  In this paper, we propose a sparsity-promoting feedback control design for
stochastic linear systems with multiplicative noise. The objective is to
identify a sparse control architecture that optimizes the closed-loop
performance while stabilizing the system in the mean-square sense. The proposed
approach approximates the nonconvex combinatorial optimization problem by
minimizing various matrix norms subject to the Linear Matrix Inequality (LMI)
stability condition. We present two design problems to reduce the number of
actuators via the static state-feedback and a low-dimensional output. A
regularized linear quadratic regulator with multiplicative noise (LQRm) optimal
control problem and its convex relaxation are presented to demonstrate the
tradeoff between the suboptimal closed-loop performance and the sparsity degree
of control structure. Case studies on power grids for wide-area frequency
control show that the proposed sparsity-promoting control can considerably
reduce the number of actuators without significant loss in system performance.
The sparse control architecture is robust to substantial system-level
disturbances while achieving mean-square stability.
