Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion
Abstract
This paper establishes the first general structural solution to the problem of state estimation for nonlinear systems driven by unknown inputs. Building upon nonlinear unknown-input observability theory, we show that every such system admits a structurally equivalent representation, referred to as the UID-induced normal form. The proposed representation decomposes the information carried by the unknown inputs into two complementary components: unknown-input directions that are structurally decoupled from the observable dynamics and observable quantities that completely represent the unknown-input information affecting the observable dynamics. As a consequence, the UID-induced normal form provides a unified structural solution to unknown-input decoupling and unknown-input reconstruction, without requiring any model or stochastic assumption on the unknown inputs. The practical significance of the proposed framework is demonstrated through a previously unexplored minimal Structure-from-Motion configuration. The proposed representation enables recursive state estimation from only three point features and a single-axis gyroscope, allowing the recovery of the three-dimensional structure and camera motion up to an unknown global scale factor. Experiments on real-world data validate the proposed framework and demonstrate the feasibility of this minimal sensing configuration.
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BibTeX
@misc{martinelli2026structural,
title = {{Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion}},
author = {Agostino Martinelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05939},
eprint = {2610.05939},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05939},
}
RIS
TY - UNPB
TI - Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion
AU - Agostino Martinelli
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05939
UR - https://arxiv.org/abs/2610.05939
AB - This paper establishes the first general structural solution to the problem of state estimation for nonlinear systems driven by unknown inputs. Building upon nonlinear unknown-input observability theory, we show that every such system admits a structurally equivalent representation, referred to as the UID-induced normal form. The proposed representation decomposes the information carried by the unknown inputs into two complementary components: unknown-input directions that are structurally decoupled from the observable dynamics and observable quantities that completely represent the unknown-input information affecting the observable dynamics. As a consequence, the UID-induced normal form provides a unified structural solution to unknown-input decoupling and unknown-input reconstruction, without requiring any model or stochastic assumption on the unknown inputs. The practical significance of the proposed framework is demonstrated through a previously unexplored minimal Structure-from-Motion configuration. The proposed representation enables recursive state estimation from only three point features and a single-axis gyroscope, allowing the recovery of the three-dimensional structure and camera motion up to an unknown global scale factor. Experiments on real-world data validate the proposed framework and demonstrate the feasibility of this minimal sensing configuration.
ER -