On the Role of Regularization in Direct Data-Driven LQR Control

The linear quadratic regulator (LQR) problem is a cornerstone of control theory and a widely studied benchmark problem. When a system model is not available, the conventional approach to LQR design is indirect, i.e., based on a model identified from data. Recently a suite of direct data-driven LQR design approaches has surfaced by-passing explicit system identification (SysID) and based on ideas from subspace methods and behavioral systems theory. In either approach, the data underlying the design can be taken at face value (certainty-equivalence) or the design is robustified to account for noise. An emerging topic in direct data-driven LQR design is to regularize the optimal control objective to account for implicit SysID (in a least-square or low-rank sense) or to promote robust stability. These regularized formulations are flexible, computationally attractive, and theoretically certifiable; they can interpolate between direct vs. indirect and certainty-equivalent vs. robust approaches; and they can be blended resulting in remarkable empirical performance. This manuscript reviews and compares different approaches to regularized direct data-driven LQR.