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On approximating the nearest Ω‐stable matrix
Numerical Linear Algebra with Applications ( IF 1.8 ) Pub Date : 2020-02-28 , DOI: 10.1002/nla.2282 Neelam Choudhary 1 , Nicolas Gillis 2 , Punit Sharma 3
Numerical Linear Algebra with Applications ( IF 1.8 ) Pub Date : 2020-02-28 , DOI: 10.1002/nla.2282 Neelam Choudhary 1 , Nicolas Gillis 2 , Punit Sharma 3
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In this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region Ω of the complex plane. More precisely, we consider three types of regions and their intersections: conic sectors, vertical strips, and disks. We refer to this problem as the nearest Ω‐stable matrix problem. This includes as special cases the stable matrices for continuous and discrete time linear time‐invariant systems. In order to achieve this goal, we parameterize this problem using dissipative Hamiltonian matrices and linear matrix inequalities. This leads to a reformulation of the problem with a convex feasible set. By applying a block coordinate descent method on this reformulation, we are able to compute solutions to the approximation problem, which is illustrated on some examples.
中文翻译：
逼近最接近的Ω稳定矩阵
在本文中，我们考虑用特征值位于复杂平面某些特定区域Ω中的矩阵逼近给定矩阵的问题。更准确地说，我们考虑三种类型的区域及其相交：圆锥形扇形，垂直条和圆盘。我们将此问题称为最近的Ω稳定矩阵问题。作为特殊情况，这包括连续和离散时间线性时不变系统的稳定矩阵。为了实现此目标，我们使用耗散哈密顿矩阵和线性矩阵不等式对这个问题进行参数化。这导致具有凸可行集的问题的重新表述。通过在这种重构上应用块坐标下降法，我们能够计算逼近问题的解，在一些示例中进行了说明。
更新日期：2020-02-28
中文翻译：
逼近最接近的Ω稳定矩阵
在本文中，我们考虑用特征值位于复杂平面某些特定区域Ω中的矩阵逼近给定矩阵的问题。更准确地说，我们考虑三种类型的区域及其相交：圆锥形扇形，垂直条和圆盘。我们将此问题称为最近的Ω稳定矩阵问题。作为特殊情况，这包括连续和离散时间线性时不变系统的稳定矩阵。为了实现此目标，我们使用耗散哈密顿矩阵和线性矩阵不等式对这个问题进行参数化。这导致具有凸可行集的问题的重新表述。通过在这种重构上应用块坐标下降法，我们能够计算逼近问题的解，在一些示例中进行了说明。
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