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,描述了从观察其输入和输出重构表征系统状态的可能性。尽管已经为研究该属性并引入能够检查动态系统是否满足该属性的分析标准做出了巨大的努力,但在动力学也由未知输入驱动的情况下自动检查状态可观测性时尚无通用的分析标准。在这里,我们介绍这个基本的问题的通用分析解,通常称为未知输入可观念性问题的解。本文提供了这个问题的通用分析解,即提供了系统过程,基于自动计算(微分和矩阵秩确定),使我们能够在未知输入存在的情况下自动检查状态可观测性(算法6.1)。这个问题的第一个解决方案在书的第二部分中介绍:“基于不变性群的可观测性:一种新理论” [45]。这篇论文提出的解决方案完善了 [45] 中的先前解决方案。特别是,新的解决方案全面考虑了不属于“与未知输入相关的典型系统”类别的系统。分析派生在[45]中引入了一些新的概念和分析结果。最后,作为在未知输入和状态可观测性问题上密切相关的问题的简单结果,我们还提供了未知输入重构问题的答案。我们通过在视觉惯导传感器融合框架下研究由两个未知输入和一个已知输入驱动的非线性系统的可观测性来说明新算法的实现。