Sparse State-Space Realizations of Linear Controllers

TL;DR

This paper introduces an exact algebraic geometry-based method to find sparse state-space realizations of linear systems with specified sparsity patterns, addressing a nonconvex problem motivated by neuroscience applications.

Contribution

It develops a novel approach using Gr"obner bases to compute sparse realizations from transfer functions, advancing control system design techniques.

Findings

The method can find exact sparse realizations for given transfer functions.

Algorithms successfully produce real- and complex-valued sparse realizations.

Demonstrations on examples validate the approach's effectiveness.

Abstract

This paper provides a novel approach for finding sparse state-space realizations of linear systems (e.g., controllers). Sparse controllers are commonly used in distributed control, where a controller is synthesized with some sparsity penalty. Here, motivated by a modeling problem in sensorimotor neuroscience, we study a complementary question: given a linear time-invariant system (e.g., controller) in transfer function form and a desired sparsity pattern, can we find a suitably sparse state-space realization for the transfer function? This problem is highly nonconvex, but we propose an exact method to solve it. We show that the problem reduces to finding an appropriate similarity transform from the modal realization, which in turn reduces to solving a system of multivariate polynomial equations. Finally, we leverage tools from algebraic geometry (namely, the Gr\"obner basis) to solve this problem exactly. We provide algorithms to find real- and complex-valued sparse realizations and demonstrate their efficacy on several examples.