cs代写|复杂网络代写complex network代考|CMSC711

cs代写|复杂网络代写complex network代考|Discussions on the convergence rate

Suppose that the distributed consensus tracking problem of CNS with followers given by (6.51) and a leader given by (6.50) can be solved by the protocol (6.52) with control parameters constructed by Algorithm 6.4. It can be seen from (6.80) to (6.82) that the convergence rate of consensus tracking in the closed-loop CNSs is characterized by $\epsilon_0=\inf _{j \in \mathbb{N}} \kappa_j$ with $\kappa_j=\beta \delta_k-\gamma \rho_j-\left(h_j-1\right) \ln \mu$. Specifically, the larger $\epsilon_0$, the faster distributed consensus tracking. For a given CNS, the convergence rate of the distributed consensus tracking can be increased by maximizing $\beta$ and minimizing $\gamma$. It can be seen from LMI $(6.56)$ that the parameter $\beta$ can be chosen as arbitrarily large if $(A, B)$ is controllable; however, for the case that $(A, B)$ is stabilizable but not controllable, $\beta$ should not be larger than $-2 \chi$, where $\chi$ is the largest real part of the uncontrollable stable eigenvalues of $A$. Also, $\gamma$ can be chosen as arbitrarily small if $A$ has no unstable eigenvalue; but if $A$ contains some unstable eigenvalues, the parameter $\gamma$ should be larger than $2 \omega$, where $\omega$ is the largest real part of the unstable eigenvalues of $A$. However, it is still unclear how to select $\beta$ and $\gamma$ such that the LMIs $(6.56)$ and $A P+P A^T<\gamma P$ share a common solution $P$ while the above-mentioned $\epsilon_0$ attains its maximum value. Nevertheless, this optimal design can be solved after $\beta$ is fixed. Specifically, let $\gamma=\gamma_0$ be fixed, and $\beta_{\max }$ and $\beta_{\min }$ be, respectively, the maximal and minimal allowable values of $\beta$ such that the LMIs (6.56) and $A P+P A^T<\gamma P$ share a common solution $P>0$ and the condition (6.63) holds. Then, the CNS with followers given by $(6.51)$ and a leader given by $(6.50)$ equipped with the protocol (6.52) constructed by Algorithm $6.4$ with $\beta=\beta_{\max }$ yields a fast convergence rate.

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

Consider a CNS consisting of $N$ agents with general linear dynamics, described by
$$\dot{x}_i(t)=A x_i(t)+B u_i(t)+D \omega_i(t),$$
where $x_i(t) \in \mathbb{R}^n$ is the state, $u_i(t) \in \mathbb{R}^m$ is the control input, $\omega_i(t) \in \mathbb{L}_2[0,+\infty)$ is the external disturbance, $A, B$, and $D$ are constant real matrices. It is assumed that matrix pair $(A, B)$ is stabilizable.

The communication topology among the $N$ agents switches at the time instants $t_1, t_2, \ldots$. It is assumed that $t_0=0$ and $t_{k+1}-t_k \geq \tau_m>0, k \in \mathbb{N}$. And it is assumed that $\mathcal{G}^{\sigma(t)} \in\left{\mathcal{G}^1, \ldots, \mathcal{G}^\kappa\right}$ with $\kappa \geq 1$ and $\kappa \in \mathbb{N}$.

To achieve consensus, the following consensus protocol based only on the relative information between agent $i$ and its neighbors is proposed:
$$u_i(t)=\alpha K \sum_{j=1}^N a_{i j}^{\sigma(t)}\left[x_j(t)-x_i(t)\right], i=1, \ldots, N,$$
where $\alpha>0$ is the coupling strength to be selected, $K \in \mathbb{R}^{m \times n}$ is the feedback gain matrix to be designed, and $\mathcal{A}^{\sigma(t)}=\left[a_{i j}^{\sigma(t)}\right]{N \times N}$ is the adjacency matrix of the graph $\mathcal{G}^{\sigma(t)}$ Substituting (7.2) into (7.1) gives that $$\dot{x}_i(t)=A x_i(t)+\alpha B K \sum{j=1}^N a_{i j}^{\sigma(t)}\left[x_j(t)-x_i(t)\right]+D \omega_i(t), i=1, \ldots, N$$
Let $x(t)=\left[x_1^T(t), \ldots, x_N^T(t)\right]^T$ and $\omega(t)=\left[\omega_1^T(t), \ldots, \omega_N^T(t)\right]^T$, one gets
$$\dot{x}(t)=\left[\left(I_N \otimes A\right)-\alpha\left(\mathcal{L}^{\sigma(t)} \otimes B K\right)\right] x(t)+\left(I_N \otimes D\right) \omega(t),$$
where $\mathcal{L}^{\sigma(t)}$ is the Laplacian matrix of the graph $\mathcal{G}^{\sigma(t)}$. Before moving forward, the following assumption is made.

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

$$\dot{x}i(t)=A x_i(t)+B u_i(t)+D \omega_i(t),$$ 在哪里 $x_i(t) \in \mathbb{R}^n$ 是状态， $u_i(t) \in \mathbb{R}^m$ 是控制输入， $\omega_i(t) \in \mathbb{L}_2[0,+\infty)$ 是外部干扰， $A, B$ ，和 $D$ 是常 数实矩阵。假设矩阵对 $(A, B)$ 是稳定的。 通信拓扑 $N$ 代理在瞬间切换 $t_1, t_2, \ldots$. 假设 $t_0=0$ 和 $t{k+1}-t_k \geq \tau_m>0, k \in \mathbb{N}$. 并且假设

$$u_i(t)=\alpha K \sum_{j=1}^N a_{i j}^{\sigma(t)}\left[x_j(t)-x_i(t)\right], i=1, \ldots, N,$$

$$\dot{x}i(t)=A x_i(t)+\alpha B K \sum j=1^N a{i j}^{\sigma(t)}\left[x_j(t)-x_i(t)\right]+D \omega_i(t), i=1, \ldots, N$$
\begin{aligned} \text { 让 } x(t)=\left[x_1^T(t), \ldots, x_N^T(t)\right]^T \text { 和 } \omega(t)=\left[\omega_1^T(t), \ldots, \omega_N^T(t)\right]^T ， \text { 个得到 } \ \dot{x}(t)=\left[\left(I_N \otimes A\right)-\alpha\left(\mathcal{L}^{\sigma(t)} \otimes B K\right)\right] x(t)+\left(I_N \otimes D\right) \omega(t), \end{aligned}

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: