Near optimal finite time identification of arbitrary linear dynamical systems
Provides the first unified finite-time analysis for arbitrary LTI systems, addressing a gap in system identification theory.
This paper derives finite time error bounds for identifying general linear time-invariant systems via least squares, covering stable, marginally stable, and explosive eigenvalue regimes. The bounds are sharp up to logarithmic factors, but the method can be statistically inconsistent under certain high signal-to-noise conditions.
We derive finite time error bounds for estimating general linear time-invariant (LTI) systems from a single observed trajectory using the method of least squares. We provide the first analysis of the general case when eigenvalues of the LTI system are arbitrarily distributed in three regimes: stable, marginally stable, and explosive. Our analysis yields sharp upper bounds for each of these cases separately. We observe that although the underlying process behaves quite differently in each of these three regimes, the systematic analysis of a self--normalized martingale difference term helps bound identification error up to logarithmic factors of the lower bound. On the other hand, we demonstrate that the least squares solution may be statistically inconsistent under certain conditions even when the signal-to-noise ratio is high.