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

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.