Inverse Problems for Matrix Exponential in System Identification: System Aliasing

TL;DR

This paper explores the challenges of identifying the system matrix in continuous-time linear systems, especially addressing system aliasing caused by slow sampling, and proposes conditions to improve identifiability using prior knowledge.

Contribution

It introduces general solvability conditions for inverse matrix exponential problems and proposes methods to mitigate system aliasing through sparsity and norm constraints.

Findings

System aliasing causes non-unique matrix logarithms.

Additional conditions can partially resolve aliasing issues.

Under certain constraints, the $A$-matrix is identifiable up to a finite class.

Abstract

This note addresses identification of the $A$-matrix in continuous time linear dynamical systems on state-space form. If this matrix is partially known or known to have a sparse structure, such knowledge can be used to simplify the identification. We begin by introducing some general conditions for solvability of the inverse problems for matrix exponential. Next, we introduce "system aliasing" as an issue in the identification of slow sampled systems. Such aliasing give rise to non-unique matrix logarithms. As we show, by imposing additional conditions on and prior knowledge about the $A$-matrix, the issue of system aliasing can, at least partially, be overcome. Under conditions on the sparsity and the norm of the $A$-matrix, it is identifiable up to a finite equivalence class.