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

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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.