论文标题
通过Riemannian优化最近的$ω$稳定矩阵
Nearest $Ω$-stable matrix via Riemannian optimization
论文作者
论文摘要
我们研究了找到最接近的$ω$ - 稳定矩阵与某个矩阵$ a $的最接近的矩阵，即，在规定的封闭套件中，最接近其所有特征值的矩阵及其所有特征值$ω$。距离在Frobenius Norm中测量。一个重要的特殊情况是找到最近在系统理论中应用的Hurwitz或Schur稳定矩阵。我们将任务的重新重新制定为对正交（或单一）矩阵的Riemannian流形的优化问题。然后，可以使用Riemannian优化理论的标准方法来解决该问题。所得算法在中小型矩阵上非常快，并且直接返回了最小化器的SCHUR分解，从而避免了与具有较高多重性能的特征值相关的数值困难。
We study the problem of finding the nearest $Ω$-stable matrix to a certain matrix $A$, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $Ω$. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.
