N} \otimes A\right) \hat{\delta}(t)+\alpha\left(I_{N} \otimes B K\right) \hat{\zeta}(t) \quad-(\bar{L} \otimes D) \tilde{d}(t)+\left[I_{N} \otimes(G A-F C-A)\right] \tilde{\delta}(t)$
$\dot{\tilde{\delta}}(t)=\left[I_{N} \otimes(G A-F C)\right] \tilde{\delta}(t)$
$\dot{\tilde{d}}(t)=\left(I_{N} \otimes W\right) \tilde{d}(t)-(\bar{L} \otimes Q D) \tilde{d}(t)+\left[I_{N} \otimes Q(G A-F C-A)\right] \tilde{\delta}(t)$
$\backslash$ rho_{N}\right } } \text { , andtheothersymbolsarethesameasthosedefinedinSection } 4 . 2 \$\text { . } ## cs代写|复杂网络代写complex network代考|Model formulation 假设所考虑的 CNS 包括$N$节点, 代理的动态$i$由 $$\dot{x} i(t)=f(x i(t), t)+\alpha \sum_{j=1}^{N} a_{i j}(t)\left(x_{j}(t)-x_{i}(t)\right),$$ 在哪里$x_{i}(t)=\left[x_{i 1}(t), \ldots, x_{i n}(t)\right]^{T} \in \mathbb{R}^{n}$为了$i=1, \ldots, N$代表代理的状态$i, \alpha>0$是耦合强度，并且$\mathrm{A}(t)=\left[a_{i j}(t)\right] N \times N$是图的邻 接矩阵$\mathrm{G}(\mathrm{A}(t))$有时$t$. 在本章中，所有函数在切换时间点的导数都应视为它们的右手导数。根据图的拉普拉斯矩阵的定义，由 (5.1) 式可 得 $$\dot{x} i(t)=f\left(x_{i}(t), t\right)-\alpha \sum_{j=1}^{N} l_{i j}(t) x_{j}(t)$$ 在哪里$\mathrm{L}(t)=\left[l_{i j}(t)\right]{N \times N}$是图的拉普拉斯矩阵$\mathrm{G}(\mathrm{A}(t))$. 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 approachs(t)什么时候t方法$+\infty$，那是，$\lim t \rightarrow \infty|x i(t)-s(t)|=0$，对所有人$i=1, \ldots, N$并且任意给定初始条件，其中 $$\dot{s}(t)=f(s(t), t)$$ 这里，$s(t)$可能是一个平衡点，一个周期轨道，甚至是一个混沌轨道。受 [74] 中作品的启发，使用线性控制器固定网络 (5.2)$-\alpha c{i}(t)\left(x_{i}(t)-s(t)\right)$代理$i$导致 $$\dot{x} i(t)=f(x i(t), t)-\alpha \sum_{j=1}^{N} l_{i j}(t) x_{j}(t)-\alpha c_{i}(t)\left(x_{i}(t)-s(t)\right)$$ 在哪里$c_{i}(t) \in 0,1$和$c_{i}(t)=1$当且仅当代理$i$的$(5.2)$被固定在时间$t$. 让$e_{i}(t)=x_{i}(t)-s(t), i=1, \ldots, N\$. 因此，从 (5.4) 可以得出
$$\dot{e} i(t)=f(x i(t), t)-f(s(t), t)-\alpha \sum_{j=1}^{N} l_{i j}(t) e_{j}(t)-\alpha c_{i}(t) e_{i}(t)$$

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assignmentutor™您的专属作业导师
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