Observability of Linear Time-Invariant Systems with Relative Measurements: A Geometric Approach

TL;DR

This paper investigates how the structure of relative measurements in linear time-invariant systems affects their observability, using a geometric approach, with applications to multi-agent networks and distributed observers.

Contribution

It introduces a geometric framework linking measurement graph structure to system observability and designs distributed observers based on relative measurements.

Findings

Graph structure determines observability conditions.

Absolute measurements enhance state estimation.

Distributed observer design is feasible with relative data.

Abstract

This paper explores the observability and estimation capability of dynamical systems using predominantly relative measurements of the system's state-space variables, with minimal to no reliance on absolute measurements of these variables. We concentrate on linear time-invariant systems, in which the observation matrix serves as the algebraic representation of a graph object. This graph object encapsulates the availability of relative measurements. Utilizing algebraic graph theory and abstract linear algebra (geometric) tools, we establish a link between the structure of the graph of relative measurements and the system-theoretic observability subspace of linear systems. Special emphasis is given to multi-agent networked systems whose dynamics are governed by the linear consensus protocol. We demonstrate the importance of absolute information and its placement to the system's dynamics in achieving full-state estimation. Finally, the analysis shifts to the synthesis of a distributed observer with relative measurements for single integrator dynamics, exemplifying the relevance of the preceding analytical findings. We support our theoretical analysis with numerical simulations.