### 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。

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