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Observability is a fundamental structural property of any dynamic system anddescribes the possibility of reconstructing the state that characterizes thesystem from observing its inputs and outputs. Despite the huge effort made tostudy this property and to introduce analytical criteria able to check whethera dynamic system satisfies this property or not, there is no general analyticalcriterion to automatically check the state observability when the dynamics arealso driven by unknown inputs. Here, we introduce the general analyticalsolution of this fundamental problem, often called the unknown inputobservability problem. This paper provides the general analytical solution ofthis problem, namely, it provides the systematic procedure, based on automaticcomputation (differentiation and matrix rank determination), that allows us toautomatically check the state observability even in the presence of unknowninputs (Algorithm 6.1). A first solution of this problem was presented in thesecond part of the book: "Observability: A New Theory Based on the Group ofInvariance" [45]. The solution presented by this paper completes the previoussolution in [45]. In particular, the new solution exhaustively accounts for thesystems that do not belong to the category of the systems that are "canonicwith respect to their unknown inputs". The analytical derivations largelyexploit several new concepts and analytical results introduced in [45].Finally, as a simple consequence of the results here obtained, we also providethe answer to the problem of unknown input reconstruction which is intimatelyrelated to the problem of state observability. We illustrate the implementationof the new algorithm by studying the observability properties of a nonlinearsystem in the framework of visual-inertial sensor fusion, whose dynamics aredriven by two unknown inputs and one known input.

Nonlinear Unknown Input Observability and Unknown Input Reconstruction: The General Analytical Solution