# cs代写|复杂网络代写complex network代考|MTH6142

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

In the case of $\omega(t) \equiv 0_{N q}$, it follows from Theorem $7.1$ that consensus in CNS (7.1) with protocol (7.2) constructed by Algorithm $7.1$ can be achieved exponentially. However, the convergence rate is not explicitly given in Algorithm 7.1, i.e., it is still unclear how fast consensus in nominal CNS (7.1) can be achieved. In this subsection, a modified algorithm is proposed to redesign protocol (7.2) such that consensus in the closed-loop nominal CNSs can be achieved with a given exponential convergence rate $c_0>0$

Algorithm 7.2 Suppose that $(A, B)$ is stabilizable and Assumption $7.1$ holds. The consensus protocol (7.2) with an exponential convergence rate $c_0$ can be designed as follows:
(1) Solve the LMI
$$A P+P A^T-B B^T+2 c_0 P<0$$ to get one feasible solution: matrix $P>0$ and scalar $c_0>0$. Then, take $K=$ $B^T P^{-1}$.
(2) Choose the coupling strength $\alpha \geq \alpha_{\mathrm{th}}$, where $\alpha_{\mathrm{th}}$ is defined in Step (2) of Algorithm 7.1.

Theorem 7.2 Suppose that Assumption 7.1 holds and LMI (7.23) is feasible. Then, consensus in the closed-loop nominal CNS (7.1) with protocol (7.2) constructed by Algorithm $7.2$ can be achieved with an exponential rate $c_0$ for any given dwell time $\tau_m>0$

## cs代写|复杂网络代写complex network代考|Selective pinning strategy

In this subsection, a graph search algorithm with linear time complexity is provided for choosing the agents to be pinned in the CNS (7.25). The important issues of at least how many and which agents should be pinned such that Assumption $7.3$ holds will be addressed.

Algorithm $7.3$ Let $\mathcal{G}(\mathcal{A})$ be the communication topology of the CNS (7.25). Then, Assumption $7.3$ will hold if the $r_0$ agents searched by the following procedures are selected and pinned.
(1) Use Tarjan’s algorithm [157] to find all the agents with zero in-degree and strongly connected components of $\mathcal{G}(\mathcal{A})$. Suppose that there are $\iota_1\left(\iota_1 \geq 0\right)$ agents with zero in-degree, labeled as $v_1, \ldots, v_{t_1}$, and $\iota_2\left(\iota_2 \geq 0\right)$ strongly connected components, represented by $\mathcal{G}\left(\mathcal{V}1, \mathcal{E}_1, \mathcal{A}_1\right), \ldots, \mathcal{G}\left(\mathcal{V}{t_2}, \mathcal{E}{t_2}, \mathcal{A}{t_2}\right)$ in $\mathcal{G}(\mathcal{A})$. Set $r_i=0$, for $i=0,1, \ldots, \iota_2$, and $g=1$.
(2) All the $\iota_1$ agents with zero in-degree should be selected and pinned. Then, update the value of $r_0$ by $r_0=r_0+\iota_1$.
(3) Check the condition $\iota_2 \neq 0$ ? If it does not hold, stop; else go to step (4).
(4) Check whether there exists at least one node in $\mathcal{V}_g$ which is reachable from a node belonging to the node set $\mathcal{V} \backslash \mathcal{V}_g$. If it holds, go to step (5); otherwise, go to step (6).
(5) Check the following condition: $g<\iota_2$ ? If it holds, let $g=g+1$ and re-perform step (4); else stop.
(6) Arbitrarily select one agent in $\mathcal{V}_g$ to be pinned, update the value of $r_0$ by $r_0=$ $r_0+1$. Check the following condition: $g<\iota_2$ ? If it holds, let $g=g+1$ and go to step (4); else stop.

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

(1) 求解LMI
$$A P+P A^T-B B^T+2 c_0 P<0$$ 得到一个可行的解决方条: 矩阵 $P>0$ 和标量 $c_0>0$. 然后，取 $K=B^T P^{-1}$.

## cs代写|复杂网络代写complex network代考|Selective pinning strategy

(1) 使用 Tarjan 算法 [157] 找到所有具有霎度数和强连通分量的智能体 $\mathcal{G}(\mathcal{A})$. 假设有 $\iota_1\left(\iota_1 \geq 0\right)$ 度数为零的 代理，标记为 $v_1, \ldots, v_{t_1}$ ，和 $\iota_2\left(\iota_2 \geq 0\right)$ 强连通分量，表示为 $\mathcal{G}\left(\mathcal{V} 1, \mathcal{E}_1, \mathcal{A}_1\right), \ldots, \mathcal{G}\left(\mathcal{V} t_2, \mathcal{E} t_2, \mathcal{A} t_2\right)$ 在 $\mathcal{G}(\mathcal{A})$. 放 $r_i=0$ ，为了 $i=0,1, \ldots, \iota_2$ ，和 $g=1$.
(2) 所有的 $\iota_1$ 应选择并锁定度数为零的代理。然后，更新的值 $r_0$ 经过 $r_0=r_0+\iota_1$.
(3) 检查条件 $\iota_2 \neq 0$ ? 如果不成立，停止；否则转到步骤 (4) 。
(4) 检查是否存在至少一个节点 $\mathcal{V}_g$ 可以从属于节点集的节点到达 $\mathcal{V} \backslash \mathcal{V}_g$. 如果成立，则进行步骤 (5)；否则 转步骤(6)。
(5) 检查下列情况: $g<\iota_2$ ? 如果它成立，让 $g=g+1$ 并重新进行步骤(4)；否则停止。
(6) 任意选择一名代理人 $\mathcal{V}_g$ 要固定，更新的值 $r_0$ 经过 $r_0=r_0+1$. 检查以下条件: $g<\iota_2$ ? 如果它成立， 让 $g=g+1$ 并进行步骤(4)；否则停止。

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