# Nonlinear Unknown Input Observability: The General Analytic Solution

## TL;DR

This paper presents a comprehensive analytical solution for nonlinear unknown input observability, enabling automatic checking of system observability even with unknown inputs through a systematic, tensor-based approach.

## Contribution

It introduces a novel, general analytical solution and an algorithm for unknown input observability, extending previous work to more complex nonlinear systems.

## Key findings

- The solution is expressed using new tensor fields invariant under a defined group.
- An algorithm for automatic observability analysis is developed.
- Application demonstrated in visual-inertial sensor fusion systems.

## Abstract

Observability is a fundamental structural property of any dynamic system and describes the possibility of reconstructing the state that characterizes the system from observing its inputs and outputs. Despite the huge effort made to study this property, there is no general analytical criterion to automatically check the state observability when the dynamics are also driven by unknown inputs. Here, we introduce the general analytical solution of this fundamental problem, often called the unknown input observability problem. We provide the systematic procedure, based on automatic computation (differentiation and matrix rank determination), that allows us to automatically check the state observability even in the presence of unknown inputs. The fundamental step to obtain this solution is the introduction of the group of invariance of observability. We introduce the group of transformations under which observability is invariant. Based on this group, we introduce new tensor fields with respect to this group of transformations. The analytical solution of the unknown input observability problem is expressed in terms of these tensors. This paper provides this solution together with the main analytical steps for its derivation and a sketch of the proof of its validity. A complete derivation, together with additional important properties, is available in [45]. On the other hand, this paper also provides a more general solution than the one presented in [45] by exhaustively dealing with the systems that do not belong to the category of the systems that are canonic with respect to their unknown inputs. This solution is also provided in the form of a new algorithm. We illustrate the implementation of the new algorithm by studying the observability properties of a nonlinear system in the framework of visual-inertial sensor fusion.