数学代写|最优化理论作业代写optimization theory代考|VARIATION OF EXTREMALS

Doug I. Jones

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数学代写|最优化理论作业代写optimization theory代考|VARIATION OF EXTREMALS

The iterative numerical technique that we shall discuss in this section is called variation of extremals, because every trajectory generated by the algorithm satisfies Eqs. (6.1-1) through (6.1-3) and hence is an extremal. To illustrate the basic concept of the algorithm, let us consider a simple example.
A First-Order Optimal Control Problem
Suppose that a first-order system
$$\dot{x}(t)=a(x(t), u(t), t)$$
is to be controlled to minimize a performance measure of the form
$$J=\int_{t_0}^{t_f} g(x(t), u(t), t) d t$$
where $x\left(t_0\right)=x_0$ is given, $t_0$ and $t_f$ are specified, and the admissible state and control values are not constrained by any boundaries. If the equation [corresponding to (6.1-3)]
$$\frac{\partial \mathscr{H}}{\partial u}=0$$
is solved for the control in terms of the state and costate and substituted in the state and costate equations, the reduced differential equations
\begin{aligned} \dot{x}(t) & =a(x(t), p(t), t) \ \dot{p}(t) & =d(x(t), p(t), t) \end{aligned}
are obtained. In general, $d$ is a nonlinear function of $x(t), p(t)$, and $t$. Since $h=0$ in the performance measure, Eq. (6.1-4b) gives $p\left(t_f\right)=0$. To determine an optimal trajectory, we must find a solution of Eq. (6.3-4) that satisfies the boundary conditions $x\left(t_0\right)=x_0, p\left(t_f\right)=0$.

数学代写|最优化理论作业代写optimization theory代考|Extensions Required for Systems of 2n Differential Equations

We have shown how the method of variation of extremals can be used to solve a two-point boundary-value problem involving two first-order differential equations. If we have $2 n$ first-order differential equations ( $n$ state equations and $n$ costate equations), the matrix generalization of Eq. (6.3-10a) is
$$\mathbf{p}^{(l+1)}\left(t_0\right)=\mathbf{p}^{(i)}\left(t_0\right)-\left[\mathbf{P}p\left(\mathbf{p}^{(i)}\left(t_0\right), t_f\right)\right]^{-1} \mathbf{p}^{(i)}\left(t_f\right),$$ where $\mathbf{P}_p\left(\mathbf{p}^{(i)}\left(t_0\right), t\right)$ is the $n \times n$ matrix of partial derivatives of the components of $\mathbf{p}(t)$ with respect to each of the components of $\mathbf{p}\left(t_0\right)$, evaluated at $\mathbf{p}^{(i)}\left(t_0\right)^{\prime}$; that is, $$\mathbf{P}_p\left(\mathbf{p}^{(i)}\left(t_0\right), t\right) \triangleq\left[\begin{array}{cccc} \frac{\partial p_1(t)}{\partial p_1\left(t_0\right)} & \frac{\partial p_1(t)}{\partial p_2\left(t_0\right)} & \cdots & \frac{\partial p_1(t)}{\partial p_n\left(t_0\right)} \ \cdot & \cdot & & \cdot \ \cdot & \cdot & & \cdot \ \cdot & \cdot & & \cdot \ \frac{\partial p_n(t)}{\partial p_1\left(t_0\right)} & \frac{\partial p_n(t)}{\partial p_2\left(t_0\right)} & \cdots & \frac{\partial p_n(t)}{\partial p_n\left(t_0\right)} \end{array}\right]{\mathbf{p}^{(n)}\left(t_0\right)}$$
The $\mathbf{P}_p$ matrix indicates the influence of changes in the initial costate on the costate trajectory at time $t$; hence, we shall call $\mathbf{P}_p$ the costate influence function matrix. Notice that (6.3-18) requires that $\mathbf{P}_p$ be known only at the terminal time $t_f$.

最优化代写

数学代写|最优化理论作业代写optimization theory代考|VARIATION OF EXTREMALS

$$\dot{x}(t)=a(x(t), u(t), t)$$

$$J=\int_{t_0}^{t_f} g(x(t), u(t), t) d t$$

$$\frac{\partial \mathscr{H}}{\partial u}=0$$

\begin{aligned} \dot{x}(t) & =a(x(t), p(t), t) \ \dot{p}(t) & =d(x(t), p(t), t) \end{aligned}

数学代写|最优化理论作业代写optimization theory代考|Extensions Required for Systems of 2n Differential Equations

$$\mathbf{p}^{(l+1)}\left(t_0\right)=\mathbf{p}^{(i)}\left(t_0\right)-\left[\mathbf{P}p\left(\mathbf{p}^{(i)}\left(t_0\right), t_f\right)\right]^{-1} \mathbf{p}^{(i)}\left(t_f\right),$$其中$\mathbf{P}_p\left(\mathbf{p}^{(i)}\left(t_0\right), t\right)$是$\mathbf{p}(t)$的各分量相对于$\mathbf{p}\left(t_0\right)$的各分量的偏导数的$n \times n$矩阵，在$\mathbf{p}^{(i)}\left(t_0\right)^{\prime}$求值;也就是$$\mathbf{P}_p\left(\mathbf{p}^{(i)}\left(t_0\right), t\right) \triangleq\left[\begin{array}{cccc} \frac{\partial p_1(t)}{\partial p_1\left(t_0\right)} & \frac{\partial p_1(t)}{\partial p_2\left(t_0\right)} & \cdots & \frac{\partial p_1(t)}{\partial p_n\left(t_0\right)} \ \cdot & \cdot & & \cdot \ \cdot & \cdot & & \cdot \ \cdot & \cdot & & \cdot \ \frac{\partial p_n(t)}{\partial p_1\left(t_0\right)} & \frac{\partial p_n(t)}{\partial p_2\left(t_0\right)} & \cdots & \frac{\partial p_n(t)}{\partial p_n\left(t_0\right)} \end{array}\right]{\mathbf{p}^{(n)}\left(t_0\right)}$$
$\mathbf{P}_p$矩阵表示在$t$时刻初始状态的变化对状态轨迹的影响;因此，我们称$\mathbf{P}_p$为协态影响函数矩阵。注意(6.3-18)要求只在终端时间$t_f$上知道$\mathbf{P}_p$。

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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。

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