### cs代写|复杂网络代写complex network代考|Numerical simulations

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|Numerical simulations

Consider the consensus tracking problem of CNS with followers’ dynamics given by (3.26) and leader’s dynamics given by (3.27). Figure $3.5$ indicates three possible switching topologies of the considered CNS, where topology $\mathcal{G}^{1}$ and $\mathcal{G}^{2}$ both contain a directed spanning trees with the leader agent being the root, while no spanning tree is involved in $\mathcal{G}^{3}$. The associated Laplacian matrix among the followers $\overline{\mathcal{L}}^{\sigma(t)}$ are given by
$$\overline{\mathcal{L}}^{1}=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ -1 & 1 & 0 & 0 \ -1 & 0 & 1 & 0 \ -1 & 0 & 0 & 1 \end{array}\right], \quad \overline{\mathcal{L}}^{2}=\left[\begin{array}{cccc} 1 & -1 & 0 & 0 \ 0 & 1 & 0 & 0 \ -1 & 0 & 1 & 0 \ -1 & 0 & 0 & 1 \end{array}\right], \quad \overline{\mathcal{L}}^{3}=\left[\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ -1 & 0 & 1 & 0 \ -1 & 0 & 0 & 1 \end{array}\right]$$
Suppose that the states of agent $i$ in the CNS are represented by $x_{i}(t)=$ $\left[x_{i 1}(t), x_{i 2}(t)\right]^{T} \in \mathbb{R}^{2}$, and their system’s matrices are given by
$$A=\left[\begin{array}{cc} -4 & 1 \ 0 & -1 \end{array}\right], B=\left[\begin{array}{l} 1 \ 0 \end{array}\right]$$

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

This chapter studies the consensus disturbance rejection problem for multiple-input multiple-output linear CNSs subject to nonvanishing disturbances. This Chapter begins by overviewing some previous works and by indicating our motivations. Section $4.2$ presents the models and proposes an unknown input observer (UIO) based on the relative outputs among neighboring agents. Section $4.3$ studies the case with static coupling and directed switching communication topologies. By using the MLFs based technique, it is shown that consensus is achieved and the disturbances are fully rejected. Section $4.4$ studies the case with dynamic couplings and directed fixed topology. As the control parameters do not depend on any global information, so the obtained consensus disturbance rejection is fully distributed. Finally, some simulations are given to validate the obtained theoretical results.

As an interesting issue continued from single systems $[21,31]$, disturbance rejection of CNSs has received more and more attention recently $[14,34,57,148,170,220]$. In [14], the authors solved the consensus problem for MIMO linear CNSs with undirected fixed topology subject to unknown disturbances which were assumed to have steady state values. Furthermore, consensus problem was solved for CNSs in the presence of harmonic nonvanishing disturbances [34]. Later, Sun et al. [148] proposed a fully distributed approach for achieving consensus disturbance rejection in linear CNSs with a directed fixed topology. More recently, the authors [170] applied the state predictor feedback method to settle the consensus disturbance rejection problem for CNSs with input delays as well as output delays.

The aforementioned literature and some references therein have broadened our knowledge on consensus control for CNSs under external disturbances. So we study the consensus disturbance rejection problem for MIMO linear CNSs under deterministic but nonvanishing disturbances, where both directed fixed and switching topologies are considered. Since only the agents’ outputs are available, a UIO is designed for each follower based upon the relative outputs to estimate the consensus error between each follower and its neighbors. With the aid of this UIO, a state estimator with static coupling strength and a disturbance estimator are designed. Based on these two estimators, a controller is designed for the considered CNSs with directed switching topologies. We show that consensus disturbance rejection can be achieved by choosing suitable control parameters if the ADT is greater than a positive constant. Furthermore, a controller which incorporates a state estimator with an adaptive coupling law and a disturbance estimator is designed for the considered CNSs with directed fixed topology. We show that consensus disturbance rejection can be achieved in a fully distributed manner.

## cs代写|复杂网络代写complex network代考|MODEL FORMULATION AND UNKNOWN INPUT OBSERVER

The CNSs under consideration have a leader and $N$ followers. For illustration convenience, we label the leader as agent 0 , and label the $N$ followers as agents $1, \ldots, N$. The dynamics of the agent $i, i=1, \ldots, N$, are described by:
\begin{aligned} &\dot{x}{i}(t)=A x{i}(t)+B u_{i}(t)+D d_{i}(t), \ &y_{i}(t)=C x_{i}(t), \end{aligned}
where $x_{i}(t) \in \mathbb{R}^{n}, u_{i}(t) \in \mathbb{R}^{m}$, and $y_{i}(t) \in \mathbb{R}^{q}$ are, respectively, the state, the control input, and the output, $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}$, and $C \in \mathbb{R}^{q \times n}$ represent, respectively, the state matrix, the control input matrix, and the output matrix, $D \in \mathbb{R}^{n \times p}$ is a constant matrix, $d_{i}(t) \in \mathbb{R}^{p}$ is the external disturbance generated by the exogenous system:
$$\dot{d}{i}(t)=W d{i}(t),$$
where $W \in \mathbb{R}^{p \times p}$ represents a known exosystem matrix. In some practical applications, the leader acts as a reference generator which provides desired trajectory for the following agents to track. So the dynamics of the agent 0 are described by:
\begin{aligned} &\dot{x}{0}(t)=A x{0}(t) \ &y_{0}(t)=C x_{0}(t) \end{aligned}
where $x_{0}(t) \in \mathbb{R}^{n}$ and $y_{0}(t) \in \mathbb{R}^{q}$ are, respectively, the leader’s state and output.
One goal of this chapter is that the disturbance $d_{i}(t)$ can be completely rejected, and the other is to make each follower evolves along the trajectory provided by the leader finally. To achieve these goals, we make the following assumptions.
Assumption 4.1 There is a constant matrix $E \in \mathbb{R}^{m \times p}$ such that $D=B E$.

## cs代写|复杂网络代写complex network代考|MODEL FORMULATION AND UNKNOWN INPUT OBSERVER

X˙一世(吨)=一个X一世(吨)+乙在一世(吨)+Dd一世(吨), 是一世(吨)=CX一世(吨),

d˙一世(吨)=在d一世(吨),

X˙0(吨)=一个X0(吨) 是0(吨)=CX0(吨)

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