Stability Analysis of Large-Scale Nonlinear Systems using Parallel Computation

随着工程系统复杂性的增长，这些系统的安全和可靠操作的难度变得更具挑战性。例如，考虑在法国卡达拉什建造的价值150亿美元的国际核聚变反应堆。尽管60年前人们就知道，通过在磁场中加热等离子体可以从核聚变中产生能量，但物理学家从未能够足够精确地控制磁场来产生大量的能量。原因是，即使是最简单的磁流体动力学模型涉及20多个耦合的非线性微分方程。虽然近年来控制算法取得了长足的进步，但对这种复杂系统的控制仍然遥不可及。该项目将设计新的控制算法，使用超级计算机和大规模并行计算，试图为大型复杂系统（如描述反应堆中的等离子体）设计安全可靠的控制器。该项目的核心是一种新的方法，使用凸优化来参数化李雅普诺夫函数（一种能量度量）。具体而言，虽然众所周知的正多项式的平方和参数化对于小规模系统是凸的、可靠的和准确的，但它不能容易地适应于超级计算机和其他形式的大规模并行计算。这个项目的本质，然后是寻找替代的数学参数化的李雅普诺夫函数是凸的，而且是适合并行计算。这样的替代存在于Handelman、Polya和伯恩斯坦的经典数学结果中。工作范围是使用这些结果来创建并行代码，可以研究多个耦合的非线性方程，并确定多项式数学语言中可能的最佳李雅普诺夫函数。该项目将在集群和并行图形处理器计算机上测试这些算法，并将能够研究多达20个状态的非线性微分方程。然后，这些算法将被应用到离散化的非线性偏微分方程表示的磁流体动力学的等离子体在核聚变反应堆，以获得一个功能的能量，然后可以用来设计和测试磁和射频控制器，这将减少或消除磁流体动力学的不稳定性。开发的算法也可以应用于任何大型非线性系统，这意味着它们可以用于改善化学反应器，基因调控网络和通信卫星等应用中的理解和控制。

As engineered systems grow in complexity, the difficulty of safe and reliable operation of these systems becomes more challenging. For example, consider the $15 billion international nuclear fusion reactor being built in Cadarache, France. Although the world has known for 60 years that it is possible to produce energy from nuclear fusion by heating plasma in a magnetic field, physicists have never been able to control the magnetic field accurately enough to produce significant amounts of power. The reason is that even the simplest models of magneto hydrodynamics involve more than 20 coupled nonlinear differential equations. Although algorithms for control have made great strides in recent years, control of systems of this complexity is still out of reach. This project will design new algorithms for control which use supercomputers and massively parallel computation in an attempt to enable the safe and reliable design of controllers for large complex systems such as describe plasma in a reactor.At the heart of the project is a new way of using convex optimization to parameterize Lyapunov functions (a measure of energy). Specifically, while the well-known sum-of-squares parameterization of positive polynomials is convex, reliable and accurate for small-scale systems, it cannot be readily adapted to supercomputers and other forms of massively parallel computation. The essence of this project, then is to look for alternative mathematical parameterizations of Lyapunov functions which are convex and furthermore are amenable to parallel computation. Such alternatives exist in classical mathematical results by Handelman, Polya and Bernstein. The scope of work is to use those results to create parallel codes, which can study multiple coupled nonlinear equations and determine the best possible Lyapunov function fit within the mathematical Language of polynomials. The project will test these algorithms on cluster and parallel graphics processor computing machines and will be able to study nonlinear differential equations with up to 20 states. These algorithms will then be applied to discretized nonlinear partial differential equation representations of the magneto-hydrodynamics of plasma in a nuclear fusion reactor to obtain a function of energy which can then be used to design and test magnetic and radio frequency controllers which will reduce or eliminate magneto hydrodynamic instabilities. The algorithms developed can also be applied to any large nonlinear system, implying they can be used to improve understanding and control in applications such as chemical reactors, gene regulatory networks and communication satellites.