## 数学代写|matlab代写|CSC113

2022年12月27日

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
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## 数学代写|matlab代写|Algorithmic Deep Learning

In this chapter, we introduce the Algorithmic Deep Learning Neural Network (ADLNN), a deep learning system that incorporates algorithmic descriptions of the processes as part of the deep learning neural network. The dynamical models provide domain knowledge. These are in the form of differential equations. The outputs of the network are both indications of failures and updates to the parameters of the models. Training can be done using simulations, prior to operations, or through operator interaction during operations.

The system is shown in Figure 5.1. This is based on work from the books by Paluszek and Thomas $[41,42]$. These books show the relationships between machine learning, adaptive control, and estimation. This model can be encapsulated in a set of differential equations. We will limit ourselves to sensor failures in this example. The output indicates what kind of failures have occurred. It indicates that either one or both of the sensors have failed.

Figure $5.2$ shows an air turbine [26]. This air turbine has a constant pressure air supply. The pressurized air causes the turbine to spin. It is a way to produce rotary motion for a drill or other purposes.

We can control the valve from the air supply, the pressure regulator, to control the speed of the turbine. The air flows past the turbine blades causing it to turn. The control needs to adjust the air pressure to handle variations in the load. The load is the resistance to turning. For example, a drill might hit a harder material while in use. We measure the air pressure $p$ downstream from the valve, and we also measure the rotational speed of the turbine $\omega$ with a tachometer.
The state vector is
$$\left[\begin{array}{l} p \ \omega \end{array}\right]$$
where $\omega$ is the tachometer rate of the turbine and $p$ is the pressure.
The dynamical model for the air turbine is
$$\left[\begin{array}{l} \dot{p} \ \dot{\omega} \end{array}\right]=\left[\begin{array}{rr} -\frac{1}{\tau_p} & 0 \ \frac{K_t}{\tau_t} & -\frac{1}{\tau_t} \end{array}\right]\left[\begin{array}{l} p \ \omega \end{array}\right]+\left[\begin{array}{l} \frac{K_p}{\tau_p} \ 0 \end{array}\right] u$$

This is a state space system
$$\dot{x}=a x+b u$$
where
$$\begin{array}{r} a=\left[\begin{array}{rr} -\frac{1}{\tau_p} & 0 \ \frac{K_t}{\tau_t} & -\frac{1}{\tau_t} \end{array}\right] \ b=\left[\begin{array}{c} \frac{K_p}{\tau_p} \ 0 \end{array}\right] \end{array}$$
The pressure downstream from the regulator is equal to $K_p u$ when the system is in equilibrium. $\tau_p$ is the regulator time constant, and $\tau_t$ is the turbine time constant. The turbine speed is $K_t p$ when the system is in equilibrium. The tachometer measures $\omega$, and the pressure sensor measures $p$. The load is folded into the time constant for the turbine.

## 数学代写|matlab代写|How It Works

The detection filter is an estimator with a specific gain matrix that multiplies the residuals.
$$\left[\begin{array}{c} \dot{\hat{p}} \ \dot{\hat{\omega}} \end{array}\right]=\left[\begin{array}{rr} -\frac{1}{\tau_p} & 0 \ \frac{K_t}{\tau_t} & -\frac{1}{\tau_t} \end{array}\right]\left[\begin{array}{l} \hat{p} \ \hat{\omega} \end{array}\right]+\left[\begin{array}{l} \frac{K_P}{\tau_p} \ 0 \end{array}\right] u+\left[\begin{array}{ll} d_{11} & d_{12} \ d_{21} & d_{22} \end{array}\right]\left[\begin{array}{l} p-\hat{p} \ \omega-\hat{\omega} \end{array}\right]$$
where $\hat{p}$ is the estimated pressure and $\hat{\omega}$ is the estimated angular rate of the turbine. The $D$ matrix is the matrix of detection filter gains. This matrix multiplies the residuals, the difference between the measured and estimated states, into the detection filter. The residual vector is
$$r=\left[\begin{array}{c} p-\hat{p} \ \omega-\hat{\omega} \end{array}\right]$$
The $D$ matrix needs to be selected so that this vector tells us the nature of the failure. The gains should be selected so that

