论文标题
通过简单杆的近似 - 第一部分：硬度空间中的密度和几何收敛速率
Approximation by Simple Poles -- Part I: Density and Geometric Convergence Rate in Hardy Space
论文作者
论文摘要
最佳的线性反馈控制设计是一个有价值但具有挑战性的问题，这是由于稳定控制器的强大空间的基础优化和无限维度的非概念性。解决最佳控制问题的强大一类技术涉及使用重新聚体化将控制设计转换为凸的但无限的维度优化。为了解决问题，历史工作集中于盖尔金型有限维度近似与Hardy空间，尤其是涉及Lorentz系列近似值的尺寸，例如有限的脉冲响应AppProximation。然而，洛伦兹系列的近似可能会导致不可行，难以合并先验知识，在有限的脉冲响应的情况下进行无力控制以及次优的增加，尤其是对于时间尺度分离较大的系统。这本分为两部分的文章的目的是基于通过选择简单极点的传输函数引入一种新的Galerkin型方法，并将此简单的极近似应用于最佳控制设计。在第一部分中，基于杆选择的几何形状提供了近似于耐力空间中任意传输函数的误差范围。结果表明，使用这些简单的极点将传递函数的空间收敛到整个耐寒空间，并且纯粹基于杆选择的几何形状提供了均匀的收敛速率。然后，这是针对基于Archimedes螺旋的特别有趣的极点选择的收敛速率的。在第二部分中，简单的极点近似与系统级合成（一种最近的重新聚集方法）结合使用，以开发一种新的控制设计方法。该技术是凸面的，可以始终可行的，可以包括先验知识，不会导致无力控制，并且适用于tim较大分离的系统
Optimal linear feedback control design is a valuable but challenging problem due to nonconvexity of the underlying optimization and infinite dimensionality of the Hardy space of stabilizing controllers. A powerful class of techniques for solving optimal control problems involves using reparameterization to transform the control design to a convex but infinite dimensional optimization. To make the problem tractable, historical work focuses on Galerkin-type finite dimensional approximations to Hardy space, especially those involving Lorentz series approximations such as the finite impulse response appproximation. However, Lorentz series approximations can lead to infeasibility, difficulty incorporating prior knowledge, deadbeat control in the case of finite impulse response, and increased suboptimality, especially for systems with large separation of time scales. The goal of this two-part article is to introduce a new Galerkin-type method based on approximation by transfer functions with a selection of simple poles, and to apply this simple pole approximation for optimal control design. In Part I, error bounds for approximating arbitrary transfer functions in Hardy space are provided based on the geometry of the pole selection. It is shown that the space of transfer functions with these simple poles converges to the full Hardy space, and a uniform convergence rate is provided based purely on the geometry of the pole selection. This is then specialized to derive a convergence rate for a particularly interesting pole selection based on an Archimedes spiral. In Part II, the simple pole approximation is combined with system level synthesis, a recent reparameterization approach, to develop a new control design method. This technique is convex and tractable, always feasible, can include prior knowledge, does not result in deadbeat control, and works well for systems with large separation of tim
