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 and to introduce analytical criteria able to check whether a dynamic system satisfies this property or not, 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. This paper provides the general analytical solution of this problem, namely, it provides 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 (Algorithm 6.1). A first solution of this problem was presented in the second part of the book: "Observability: A New Theory Based on the Group of Invariance" [45]. The solution presented by this paper completes the previous solution in [45]. In particular, the new solution exhaustively accounts for the systems that do not belong to the category of the systems that are "canonic with respect to their unknown inputs". The analytical derivations largely exploit several new concepts and analytical results introduced in [45]. Finally, as a simple consequence of the results here obtained, we also provide the answer to the problem of unknown input reconstruction which is intimately related to the problem of state observability. 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, whose dynamics are driven by two unknown inputs and one known input.

翻译：可观性是任何动态系统的基本结构属性，描述了通过观察系统输入和输出重构系统状态的可能性。尽管已有大量研究致力于探讨该属性并引入能够检验动态系统是否满足该属性的解析准则，但当系统动力学也受未知输入驱动时，仍缺乏自动检验状态可观性的一般解析准则。本文提出了这一根本问题（通常称为未知输入可观性问题）的一般解析解。本文提供了该问题的一般解析解，即基于自动计算（微分和矩阵秩判定）的系统化流程，允许我们在存在未知输入的情况下自动检验状态可观性（算法6.1）。该问题的首个解发表于专著《可观性：基于不变性群的新理论》[45]的第二部分。本文所提出的解完善了文献[45]中的先前解。具体而言，新解全面涵盖了不属于“关于其未知输入为规范形式”的系统类别。解析推导大量利用了文献[45]中引入的新概念和解析结果。最后，作为本文结果的直接推论，我们还给出了与状态可观性问题密切相关的未知输入重构问题的答案。我们通过研究视觉-惯性传感器融合框架中一个非线性系统的可观性性质（其动力学由两个未知输入和一个已知输入驱动），说明了新算法的实现。