Numerical Nonlinear and Optimal Control Using Wavelets

派的名字：Panagiotis Tsiotras机构：佐治亚理工学院航空航天工程学院提案编号：0084954提案标题：数字非线性与小波最优控制摘要：在过去的半个世纪里，非线性系统的最优反馈控制律的发展一直是密集研究的主题。该理论的基石是所谓的Hamilton-Jacobi-Bellman偏微分方程，它的解为最优反馈控制提供了一个解析表达式。不幸的是，求哈密顿-雅可比-贝尔曼方程的解析解是一项艰巨的任务。只有在一些非常低维的特殊情况下才能得到解。即使是求解哈密尔顿-雅各比-贝尔曼方程的传统数值方法，也常常由于“维度诅咒”而不能充分发挥作用。也就是说，所涉及的计算量随着系统状态的变化而大大增加。在这个项目中，我们建议使用小波来数值求解Hamilton-Jacobi-Bellman方程。小波是使用多分辨率概念的正交基函数。也就是说，它们能够捕获信号在频率和时间上的局部行为。通过平移和膨胀，它们将解空间分解为越来越细微的子空间，最终的解被计算为这些细子空间中的解的聚集。在最终的傅立叶/小波级数展开中，只有对解有重要贡献的子空间才具有非零系数。因此，给定解的精度水平，只需要很少的非零傅立叶系数来捕捉精确解。因此，与其他方法相比，可以更有效地获得解。这项研究将对嵌入式控制系统领域产生立竿见影的影响。运行开发的算法的专用计算机芯片将能够在线计算最优反馈控制律，从而在汽车控制、飞机导航、自主移动机器人控制等应用中获得前所未有的智能、自主性和多功能性。此外，小波展开的多分辨率特性意味着解在不同分辨率水平上的解耦。因此，基于小波的解决方案特别适合于计算机实现的并行处理。

PI's Name: Panagiotis TsiotrasInstitution: School of Aerospace Engineering, Georgia Institute of TechnologyProposal Number: 0084954Proposal Title: Numerical Nonlinear and Optimal Control Using WaveletsAbstract: The development of optimal feedback control laws for nonlinear systems has been the topic of intense research over the past half a century. The cornerstone of this theory is the so-called Hamilton-Jacobi-Bellman partial differential equation, the solution of which provides an analytic expression for the optimal feedback control. Unfortunately, the analytic solution of the Hamilton-Jacobi-Bellman equation is a formidable task. Solutions can be obtained only for some special cases of very low dimension. Even traditional numerical methods for solving the Hamilton-Jacobi-Bellman equation are often inadequate due to the ``curse of dimensionality''. That is, the number of computations involved increases tremendously with the state of the system. In this project, we propose to use wavelets for solving the Hamilton-Jacobi-Bellman equation numerically. Wavelets are orthogonal basis functions that use the concept of multiresolution. Namely, they are able to capture the local behaviour of signals both in frequency and time. Using translations and dilations, they decompose the solution space into finer and finer subspaces and the final solution is computed as an aggregation of the solutions in these finer subspaces. Only the subspaces which significantly contribute to the solution have non-zero coefficients in the final Fourier/Wavelet series expansion. Therefore, given a level of accuracy for the solution, only few non-zero Fourier coefficients are needed to capture the exact solution. As a result, solutions can be obtained much more efficiently than with other methods. This research will have an immediate impact on the area of embedded control systems. Dedicated computer chips running the developed algorithms will be able to calculate optimal feedback control laws on-line, thus achieving unprecedented levels of intelligence, autonomy and versatility for several applications such as, automobile control, aircraft navigation, autonomous mobile robot control, etc. In addition, the multiresolution property of wavelet expansions implies a decoupling of the solutions in the different resolution levels. Therefore, wavelet-based solutions are uniquely suited to parallel processing for computer implementation.