PYI: Nonlinear Control and Order Reduction

该PYI奖将资助开发非线性系统降阶方法的研究，该方法(i)尊重并保留原始模型的非线性，（ii）不需要精确了解各种系统变量的相对速度，以及（iii）结合外生控制输入的影响。研究结果将对具有快速和慢速模式的系统的非线性控制器设计具有指导意义。发展非线性模型降阶技术有两个主要原因。第一个是所有大型动力工程系统所共有的：大量的控制方程。第二个原因是非线性效应不能被忽视，因为非线性效应对许多先进工程系统的性能有决定性的影响，如高性能飞机、快速机动的航天器和化学反应器。在首先对动力学方程进行线性化，然后使用线性模型降阶技术的标准技术中，只考虑了第一个问题。显然，使用这种方法，任何非线性效应（如分岔，甚至非线性瞬态响应）都将被忽略。本研究的主要分析框架是多参数奇异摄动理论和中心流形理论。这两个强大的工具将被合并和扩展。使用泛函分析的结果，如所谓的硬隐函数定理。该PYI奖支持的研究涉及非线性系统模型降阶方法的发展。该方法将尝试保持原始系统的非线性性质，因为非线性可能是许多先进工程系统的决定性影响。

This PYI award will fund research into the development of a methodology for order reduction of nonlinear systems which (i) respects and preserves the nonlinearity of the original model, (ii) does not require exact knowledge of the relative speeds of various system variables, and (iii) incorporates the effect of an exogenous control input. The results will have implications for nonlinear controller design of systems with fast and slow modes. There are two major reasons for developing a nonlinear model order reduction technique. The first is common to all large dynamical engineering systems: the large number of governing equations. The second reason concerns the fact that nonlinear effects can not be neglected since they will often be the deciding influence in the performance of many advanced engineering systems, such as high-performance aircraft, rapidly maneuvering spacecraft, and chemical reactors. Only the first of these issues is considered in the standard technique of first linearizing the dynamical equations and then using a linear model order reduction technique. Clearly, any nonlinear effect (such as bifurcations, or even nonlinear transient response) will be lost using this approach. The main analytical framework for this study consists of multiparameter singular perturbation theory and center manifold theory. These two powerful tools will be merged and extended. Use of functional-analytic results such as the so-called hard implicit function theorem will be made. The research supported under this PYI award concerns the development of a methodology for model order reduction of nonlinear systems. The approach will attempt to preserve the nonlinear nature of the original system since the nonlinearities may be the deciding influence in many advanced engineering systems.