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：可观测性是任何动态系统的基本结构性质，描述了通过观测系统输入和输出来重构表征系统状态的可能性。尽管已有大量研究致力于这一性质并引入了解析准则来检验动态系统是否满足该性质，但当前仍缺乏能自动检验由未知输入驱动的动态系统状态可观测性的一般解析准则。本文提出了这一基本问题（通常称为未知输入可观测性问题）的一般解析解。具体而言，本文给出了该问题的系统化求解方法——基于自动计算（微分与矩阵秩判定）的规范流程，可自动检验含未知输入情况下的状态可观测性（算法6.1）。该问题的首个求解方案见于专著《可观测性：基于不变性群的新理论》[45]的第二部分。本文提出的方案是对文献[45]前期工作的完善。特别地，新方案全面涵盖了不属于“关于未知输入呈典范形式”的系统类别。解析推导大量运用了文献[45]中引入的新概念与分析结果。最后，作为本文结论的直接推论，我们还给出了与状态可观测性问题密切相关的未知输入重构问题的解答。通过研究视觉-惯性传感器融合框架下非线性系统的可观测性性质（该系统由两个未知输入和一个已知输入驱动），我们展示了新算法的具体实现。