Reduced Order Modeling of High Dimensional Nonlinear Control Systems

对于低维非线性系统的控制和估计，已有成熟的方法。 也有发达的方法来控制和估计高维或无限维线性系统。 但对于高维或无穷维非线性系统的控制和估计方法还很不成熟。 本研究计画的目标是发展高维或无穷维非线性控制系统的模型降阶技术，以便可以利用低维方法和直觉。 该项目将基于平衡实现和适当正交分解的非线性系统的新概括。世界正变得越来越复杂，用于描述，控制和测量它的数学模型也变得越来越复杂。不仅模型的规模在增长，而且它们的复杂非线性行为也在增长。 我们需要用更简单的描述来捕捉它们的基本行为。 一个例子是区域一级的气候建模。 这种现象的时间和三个空间变量的非线性偏微分方程模型。 此外，还有来自其他气候区域、海洋等的输入。此外，只能在离散的时间和地点测量气候。 这样的模型被离散化成非常高维的控制系统，这会使我们目前的计算能力和解释结果数据的能力紧张。 我们需要的是降低模型复杂性的方法，同时保持它们反映现实的能力。 这个项目的目标就是开发这样的方法。

There are well-developed methods for the control and estimation of low dimensional nonlinear systems. There are also well-developed methods for the control and estimation of high or infinite dimensional linear systems. But methods for the control and estimation of high or infinite dimensional nonlinear systems are much less developed. The goal of this research project is to develop model reduction techniques for high or infinite dimensional nonlinear control systems so that low dimensional methodologies and intuition can be utilized. The project will be based on new generalizations to nonlinear systems of balanced realizations and proper orthogonal decompositions.The world is growing more complex as are the mathematical models that are used to describe, control and measure it. Not only are the sizes of the models growing but also their complex nonlinear behaviors. We need methods to capture their essential behaviors in simpler descriptions. An example is modeling of climate on a regional level. Such phenomena are modeled by nonlinear partial differential equations in time and three space variables. Moreover there are inputs from other climate regions, oceans, etc. Moreover it is only possible to measure the climate at discrete times and locations. Such models are discretized into very high dimensional control systems that strain our current computing capacity and our abilities to interpret the resulting data. What is needed are methods for reducing the complexity of models while maintaining their ability to mirror reality. The goal of this project is to develop such methods.