Nonlinear Control, HJB Equations, and the Max-Plus Algebra

非线性控制、HJB 方程和 Max-Plus 代数

• 批准号: 0307229
• 负责人: William McEneaney
• 金额: --
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0307229&HistoricalAwards=false
• 关键词: Nonlinear; Control; HJB; Equations; Max

The project focuses on the use of the max-plus algebra as a tool for the solution of nonlinear control and estimation problems. The main classes of problems addressed are those for which the associated dynamic programming equation takes the form of a nonlinear Hamilton-Jacobi-Bellman partial differential equation (HJB PDE). The semigroups associated with such problems are time-indexed operators which are max-plus linear. The max-plus linearity may be exploited to develop numerical methods for HJB PDEs, which might be described as (max-plus) spectral methods. These form a completely new class of numerical methods for HJB PDEs. The project will also consider approaches where one can construct complex operators in the semiconvex dual space from operators for simpler problems such as linear-quadratic problems. This allows one to avoid the curse-of-dimensionality in the most computationally intensive portion of the computations in max-plus based methods for problems whose operators may be approximated by such constructions in the dual-space.Control Theory is useful for any system where one desires to estimate the true state of the system and/or to control its future behavior. The methods of control theory apply to a tremendous variety of real-world systems such as aircraft dynamics, spacecraft dynamics, portfolio optimization, option pricing, and collections of robotic vehicles. Although the control of systems whose behavior is close to linear has been quite successful, there are many problems where the system behavior may be highly nonlinear, and the number of such problems is on the rise. The solution of nonlinear control problems is quite difficult, and not computationally tractable for systems whose states are described by more than just two or three scalar variables. The solution of such problems is most often obtained by the solution of an associated partial differential equation. The most common approach has been to adopt finite element methods (often used to solve problems such as fluid flow) in order to solve such partial differential equations, and consequently, the control problems. However, the computational requirements grow exponentially fast as a function of the number of scalar state variables.This is commonly referred to as the curse-of-dimensionality. Due to this exponential growth in computational cost, we cannot hope that faster computers will lead to solution of reasonably large problems in the foreseeable future. Therefore, we must explore alternative approaches to the solution of such problems. In this project, we apply a new class of methods to such nonlinear control problems. These methods exploit the fact that these nonlinear problems are linear over a different set of algebraic operations known as the max-plus algebra. By employing this max-plus linearity, one can obtain new numerical methods that appear to provide computational savings. Although one cannot hope to completely remove the curse-of-dimensionality, these methods should attenuate its effects.

该项目的重点是使用最大代数作为非线性控制和估计问题的解决方案的工具。主要解决的问题是那些相关的动态规划方程的形式的非线性哈密尔顿-雅可比-贝尔曼偏微分方程（HJB PDE）。与这类问题相关的半群是时间指标算子，它们是极大线性的。最大加线性度可用于开发HJB偏微分方程的数值方法，其可被描述为（最大加）谱方法。这些形成了一类全新的数值方法HJB偏微分方程。该项目还将考虑的方法，其中一个可以构建复杂的运营商在双凸对偶空间从运营商的简单问题，如线性二次问题。这使得人们可以避免在最大加为基础的方法的问题，其运营商可以近似这样的建设在dual-spaces.Control理论是有用的任何系统，其中一个愿望，以估计系统的真实状态和/或控制其未来的行为的计算最密集的部分的维数曲线。控制理论的方法适用于各种各样的现实世界的系统，如飞机动力学，航天器动力学，投资组合优化，期权定价和机器人车辆的集合。虽然行为接近线性的系统的控制已经相当成功，但是存在许多问题，其中系统行为可能是高度非线性的，并且这样的问题的数量正在增加。非线性控制问题的解决是相当困难的，并且对于其状态由多于两个或三个标量变量描述的系统来说，在计算上是不容易处理的。这类问题的解通常是通过求解相关的偏微分方程得到的。最常见的方法是采用有限元方法（通常用于解决流体流动等问题），以解决此类偏微分方程，从而解决控制问题。然而，计算需求随着标量状态变量的数量呈指数级快速增长，这通常被称为维数灾难。由于计算成本的指数增长，我们不能指望更快的计算机在可预见的未来能够解决相当大的问题。因此，我们必须探讨解决这些问题的其他办法。在这个项目中，我们应用一类新的方法，这样的非线性控制问题。这些方法利用了这样一个事实，即这些非线性问题在称为最大代数的不同代数运算集上是线性的。通过采用这种最大加线性，人们可以获得新的数值方法，似乎提供计算节省。虽然不能指望完全消除维度诅咒，但这些方法应该会减弱其影响。