Two-Timescale Optimization Framework for Sparse-Feedback Linear-Quadratic Optimal Control

A H2-guaranteed sparse-feedback linear-quadratic (LQ) optimal control with convex parameterization and convex-bounded uncertainty is studied in this paper, where ℓ0-penalty is added into the H2 cost to penalize the number of communication links among distributed controllers. Then, the sparse-feedback gain is investigated to minimize the modified H2 cost together with the stability guarantee, and the corresponding main results are of three parts. First, the ℓ1 relaxation sparse-feedback LQ problem is of concern, and a two-timescale algorithm is developed based on proximal coordinate descent and primal-dual splitting approach. Second, piecewise quadratic relaxation sparse-feedback LQ control is investigated, which exhibits an accelerated convergence rate. Third, sparse-feedback LQ problem with ℓ0-penalty is directly studied through BSUM (Block Successive Upper-bound Minimization) framework, and precise approximation method and variational properties are introduced.