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Sparse State-Space Realizations of Linear Controllers

Yaozhi Du, Jing Shuang Li

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 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.

Sparse State-Space Realizations of Linear Controllers

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 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.

Paper Structure

This paper contains 9 sections, 8 theorems, 37 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Equivalence of sparse realization to multivariate polynomial equations
  4. Solving multivariate polynomial equations with Gröbner basis
  5. Computing complex-valued solutions
  6. Computing real-valued solutions
  7. Finding sparse realizations
  8. Numerical Examples
  9. Conclusions and future work

Key Result

Theorem 1

For two minimal realizations of $\mathbf{K}(s)$ written as $\Sigma = (A,B,C,D)$ and $\Tilde{\Sigma} = (\Tilde{A},\Tilde{B},\Tilde{C},D)$, there exists a unique invertible matrix $T \in \mathbb{C}^{n \times n}$ s.t.

Figures (1)

  • Figure 2: Controller structures from numerical examples

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1: de2000minimal
  • Corollary 1
  • Lemma 1
  • Corollary 2
  • Theorem 2
  • proof
  • ...and 7 more