最优控制问题求解方法综述(中英双语)

最优控制问题求解方法综述

Summary of approaches of optimal control problem

摘要：最优控制问题就是依据各种不同的研究对象以及人们预期达到的目的，寻找一个最优控制规律或设计出一个最优控制方案或最优控制系统。解决最优问题的主要方法有变分法、极小值原理和动态规划法，本文重点阐述了各种方法的特点、适应范围、可求解问题的种类和各种方法之间的互相联系。

Abstract：Optimal control problems are to find an optimal control law or design a optimal control program or system according to various kinds of different research objects and the aim people want. The approaches to solve optimal control problems generally contain variational method, the pontryagin minimum principle and dynamic programming. This paper mainly states characteristics, range of application, kinds of the solvable problems of each approach and the association between these three methods.

关键词：最优控制、变分法、极小值、动态规划

Keywords: optimal control , classical variational method , the pontryagin minimum principle , dynamic programming

正文：

最优控制理论是现代控制理论的一个主要分支，着重于研究使控制系统的性能指标实现最优化的基本条件和综合方法。最优控制理论是研究和解决从一切可能的控制方案中寻找最优解的一门学科。它所研究的问题可以概括为：对一个受控的动力学系统或运动过程，从一类允许的控制方案中找出一个最优的控制方案，使系统的运动在由某个初始状态转移到指定的目标状态的同时，其性能指标值为最优。这类问题广泛存在于技术领域或社会问题中。

Optimal control theory is a main branch of modern control theory, which focuses on studying basic conditions and synthetic approaches of optimizing systematic performance index. Optimal control theory is a subject studying and solving for the optimal solution from all possible control solutions. What it study can be summarized in this way: given a manipulated dynamic system or motor process, we are supposed to find a optimal control solution from allowable solutions of the same category, making the systematic movement transfer to the appointed state from a original state and getting a optimal performance index at the same time. And this kind of problems exist in technology field or social problems.

为了解决最优控制问题，必须建立描述受控运动过程的运动方程，给出控制变量的允许取值范围，指定运动过程的初始状态和目标状态，并且规定一个评价运动过程品质优劣的性能指标。通常，性能指标的好坏取决于所选择的控制函数和相应的运动状态。系统的运动状态受到运动方程的约束，而控制函数只能在允许的范围内选取。因此，从数学上看，确定最优控制问题可以表述为：在运动方程和允许控制范围的约束下，对以控制函数和运动状态为变量的性能指标函数（泛函）求取极值（极大值或极小值）。解决最优控制问题的主要方法有古典变

分法、极小值原理和动态规划。

For suppose of solving optimal control problems, we must build motion equations describing the manipulated motor process, give allowable value range of the control variables, designate the original state and target state of the motor process and stipulate a performance index to evaluate merits of the quality in the motor process. In general, the merits of a performance index depend on the control function and homologous motion state that we choose. Thus, the optimal control problems can be formulated from mathematical point of view as follows: solving for extremum (maximum or minimum) of the performance index function (functional) based on control function and the motion state under the constraint of motion equation and the allowable control range. The main approaches of solving optimal control problems includes classical variational method, the pontryagin minimum principle and dynamic programming.

一、变分法

First. Variational method

变分法是处理泛函的数学方法，和处理函数的普通微积分相对，譬如，这样的泛函可以通过未知函数的积分和它的导数来构造。变分法最终寻求的是极值函数：它们使得泛函取得极大或极小值。有些曲线上的经典问题采用这种形式表达：一个经典的例子是最速降线，在重力作用下一个粒子沿着该路径可以在最短时间从点A到达不直接在它地下的一点B。在所有从A到B的曲线中必须极小化的是下降时间的表达式。变分法的关键定理是欧拉——拉格朗日方程，它对应于泛函的临界点。在寻找函数的极大和极小值时，在一个解附近的微小变化的分析给出一阶的一个近似。它不能分辨是找到了最大值或最小值（或者都不是）。

Variational method is a mathematical method to conduct functional, just as the ordinary calculus dealing with functions. For instance, such functional can be constructed by unknown functional calculus and its differential. Variational method aims at findin extreme functions that make functional obtain maximum or minimum. Some classical problems on curved lines always adopt this kind of expression: a classical example is brachistochrone, along which a granule can get to B (not under A directly) from A in the minimum duration under the effect of gravity. In brachistochrone, what is supposed to be minimum is the expression of fall time among all these curved lines from A to B. The key theorem of variational method is Euler——Lagrangian equation, which is correspondent to the functional critical point. While we can’t distinguish the maximum or minimum (or neither) when we are finding the functional extremum through giving a first order approximation of a small change around a solution.

用变分法求解连续系统最优控制问题，实际上就是具有等式约束条件的泛函极值问题，只要把受控系统的数学模型看成是最优轨线)(t

x应满足的等式约束条件即可。变分法中的三类基本问题：拉格朗日（Lagrange）问题、梅耶（Mayer）问题、波尔扎（Bolza）问题。

Solving the optimal control problems of continuous systems with variational method is the functional extremum problem with conditions of equality constraint.We just need to regard the methematical models of manipulated systems as