New Methods in Modeling and Control of Dynamical Systems

这是对动态系统建模和控制的某些方面的研究。特别是，将开发一种基于图的概念的线性动态系统建模和控制的综合方法。动力系统图的概念和间隙度量的概念是系统理论中的重要新概念。间隙度量是适用于鲁棒镇定和控制的系统之间的距离的概念。利用间隙度量可以实现对线性动力系统模型的逼近。这是因为需要使用“低阶”模型来合成控制器。建模误差的描述在给定系统的图中以不确定性的形式自然地出现--图是系统的所有有界输入-输出对的集合。研究的第一部分将是对这种建模理论的研究。在研究的第二部分，将开发一个使用给定信号的自相关函数估计的建模框架。特别是，未知线性系统的含噪声输入输出数据的测量可以归入该框架。提出了一种基于自相关(协方差)数据的递归谱估计方法。间隙度量将被用来获得关于谱估计和对所寻找模型的图形中的不确定性的估计的先验界限。研究的第三部分将继续发展关于谱因式分解的新方法的最新结果。这是基于给定线性系统的图形上的一系列变换。该技术基于迭代方案，具有产生快速高效的谱分解和递归滤波算法的潜力。

This is research on certain aspects of modeling and control of dynamical systems. In particular, an integrated approach to modeling and control of linear dynamical systems will be developed based on the notion of a graph. The notion of a graph of a dynamical system and the notion of the gap metric are important new ideas in systems theory. The gap metric is a notion of distance between systems suitable for use in the context of robust stabilization and control. Approximation of models of linear dynamical systems can be done with use of the gap metric. This is motivated by the need to synthesize controllers using "low" order models. The description of the modeling error appears naturally in the form of uncertainty in the graph of a given system -- graph being the set of all bounded input-output pairs of the system. The first part of the research will be on such modeling theory. In the second part of the research, a framework of modeling which uses estimates of the autocorrelation function of a given signal will be developed. In particular, measuring noisy input-output data of an unknown linear system can be cast into this framework. An approach to recursive spectral estimation based on autocorrelation (covariance) data is proposed. The gap metric will be used to obtain a-priori bounds on spectral estimates and on estimates of the uncertainty in the graph of the sought model. The third part of the research will be to continue the development of recent results on a new approach to spectral factorization. This is based on a sequence of transformations on the graph of a given linear system. The technique is based on an iterative scheme and holds the potential of leading to fast and efficient algorithms for spectral factorization and recursive filtering.