CRS--EHS: Collaborartive Research: An Algebraic Geometric Approach to Hybrid Systems Identification

CRS--EHS：协作研究：混合系统识别的代数几何方法

• 批准号: 0509151
• 负责人: Yi Ma
• 金额: $ 15万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2009-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0509151&HistoricalAwards=false
• 关键词: CRS; EHS; Collaborartive; Research; Algebraic

Title: An Algebraic Geometric Approach to Hybrid System IdentificationThe past few years have seen tremendous advances on the analysis, verification, stability, and controllability of systems exhibiting interacting continuous and discrete behavior. However, these advances are strongly based on the assumption that accurate quantitative models are readily available. Relatively less attention has been paid to identification of the model parameters and estimation of the continuous and discrete states of such hybrid systems from input-output data. The intellectual merit of this project is to study fundamental theoretical and computational issues in hybrid system identification from a novel algebraic geometric perspective. The configuration space of a hybrid system is represented as an algebraic variety and its constituent systems as the irreducible components of this variety. Hence, hybrid system identification becomes equivalent to the estimation and decomposition of this variety. This formulation leads to novel batch algorithms for the identification of switched ARX models, jump linear systems, piecewise affine hybrid systems, and classes of nonlinear hybrid systems, together with algebraic conditions under which the provided solutions are unique. The broader impact of the proposed research includes various inference problems across many scientific and engineering fields, such as recognition of human gaits, segmentation of dynamic textures, image/video segmentation, compression, and classification, simultaneous mapping and localization for multiple robots, and modeling of biological networks. Therefore, it is expected that this project will have a broad impact not only in control and systems identification theory, but also in various other disciplines such as algebraic geometry, machine learning, computer vision, and biomedical engineering.

标题：混合系统辨识的代数几何方法在过去的几年里，在分析、验证、稳定性和可控性方面取得了巨大的进展，这些系统表现出相互作用的连续和离散行为。然而，这些进展在很大程度上是基于这样一个假设，即准确的定量模型是现成的。对模型参数的辨识以及从输入输出数据估计这类混合系统的连续状态和离散状态的研究相对较少。该项目的智力优势是从一个新的代数几何角度研究混杂系统辨识中的基本理论和计算问题。混合系统的构型空间被表示为一个代数簇，其组成系统被表示为该簇的不可约分支。因此，混杂系统辨识等价于这种变化的估计和分解。这一形式导致了新的批量算法，用于辨识切换ARX模型、跳跃线性系统、分段仿射混杂系统和一类非线性混杂系统，以及所提供的解唯一的代数条件。这项研究的更广泛影响包括许多科学和工程领域的各种推理问题，如人体步态识别、动态纹理分割、图像/视频分割、压缩和分类、多机器人的同时映射和定位以及生物网络的建模。因此，预计该项目不仅将在控制和系统辨识理论方面产生广泛影响，而且还将在代数几何、机器学习、计算机视觉和生物医学工程等其他学科中产生广泛影响。