A Practical Approach to Nonlinear Robust Control

9732917 BeardLinear H ∞控制是一种设计控制律的技术，该控制律对于系统参数的有界变化和有界外部干扰是鲁棒的。 近年来，线性H ∞技术已扩展到非线性系统。 非线性H ∞控制问题由两个一阶非线性偏微分方程（Hamilton-Jacobi-Isaacs方程）求解。 这些方程是非常困难的解决，除非在非常特殊的情况下，他们很难近似的标准技术，除了低阶系统。 无法解决这些方程导致了未能实现的潜力，非线性H-∞控制techniques.This proposal概述了一个计划，调查创新和计算实用的技术，近似闭环非线性H-∞最优控制律。 该方法是近似的Hamilton-Jacobi-Isaacs方程在两个步骤的过程。 首先，在策略空间中的迭代被用来减少非线性偏微分方程的线性偏微分方程的无穷序列。 第二步用Galerkin谱方法逼近每个线性偏微分方程。 这个过程的结果是一个反馈控制律，具有一个定义良好的吸引区域，收敛到非线性H-无穷解的近似阶数增加。 上面概述的两步过程遭受贝尔曼的“维数灾难”，即，所需的计算和存储器随着系统的阶数呈指数增长。 这种“维数灾难”是克服利用的结构问题的一个相当普遍的类nonlinear systems.The拟议的研究将推进国家的艺术提供了一个实用的方法来逼近非线性H ∞控制律。 这一结果将有助于研究人员在该领域的工作，使他们能够测试他们的想法对真实的世界的问题，并实践工程师谁希望实现控制理论研究的最新发展。 所提出的研究还通过说明如何利用某类非线性系统的最优控制结构来减轻“维数灾难”来推进知识状态。 ***

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. ***