State Representations From Finite Time Series

Abstract

We present two algorithms for construction of a state sequence of a linear time-invariant system from a finite exact trajectory of that system. The first algorithm uses the classical in subspace identification splitting of the data into "past" and "future" and computes a bases for the past-future intersection. The second algorithm, based on the shift-and-cut map, operates on a half deeper Hankel matrix.Index Terms-Subspace identification, state construction, shift-and-cut map, persistency of excitation, behavioral systems.

Formulas for Data-Driven Control: Stabilization, Optimality, and Robustness

“…This is the key property that enables one to replace a parametric description of the system with data. Lemma 2 has been originally proven in [27, Theorem 1] using the behavioral language, and it was later referred to in [33] as the fundamental lemma to describe a linear system through a finite collection of its input/output data. Here, for making the paper as self-contained as possible, we gave a proof of this result using state-space descriptions, as they will recur often in the reminder of this paper.…”

Willems’ Fundamental Lemma for State-Space Systems and Its Extension to Multiple Datasets

“…Therefore, our remaining task is to computeȳ [0,4] . Inspired by [2], we will compute this trajectory iteratively by computing multiple length 3 trajectories as linear combinations of the columns of (16). To begin with, we compute the first unknown in (18), which isȳ(0).…”

“…In the seminal work by Willems and coauthors [1], it was shown that a single, sufficiently exciting trajectory of a linear system can be used to parameterize all trajectories that the system can produce. This result has later been named the fundamental lemma [2], [3], and plays an important role in the learning and control of dynamical systems on the basis of measured data.…”

On Persistency of Excitation and Formulas for Data-driven Control

“…In turn, condition (3) makes it possible to represent any input/output trajectory of the system as a linear combination of previously collected input/output data. Lemma 2 has been originally proven in [22, Theorem 1] using the behavioral language, and it was later referred to as the fundamental lemma [28] to describe a linear system through a finite collection of its input/output data.…”