Advances in Robust Control; and in High Resolution Spectral Estimation

本文提出的研究代表了我们目前在开发动力系统建模和控制的理论和计算工具方面的计划的延续。我们的计划是调查两个主要主题。1. 非线性系统的鲁棒控制。我们的方法依赖于图的概念。模型的不确定性是用测量潜在图扰动的间隙度量的适当泛化来量化的。给出了鲁棒裕度的显式鲁棒控制理论。此外，我们在最近的工作中提出了一种计算方法。建议的研究重点是如何(i)提高计算工具的有效性并寻求更有效的工具，（ii）将基本鲁棒性理论扩展到关于平衡或预定轨迹的基本稳定性范式之外。特别是，我们想要评估振荡系统的鲁棒性，以及存在不确定偏差的系统。高分辨率光谱估计。最近与Chris Byrnes和Anders Linquist的合作为这一方向提供了动力。我们介绍了滤波器组，以及随后的一般输入到状态（IS）滤波器，用于提供所选频谱间隔的谐波放大。我们正在完善一个通用理论，利用这种滤波器的选择性特性，并允许非常高分辨率的光谱估计。特别是，对于识别噪声中的正弦波，这是雷达和声纳应用中的一个关键问题，我们的方法已经比标准的最先进的MUSIC和ESPRIT算法具有更好的分辨率。

The research proposed herein represents a continuation of our current program in developing theoretical and computational tools for modeling and control of dynamical systems. Our plan is to investigate two main topics. 1. Robust control of nonlinear systems.Our approach relies on the concept of the graph. Model uncertainty is quantified using a suitable generalization of the gap metric which measures potential graph perturbations. A theory of robust control with explicit expressions for robustness margin is in place. Also, a computational approach has been presented in our recent work. The proposed research focuses on how to (i) enhance the effectiveness of the computational tools as well as seek more effective ones, (ii) extend the basic robustness theory outside the basic paradigm of stability about an equilibrium or a prespecified trajectory. In particular, we want to assess robustness of oscillatory systems, and systems in the presence of uncertain biases.2. High-resolution spectral estimation.The impetus for this direction was provided by recent joint work with Chris Byrnes and Anders Linquist. We have introduced filter banks, and subsequently general input-to-state (IS) filters, for providing harmonic amplification of selected spectral intervals. We are in the process of completing a general theory which takes advantage of the selectivity properties of such filters and allows very high resolution spectral estimation. In particular, for identifying sinusoids in noise, which is a key problem in radar and sonar applications, our approach has already permitted significantly better resolution than standard state-of-the-art MUSIC and ESPRIT algorithms.