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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 consisting of a leader and $N$ followers, where the leader is labelled as agent 0 , and the followers are respectively labelled as agents $1, \ldots, N$. The dynamics of agent $i, i=1, \ldots, N$, are described by
$$\dot{x}_i(t)=A x_i(t)+B u_i(t),$$
where $x_i(t) \in \mathbb{R}^n$ and $u_i(t) \in \mathbb{R}^m$ are respectively the state and the control input, $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$ are respectively the system and input matrices. Assume that $(A, B)$ is stabilizable. It is further assumed that the leader has no neighbors throughout this section, i.e., the leader’s dynamics will not be affected by those of any followers. Then the dynamics of agent 0 are described by
$$\dot{x}_0(t)=A x_0(t)+B f\left(x_0(t), t\right),$$
where $x_0(t) \in \mathbb{R}^n$ is the leader’s state, and $f\left(x_0(t), t\right) \in \mathbb{R}^m$ is an unknown nonlinear function describing external control inputs acting on the leader.

It is assumed that the communication graph for the $N+1$ agents is switched within a finite graph set $\widehat{\mathcal{G}}=\left{\mathcal{G}^1, \ldots, \mathcal{G}^\kappa\right}, \kappa>1$ and $\kappa \in \mathbb{N}$. Since the leader has no neighbors, the associated Laplacian matrix can be rewritten as
$$\mathcal{L}^{\sigma(t)}=\left[\begin{array}{cc} 0 & \mathbf{0}N^T \ \mathbf{P} & \overline{\mathcal{L}}^{\sigma(t)} \end{array}\right]$$ $$\overline{\mathcal{L}}^{\sigma(t)}=\left[\begin{array}{cccc} \sum{j \in \mathcal{N}1} a{1 j}^{\sigma(t)} & -a_{12}^{\sigma(t)} & \ldots & -a_{1 N}^{\sigma(t)} \ -a_{21}^{\sigma(t)} & \sum_{j \in \mathcal{N}2} a{2 j}^{\sigma(t)} & \ldots & -a_{2 N}^{\sigma(t)} \ \vdots & \vdots & \ddots & \vdots \ -a_{N 1}^{\sigma(t)} & -a_{N 2}^{\sigma(t)} & \ldots & \sum_{j \in \mathcal{N}N} a{N j}^{\sigma(t)} \end{array}\right]$$
where $\mathbf{P}=-\left[a_{10}^{\sigma(t)}, \ldots, a_{N 0}^{\sigma(t)}\right]^T$, the piecewise constant function $\sigma(t):[0,+\infty) \mapsto$ ${1, \ldots, \kappa}$ represents the switching signal and satisfies the ADT condition (2.23).

## cs代写|复杂网络代写complex network代考|Main results for an autonomous leader case

In this subsection, the leader is assumed to be autonomous. That is, the dynamics of the leader are described by (3.27) with $f\left(x_0(t), t\right)=\mathbf{0}_m$. To achieve consensus tracking, a distributed protocol is proposed as follows
$$u_i(t)=c K \delta_i(t), \quad i=1, \ldots, N,$$
where $\delta_i(t)$ is given in $(3.29), c>0$ and $K \in \mathbb{R}^{m \times n}$ are the control parameters to be designed later.

Let $e(t)=\left[e_1^T(t), \ldots, e_N^T(t)\right]^T$, where $e_i(t)=x_i(t)-x_0(t), i=1, \ldots, N$. Combining (3.26), (3.27) together with (3.30) yields
$$\dot{e}(t)=\left[I_N \otimes A-c\left(\overline{\mathcal{L}}^{\sigma(t)} \otimes B K\right)\right] e(t) .$$
Obviously, the consensus tracking problem is solved if and only if $\lim _{t \rightarrow+\infty}|e(t)|=0$. That is, consensus tracking in the considered CNSs will be achieved if and only if the zero fixed point of switched systems (3.31) is globally attractive. Throughout the section, the derivatives of all signals at switching time instants should be considered as their right derivatives.
Before moving on, the following Assumption is made.
Assumption 3.2 For each $i \in{1, \ldots, \kappa}$, the directed graph $\mathcal{G}^i$ contains a directed spanning tree rooted at node 0 (i.e., the leader).

Under Assumption 3.2, it can be obtained from Lemma $2.14$ that all the eigenvalues of $\overline{\mathcal{L}}^i$ have positive real parts, i.e., $\overline{\mathcal{L}}^i$ is anti-stable. Thus, the Lyapunov inequalities
$$Q^i \overline{\mathcal{L}}^i+\left(\overline{\mathcal{L}}^i\right)^T Q^i>0$$
are simultaneously feasible for some positive definite matrices $Q^i, i \in{1, \ldots, \kappa}$. Since the righ-hand side of (3.32) is homogeneous for $Q^i$ for each $i \in{1, \ldots, \kappa}$, one gets that the matrix inequalities
$$\bar{Q}^i \overline{\mathcal{L}}^i+\left(\overline{\mathcal{L}}^i\right)^T \bar{Q}^i>0, \bar{Q}^i \leq I_N, \text { and } \bar{Q}^i>0,$$
are simultaneously feasible for some positive definite matrices $\bar{Q}^i$. To arrive at a less conservative estimation for the minimum allowable ADT for achieving consensus tracking, the following optimization algorithm is proposed.

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

$$\dot{x}i(t)=A x_i(t)+B u_i(t),$$ 这部分没有邻居，即领导者的动态不会受到任何追随者的影响。那么agent 0 的动态描述为 $$\dot{x}_0(t)=A x_0(t)+B f\left(x_0(t), t\right),$$ 在哪里 $x_0(t) \in \mathbb{R}^n$ 是领导者的状态，并且 $f\left(x_0(t), t\right) \in \mathbb{R}^m$ 是描述作用于领导者的外部控制输入的末知非线性函数。 假设通信图为 $N+1$ 代理在有限图集中切换 $1 \mathrm{eft}$ 的分隔符缺失或无法识别 和 $\kappa \in \mathbb{N}$. 由于领导者没有邻居，因此相关的拉普拉斯矩阵可以重写为 $\mathcal{L}^{\sigma(t)}=\left[\begin{array}{lll}0 & \mathbf{0} N^T \mathbf{P} & \overline{\mathcal{L}}^{\sigma(t)}\end{array}\right]$ 在哪里 $\mathbf{P}=-\left[a{10}^{\sigma(t)}, \ldots, a_{N 0}^{\sigma(t)}\right]^T$, 分段常数函数 $\sigma(t):[0,+\infty) \mapsto 1, \ldots, \kappa$ 表示开关信号，满足 ADT 条件 (2.23)。

## cs代写|复杂网络代写complex network代考|Main results for an autonomous leader case

$$u_i(t)=c K \delta_i(t), \quad i=1, \ldots, N,$$

$$\dot{e}(t)=\left[I_N \otimes A-c\left(\overline{\mathcal{L}}^{\sigma(t)} \otimes B K\right)\right] e(t) .$$

㩺设 $3.2$ 对于每个 $i \in 1, \ldots, \kappa_1$, 有向图 $\mathcal{G}^i$ 包含一个以节点 0 为根的有向生成树（即领导者）。

$$Q^i \overline{\mathcal{L}}^i+\left(\overline{\mathcal{L}}^i\right)^T Q^i>0$$
$$\bar{Q}^i \overline{\mathcal{L}}^i+\left(\overline{\mathcal{L}}^i\right)^T \bar{Q}^i>0, \bar{Q}^i \leq I_N, \text { and } \bar{Q}^i>0,$$

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assignmentutor™您的专属作业导师
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