Algebraic Criterion for Nonlinear Unknown Input Observability in Systems Without Control Inputs

This paper investigates nonlinear systems driven solely by unknown inputs. A straightforward definition of observability is introduced for such systems, and based on this, the paper derives the algebraic criterion that characterizes state observability. The criterion is easy to apply and well-suited for complex scenarios, as demonstrated by its ability to consistently yield new insights across various domains. Specifically, the paper applies this criterion to revisit the structure-from-motion problem in computer vision. Remarkably, the straightforward application of the algebraic criterion provides results that are consistent with, and in some cases more general than, the state-of-the-art results in computer vision, all while proceeding algebraically and without relying on epipolar geometry, which is a key mathematical foundation for structure from motion. Furthermore, in the context of two important biological models, the criterion uncovers new findings that not only refine but also correct previous conclusions in the existing literature.