用Frobenius范数有界摄动逼近实结构稳定半径
Approximating the Real Structured Stability Radius with Frobenius Norm Bounded Perturbations
摘要：
提出了一种快速逼近具有输出反馈的线性动力系统的实稳定半径的方法，其中扰动被限制为实值且关于Frobenius范数有界.我们的工作建立在一些可扩展的算法，近年来已经提出，从方法，近似复杂或真正的伪谱横坐标和半径的大型稀疏矩阵（和概括这些方法的伪谱值集）的算法近似复杂的稳定半径（$H_\infty$规范的倒数）。虽然我们的算法保证只找到真正的稳定半径的上界，它似乎在实践中是相当有效的。据我们所知，这是第一个解决这个问题的Frobenius范数版本的算法。由于该算法的代价主要是计算矩阵序列的连续时间系统的最大实部特征值（或离散时间系统的模），因此只要系统矩阵稀疏，该算法对大规模系统仍然非常有效。
英文摘要：
We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the $H_\infty$ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost mainly consists of computing the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse.
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