War on Boundary Conditions - A Control-Oriented Framework for Partial Differential Equations

边界条件之战 - 偏微分方程的面向控制的框架

基本信息

批准号：
1935453
负责人：
Matthew Peet
金额：
$ 26万
依托单位：
Arizona State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1935453&HistoricalAwards=false
关键词：
War Boundary Conditions Control Oriented

项目摘要

The joke goes like this: "Fusion power is just 20 years away and always will be". Why 20 years? Because that is how long it takes to build a tokamak fusion reactor. Why always? Because for a tokamak to produce energy, it must control a 10M degree plasma using a magnetic field, a dynamic process governed by six coupled nonlinear Partial-Differential Equations (PDEs) in two spatial variables and no real progress has ever been made on feedback control of such systems. The goal of the project, then, is to make substantial progress on the control of coupled PDEs. Such progress is through foundational advances in mathematics and software infrastructure which will enable the broader PDE controls community to grow. Specifically, by developing a universal, easily-understood and applied computational framework for the control of PDEs, this research will provide the tools needed to allow controls engineers with limited or no PDE experience to design reliable and effective controllers for these systems. Since nuclear fusion reactors have no long-term radioactive waste, and since fuel for these reactors can be extracted from sea water, progress in this area has the potential for creating an unlimited supply of clean energy. Furthermore, this work has the potential to advance several applications beyond nuclear fusion including hypersonic vehicles, traffic management, soft robotics, cavitation, flow control, and vibration suppression.The goal of the project is to replace Partial Differential Equations (PDEs) with Partial Integral Equations (PIEs). Historically, PDEs are defined by three sometimes contradictory sets of equations and constraints: the PDE itself, which moves the state; the Boundary Conditions (BCs), which implicitly constrain the motion of the state; and the continuity conditions, which couple the BCs to constraints on the motion of the state. By contrast, PIEs combine PDE, BCs, and continuity conditions into a single equation, defined by bounded Partial-Integral (PI) operators, any solution to which satisfies the original PDE and requires no boundary conditions or continuity constraints. PI operators are bounded, form an algebra, and can be parameterized using matrices. Further, positivity of PI operators can be enforced using Linear Matrix Inequalities (LMIs). This means that algorithms developed for control of ODEs using LMIs can be generalized to control of PIEs with relatively little effort. This project will pursue the generalization of several such LMIs to PDEs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

笑话是这样的：“融合力量只有20年，而且永远都是”。为什么要20年？因为那是构建Tokamak Fusion反应堆需要多长时间。为什么总是？因为要为Tokamak产生能量，它必须使用磁场控制10m度的血浆，这是由两个空间变量中的六个耦合的非线性部分差异方程（PDE）控制的动态过程，并且在此类系统的反馈控制方面从未取得任何实际进度。因此，该项目的目的是在控制耦合PDE的控制方面取得重大进展。这种进步是通过数学和软件基础架构的基本进步，这将使更广泛的PDE控制社区发展。具体而言，通过为控制PDE的控制开发通用，易于理解和应用的计算框架，本研究将提供所需的工具，以允许控制具有有限或没有PDE经验的工程师，以设计这些系统的可靠和有效的控制器。由于核融合反应堆没有长期的放射性废物，并且可以从海水中提取这些反应器的燃料，因此该地区的进度有可能产生无限的清洁能量。此外，这项工作有可能将多个应用超越核融合，包括高超音速器，交通管理，软机器人技术，气态，流动控制和振动抑制。该项目的目的是用部分积分方程（PIES）替换部分微分方程（PDE）。从历史上看，PDE是由三个有时矛盾的方程式和约束组来定义的：PDE本身，它可以移动状态；边界条件（BCS）隐含地限制了状态的运动；连续性条件将BCS与国家运动的限制结合在一起。相比之下，派对将PDE，BCS和连续性条件组合到一个单一方程式中，该方程是由有限的部分综合（PI）操作员定义的，这些解决方案的任何解决方案都满足原始PDE并且不需要边界条件或连续性约束。 PI运算符是界限的，形成一个代数，可以使用矩阵进行参数化。此外，可以使用线性基质不等式（LMI）来实现PI操作员的阳性。这意味着开发用于使用LMI控制ODE的算法可以推广以相对较少的精力控制派。该项目将追求对PDE的几个这样的LMI的概括。该奖项反映了NSF的法定任务，并认为使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估。