Reduced Order Modeling of High Dimensional Nonlinear Control Systems

对于低维非线性系统的控制和估计，已经有了成熟的方法。对于高维或无限维线性系统的控制和估计，也有成熟的方法。但是，对于高维或无限维非线性系统的控制和估计的方法却发展得很少。该研究项目的目标是开发高维或无限维非线性控制系统的模型降阶技术，以便能够利用低维方法和直觉。该项目将基于对平衡实现和适当的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越来越复杂。不仅模型的规模在不断增大，而且其复杂的非线性行为也在不断增加。我们需要用更简单的描述来捕捉它们的基本行为的方法。一个例子是区域层面的气候建模。这种现象可用时间和空间三个变量的非线性偏微分方程组来模拟。此外，还有来自其他气候区域、海洋等的投入。此外，只能在离散的时间和地点测量气候。这样的模型被离散到非常高维的控制系统中，这会使我们目前的计算能力和解释结果数据的能力变得紧张。需要的是在保持模型反映现实的能力的同时降低模型的复杂性的方法。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.