Feedback Synthesis for Infinite Dimensional Systems Dominated by Nonlinearities

这项建议概述了在一个新兴的非线性分布参数系统控制领域的综合方案。激励性的实际问题来自不稳定的流体流动的稳定，不稳定的流体流动在当今的先进消费品和汽车、航空航天和海洋车辆中普遍存在。这些应用充斥着超线性增长的非线性和空间域内部的不稳定性，其中控制只能从边界起作用。这些问题已经成功地在有限维环境中处理，并且对于分布参数系统而言具有独特的地位。首先，基于反推的方法，它要么利用空间域内部的状态测量，要么使用非线性观测器，用于具有边界控制的高度不稳定系统的镇定。控制作用通常通过许多分布参数系统中存在的粘性/扩散机制从边界传播到区域内部。第二种方法基于仅涉及边界测量的简单反馈。这种反馈的形式是Jurdjevic-Quinn反馈律，它为具有已知控制Lyapunov函数的系统的渐近镇定增加了阻尼。给出了一个流体流动问题的算例，其中J-Q型控制器适用于低雷诺数，并证明了该控制器在较高的实际雷诺数，比设计值高出许多数量级时的有效性。在同样的背景下，这一建议提出超越均衡稳定化，实施非平衡准稳态运动。这样的控制目标的一个例子是在流体中混合，其中人们希望通过确定性的手段实现颗粒的随机运动。这项提议计划极大地改变解决这个问题的传统方法，这些方法都是开环/运动规划类型的。在不确定的非线性系统的稳定化(产生了几个最佳论文奖和一本关于这一主题的新书)以及过渡(我们的控制算法的工业实验和一项专利授权给一家领先的航空发动机制造商)方面取得了几项理论上的进展。

This proposal outlines a comprehensive program in an emerging area of control of nonlinear distributed parameter systems. The motivating practical problems come from stabilization of unstable fluid flows, ubiquitous in today's advanced consumer products and automotive, aerospace, and marine vehicles. These applications abound with nonlinearities of superlinear growth and instabilities in the interior of a spatial domain where controls can act only from the boundary. These issues have successfully been dealt with in a finite-dimensional setting, and are uniquely positioned to address them for distributed parameter systems.This proposal advocates the development of two types of methods. First, methods based on backstepping, which employ either the measurement of the state in the interior of the spatial domain, or a nonlinear observer, for stabilization of highly unstable systems with boundary controls. The control action is typically propageted from the boundary into the interior of the domain by viscosity/diffusion mechanisms present in many disstributed parameter systems.The second method is based on simple feedbacks involving only boundary measurements. Such feedbacks are in the form of Jurdjevic-Quinn feedback laws that add damping for asymptotic stabilization of systems with known control Lyapunov functions. An example is shown of a fluid flow problem in which a J-Q type controller is developed for low Reynolds number and demonstrate its effectiveness for a high, realistic value of Reynolds number, many orders of magnitude higher than the design value. In the same setting, this proposal proposes to go beyond equilibrium stablization, to enforcement of non-equilibrium quasi-steady motion. An example of such a control objective is mixing in fluids where one wants to achieve random motion of particles by deterministic means. This proposal plans to dramatically depart from conventional approaches to this problem which are all of open-loop/motion-planning type. A feedback law will be shown that raises both the turbulent kinetic energy and the vorticity of the fluid (uniformly in space).Several theoretical advances have been produced in stabilization of uncertain nonlinear systems (resulting in several best paper awards and a new book on this topic), as well as transition (industrial experiments with our control algorithms and a patent offered for licensing consideration to a leading aeroengine manufacturer.