1. The filter is stable.
2. If the pressure regulator fails, the first residual, $p-\hat{p}$, is nonzero, but the second remains zero.
3. If the turbine fails, the second residual $\omega-\hat{\omega}$ is nonzero, but the first remains zero.
The gain matrix is
$$D=a+\left[\begin{array}{cc} \frac{1}{\tau_1} & 0 \ 0 & \frac{1}{\tau_2} \end{array}\right]$$
We can see this by substituting this $D$ into Equation 5.6:
$$\left[\begin{array}{l} \dot{\hat{p}} \ \dot{\hat{\omega}} \end{array}\right]=\left[\begin{array}{rr} a_{11} & a_{12} \ a_{21}+\frac{K_t}{\tau_t} & a_{22} \end{array}\right]\left[\begin{array}{l} \hat{p} \ \hat{\omega} \end{array}\right]+\left[\begin{array}{l} \frac{K_P}{\tau_p} \ 0 \end{array}\right] u+D\left[\begin{array}{l} p \ \omega \end{array}\right]$$
The time constant $\tau_1$ is the pressure residual time constant. The time constant $\tau_2$ is the tachometer residual time constant. In effect, we cancel out the dynamics of the plant and replace them with decoupled detection filter dynamics. These time constants should be shorter than the time constants in the dynamical model so that we detect failures quickly. However, they need to be at least twice as long as the sampling period to prevent numerical instabilities.

# matlab代写

## 数学代写|matlab代写|Algorithmic Deep Learning

$$[p \omega]$$

$$[\dot{p} \dot{\omega}]=\left[\begin{array}{lll} -\frac{1}{\tau_p} & 0 \frac{K_t}{\tau_t} & -\frac{1}{\tau_t} \end{array}\right][p \omega]+\left[\frac{K_p}{\tau_p} 0\right] u$$

$$\dot{x}=a x+b u$$

$$a=\left[\begin{array}{lll} -\frac{1}{\tau_p} & 0 \frac{K_t}{\tau_t} & -\frac{1}{\tau_t} \end{array}\right] \quad b=\left[\frac{K_p}{\tau_p} 0\right]$$

## 数学代写|matlab代写|How It Works

$$[\dot{\hat{p}} \dot{\hat{\omega}}]=\left[\begin{array}{lll} -\frac{1}{\tau_p} & 0 \frac{K_t}{\tau_t} & -\frac{1}{\tau_t} \end{array}\right][\hat{p} \hat{\omega}]+\left[\frac{K_P}{\tau_p} 0\right] u+\left[\begin{array}{ll} d_{11} & d_{12} \end{array}\right.$$

$$r=[p-\hat{p} \omega-\hat{\omega}]$$

1. 过滤器稳定。
2. 如果压力调节器出现故障，首先残留， $p-\hat{p}$, 是非零的，但第 二个保持为零。
3. 如果涡轮机发生故障，第二个残差 $\omega-\hat{\omega}$ 是非零的，但第一个 保持零。
增益矩阵是
$$D=a+\left[\begin{array}{llll} \frac{1}{\tau_1} & 0 & 0 & \frac{1}{\tau_2} \end{array}\right]$$
我们可以通过替换这个来看到这一点 $D$ 进入公式 5.6:
时间常数 $\tau_1$ 是压力残余时间常数。时间常数 $\tau_2$ 是转速表剩余时 间常数。实际上，我们抵消了被控对象的动态并用解耦检测滤 波器动态替换它们。这些时间常数应该比动态模型中的时间常 数短，以便我们快速检恻到故障。但是，它们需要至少是采样 周期的两倍，以防止数值不稳定。

## 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。