Solving Large Sum-of-Squares Optimization Problems in Control by Exploiting the Parallel Structure of Polya's Algorithm

这个项目的目标是开发新的并行算法，用于非线性和不确定系统的控制。计算机体系结构正在发生变化，多核芯片和显卡取代了过去几年基于CPU的台式机。随着这种变化，大量的控制系统技术正在变得过时。问题是，控制工程师正在使用的优化算法并不是为我们今天遇到的并行处理环境构建的。这将越来越限制我们控制用于描述聚变能源和生物免疫等现象的大型复杂模型的能力。对于这个项目，我们已经确定了一种基于Polya引理的基于最优化的非线性或不确定动力学控制方法。这种方法的独特之处在于，当优化算法应用于Polya引理时，几乎是完全并行的。这意味着我们开发的算法可以在几乎任何类型的并行计算体系结构上运行，包括集群计算机和超级计算机。考虑到目前这些平台上可用的计算能力是单核台式机上可用计算能力的200多万倍，这个项目的结果将是我们可以控制的系统的复杂性增加了一个数量级。项目组织由三部分组成。I)利用Polya引理发展并行算法，将单纯形上的鲁棒控制问题转化为半定规划问题；ii)发展并行原-对偶内点算法，解决集群计算和多核体系结构上的鲁棒控制问题。Iii)将范围从稳健控制扩展到非线性分析和更一般类型的系统不确定性，并包括其他计算平台，如GPU计算和超级计算。在这个项目中开发的算法将发布在网上，使用公共许可证免费分发。可控系统复杂性的数量级增加具有重要意义。例如，尽管聚变反应堆中的等离子体的详细模型是可用的，但这些模型太复杂了，无法使用现有算法进行有效控制。其结果是糟糕的等离子体限制和低效的能源生产。如果能够使用更详细的模型来改善等离子体限制，这将对能源生产产生深远的影响。此外，免疫系统中细胞间相互作用的生物模型包含许多不同的因素，并且是非线性的和高度不确定的。分析和控制这些模型的能力的提高可能会导致癌症等疾病的创新治疗形式，这些疾病被认为是由于免疫系统未能自我调节而引起的。国际和平研究所在融合研究和免疫学领域都有正在进行的项目，这项研究将被纳入这些项目。最后，该项目还有一个与巴西坎皮纳斯大学合作的国际部分，包括在坎皮纳斯和芝加哥扩大教学交流。这将加强这两所学校之间的合作关系，并为两所学校的学生提供国际视野和教育机会。

The goal of this project is to develop new parallel algorithms for control of nonlinear and uncertain systems. Computer architectures are changing, with multi-core chips and graphics cards replacing the CPU-based desktop powerhouses of years past. And with this change, a great deal of control systems technology is becoming obsolete. The problem is that the optimization algorithms being used by controls engineers are not built for the parallel processing environments we are encountering today. Increasingly, this will limit our ability to control the large and complex models we use to describe such phenomena as fusion energy and biological immunity. For this project, we have identified an approach to control of nonlinear or uncertain dynamics based on optimization using Polya's lemma. The unique feature of this approach is that the optimization algorithms when applied to Polya's lemma become almost perfectly parallel. This means that the algorithms we develop can run on almost any type of parallel computing architecture, including cluster computers and supercomputers. Considering that the computing power available on these platforms is currently more than 2,000,000 times great than that available on a single-core desktop, the result of this project will be an order of magnitude increase in the complexity of systems we can control. The project organization has three parts. i) develop parallel algorithms to formulate robust control problems on the simplex as semidefinite-programming problems via Polya's lemma ii) develop parallel primal-dual interior-point algorithms for the problem of robust control on cluster computing and multi-core architectures. Iii) expand the scope beyond robust control to nonlinear analysis and more general types of system uncertainty and include additional computing platforms such as GPU computing and supercomputing. The algorithms developed in this project will be posted online for free distribution using a public license.An order-of-magnitude increase in the complexity of systems that can be controled has important implications. For example, although detailed models of plasma in fusion reactors are available, those models are too complex to control efficiently using existing algorithms. The result is poor plasma confinement and inefficient energy production. If more detailed models can be used to improve plasma confinement, this has far-reaching implications for energy production. Additionally, biological models of interaction between cells in the immune system contain many different actors and are nonlinear and highly uncertain. An improved ability to analyze and control these models may lead to innovative forms of treatment for diseases such as cancer which is believed to be caused by a failure of the immune system to self-regulate. The PI has ongoing projects in both the areas of fusion research and immunology and this research will be integrated into these projects. Finally, this project has an international component with the University of Campinas in Brasil, including extended teaching exchanges in both Campinas and Chicago. This will strengthen the collaborative relationship between these two institutions and provide an international perspective and educational opportunity for students at both schools.