Observability of linear systems on the Heisenberg Lie group
Thiago Matheus Cavalheiro, Alexandre José Santana, Victor Ayala
TL;DR
The paper addresses the problem of determining observability for linear control systems on the 3D Heisenberg group $\mathcal{H}$. It adopts a quotient-output framework by considering homomorphisms from $\mathcal{H}$ to its simply connected subgroups with kernels $K$ and outputs given by $\pi_K$, leveraging the Ayala-Haci criteria for observability on Lie groups. A detailed analysis of the linear vector fields on $\mathcal{H}$ yields the flow, fixed-point structure, and subgroups $H_i$ (H1–H9); the authors derive sufficient observability conditions, identify non-observable configurations, and provide explicit kernel-based criteria for each subgroup case. The work extends observability theory from Euclidean spaces to a nontrivial nilpotent Lie group setting and outlines a path toward applying these methods to broader classes of solvable affine 3D Lie groups, with practical implications for state reconstruction from quotient outputs.
Abstract
In control theory, understanding the observability property of a system is crucial for effectively managing and controlling dynamical systems. This property empowers us to deduce the internal state of a system from its outputs over time, even when direct measurements are impossible. By harnessing observability, we can accurately estimate the complete state of a system and reconstruct its dynamics using just limited information. In this work, we will find conditions for observability of linear systems in the three dimensional Heisenberg group $\mathcal{H}$. Considering the homomorphisms between the group and its simply connected subgroups, whose kernel is denoted by $K$, we will find sufficient conditions for observability on the system using a quotient space $\mathcal{H}/K$ as the output.