### cs代写|复杂网络代写complex network代考|TSKS33

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|Multiple Lyapunov functions

To proceed, the notion of time dependent switching is introduced.
As a special kind of hybrid dynamic system, switched system has been studied for quite some time by researchers from applied mathematics, systems and control fields. Roughly speaking, a switched system is a dynamic system that consists of a number of subsystems and a switching rule that determines switches among these subsystems. Suppose the switched system is generated by the following family of subsystems
$$\dot{x}(t)=f_{p}(t, x(t)), x(t) \in \mathbb{R}^{n}, p \in{1, \ldots, \kappa},$$
together with a switching signal $\sigma(t):\left[t_{0},+\infty\right) \mapsto{1, \ldots, \kappa}$. Note that $\sigma(t)$ is a piecewise constant function that switches at the switching time instants $t_{1}, t_{2}, \ldots$, and is constant on the time interval $\left[t_{k}, t_{k+1}\right), k=0,1, \ldots$. In this book, we assume $\sigma(t)$ is right continuous, i.e., $\sigma(t)=\lim {\iota} t \sigma(\iota)$, and $\inf {k \in \mathbb{N}}\left(t_{k+1}-t_{k}\right) \geq \tau_{m}$ for some given positive scalar $\tau_{m}$ where inf represents the infimum. Please see Figure $2.2$ for an example. Thus the switched systems with time-dependent switching signal $\sigma(t)$ can be described by the equation
$$\dot{x}(t)=f_{\sigma(t)}(t, x(t)) .$$
According to Theorem 2.1, each subsystem has a unique solution over arbitrary interval $\left[t_{k}, t_{k+1}\right), k=0,1, \ldots$, with arbitrary initial value $x\left(t_{k}\right) \in \mathbb{R}^{n}$ if the function $f_{p}$, for each $p=1, \ldots, \kappa$, is globally Lipschitz in $x(t)$ uniformly over $t$. Thus the switched system (2.10) is well defined for arbitrary switching signal $\sigma(t)$ defined above and any given initial value $x\left(t_{0}\right) \in \mathbb{R}^{n}$. Throughout this chapter, we assume that such a globally Lipschitz condition holds for the subsystems, and thus the well-definedness of the switched system is guaranteed. We further assume that $f_{p}\left(t, \mathbf{0}{n}\right)=\mathbf{0}{n}$ for each $p=1, \ldots, \kappa$. Thus, the zero vector is an equilibrium point of the switched system (2.10). Next, some stability notions for the zero equilibrium point of switched systems are introduced.

## cs代写|复杂网络代写complex network代考|CONSENSUS OF LINEAR CNSS WITH DIRECTED SWITCHING TOPOLOGIES

In the past decade, the consensus problem of general linear CNSs has received a lot of attention $[76,146,162,185,186,224]$. Specifically, the consensus problem of linear CNSs under a directed fixed communication topology has been addressed in $[76,224]$. In [162], the robust consensus of linear CNSs with additive perturbations of the transfer matrices of the nominal dynamics was studied. In [163] and a number of subsequent papers, the robust consensus was analyzed from the viewpoint of the $\mathcal{H}_{\infty}$ control theory. Among other relevant references, we mention [146] where, while assuming that the open loop systems are Lyapunov stable, the consensus problem of linear CNSs with undirected switching topologies has been investigated. In the situation where the CNS is equipped with a leader and the topology of the system belongs to the class of directed switching topologies, the consensus tracking problem has been studied in $[185,186]$. One feature of the results in these references is that the open loop agents’ dynamics do not have to be Lyapunov stable. Note that the presence of the leader in the CNSs considered in these references facilitate the derivations and the direct analyses of the consensus error system. However, when the open loop systems are not Lyapunov stable and/or there is no designated leader in the group, the consensus problem for linear CNSs with directed switching topologies remains challenging.

Motivated by the above discussion, this section aims to study the consensus problem for linear CNSs with directed switching topologies. Several aspects of the current study are worth mentioning. Firstly, some of the assumptions in the existing works are dismissed, e.g., the open loop dynamics of the agents do not have to be Lyapunov stable in this chapter. Furthermore, the CNSs under consideration are not required to have a leader. Compared with the consensus problems for linear CNSs with a designated leader, the point of difference here concerns the assumption on the system’s communication topology. In the previous work on the consensus tracking of linear CNSs such as [185], each possible augmented system graph was required to contain a directed spanning tree rooted at the leader. Compared with that work, the switching topologies in this section are allowed to have spanning trees rooted at different nodes. This is a significant relaxation of the previous conditions since it enables the system to be reconfigured if necessary (e.g., to allow different nodes to serve as the formation leader). This also has a potential to make the system more reliable.

## cs代写|复杂网络代写complex network代考|Multiple Lyapunov functions

$$\dot{x}(t)=f_{p}(t, x(t)), x(t) \in \mathbb{R}^{n}, p \in 1, \ldots, \kappa$$

$$\dot{x}(t)=f_{\sigma(t)}(t, x(t)) .$$

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## MATLAB代写

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