论文标题
计算时间延迟系统的伪造演
Computing the Pseudospectral Abscissa of Time-Delay Systems
论文作者
论文摘要
线性时间变变系统的伪谱是当系统矩阵与给定上限的所有可能的扰动遭受所有可能的扰动时。伪横肌被定义为伪谱中特征根的最大实际部分，因此，从强大的稳定性角度来看，它们很重要。在本文中，我们提出了一种数值方法，用于计算具有离散延迟的智障延迟微分方程的伪横梁。我们的方法是基于伪谱之间的连接和适当定义的复杂函数的水平集。这些连接导致我们采用了计算伪横肌的一分配算法，每个步骤都依赖于检查适当定义的操作员的假想轴特征值的存在。因为该操作员是无限维的，因此采用了预测指标方法。在预测器步骤中，应用了将操作员离散为矩阵，将分配算法应用于矩阵，从而对伪造脱超囊产生近似值。离散化的影响在论文中充分表征。在校正步骤中，通过求解一组表征伪孔轮廓中极端点的非线性方程，将近似的伪横梁纠正到任何给定的精度。
The pseudospectra of a linear time-invariant system are the sets in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa are defined as the maximum real part of the characteristic roots in the pseudospectra and, therefore, they are for instance important from a robust stability point of view. In this paper we present a numerical method for the computation of the pseudospectral abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectra and the level sets of an appropriately defined complex function. These connections lead us to a bisection algorithm for the computation of the pseudospectral abscissa, where each step relies on checking the presence of imaginary axis eigenvalues of an appropriately defined operator. Because this operator is infinite-dimensional a predictor-corrector approach is taken. In the predictor step the bisection algorithm is applied where the operator is discretized into a matrix, yielding approximations for the pseudospectral abscissa. The effect of the discretization is fully characterized in the paper. In the corrector step, the approximate pseudospectral abscissa are corrected to any given accuracy, by solving a set of nonlinear equations that characterize extreme points in the pseudospectra contours.
