Posted on Categories:Optimization Theory, 优化理论, 数学代写, 最优化

# 数学代写|最优化作业代写optimization theory代考|CONTINUOUS LINEAR REGULATOR PROBLEMS

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|最优化作业代写optimization theory代考|CONTINUOUS LINEAR REGULATOR PROBLEMS

Problems like Example 3.11-1 with linear plant dynamics and quadratic performance criteria are referred to as linear regulator problems. In this section we investigate the use of the Hamilton-Jacobi-Bellman equation as a means of solving the general form of the continuous linear regulator problem.t
The process to be controlled is described by the state equations
$$\dot{\mathbf{x}}(t)=\mathbf{A}(t) \mathbf{x}(t)+\mathbf{B}(t) \mathbf{u}(t)$$
and the performance measure to be minimized is
$$J=\frac{1}{2} \mathbf{x}^T\left(t_f\right) \mathbf{H} \mathbf{x}\left(t_f\right)+\int_{t_0}^{t_f} \frac{1}{2}\left[\mathbf{x}^T(t) \mathbf{Q}(t) \mathbf{x}(t)+\mathbf{u}^T(t) \mathbf{R}(t) \mathbf{u}(t)\right] d t$$
$\mathbf{H}$ and $\mathbf{Q}$ are real symmetric positive semi-definite matrices, $\mathbf{R}$ is a real, symmetric positive definite matrix, the initial time $t_0$ and the final time $t_f$ are specified, and $\mathbf{u}(t)$ and $\mathbf{x}(t)$ are not constrained by any boundaries.

To use the Hamilton-Jacobi-Bellman equation, we first form the Hamiltonian:
$$\begin{array}{cc} \mathscr{H}\left(\mathbf{x}(t), \mathbf{u}(t), J_{\mathbf{x}}^, t\right)= & \frac{1}{2} \mathbf{x}^T(t) \mathbf{Q}(t) \mathbf{x}(t)+\frac{1}{2} \mathbf{u}^T(t) \mathbf{R}(t) \mathbf{u}(t)+J_{\mathbf{x}}^{ T}(\mathbf{x}(t), t) \ & \cdot[\mathbf{A}(t) \mathbf{x}(t)+\mathbf{B}(t) \mathbf{u}(t)] \end{array}$$

## 数学代写|最优化作业代写optimization theory代考|THE HAMILTON-JACOBI-BELLMAN EQUATION-SOME OBSERVATIONS

We have derived the Hamilton-Jacobi-Bellman equation and used it to solve two examples of the linear regulator type. Let us now make some observations concerning the H-J-B functional equation.
Boundary Conditions
In our derivation we have assumed that $t_f$ is fixed; however, the results still apply if $t_f$ is free. For example, if $S$ represents some hypersurface in the state space and $t_f$ is defined as the first time the system’s trajectory intersects $S$, then the boundary condition is
$$J^\left(\mathbf{x}\left(t_f\right), t_f\right)=h\left(\mathbf{x}\left(t_f\right), t_f\right)$$ A Necessary Condition The results we have obtained represent a necessary condition for optimality; that is, the minimum cost function $J^(\mathbf{x}(t), t)$ must satisfy the Hamilton-Jacobi-Bellman equation.

A Sufficient Condition
Although we have not derived it here, it is also true that if there is a cost function $J^{\prime}(\mathbf{x}(t), t)$ that satisfies the Hamilton-Jacobi-Bellman equation, then $J^{\prime}$ is the minimum cost function; i.e.,
$$J^{\prime}(\mathbf{x}(t), t)=J^*(\mathbf{x}(t), t)$$
Rigorous proofs of the necessary and sufficient conditions embodied in the H-J-B equation are given in [K-5] and also in [A-2], which contains several examples.
Solution of the Hamilton-Jacobi-Bellman Equation
In both of the examples that we considered, a solution was obtained by guessing a form for the minimum cost function. Unfortunately, we are normally unable to find a solution so easily. In general, the H-J-B equation must be solved by numerical techniques-see [F-1], for example. Actually, a numerical solution involves some sort of a discrete approximation to the exact optimization relationship [Eq. (3.11-10)]; alternatively, by solving the recurrence relation [Eq. (3.7-18)] we obtain the exact solution to a discrete approximation of the Hamilton-Jacobi-Bellman functional equation.

## 数学代写|最优化作业代写optimization theory代考|CONTINUOUS LINEAR REGULATOR PROBLEMS

$$\dot{\mathbf{x}}(t)=\mathbf{A}(t) \mathbf{x}(t)+\mathbf{B}(t) \mathbf{u}(t)$$

$$J=\frac{1}{2} \mathbf{x}^T\left(t_f\right) \mathbf{H} \mathbf{x}\left(t_f\right)+\int_{t_0}^{t_f} \frac{1}{2}\left[\mathbf{x}^T(t) \mathbf{Q}(t) \mathbf{x}(t)+\mathbf{u}^T(t) \mathbf{R}(t) \mathbf{u}(t)\right] d t$$
$\mathbf{H}$和$\mathbf{Q}$是实对称正半定矩阵，$\mathbf{R}$是实对称正定矩阵，指定了初始时间$t_0$和最终时间$t_f$, $\mathbf{u}(t)$和$\mathbf{x}(t)$不受任何边界约束。

$$\begin{array}{cc} \mathscr{H}\left(\mathbf{x}(t), \mathbf{u}(t), J_{\mathbf{x}}^, t\right)= & \frac{1}{2} \mathbf{x}^T(t) \mathbf{Q}(t) \mathbf{x}(t)+\frac{1}{2} \mathbf{u}^T(t) \mathbf{R}(t) \mathbf{u}(t)+J_{\mathbf{x}}^{ T}(\mathbf{x}(t), t) \ & \cdot[\mathbf{A}(t) \mathbf{x}(t)+\mathbf{B}(t) \mathbf{u}(t)] \end{array}$$

## 数学代写|最优化作业代写optimization theory代考|THE HAMILTON-JACOBI-BELLMAN EQUATION-SOME OBSERVATIONS

$$J^\left(\mathbf{x}\left(t_f\right), t_f\right)=h\left(\mathbf{x}\left(t_f\right), t_f\right)$$必要条件我们得到的结果是最优性的必要条件;即最小代价函数$J^(\mathbf{x}(t), t)$必须满足Hamilton-Jacobi-Bellman方程。

$$J^{\prime}(\mathbf{x}(t), t)=J^*(\mathbf{x}(t), t)$$
[K-5]和[A-2]给出了H-J-B方程所包含的充分必要条件的严格证明，并给出了几个例子。
Hamilton-Jacobi-Bellman方程的解

## 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。