A Practical Approach to Nonlinear Robust Control

Beard线性H_无穷控制是一种用于设计控制律的技术，该控制律对于系统参数的有界变化和有界的外部干扰是鲁棒的。近年来，线性H_无穷技术已扩展到非线性系统。利用两个一阶非线性偏微分方程组--Hamilton-Jacobi-Isaacs方程来求解非线性H_∞控制问题。这些方程除了在非常特殊的情况下很难求解，而且除了低阶系统外，很难用标准的方法来逼近。不能求解这些方程导致无法实现非线性H_无穷大控制技术的潜力。该提议概述了一项计划，以研究用于逼近闭环系统的非线性H_无穷最优控制律的创新和计算实用技术。方法是分两步逼近哈密顿-雅可比-艾萨克斯方程。首先，在策略空间中进行迭代，将非线性偏微分方程化成一个无穷序列的线性偏微分方程组。第二步用Galerkin谱方法逼近每个线性偏微分方程组。这个过程的结果是反馈控制律，具有明确的吸引域，随着逼近阶数的增加而收敛到非线性H-无穷大解。上面概述的两步过程受到Bellman的“维度诅咒”的影响，即所需的计算量和内存随着系统的阶数呈指数级增加。对于一类相当一般的非线性系统，通过利用问题中的结构，可以克服这种“维度诅咒”。所提出的研究将为逼近非线性H_无穷控制律提供一种实用的方法，从而提高最新技术水平。这一结果将有助于在该领域工作的研究人员，使他们能够在现实世界的问题上测试他们的想法，以及希望实施控制理论研究的最新发展的实践工程师。这项研究还通过说明如何利用一类非线性系统的最优控制结构来缓解“维度诅咒”，从而提高了知识的状态。***

9732917BeardLinear H-infinity control is a technique for designing control laws that are robust with respect to bounded variations in system parameters and bounded external disturbances. In recent years, linear H-infinity techniques have been extended to nonlinear system. The nonlinear H-infinity control problem is solved by two first order nonlinear partial differential equations called Hamilton-Jacobi-Isaacs equations. These equations are extremely difficult to solve except in very special cases and they are difficult to approximate by standard techniques except for low order systems. The inability to solve these equations has lead to a failure to realize the potential of nonlinear H-infinity control techniques.This proposal outlines a plan to investigate innovative and computationally practical techniques for approximating closed-loop nonlinear H-infinity optimal control laws. The approach is to approximate the Hamilton-Jacobi-Isaacs equation in a two step procedure. First, an iteration in policy space is used to reduce the nonlinear partial differential equation to an infinite sequence of linear partial differential equations. The second step uses the Galerkin spectral method to approximate each linear partial differential equation. The result of this process is a feedback control law, with a well defined region of attraction, that converges to the nonlinear H-infinity solution as the order of approximation increases. The two step process outlined above suffers from Bellman's "curse of dimensionality," i.e., the required computations and memory increase exponentially with the order of the system. This "curse of dimensionality" is overcome by exploiting the structure in the problem for a fairly general class of nonlinear systems.The proposed research will advance the state of the art by supplying a practical method for approximating nonlinear H-infinity control laws. This result will be helpful to researchers working in the area, allowing them to test their ideas on real world problems, and to practicing engineers who wish to implement the latest developments in control theory research. The proposed research also advances the state of knowledge by illustrating how to mitigate the "curse of dimensionality" by exploiting the structure of optimal control for a certain class of nonlinear systems. ***