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

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## 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
$$\mathcal{L}^{\sigma(t)}=\left[\begin{array}{cc} 0 & \mathbf{0}{N}^{T} \ \mathbf{P} & \overline{\mathcal{L}}^{\sigma(t)} \end{array}\right]$$ 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).
Within the context of CNSs, each agent only communicates with its neighbors. Then the relative state information of other agents with respect to agent $i$ is given by
$$\delta_{i}(t)=\sum_{j=0}^{N} a_{i j}^{\sigma(t)}\left(x_{j}(t)-x_{i}(t)\right), i=1, \ldots, N$$
Note that $\delta_{i}(t)$ is a local information and thereby can be used for the controller design.

## 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|Main results for a nonautonomous leader case

In this subsection, the leader is assumed to be a nonautonomous agent. To achieve consensus tracking, a distributed protocol is proposed as follows
$$u_{i}(t)=d_{1} F \delta_{i}(t)+d_{2} \operatorname{sgn}\left(F \delta_{i}(t)\right), \quad i=1, \ldots, N,$$
where $\delta_{i}(t)$ is given in (3.29), $d_{1}>0, d_{2}>0$, and $F \in \mathbb{R}^{m \times n}$ are the control parameters to be designed later, $\operatorname{sgn}(\cdot)$ denotes the element-wise sign function. For brevity, let $e(t)=\left[e_{1}^{T}(t), \ldots, e_{N}^{T}(t)\right]^{T}$ with $e_{i}(t)=x_{i}(t)-x_{0}(t)$. The tracking error system for the CNS (3.26) under protocol (3.59) with a nonautonomous leader (3.27) can be found to be
\begin{aligned} \dot{e}(t)=& {\left[I_{N} \otimes A-d_{1}\left(\overline{\mathcal{L}}^{\sigma(t)} \otimes B F\right)\right] e(t) } \ &-d_{2}\left(I_{N} \otimes B\right) \cdot \operatorname{sgn}\left(\left(\overline{\mathcal{L}}^{\sigma(t)} \otimes F\right) e(t)\right)-\left(\mathbf{1}{N} \otimes B\right) f\left(x{0}(t), t\right), \end{aligned}
where $\overline{\mathcal{L}}^{\sigma(t)}$ is given in (3.28). Note that the subsequent analysis is performed based on Assumption $3.2$ and the following assumption.

Assumption 3.5 There exists a positive scalar $d_{0}$ such that $\left|f\left(x_{0}(t), t\right)\right|_{\infty} \leq d_{0}$ for all $t \geq 0$.

It is worth noticing that Assumption $3.5$ provides an assurance preventing the actuators from blowing up physically. Note also that the explicit form of nonlinear function $f\left(x_{0}(t), t\right)$ is unknown to any follower. For notational brevity, let
$$\bar{\kappa}{0}=\max {i, j \in{1, \ldots, \kappa}, i \neq j}\left{\bar{\chi}^{i} / \chi^{j}\right},$$
where $\bar{\chi}^{i}=\lambda_{\max }\left(\left(\overline{\mathcal{L}}^{i}\right)^{T} \Phi^{i} \overline{\mathcal{L}}^{i}\right), \chi^{j}=\lambda_{\min }\left(\left(\overline{\mathcal{L}}^{i}\right)^{T} \Phi^{i} \overline{\mathcal{L}}^{i}\right)$, and matrices $\Phi^{i}, i \in{1, \ldots, \kappa}$, are determined in Algorithm $3.1$ by restricting $\bar{Q}^{i}$ and $\Omega^{i}$ to be positive definite and diagonal matrices. In this case, one knows that $\Phi^{i}, i \in{1, \ldots, \kappa}$, are all positive definite and diagonal matrices. Obviously, $\bar{\kappa}{0} \geq 1$. Furthermore, introduce $$\chi{0}=\min {i \in{1, \ldots, \kappa}}\left{\lambda{\min }\left(\overline{\mathcal{L}}^{i}+\left(\Phi^{i}\right)^{-1}\left(\overline{\mathcal{L}}^{i}\right)^{T} \Phi^{i}\right)\right} .$$

## 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)+乙在一世(吨)

X˙0(吨)=一个X0(吨)+乙F(X0(吨),吨),

d一世(吨)=∑j=0ñ一个一世jσ(吨)(Xj(吨)−X一世(吨)),一世=1,…,ñ

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

$$\bar{\kappa}{0}=\max {i, j \in{1, \ldots, \kappa}, i \neq j}\left{\bar{\chi}^ {i} / \chi ^{j}\right},$$

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