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特性，它描述了通过观测系统输入和输出重构系统状态的可能性。尽管学界已投入巨大努力研究该特性并建立了可判定动态系统是否满足该性质的解析准则，但当系统动力学同时受未知输入驱动时，尚不存在通用的解析准则来自动检验状态可观性。本文提出了这一基础问题（常称为未知输入可观性问题）的通用解析解。本论文提供了该问题的通用解析解，即基于自动计算（微分与矩阵秩判定）的系统化流程，使我们即使在存在未知输入的情况下也能自动检验状态可观性（算法6.1）。该问题的初步解法已发表于专著《可观性：基于不变群的新理论》[45]的第二部分。本文提出的解法是对[45]中已有解的完善，特别针对那些不属于"关于未知输入为典范系统"范畴的系统进行了完备性处理。本文的解析推导充分运用了[45]中提出的若干新概念与解析结果。最后，基于所得结果的自然延伸，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解答。通过研究视觉-惯性传感器融合框架下受两个未知输入和一个已知输入驱动的非线性系统的可观性特性，我们展示了新算法的实施过程。