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

TL;DR

This paper introduces a convex optimization-based method for designing sparse feedback controllers for stochastic linear systems with multiplicative noise, balancing performance and actuator reduction.

Contribution

It develops a novel LMI-based approach to promote sparsity in control architectures for stochastic systems, including regularized LQRm and output feedback designs.

Findings

Significant actuator reduction achieved with minimal performance loss.

Sparse control architectures maintain robustness and stability.

Method validated on power grid frequency control case studies.

Abstract

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.