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

TL;DR

The general analytical solution of the unknown input observability problem is provided, 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.

Abstract

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.

Figures (3)

• Figure 1: Wheeled robot moving on a plane.
• Figure 2: In $(a)$, the three initial robot configurations are compatible with the same initial observation ($\beta$). In $(b)$, the two initial positions ($A$ and $B$) do not reproduce the same observations ($\alpha\neq \gamma$). In $(c)$ the two indicated trajectories provide the same bearing observations at every time.
• Figure 3: The global frame, the body frame and the observation provided by the $\mathcal{V}$ sensor ($\beta$).

Theorems & Definitions (17)

• Definition 2.1: Observable Function
• Definition 2.2: Unknown input reconstructability matrix
• Definition 2.3: Unknown input degree of reconstructability
• Definition 2.4: Canonic system wrt the UIs
• Definition 2.5: System in canonical form wrt the UIs
• Theorem 2.1
• Definition 3.1: Autobracket
• Theorem 3.1
• Lemma 3.1
• Lemma 3.2