Sparse Optimal Control of LTI Systems under Sparsity-Dependent Delays

We address the problem of sparse H
₂
control design for linear time invariant (LTI) systems in the presence of feedback delays. Depending on the network configuration, the delays may be constant, or an explicit function of the sparsity itself. We first state a sufficient condition for delay stability, and show that it appears as a non-convex bilinear matrix inequality (BMI) constraint for the H
₂
-minimization problem. We relax this BMI as a set of convex semidefinite programs that result in a two loop optimization algorithm. The inner-loop promotes sparsity of the state feedback matrix constrained by the set of stabilizing gains for the delayed system. The outer-loop aligns the direction of sparsity with the minimizing H
₂
norm direction. We solve this two loop optimization using the alternating direction method of multipliers (ADMM). Simulation results illustrate the effectiveness of the proposed approach in finding sparse feedback matrices that achieve a low H
₂
norm while guaranteeing stability of the closed-loop delayed system.