Two-Timescale Optimization Framework for Decentralized Linear-Quadratic Optimal Control

A $\mathcal{H}_2$-guaranteed decentralized linear-quadratic optimal control with convex parameterization and convex-bounded uncertainty is studied in this paper, where several sparsity promoting functions are added, respectively, into the $\mathcal{H}_2$ cost to penalize the number of communication links among decentralized controllers. Then, the sparse feedback gain is investigated to minimize the modified $\mathcal{H}_2$ cost together with the stability guarantee, and the corresponding main results are of three parts. First, the weighted-$\ell_1$ sparsity promoting function is of concern, and a two-timescale algorithm is developed based on the BSUM (Block Successive Upper-bound Minimization) framework and a primal-dual splitting approach. Second, the optimization problem induced by piecewise quadratic sparsity penalty is investigated, which exhibits an accelerated convergence rate. Third, the nonconvex sparse optimization problem with $\ell_0$-penalty is studied, which can be approximated by successive coordinatewise convex optimization problems.