Sparse Data-Driven LQR Via Augmented Lagrangian-Based Policy Search

In this paper, we investigate a data-driven framework to solve infinite-horizon Linear Quadratic Regulator (LQR) problems when the dynamics are partially unknown, with the additional challenge of seeking a linear control policy that must simultaneously stabilize the (possibly unstable) system while satisfying some sparsity constraints, i.e., some entries of the gain matrix must be zero. We reformulate this sparse LQR gain design as a direct policy search and propose a data-driven approach that exploits data (system trajectories) to handle the uncertainty about the dynamics. The proposed algorithmic strategy, termed SPARK-LQR, computes the optimal gain with the desired structure by combining policy iteration and the augmented Lagrangian approach. This methodology is then extended to robustly address the case of noisy trajectories by employing a regularization approach based on Linear Matrix Inequalities (LMIs). The effectiveness of the proposed strategies is corroborated by numerical simulations, where SPARK-LQR, and its regularized version are applied on the gain design problem for a spatially distributed system, both in the deterministic setting and in the scenario with noisy data.