Beyond the fundamental lemma: from finite time series to linear system
Kanat Camlibel, Paolo Rapisarda
TL;DR
The work establishes necessary and sufficient conditions under which finite input-output data uniquely identify a minimal input-state-output linear system up to state isomorphism, given lower/upper bounds on lag and state dimension. It introduces data informativity and derives data-dependent invariants from Hankel matrices, notably the deltas $\\delta_k$, to bound the shortest lag $\\ell_{ ext{min}}$ and the minimum state count $n_{ ext{min}}$, with explicit constructions of an explaining system from data. A key result shows that a depth-$L^{a}_+$ Hankel rank condition is both necessary and sufficient for informativity within the specified model class, and these conditions also yield sharpened lag bounds and a concrete online-state-construction method. The analysis leverages left-kernel decompositions, state reconstruction from data, and the relation between lag structures and observability/controllability to deliver a principled, data-driven identification framework that generalizes Willems’ fundamental lemma to finite data scenarios. The findings have potential implications for online experiment design and data-driven controller design within deterministic linear settings, and hint at extensions to broader model classes in the behavioral framework.
Abstract
We state necessary and sufficient conditions to uniquely identify (modulo state isomorphism) a linear time-invariant minimal input-state-output system from finite input-output data and upper- and lower bounds on lag and state space dimension.