  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.
