### cs代写|复杂网络代写complex network代考|Consensus tracking of CNSs with higher-order dynamics

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## cs代写|复杂网络代写complex network代考|directed switching topologies

This chapter studies the consensus tracking of CNSs with higher-order dynamics and directed switching topologies. This chapter begins by overviewing some previous works and by indicating our motivations. Section $6.2$ firstly studies the case with Lipschitz nonlinear dynamics and directed fixed topology. Then we extend the results to directed switching topologies with each topology contains a directed spanning tree. This section finally studies the case with directed switching topologies that frequently contain a directed spanning tree. Section $6.3$ studies the case with general linear dynamics and occasionally missing control inputs. This section presents some sufficient criteria for achieving consensus tracking. Moreover, the convergence rate is discussed. Finally, some simulations are given to validate the theoretical results.

In contrast to CNSs with first-order nonlinear dynamics, CNSs with second-order nonlinear dynamics are more interesting as it can describe a large class of real networked systems, including coupled pendulums [5] and coupled point-mass systems with or without nonlinear disturbances [165]. Leaderless consensus problem for CNSs with second-order nonlinear dynamics and a fixed weakly connected topology was investigated in [217]. In [137], the consensus tracking problem for CNSs with second-order nonlinear dynamics in the presence of a leader under an arbitrarily given directed topology was studied from pinning control approach. Furthermore, consensus tracking problem for CNSs with higher-order Lipschitz type agent dynamics and a fixed topology was studied in [79].

In the existing literature on the consensus tracking problem for CNSs with nonlinear dynamics, it is commonly assumed that the communication topology is fixed.

## cs代写|复杂网络代写complex network代考|CONSENSUS TRACKING OF CNSS WITH HIGHER-ORDER NONLINEAR DYNAMICS

Consider a CNS consisting of a leader and $N$ followers, where the leader is labelled as agent 0 and the followers are labelled as agents $1, \ldots, N$. The dynamics of agent $i, i=0,1, \ldots, N$, are given by
$$\dot{x}{i}(t)=A x{i}(t)+C f\left(x_{i}(t), t\right)+B u_{i}(t),$$
where $x_{i}(t) \in \mathbb{R}^{n}$ represent the states of agent $i, f(\cdot, \cdot): \mathbb{R}^{n} \times[0,+\infty) \mapsto \mathbb{R}^{p}$ is a continuously differentiable vector-valued function representing the intrinsic nonlinear dynamics, $u_{i}(t) \in \mathbb{R}^{m}$ is the control input to be designed, $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}$, and $C \in \mathbb{R}^{n \times p}$ are constant real matrices. It is assumed that the matrix pair $(A, B)$ is stabilizable. In this section, it is assumed that the leader will not being affected by any followers, i.e. $u_{0}(t) \equiv \mathbf{0}{m}$ in CNS (6.1). Before moving on, the following assumption is made. Assumption 6.1 There exists a nonnegative constant $\varrho$, such that $$|f(y, t)-f(z, t)| \leq \varrho|y-z|, \forall y, z \in \mathbb{R}^{n}, t \geq 0$$ To achieve consensus tracking, the following distributed consensus tracking protocol is proposed for each follower $i$ : $$u{i}(t)=\alpha F \sum_{j=0}^{N} a_{i j}(t)\left(x_{j}(t)-x_{i}(t)\right), \quad i=1, \ldots, N,$$
where $\alpha>0$ represents the coupling strength, $F \in \mathbb{R}^{m \times n}$ is the feedback gain matrix to be designed, and $\mathcal{A}(t)=\left[a_{i j}(t)\right]_{(N+1) \times(N+1)}$ is the adjacency matrix of graph $\mathcal{G}(t)$. Here, $\mathcal{G}(t)$ describes the underlying communication topology among the $N+1$ agents at time $t$.

## cs代写|复杂网络代写complex network代考|Main results for fixed topology containing a directed spanning tree

In this section, distributed consensus tracking is addressed for CNS (6.1) with a fixed communication topology containing a directed spanning tree.

Without loss of generality, let $\mathcal{G}(t)=\mathcal{G}$ for all $t \geq 0$ since the communication topology is assumed to be fixed in this subsection. To derive the main results, the following assumption is needed.

Assumption 6.2 The communication topology $\mathcal{G}$ contains a directed spanning tree with agent 0 (i.e. the leader) as the root.

