On real structured controllability/stabilizability/stability radius: Complexity and unified rank-relaxation based methods

TL;DR

This paper investigates the computational complexity of real structured controllability, stabilizability, and stability radii, proving NP-hardness, and introduces unified rank-relaxation algorithms that effectively approximate solutions for these problems.

Contribution

It proves NP-hardness of computing structured controllability and stability radii and develops unified rank-relaxation algorithms applicable to various norms and problem types.

Findings

NP-hardness of computing structured radii established

Unified rank-relaxation algorithms effectively approximate solutions

Algorithms perform comparably to existing methods in simulations

Abstract

This paper addresses the real structured controllability, stabilizability, and stability radii (RSCR, RSSZR, and RSSR, respectively) of linear systems, which involve determining the distance (in terms of matrix norms) between a (possibly large-scale) system and its nearest uncontrollable, unstabilizable, and unstable systems, respectively, with a prescribed affine structure. This paper makes two main contributions. First, by demonstrating that determining the feasibilities of RSCR and RSSZR is NP-hard when the perturbations have a general affine parameterization, we prove that computing these radii is NP-hard. Additionally, we prove the NP-hardness of a problem related to the RSSR. These hardness results are independent of the matrix norm used. Second, we develop unified rank-relaxation based algorithms for these problems, which can handle both the Frobenius norm and the $2$-norm based problems and share the same framework for the RSCR, RSSZR, and RSSR problems. These algorithms utilize the low-rank structure of the original problems and relax the corresponding rank constraints with a regularized truncated nuclear norm term. Moreover, a modified version of these algorithms can find local optima with performance specifications on the perturbations, under appropriate conditions. Finally, simulations suggest that the proposed methods, despite being in a simple framework, can find local optima as good as several existing methods.