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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Cite this article
APA 7
Martinelli, A. (2026). Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion. https://omanscience.com/en/articles/structural-foundations-of-nonlinear-systems-with-unknown-inputs-the-uid-induced-normal-form-and-minimal-sensing-structure-from-motion
MLA 9
Martinelli, Agostino. "Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion." https://omanscience.com/en/articles/structural-foundations-of-nonlinear-systems-with-unknown-inputs-the-uid-induced-normal-form-and-minimal-sensing-structure-from-motion.
Chicago (author–date)
Martinelli, Agostino. 2026. "Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion." https://omanscience.com/en/articles/structural-foundations-of-nonlinear-systems-with-unknown-inputs-the-uid-induced-normal-form-and-minimal-sensing-structure-from-motion.
Harvard
Martinelli, A. (2026) 'Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion', Available at: https://omanscience.com/en/articles/structural-foundations-of-nonlinear-systems-with-unknown-inputs-the-uid-induced-normal-form-and-minimal-sensing-structure-from-motion.
Vancouver
Martinelli A. Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion. https://omanscience.com/en/articles/structural-foundations-of-nonlinear-systems-with-unknown-inputs-the-uid-induced-normal-form-and-minimal-sensing-structure-from-motion
IEEE
A. Martinelli, "Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion," https://omanscience.com/en/articles/structural-foundations-of-nonlinear-systems-with-unknown-inputs-the-uid-induced-normal-form-and-minimal-sensing-structure-from-motion.