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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. Thispaper makes two main contributions. First, by demonstrating that determiningthe feasibilities of RSCR and RSSZR is NP-hard when the perturbations have ageneral affine parameterization, we prove that computing these radii isNP-hard. Additionally, we prove the NP-hardness of a problem related to theRSSR. These hardness results are independent of the matrix norm used. Second,we develop unified rank-relaxation based algorithms for these problems, whichcan handle both the Frobenius norm and the $2$-norm based problems and sharethe same framework for the RSCR, RSSZR, and RSSR problems. These algorithmsutilize the low-rank structure of the original problems and relax thecorresponding rank constraints with a regularized truncated nuclear norm term.Moreover, a modified version of these algorithms can find local optima withperformance specifications on the perturbations, under appropriate conditions.Finally, simulations suggest that the proposed methods, despite being in asimple framework, can find local optima as good as several existing methods.

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