Numerical Nonlinear and Optimal Control Using Wavelets

PI姓名：Panagiotis Tsiotras机构：格鲁吉亚理工学院航空航天工程学院提案编号：0084954提案标题：使用小波的数值非线性和最优控制摘要：非线性系统最优反馈控制律的开发一直是过去半个世纪来深入研究的主题。该理论的基石是所谓的Hamilton-Jacobi-Bellman偏微分方程，其解为最优反馈控制提供了解析表达式。不幸的是，Hamilton-Jacobi-Bellman方程的解析解是一项艰巨的任务。只有在维数很低的特殊情况下才能得到解。即使是传统的数值方法求解汉密尔顿-雅可比-贝尔曼方程往往是不够的，由于“灾难的维数”。也就是说，所涉及的计算数量随着系统的状态而极大地增加。在这个项目中，我们建议使用小波数值求解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。使用平移和伸缩，他们分解的解决方案空间到更精细的子空间和最终的解决方案计算为这些更精细的子空间的解决方案的聚合。 只有对解有显著贡献的子空间在最终的傅立叶/小波级数展开中具有非零系数。因此，给定解的精度水平，仅需要很少的非零傅立叶系数来捕获精确解。 因此，可以比其他方法更有效地获得解决方案。这项研究将对嵌入式控制系统领域产生直接影响。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，从而实现前所未有的智能水平，自主性和多功能性的几个应用，如汽车控制，飞机导航，自主移动的机器人控制等。此外，小波展开的多分辨率属性意味着在不同的分辨率水平的解决方案的解耦。因此，基于小波的解决方案是唯一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.