Yaozhi Du

Yaozhi Du, Jing Shuang Li

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 Gröbner basis) to solve this problem exactly. We provide algorithms to find real- and complex-valued sparse realizations and demonstrate their efficacy on several examples.

Yaozhi Du, Jing Shuang Li

This paper designs H2 and H-infinity distributed controllers with local communication and local disturbance rejection. We propose a two-step procedure: first, select closed-loop poles; then, optimize over parameterized controllers. We build on the system level synthesis (SLS) parameterization -- primarily used in the discrete-time setting -- and extend it to the general continuous-time setting. We verify our approach in simulation on a 9-node grid governed by linearized swing equations, where our distributed controllers achieve performance comparable to that of optimal centralized controllers while facilitating local disturbance rejection.