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

在过去的几年里，我们已经看到了巨大的进步的分析，验证，稳定性和可控性的系统表现出相互作用的连续和离散的行为。然而，这些进展都是建立在假设准确的定量模型是现成的基础上。相对较少的注意力已经支付的模型参数的识别和估计的连续和离散状态的混合系统的输入输出数据。这个项目的智力价值是从一个新的代数几何的角度研究混合系统识别的基本理论和计算问题。混合系统的位形空间表示为一个代数簇，其组成系统表示为这个代数簇的不可约分量。因此，混合系统的识别成为等价的估计和分解，这一品种。该配方导致新的批处理算法的切换ARX模型，跳跃线性系统，分段仿射混合动力系统，和类的非线性混合动力系统，连同代数条件下，所提供的解决方案是唯一的识别。拟议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，例如人类步态识别，动态纹理分割，图像/视频分割，压缩和分类，多个机器人的同步映射和定位，以及生物网络建模。因此，预计该项目将产生广泛的影响，不仅在控制和系统识别理论，而且在各种其他学科，如代数几何，机器学习，计算机视觉和生物医学工程。

The 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.