Sparse Structure Design for Stochastic Linear Systems via a Linear Matrix Inequality Approach

We propose a sparsity-promoting feedback control design for stochastic linear systems with multiplicative noise. The objective is to identify an optimal sparse control architecture and optimize the closed-loop performance while stabilizing the system in the mean-square sense. Our 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 and the number of sensors via a low-dimensional output. A regularized linear quadratic regulator with multiplicative (LQRm) noise 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 sensors and actuators without significant loss in system performance. The sparse control architecture is robust to substantial system-level disturbances while achieving mean-square stability.