Posted on Categories:EE代写, 电子代写, 连续线性系统

# EE代写|连续线性系统代写Continous Time Linear System代考|EE235 Invariant sets

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## EE代写|连续线性系统代写Continous Time Linear System代考|Invariant sets

Invariant sets can be seen as the most basic building blocks for the understanding of the behaviour of a given dynamical system. These sets have the following property: trajectories starting in the invariant set, remain in the invariant set, for all of their future, and all of their past.

Definition 2.2.1. Let $S \subset \mathbb{R}^n$ be a set. Then $S$ is said to be invariant under the dynamics of (3) if for any $\boldsymbol{x}_0 \in S$ we have $\phi^t\left(\boldsymbol{x}_0\right) \in S$ for all $t \in \mathbb{R}$. If we restrict ourselves to positive times $t$ then we refer to $S$ as a positively invariant set and, for negative time, as a negatively invariant set.

We will now discuss some special invariant sets: orbits, equilibria, periodic orbits, and limit sets. We will introduce other invariant sets later in this notes.

Let $\boldsymbol{x}0 \in \mathbb{R}^n$ be a point in the phase space of (3). The orbit through $\boldsymbol{x}{\mathbf{o}}$, which we denote by $\mathcal{O}\left(\boldsymbol{x}0\right)$, is the set of points in phase space that lie on a trajectory of (3) passing through $\boldsymbol{x}_0$ : $$\mathcal{O}\left(\boldsymbol{x}_0\right)=\left{\phi^t\left(\boldsymbol{x}{\mathbf{o}}\right): t \in \mathbb{R}\right}$$
The positive semiorbit through $\boldsymbol{x}0$ is the set $$\mathcal{O}^{+}\left(x_0\right)=\left{\phi^t\left(x{\mathbf{o}}\right): t \geq 0\right} .$$
and the negative semiorbit through $\boldsymbol{x}0$ is the set $$\mathcal{O}^{-}\left(\boldsymbol{x}_0\right)=\left{\phi^t\left(\boldsymbol{x}{\mathbf{o}}\right): t \leq 0\right} .$$
Note that for any $t \in \mathbb{R}$ we have that $\mathcal{O}\left(\phi^t\left(\boldsymbol{x}_0\right)\right)=\mathcal{O}\left(\boldsymbol{x}_0\right)$.
A point $\boldsymbol{p} \in \mathbb{R}^n$ is an equilibrium for the flow of (3) if $\phi^t(\boldsymbol{p})=\boldsymbol{p}$ for all $t \in \mathbb{R}$. Since the flows we consider here are solutions of differential equations, we obtain that an equilibrium $\boldsymbol{p}$ for the flow $\phi^t$ of the differential equation (3) must satisfy $f(\boldsymbol{p})=\mathbf{0}$. Equilibria are solutions that do not change in time, thus providing the most simple example of invariant sets.

A point $\boldsymbol{p} \in M$ is a periodic point of period $T$ for the flow of (3) if there exists some positive number $T \in \mathbb{R}$ such that $\phi^T(\boldsymbol{p})=\boldsymbol{p}$ and $\phi^t(\boldsymbol{p}) \neq \boldsymbol{p}$ for every $0<t<T$. The orbit $\mathcal{O}(\boldsymbol{p})$ of a periodic point is called a periodic orbit.

## EE代写|连续线性系统代写Continous Time Linear System代考|Stability

We will now discuss the notions of Lyapunov stability and asymptotic stability. Such notions can be intuitively stated as follows: if $\boldsymbol{p}$ is a Lyapunov stable point then for every point $\boldsymbol{q}$ close enough to $\boldsymbol{p}$ its orbit stays close to the orbit of $\boldsymbol{p}$; if $\boldsymbol{p}$ is asymptotically stable if it is Lyapunov stable and for every point $\boldsymbol{q}$ close enough to $\boldsymbol{p}$ the forward orbit of $\boldsymbol{q}$ will converge to the forward orbit of $\boldsymbol{p}$.

Definition 2.3.1 (Lyapunov stability). The orbit of a point $\boldsymbol{p} \in \mathbb{R}^n$ is Lyapunov stable by the flow $\phi^t$ of (3) if for any $\epsilon>0$ there is $\delta>0$ such that if $|\boldsymbol{q}-\boldsymbol{p}|<\delta$, then $$\left|\phi^t(\boldsymbol{q})-\phi^t(\boldsymbol{p})\right|<\epsilon$$ for all $t \geq 0$. The orbit of a point $\boldsymbol{p} \in \mathbb{R}^n$ which is not stable is said to be unstable. Definition 2.3.2 (Asymptotic stability). The orbit of a point $\boldsymbol{p} \in \mathbb{R}^n$ is asymptotically stable by the flow $\phi^t$ of (3) if it is Lyapunov stable and there exists a neighbourhood $V$ of $\boldsymbol{p}$ such that for every $\boldsymbol{q} \in V$, $$\left|\phi^t(\boldsymbol{q})-\phi^t(\boldsymbol{p})\right| \rightarrow 0,$$ as $t$ tends to infinity. Example 2.3.3. Consider again the logistic model for population growth: $$\dot{x}=a x\left(1-\frac{x}{K}\right), \quad x \in \mathbb{R}, a, K>0 .$$
From the analysis of its phase portrait, we obtain that $x=0$ is an asymptotically stable equilibrium while $x=K$ is an unstable equilibrium.

For an example of a Lyapunov stable equilibrium which is not asymptotically stable, consider the two dimensional system
$$\left{\begin{array}{l} \dot{u}=v \ \dot{v}=-u, \quad(u, v) \in \mathbb{R} \times \mathbb{R} . \end{array}\right.$$

## EE代写|连续线性系统代写Continous Time Linear System代考|Invariant sets

、left 缺少或无法识别的分隔符

《left 缺少或无法识别的分隔符

、left 缺少或无法识别的分隔符

## EE代写|连续线性系统代写Continous Time Linear System代考|Stability

\$\$
$\backslash$ left {
$$\dot{u}=v \dot{v}=-u, \quad(u, v) \in \mathbb{R} \times \mathbb{R} .$$
、正确的。
$\$ \

EE代写|连续线性系统代写Continous Time Linear System代考 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。