数学代写|matlab代写|BMS13

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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数学代写|matlab代写|Numerical Model

For our example, we need a numerical model of disruptions [8], [7], [37]. Ideally, our model would include all of the effects in the list given earlier. We use the model in Scibile [38]. We will only consider vertical movement.

The equilibrium force on the plasma is induced by the magnetic field and current density in the plasma:
$$J \times B=\nabla p$$
where $J$ is the current density, $B$ is the magnetic field, and $p$ is the pressure. $J, B$, and $p$ are all three-element vectors, and that $\nabla$ is a cross-product operation. Pressure is the force per unit area on the plasma. The momentum balance is [1]
$$\rho \frac{d v}{d t}=J \times B-\nabla p$$
where $v$ is the plasma velocity, and $\rho$ is the plasma density. The imbalance causes plasma motion, that is, when $J \times B \neq \nabla p$. If we neglect the plasma mass, we get
$$L_P^T I+A_{P P} z I_p=F_p$$
where $L_p$ is the mutual change inductance matrix of the coils. $I$ is the vector of currents in the Tokamak coils and the conducting shell around the plasma. $F_p$ is the external force normalized to the plasma current $I_p$, and $A_{P P}$ is the normalized destabilizing force. If we lump currents into active currents, driven by an external voltage, and passive currents, we get a simplified model of the plasma. We need to add Kirchhoff’s voltage law to get a dynamical model.
$$L \dot{I}+R I+L_P \dot{z} I_p=\Gamma V$$
$\Gamma$ couples voltages to the currents, $L$ is the coil inductance matrix, and $R$ is the coil resistance. If we combine these, we get the state space matrices shown in the following. The lumped model is shown in Figure 6.3. Table $6.1$ gives the model parameters.

数学代写|matlab代写|Disturbances

The disturbances are due to Edge Localized Modes (ELM). An Edge Localized Mode is a disruptive magnetohydrodynamic instability that occurs along the edges of a Tokamak plasma due to steep plasma pressure gradients [23]. The strong pressure gradient is called the edge pedestal. The edge pedestal improves plasma confinement time by a factor of two over the low-confinement mode. This is now the preferred mode of operation for Tokamaks. A simple model for an ELM is
$$d=k\left(e^{-\frac{t}{\tau_1}}-e^{-\frac{t}{\tau_2}}\right)$$
$d$ is the output of the ELM. It can be scaled by $k$ based on the usage. For example, in [38] it is scaled to show the output of a sensor. In our simulation, it is scaled to produce a driving force on the plasma.
$\tau_1>\tau_2$ with the ELMs appearing randomly. The function ELM produces one ELM. The simulation must call it with a new sequence of times to get a new ELM. Figure $6.5$ shows the results of the built-in demo in ELM. The function also computes the derivative since the derivative of the disturbance is also an input.

We will use a controller to control the vertical position of the plasma, which otherwise is unstable as shown earlier. The controller will be a state space system using full state feedback. The states are the two currents. Position, $z$, is controlled indirectly. We will use a quadratic regulator. We will use a continuous version. This just means that we need to sample much faster than the range of frequencies for the control.

matlab代写

数学代写|matlab代写|Numerical Model

$$J \times B=\nabla p$$

$$\rho \frac{d v}{d t}=J \times B-\nabla p$$

$$L_P^T I+A_{P P} z I_p=F_p$$

$$L \dot{I}+R I+L_P \dot{z} I_p=\Gamma V$$
$\Gamma$ 将电压耦合到电流， $L$ 是线圈电感矩阵，和 $R$ 是线圈电阻。如果我们将文些结合起来，我们将得到如下所示的 状态空间矩阵。集总模型如图 $6.3$ 所示。桌子6.1给出模型参数。

数学代写|matlab代写|Disturbances

$$d=k\left(e^{-\frac{t}{\tau_1}}-e^{-\frac{t}{\tau_2}}\right)$$
$d$ 是 ELM 的输出。它可以按比例缩放 $k$ 根据使用情况。 例如，在 [38] 中它被缩放以显示传感器的输出。在我们 的模拟中，它被缩放以在等离子体上产生驱动力。
$\tau_1>\tau_2$ ELM 随机出现。函数 ELM 产生一个ELM。模 拟必须使用新的时间序列调用它以获得新的 ELM。数字 $6.5$ 显示 ELM 中内置演示的结果。该函数还计算导数， 因为扰动的导数也是输入。

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。