Recursive formulation of the direct data-based LQR parameterization

Determine a recursive formulation for the direct data-driven LQR design that parameterizes the controller via a data-based matrix G with constraints X0G = I_n and Σ = I_n + X1 G Σ G^{ op} X1^{ op} (with K = U0G), so that the resulting update is suited for online closed-loop adaptation and does not scale with the data length t.

Background

The paper describes a direct data-based LQR parameterization in which the controller gain is expressed through a matrix G constrained by subspace relations derived from persistently exciting data. In this formulation, the dimension of G grows with the data length t.

The authors explicitly state that it is unclear how to convert this data-based LQR parameterization into a recursive form suitable for online adaptation, highlighting a concrete technical obstacle to real-time direct data-driven control.