关于实数结构化可控性/可镇定性/稳定性半径：复杂度与基于统一秩松弛的方法
On real structured controllability/stabilizability/stability radius: Complexity and unified rank-relaxation based methods
AI总结：
本文证明了实数结构化可控性、可镇定性和稳定性半径计算的NP难性，并提出统一的秩松弛算法框架，可有效求解Frobenius范数和2-范数下的局部最优解。
摘要：
本文研究线性系统的实数结构化可控性半径（RSCR）、可镇定性半径（RSSZR）和稳定性半径（RSSR），这些问题涉及在给定仿射结构下，确定一个（可能大规模的）系统与其最近不可控、不可镇定和不稳定系统之间的距离（以矩阵范数度量）。本文有两个主要贡献。第一，通过证明当扰动具有一般仿射参数化时，确定RSCR和RSSZR的可行性是NP难的，我们证明了计算这些半径是NP难的。此外，我们证明了一个与RSSR相关的问题的NP难性。这些困难性结果与所使用的矩阵范数无关。第二，我们为这些问题开发了基于统一秩松弛的算法，这些算法既能处理基于Frobenius范数的问题，也能处理基于$2$-范数的问题，并且对RSCR、RSSZR和RSSR问题共享同一框架。这些算法利用原始问题的低秩结构，并用正则化截断核范数项松弛相应的秩约束。此外，这些算法的改进版本在适当条件下可以找到满足扰动性能指标的局部最优解。最后，仿真结果表明，所提出的方法尽管框架简单，但能够找到与若干现有方法同样好的局部最优解。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
