### 数学代写|matlab代写|Linear Dynamic Systems

matlab是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

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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

## 数学代写|matlab代写|Dynamic Systems Represented by Differential Equations

A system is an assemblage of interrelated entities that can be considered as a whole. If the attributes of interest of a system are changing with time, then it is called a dynamic system. A process is the evolution over time of a dynamic system.

Our solar system, consisting of the sun and its planets, is a physical example of a dynamic system. The motions of these bodies are governed by laws of motion that depend only upon their current relative positions and velocities. Sir Isaac Newton (1642-1727) discovered these laws and expressed them as a system of differential equations-another of his discoveries. From the time of Newton, engineers and scientists have learned to define dynamic systems in terms of the differential equations that govern their behavior. They have also learned how to solve many of these differential equations to obtain formulas for predicting the future behavior of dynamic systems.

EXAMPLE $2.1$ (below, left): Newton’s Model for a Dynamic System of $\boldsymbol{n}$ Massive Bodies For a planetary system with $n$ bodies (idealized as point masses), the acceleration of the $i$ th body in any inertial (i.e., non-rotating and non-accelerating) Cartesian coordinate system is given by Newton’s third law as the second-order differential equation
$$\frac{d^{2} r_{i}}{d t^{2}}=C_{g} \sum_{\substack{j=1 \ j / i}}^{n} \frac{m_{j}\left[r_{j}-r_{i}\right]}{\left|r_{j}-r_{i}\right|^{3}}, 1 \leq i \leq n$$
where $r_{j}$ is the position coordinate vector of the $j$ th body, $m_{j}$ is the mass of the $j$ th body, and $C_{g}$ is the gravitational constant. This set of $n$ differential equations, plus the associated initial conditions of the bodies (i.e., their initial positions and velocities) theoretically determines the future history of the planetary system.

## 数学代写|matlab代写|State Variables and State Equations

The second-order differential equation of the previous example can be transformed to a system of two first-order differential equations in the two dependent variables $x_{1}=\delta$ and $x_{2}=d \delta / d t$. In this way, one can reduce the form of any system of higher order differential equations to an equivalent system of first-order differential equations. These systems are generally classified into the types shown in Table 2.1, with the most general type being a time-varying differential equation for representing a dynamic system with time-varying dynamic characteristics. This is represented in vector form as
$$\dot{x}(t)=f(t, x(t), u(t)),$$
where Newton’s “dot” notation is used as a shorthand for the derivative with respect to time, and a vector-valued function $f$ to represent a system of $n$ equations
\begin{aligned} \dot{x}{1} &=f{1}\left(t, x_{1}, x_{2}, x_{3}, \ldots, x_{n}, u_{1}, u_{2}, u_{3}, \ldots, u_{r}, t\right), \ \dot{x}{2} &=f{2}\left(t, x_{1}, x_{2}, x_{3}, \ldots, x_{n}, u_{1}, u_{2}, u_{3}, \ldots, u_{r}, t\right), \ \dot{x}{3} &=f{3}\left(t, x_{1}, x_{2}, x_{3}, \ldots, x_{n}, u_{1}, u_{2}, u_{3}, \ldots, u_{r}, t\right), \ & \vdots \ \dot{x}{n} &=f{n}\left(t, x_{1}, x_{2}, x_{3}, \ldots, x_{n}, u_{1}, u_{2}, u_{3}, \ldots, u_{r}, t\right) \end{aligned}
in the independent variable $t$ (time), $n$ dependent variables $\left{x_{i} \mid 1 \leq i \leq n\right}$, and $r$ known inputs $\left{u_{i} \mid 1 \leq i \leq r\right}$. These are called the state equations of the dynamic system.

## 数学代写|matlab代写|Continuous Time and Discrete Time

The dynamic system defined by Equation $2.2$ is an example of a continuous system, so called because it is defined with respect to an independent variable $t$ that varies continuously over some real interval $t \in\left[t_{0}, t_{f}\right]$. For many practical problems, however, one is only interested in knowing the state of a system at a discrete set of times $t \in\left{t_{1}, t_{2}, t_{3}, \ldots\right}$. These discrete times may, for example, correspond to the times at which the outputs of a system are sampled (such as the times at which Piazzi recorded the direction to Ceres). For problems of this type, it is convenient to order the times $t_{k}$ according to their integer subscripts:
$$t_{0}<t_{1}<t_{2}<\cdots t_{k-1}<t_{k}<t_{k+1}<\cdots .$$

That is, the time sequence is ordered according to the subscripts, and the subscripts take on all successive values in some range of integers. For problems of this type, it suffices to define the state of the dynamic system as a recursive relation,
$$x\left(t_{k+1}\right)=f\left(x\left(t_{k}\right), t_{k}, t_{k+1}\right),$$
by means of which the state is represented as a function of its previous state. This is a definition of a discrete dynamic system. For systems with uniform time intervals $\Delta t$
$$t_{k}=k \Delta t .$$

## 数学代写|matlab代写|Dynamic Systems Represented by Differential Equations

d2r一世d吨2=CG∑j=1 j/一世n米j[rj−r一世]|rj−r一世|3,1≤一世≤n

## 数学代写|matlab代写|State Variables and State Equations

X˙(吨)=F(吨,X(吨),在(吨)),

X˙1=F1(吨,X1,X2,X3,…,Xn,在1,在2,在3,…,在r,吨), X˙2=F2(吨,X1,X2,X3,…,Xn,在1,在2,在3,…,在r,吨), X˙3=F3(吨,X1,X2,X3,…,Xn,在1,在2,在3,…,在r,吨), ⋮ X˙n=Fn(吨,X1,X2,X3,…,Xn,在1,在2,在3,…,在r,吨)

## 数学代写|matlab代写|Continuous Time and Discrete Time

X(吨ķ+1)=F(X(吨ķ),吨ķ,吨ķ+1),

## 有限元方法代写

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## 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。