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Inverse Problems for Matrix Exponential in System Identification: System Aliasing

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.