### 数学代写|matlab代写|ON THE NOTATION USED IN THIS BOOK

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|Symbolic Notation

The fundamental problem of symbolic notation, in almost any context, is that there are never enough symbols to go around. There are not enough letters in the Roman alphabet to represent the sounds of standard English, let alone all the variables in Kalman filtering and its applications. As a result, some symbols must play multiple roles. In such cases, their roles will be defined as they are introduced. It is sometimes confusing, but unavoidable.

“Dot”‘ Notation for Derivatives. Newton’s notation using $\vec{f}(t), \vec{f}(t)$ for the first two derivatives of $f$ with respect to $t$ is used where convenient to save ink.

Standard Symbols for Kalman Filter Variables. There appear to be two “standard” conventions in technical publications for the symbols used in Kalman filtering. The one used in this book is similar to the original notation of Kalman [179]. The other standard notation is sometimes associated with applications of Kalman filtering in control theory. It uses the first few letters of the alphabet in place of the Kalman notation. Both sets of symbol usages are presented in Table 1.2, along with the original (Kalman) notation.

State Vector Notation for Kalman Filtering. The state vector $x$ has been adorned with all sorts of other appendages in the usage of Kalman filtering. Table $1.3$ lists the notation used in this book (left column) along with notations found in some other sources (second column). The state vector wears a “hat” as the estimated value, $\hat{x}$, and subscripting to denote the sequence of values that the estimate assumes over time. The problem is that it has two values at the same time: the a priori ${ }^{17}$ value (before the measurement at the current time has been used in refining the estimate) and the a posteriori value (after the current measurement has been used in refining the estimate). These distinctions are indicated by the signum. The negative sign $(-)$ indicates the a priori value, and the positive sign $(+)$ indicates the a posteriori value.

## 数学代写|matlab代写|SUMMARY

The Kalman filter is an estimator used to estimate the state of a linear dynamic system perturbed by Gaussian white noise using measurements that are linear functions of the system state but corrupted by additive Gaussian white noise. The mathematical model used in the derivation of the Kalman filter is a reasonable representation for many problems of practical interest, including control problems as

well as estimation problems. The Kalman filter model is also used for the analysis of measurement and estimation problems.

The method of least squares was the first “optimal” estimation method. It was discovered by Gauss (and others) around the end of the eighteenth century, and it is still much in use today. If the associated Gramian matrix is nonsingular, the method of least squares determines the unique values of a set of unknown variables such that the squared deviation from a set of constraining equations is minimized.

Observability of a set of unknown variables is the issue of whether or not they are uniquely determinable from a given set of constraining equations. If the constraints are linear functions of the unknown variables, then those variables are observable if and only if the associated Gramian matrix is nonsingular. If the Gramian matrix is singular, then the unknown variables are unobservable.

The Wiener-Kolmogorov filter was derived in the $1940 \mathrm{~s}$ by Norbert Wiener (using a model in continuous time) and Andrei Kolmogorov (using a model in discrete time) working independently. It is a statistical estimation method. It estimates the state of a dynamic process so as to minimize the mean-squared estimation error. It can take advantage of statistical knowledge about random processes in terms of their power spectral densities in the frequency domain.

The “state-space” model of a dynamic process uses differential equations (or difference equations) to represent both deterministic and random phenomena. The state variables of this model are the variables of interest and their derivatives of interest. Random processes are characterized in terms of their statistical properties in the time domain, rather than the frequency domain. The Kalman filter was derived as the solution to the Wiener filtering problem using the state-space model for dynamic and random processes. The result is easier to derive (and to use) than the WienerKolmogorov filter.

Square-root filtering is a reformulation of the Kalman filter for better numerical stability in finite-precision arithmetic. It is based on the same mathematical model, but it uses an equivalent statistical parameter that is less sensitive to roundoff errors in the computation of optimal filter gains. It incorporates many of the more numerically stable computation methods that were originally derived for solving the least-squares problem.

## 数学代写|matlab代写|CHAPTER FOCUS

Models for Dynamic Systems. Since their introduction by Isaac Newton in the seventeenth century, differential equations have provided concise mathematical models for many dynamic systems of importance to humans. By this device, Newton was able to model the motions of the planets in our solar system with a small number of variables and parameters. Given a finite number of initial conditions (the initial positions and velocities of the sun and planets will do) and these equations, one can uniquely determine the positions and velocities of the planets for all time. The finite-dimensional representation of a problem (in this example, the problem of predicting the future course of the planets) is the basis for the so-called state-space approach to the representation of differential equations and their solutions, which is the focus of this chapter. The dependent variables of the differential equations become state variables of the dynamic system. They explicitly represent all the important characteristics of the dynamic system at any time.

The whole of dynamic system theory is a subject of considerably more scope than one needs for the present undertaking (Kalman filtering). This chapter will stick to just those concepts that are essential for that purpose, which is the development of the statespace representation for dynamic systems described by systems of linear differential equations. These are given a somewhat heuristic treatment, without the mathematical rigor often accorded the subject, omitting the development and use of the transform methods of functional analysis for solving differential equations when they serve no purpose in the derivation of the Kalman filter. The interested reader will find a more formal and thorough presentation in most upper-level and graduate-level textbooks on ordinary differential equations. The objective of the more engineering-oriented treatments of dynamic systems is usually to solve the controls problem, which is the problem of defining the inputs (i.e., control settings) that will bring the state of the dynamic system to a desirable condition. That is not the objective here, however.

## 数学代写|matlab代写|SUMMARY

Wiener-Kolmogorov 滤波器由1940 s由 Norbert Wiener（使用连续时间的模型）和 Andrei Kolmogorov（使用离散时间的模型）独立工作。它是一种统计估计方法。它估计动态过程的状态，以最小化均方估计误差。它可以利用随机过程在频域中的功率谱密度方面的统计知识。

## 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。