Closed-form and robust expressions for data-driven LQ control

This article provides an overview of certain direct data-driven control results, where control sequences are computed from (noisy) data collected during offline control experiments without an explicit identification of the system dynamics. For the case of noiseless datasets, we derive several closed-form data-driven expressions that solve a variety of optimal control problems for linear systems with quadratic cost functions of the state and input (including the linear quadratic regulator problem, the minimum energy control problem, and the linear quadratic control problem with terminal constraints), discuss their advantages and drawbacks with respect to alternative data-driven and model-based approaches, and showcase their effectiveness through a number of numerical studies. Interestingly, these results provide an alternative and explicit way of solving classic control problems that, for instance, does not require the solution of an implicit and recursive Riccati equation as in the model-based setting. For the case of noisy datasets, we show how the closed-form expressions derived in the noiseless setting can be modified to compensate for the bias induced by noise, and perform a sensitivity analysis to reveal favorable asymptotic robustness properties of the derived data-driven controls. We conclude the paper with some considerations and a discussion of outstanding questions and directions of future investigation.