# Sparsity Invariance for Convex Design of Distributed Controllers

## TL;DR

This paper introduces a novel convex framework called Sparsity Invariance (SI) for designing optimal distributed LTI controllers with sparsity constraints, extending beyond quadratic invariance and ensuring global optimality in many cases.

## Contribution

The paper develops the concept of Sparsity Invariance (SI), enabling convex design of distributed controllers that surpasses quadratic invariance limitations and guarantees global optimality when applicable.

## Key findings

- SI always produces convex restrictions for sparsity-constrained control design.
- SI guarantees global optimality when quadratic invariance holds.
- Numerical examples demonstrate SI's superior performance and optimality in non-QI cases.

## Abstract

We address the problem of designing optimal linear time-invariant (LTI) sparse controllers for LTI systems, which corresponds to minimizing a norm of the closed-loop system subject to sparsity constraints on the controller structure. This problem is NP-hard in general and motivates the development of tractable approximations. We characterize a class of convex restrictions based on a new notion of Sparsity Invariance (SI). The underlying idea of SI is to design sparsity patterns for transfer matrices Y(s) and X(s) such that any corresponding controller K(s)=Y(s)X(s)^-1 exhibits the desired sparsity pattern. For sparsity constraints, the approach of SI goes beyond the notion of Quadratic Invariance (QI): 1) the SI approach always yields a convex restriction; 2) the solution via the SI approach is guaranteed to be globally optimal when QI holds and performs at least as well as considering a nearest QI subset. Moreover, the notion of SI naturally applies to designing structured static controllers, while QI is not utilizable. Numerical examples show that even for non-QI cases, SI can recover solutions that are 1) globally optimal and 2) strictly more performing than previous methods.