Under Assumption 6.2, , the Laplacian matrix of directed graph $\mathcal{G}$ can be written as
$$\mathcal{L}=\left[\begin{array}{cc} 0 & \mathbf{0}{N}^{T} \ \mathbf{P} & \overline{\mathcal{L}} \end{array}\right], \quad \overline{\mathcal{L}}=\left[\begin{array}{cccc} \sum{j \in N_{1}} a_{1 j} & -a_{12} & \cdots & -a_{1 N} \ -a_{21} & \sum_{j \in \mathcal{N}{2}} a{2 j} & \cdots & -a_{2 N} \ \vdots & \vdots & \ddots & \vdots \ -a_{N 1} & -a_{N 2} & \cdots & \sum_{j \in \mathcal{N}{N}} a{N j} \end{array}\right]$$
where $\mathbf{P}=-\left[a_{10}, \ldots, a_{N 0}\right]^{T}$. It can be thus obtained from Lemma $2.15$ that there exists a positive definite diagonal matrix $\Phi=\operatorname{diag}\left{\phi_{1}, \ldots, \phi_{N}\right}$ such that $\overline{\mathcal{L}}^{T} \Phi+$ $\Phi \overline{\mathcal{L}}>0$. One such $\phi=\left[\phi_{1}, \ldots, \phi_{N}\right]^{T}$ can be obtained by solving the matrix equation $\overline{\mathcal{L}}^{T} \phi=\mathbf{1}{N}$. Since $u{0}(t) \equiv \mathbf{0}{m}$, one has $$\dot{x}{0}(t)=A x_{0}(t)+C f\left(x_{0}(t), t\right) .$$
Furthermore, substituting (6.2) into (6.1) gives a closed-loop system:
$$\dot{x}{i}(t)=A x{i}(t)+C f\left(x_{i}(t), t\right)+\alpha B F \sum_{j=0}^{N} a_{i j}\left(x_{j}(t)-x_{i}(t)\right), i=1, \ldots, N,$$
where $\mathcal{A}=\left[a_{i j}\right]{(N+1) \times(N+1)}$ is the adjacency matrix of graph $\mathcal{G}$. Define $e{i}(t)=x_{i}(t)-x_{0}(t), i=1, \ldots, N$, and $e(t)=\left[e_{1}^{T}(t), \ldots, e_{N}^{T}(t)\right]^{T}$. Based on the above analysis, one has the following error dynamical system:
$$\dot{e}{i}(t)=A e{i}(t)+C\left(f\left(x_{i}(t), t\right)-f\left(x_{0}(t), t\right)\right)-\alpha B F \sum_{j=1}^{N} \bar{l}{i j}(t) e{j}(t)$$
Rewriting (6.5) into a compact form, one has
$$\dot{e}(t)=\left(I_{N} \otimes A\right) e(t)+\left(I_{N} \otimes C\right) \tilde{f}(x(t), t)-\alpha(\overline{\mathcal{L}} \otimes B F) e(t)$$

where $\tilde{f}(x(t), t)=\left[\left(f\left(x_{1}(t), t\right)-f\left(x_{0}(t), t\right)\right)^{T}, \ldots,\left(f\left(x_{N}(t), t\right)-f\left(x_{0}(t), t\right)\right)^{T}\right]^{T}$ and $x(t)=\left[x_{0}^{T}(t), x_{1}^{T}(t), \ldots, x_{N}^{T}(t)\right]^{T} .$

Before moving on, a multi-step design procedure is given for selecting the control parameters in protocol (6.2) under a fixed topology $\mathcal{G}$.

## cs代写|复杂网络代写complex network代考|CONSENSUS TRACKING OF CNSS WITH HIGHER-ORDER NONLINEAR DYNAMICS

X˙一世(吨)=一个X一世(吨)+CF(X一世(吨),吨)+乙在一世(吨),

|F(是,吨)−F(和,吨)|≤ϱ|是−和|,∀是,和∈Rn,吨≥0为了实现共识跟踪，为每个追随者提出了以下分布式共识跟踪协议一世 :

## cs代写|复杂网络代写complex network代考|Main results for fixed topology containing a directed spanning tree

X˙0(吨)=一个X0(吨)+CF(X0(吨),吨).

X˙一世(吨)=一个X一世(吨)+CF(X一世(吨),吨)+一个乙F∑j=0ñ一个一世j(Xj(吨)−X一世(吨)),一世=1,…,ñ,

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