## 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}

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