Rank Robustness of Complex Matrices with Respect to Real Perturbations
Rank Robustness of Complex Matrices with Respect to Real Perturbations
复制标题
复杂矩阵相对于真实扰动的秩鲁棒性
DOI：
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发表时间：
1994
影响因子：
1.5
通讯作者：
R. Decarlo
中科院分区：
文献类型：
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作者：
M. Wicks;R. Decarlo
本文研究了计算最小范数真实的矩阵扰动，导致一般复杂（系统）矩阵降秩的问题。给定描述线性时不变系统的状态模型，该矩阵扰动的范数有助于确定几个系统特性相对于真实的参数变化的鲁棒性。这种扰动的范数，或复矩阵的实限制奇异值，已知是复矩阵的不连续函数。本文给出了复矩阵消除这种不连续性的一个简单条件。具体地说，只要复矩阵的虚部满秩，最小真实的降秩扰动的大小是复矩阵的连续函数.本文件审查了问题连续性的其他方面。它还提出了一种算法，收敛到一个点，满足一个必要条件，以获得最小的真实的降秩矩阵扰动。一个李雅普诺夫函数的方法是用来建立算法的收敛性。一些数值例子说明了该方法的准确性。
This paper examines the problem of computing a minimum norm real matrix perturbation that causes a general complex (system) matrix to drop rank. Given the state model describing a linear time-invariant system, the norm of this matrix perturbation helps to determine the robustness of several system properties with respect to real parameter variations. The norm of this perturbation, or the real-restricted singular value of the complex matrix, is known to be a discontinuous function of the complex matrix. The paper presents a simple condition on the complex matrix that eliminates this discontinuity. Specifically, the paper shows that the size of the smallest real rank-reducing perturbation is a continuous function of the complex matrix as long as the imaginary part of the complex matrix has full rank. The paper examines other aspects of the continuity of the problem. It also presents an algorithm that converges to a point satisfying a necessary condition for obtaining the smallest real rank-reducing matrix perturbation. A Lyapunov function approach is used to establish convergence of the algorithm. Some numerical examples are included illustrating the accuracy of the approach.